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• MIT designers take different tacks for America's Cup

MIT designers take different tacks for America's Cup

MIT technology is once again driving the quest for speed on the water in the ancient art of sailing.

Two of the three American syndicates vying for the honor of defending the America's Cup have MIT ocean engineering department expertise represented in the design of their boats.

Professors Paul Sclavounos and Jerome Milgram have different ideas about what combinations will yield the fastest boat under all conditions.

Professor Sclavounos, a member of the Young America/PACT 95 design team, described the innovative winged rudder on Young America as the innovation of this America's Cup. "The data out of our computer tell us it will really increase boat speed," he said.

Professor Milgram, design director of Mighty Mary/America3 (A3), has a radical "whale's tail" design for the keel, the bulbous stabilizing element attached to the bottom of a racing yacht's hull, and a very long, 13-foot rudder.

While Professor Milgram agrees that rudder wings show enough promise to be worthy of a full-scale experiment, the evaluations and tests by A3 indicated that gain or loss was small either way.

The two boats have set different priorities. A3 has concentrated on improving light-air capabilities after being the clear winner in heavy conditions in 1992. (Professor Sclavounos, now with PACT 95, was responsible for the research on rough seas that aided the A3 effort in 1992.) PACT 95, on the other hand, has concentrated on performing well in rough water.

Professor Sclavounos and his cadre at MIT are also responsible for a computer program called Ship Wave Analysis (SWAN), which allows measurement of the water flow around a ship's hull. Being able to measure the performance of a boat in waves allowed them to test hundreds of hull designs by computer simulation. He said a one percent difference in speed translates to one minute on the race course.

Two weeks ago, the veil of secrecy around the ships' designs was dropped and the America's Cup contenders showed their keels, which are also known as bulbs. This was done to alleviate the intelligence and counterintelligence subterfuge that had grown as technology assumed greater importance. The results surprised many.

New Zealand, which has been steamrolling the foreign competition (the challengers) and was thought to have some new secret appendage, has a rather conventional design, including its bulb. According to Professor Sclavounos, it shares the "minimalist" approach with PACT 95's Young America. The most visible design difference is that New Zealand has large "unswept" winglets extending straight out from the bulb, whereas Young America has "swept" winglets angled toward the stern of the boat to allow kelp to slide away.

A3 and Australia were described as more "maximalist" by Professor Sclavounos. Professor Milgram agreed that they are a little more exotic, but only because of their appendages. A3 has a very long rudder-at 13 feet it was the maximum allowed. Initially, it also had the first-ever all-women's team, which has now been joined by tactician Dave Dellenbaugh.The women, while top-notch athletes and dinghy (small boat) racers, have not had experience in big-boat racing, which offers different challenges.

"As designers, we needed to accommodate to a less experienced sailing team," Professor Milgram said. "We designed a deeper rudder to be efficient at the unusually large rudder angles these particular sailors seemed to need to steer well. Mighty Mary, with its sleek narrow hull and relatively small keel, has terrific maneuvering, but keeping it aimed correctly is a challenge.

"We have a radical bulb, " he said. "It's short, shaped like a whale's tail. It has the best faring of winglets to keel, we think. Whales do their tails very nicely-they've had so long to evolve."

Since two designers formerly with A3 went to Australia this time, it is not surprising that the boats would more closely resemble each other. The Australian team is headed by John Bertrand, who won the America's Cup in 1983 and is a former student of Professor Milgram and a 1972 graduate of MIT.

"We've been thinking we should charge Australia a copyright fee," Professor Milgram joked. "But we're a lot stronger. Remember, New Zealand and A3 are the only boats remaining which haven't nearly sunk," he added, referring to the sinking of Australia's new boat. Pushing the envelope of design for speed has made many boats more fragile than in the past.

The third syndicate-that of Dennis Conner-has relatively little technology in its program. Its bulb has an unusual dip in the middle, which is difficult to see from the photo angles that were permitted at the unveiling. It is the slowest boat by all reports, yet continues to surprise with wins and brilliant tactical sailing.

So how much of the art is intuitive and emotional, and how much is technology? Both are needed these days, say the experts.

"A difficult challenge is getting the fruits of the science put into practice," Professor Milgram said. "The sailors need to learn to coordinate their intuition with the technical findings, and the scientists and engineers need to learn to make the boat perform optimally when the sailors are using their intuition."

For example, he said, the sailors like the feel of the boat at excessive heel angles. Heeling is when the boat leans far over to one side, bringing the center of gravity higher and thereby slowing the boat. The designers are thus pressed to distribute the weight in the boat for less than the optimum heel stability. This robs the sails of power. It takes time in the office and on the water to bring all elements together to the point where the crew is comfortable sailing a boat going as fast as it can, he said.

The entire effort is a massive system project involving management, fund-raising, science, technology, and most importantly, the skills of the team on the water, Professor Milgram said. Although science and technology can improve the overall result, it adds to the overall time required to do the project. If the managers of the projects started sooner, it would greatly increase their chances of success, he said.

Professor Milgram thinks that good sail shapes, good sail control and good sailing will decide the winner. Since most of the boats' hulls are similar to the A3 Cup winner of 1992, he thinks that boat design will be less of a factor this time. Professor Sclavounos, with a grin, said, "We'll see."

A version of this article appeared in MIT Tech Talk on April 26, 1995.

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On a chilly day in the mid-1930s, Erwin Schell ‘12, head of MIT’s business administration course, looked out his office window and noticed some MIT students sailing a frostbite race on the Charles. The sight inspired him. Soon after, he and George Owen, Class of 1894, who headed the Department of Naval Architecture, went to see Walter Cromwell “Jack” Wood ‘17 to discuss starting an MIT sailing program. Owen drew up plans for a boat eventually dubbed the “Tech Dinghy,” and Schell set about raising funds to build a small fleet of them. As MIT’s first sailing master, Wood then presided over that 48-boat fleet at a new pavilion, and MIT hosted its first intercollegiate sailing competition in 1937.

Owen’s design for the Tech Dinghy was ingenious, incorporating both cat and sloop rigging to maximize performance for highly skilled sailors while providing stability for novices. Although the boat has evolved–it’s now in its fifth generation–its great balance remains its defining feature. In 1953, the second generation of the dinghy hit the water, a fiberglass hull replacing Owen’s wooden one. For the third generation, Halsey Herreshoff, SM ‘60, increased the height of its sides to prevent it from taking on water; for the fourth, he heightened the mast. And in 2004, MIT sailors launched the fifth generation of the dinghy, which has flotation tanks that make it easier to right when it capsizes.

In typical MIT fashion, the Tech Dinghy has featured in several hacks–appearing fully rigged on the small dome of Building 7, in the Alumni Pool, and in the campus chapel’s moat. Institute presidents including Karl Compton, Paul Gray, and Susan Hockfield have sailed Tech Dinghies. And every year, 1,200 to 1,400 students take sailing lessons, and even more take the 37 boats now in the fleet out on the Charles.

Perhaps the most famous student sailor to take the helm of a Tech Dinghy is John Bertrand, SM ‘72, who won the 1983 America’s Cup for an Australian team, ending a 132-year U.S. reign. ­Bertrand had lost his first attempt at the cup in 1970, the summer before he came to MIT. Studying naval architecture under Jerry ­Milgram ‘61, SM ‘62, PhD ‘65–whom he has called “brilliant” and a “crazy man”–broadened Bertrand’s knowledge of fluid dynamics and tactical dinghy sailing.

“He was a crew member and a student of mine,” Milgram recalls. “I supervised his thesis. We worked one on one about how you could apply that theoretical knowledge to actual sailing. We would talk about the relationship between the hull resistance and the generation of side forces related to the force made by the sails and say, What’s the best thing for you to do after you tack, when you’re down in low speeds and you need to accelerate up to full speed?”

Bertrand remembers being surprised by the local talent on the Charles. “It felt initially like lambs to the slaughter; the local wind shifts and local knowledge were all-consuming until I started to get the hang of it,” he recalls. “The Tech Dinghy, although old-fashioned and slow, was a superb training and racing boat, since it was easy to rig and sail.

“MIT showed me many ways to think about and solve problems,” he continues. “There was a can-do mentality that was part of the culture. I also learned how to work with creative geniuses like Jerry, which was fundamental in my ability to later put together and work with a world-class group for our successful America’s Cup challenge in 1983.”

Milgram was asked to teach his legendary sailboat design course one last time before he retires this August. But he warns that the class is not all fun and games. “That’s what too many people think,” he says. “Then they come in and find out it’s a real MIT course, with real MIT learning, real tough stuff, and half of them drop out. They thought it was going to be fun, but a sailboat is a very complex system with complex engineering.”

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Mit alumni in the olympics: a brief history.

• Jul 19, 2021
• Slice of MIT

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At a school known for its Nobel Prize winners , it’s easy for athletes–even Olympians–to get overshadowed at MIT. But the Institute has a long Olympic history, from the first modern Olympic Games in Athens in 1896 to this year’s Summer Olympics in Tokyo.

According to research compiled by the Alumni Association, at least 40 MIT alumni representing 13 countries have qualified, participated, or served as an alternate in 13 sports at 29 Olympic Games, including two alumnae who are first-time Olympians competing in Tokyo.

Alexis Sablone MArch ’16 will represent the US in skateboarding, and Veronica Toro ’16 is Puerto Rico’s first-ever female Olympic rower.

In total, based on Association research, MIT alumni Olympians have won three gold, five silver, and five bronze medals. Scroll down to see more MIT Olympic facts and the most up-to-date list of alumni Olympians.

Facts and Figures: MIT Alumni at the Olympics  

• MIT’s first Olympian, Thomas Curtis 1894, won the gold medal for the US in the 110-meter hurdles at the first Olympics in 1896. Curtis recounted his Olympic experience in a July 1924 issue of MIT Technology Review .  
• Sablone is a member of the first-ever US Olympic skateboarding team, and she has been featured in the New York Times , Rolling Stone , and GQ .  
• In addition to being Puerto Rico’s first-ever female rower in the Olympics, Toro was named its rowing athlete of the year in 2014, 2015, 2016, and 2018.
• Prior to this year, the most recent MIT alumni to participate in the Games were AJ Edelman ’14, who competed for Israel in the men's skeleton, and Mahmoud Shaker Al-Abood ’04, MBA '08, who represented Monaco as an alternate in the two-man bobsleigh, both at the 2018 Winter Olympics.  
• American fencer Joseph Levis ’26 was MIT's first multiple medal winner, winning the silver medal in men's individual foil and the bronze medal in men's team foil in 1932. His son, Roberto Levis ’64, fenced for Puerto Rico at the 1972 games.  
• American short track speed skater Jordan Malone ’18 is the Institute’s other two-time medalist, having won bronze (2010) and silver (2014) before matriculating to MIT in 2015.  
• Three MIT alumni have competed in three separate Olympic Games: Joseph Levis (1928, 1932, 1936); Bermudian sailor Paula Lewin ’93 (1992, 1996, 2004), and Cypriot skier Alexis Photiades ’91, SM ’92 (1984, 1988, 1992).  
• In total, MIT alumni have earned 13 medals at the Olympics, including three gold medals: Curtis; Swedish fencer Johan Harmeberg '81 (1980); and American rower Alden "Zeke" Sanborn SM ’28 (1920).  
• Mark Smith '78 was one of the final runners carrying the Olympic Torch during the Opening Ceremonies when United States hosted the 1996 Summer Olympic Games in Atlanta.  
• Since 1992, MIT alumni have represented at least nine different countries: Bermuda; Bosnia and Herzegovina;  Cyprus; Great Britain; Israel; Lebanon; Puerto Rico; Trinidad and Tobago; and the United States.  
• The most common Olympic sports for alumni: rowing (11 alumni) and sailing (7).

MIT Alumni in the Olympic Games

Nate Ackerman PhD '04

Great Britain

Wrestling Men's Middleweight, Freestyle

Athens, 2004

Pat Antaki '84

Men's Skeleton

Turin, 2006

" 10 Questions with MIT Olympian Patrick Antaki ’84 "

John Bertrand SM '72

Sailing Mixed One Person Dinghy

Munich, 1972 Montreal, 1976 ( Bronze Medal )

Harry Blieden '57

Rome, 1960 (alternate)

Elizabeth Bradley '83, SM '86, PhD '92

United States

Rowing Women's Coxed Fours

Seoul, 1988

Peter Camejo '62

Sailing Mixed Two Person Keelboat

Thomas Curtis 1894

Men's 100 meters Men's 110 meters hurdles

Athens, 1896   ( Gold Medal )

" Olympic color from 1896 "

AJ Edelman '14

2018, PyeongChang

" MIT Alumnus Competes in Winter Olympics "

Ralph Evans '48

London, 1948 ( Silver Medal )

John Everett '76, PhD '91

Rowing Men's Coxed Eights

Montreal, 1976 Moscow, 1980

Jack Frailey '44

Rowing (coach)

Mexico, 1968 Montreal, 1976

Nicole Freedman '94

Cycling Women's Road Race

Sydney, 2000

Michelle Guerette MBA '12

Rowing Women's Quadruple Sculls Women's Single Sculls

Athens, 2004 Beijing, 2008 ( Silver Medal )

Janet Goldman '89

Speed Skating Women's 1,500 meters Women's 3,000 meters Women's 5,000 meters

Sarajevo, 1984 Calgary, 1988

Johan Harmenberg '81

Men's Fencing

Moscow, 1980 ( Gold Medal )

Larry Hough SM ’72

Rowing Coxless Pairs, Men

Mexico City, 1968 Munich, 1972 ( Silver Medal )

Joseph Levis '26

Fencing Men's Foil

Amsterdam, 1928 Los Angeles, 1932 ( Silver Medal, Bronze Medal ) Berlin, 1936 

Roberto Levis '64

Puerto Rico

Munich, 1972

Nedžad Lomigora SM '98

Bosnia and Herzegovina

Luge Singles, Men Lillehammer, 1994

Paula Lewin '93

Sailing Women's One Person Dinghy Women's Three Person Keelboat

Barcelona, 1992 Atlanta, 1996 Athens, 2004

Mary (Kellogg) Lyman SM '78

Montreal, 1976

Jordan Malone '18

Short Track Speed Skating Men's 5,000 Meters Relay

Vancouver, 2010 ( Bronze Medal ) Sochi, 2014 ( Silver Medal )

" 10 Questions with MIT Olympian Jordan Malone ’18 "

John Marvin '49

Melbourne, 1956 ( Bronze Medal )

Ed Melaika '53

Helsinki, 1952

Eric Olsen '41

Sailing Mixed Two Person Heavyweight Dinghy

Melbourne, 1956

Chinedum Osuji PhD '01

Trinidad and Tobago

Taekwondo Men's Welterweight

Alexis Photiades '91, SM '92

Alpine Skiing Men's Giant Slalom Men's Super G

Sarajevo, 1984 Calgary, 1988 Albertville, 1992

" 10 Questions with MIT Olympian Alexis Photiades ’91, SM ’92 "

Gary Piantedosi '76

Rowing Men's Coxless Fours     

Chester Riley '62

Tokyo, 1964

Alexis Sablone MArch ’16

Skateboarding Skateboard Street

Tokyo, 2021

Mahmoud Shaker Al-Abood '04, MBA '08

Bobsleigh Two-Man  (alternate)

Captain Alden "Zeke" Sanborn SM '28

Antwerp, 1920 ( Gold Medal )

Mark Smith '78

Fencing Men's Foil, Team

Moscow, 1980 Los Angeles, 1984

Ilkka Suvanto '68

Swimming Men's 400 Meters Freestyle Men's 1,500 Meters Freestyle Men's 100 Meters Medley Relay Men's 200 Meters Freestyle Relay Men's 400 Meters Individual Medley

Rome, 1960 Tokyo, 1964

Veronica Toro ’16

Puerto Rico Rowing Women’s Single Scull

“ From Rowing on the Charles to Rowing for Puerto Rico ”

Steve Tucker '91

Rowing Men's Lightweight Double Sculls

Sydney, 2000 Athens, 2004

Erland Van Lidth De Jeude '77

Men's Wrestling

Montreal, 1976 (alternate)

Nancy Vespoli SM '79

Moscow, 1980

Herb Voelcker '48

Shooting Men's Free Rifle, Three Positions, 300 meters  

Andrew Weaver MArch '86

Cycling Men's 100 Kilometers (Bronze Medal)

Los Angeles, 1984

This list includes MIT alumni who were Olympic coaches, Olympic alternates, and alumni who were members of the 1980 US Olympic team, which did not participate in the Games that year. MIT’s alumni Olympic records may be inexact. If there is alumnus Olympian that is not included in the list, please notify Slice of MIT in the comments below.

Updates: A version of this story was originally published in July 2012 and was updated in 2016, 2018, and 2021.

• Log in to post comments

Sat, 02/22/2014 8:20pm

How about some designation for those who competed for MIT? Most of the Olympian alums with only graduate degrees never wore an MIT uniform.

Sat, 02/22/2014 8:13pm

Erland Van Lidth De Jeude may have been an alternate, but he was not on the '76 Olympic Team.

Sat, 02/08/2014 8:12pm

Nice article, Jay. Would also have been nice to see Henry Steinbrenner's name mentioned in the article. He is on the MIT full list of Olympians, though his class year ('27) is missing. I had the pleasure of meeting Henry at an alumni reunion in Boston the weekend Steinbrenner Stadium was being dedicated, in 1978. Had a great conversation with him that day! Here's an article from 1996: http://web.mit.edu/newsoffice/1996/olymp1896-curtis.html You may want to consider finding stories from The Tech at the time these men and women participated in the Olympics and update your 2014 article with those links.

Fri, 02/07/2014 7:51pm

Didn't the US boycott the 1980 olympics?

Wed, 08/22/2012 6:51pm

Thank you for doing an article on this topic. Look for me, Gwendolyn Sisto SM 2010, in 2016 in women's weightlifting. I competed at the 2010 World University Championships, while a student at MIT. I placed in the top 3 63kg women at the 2012 US National Championships (fun fact: My results at the 2012 National Championships would have placed me 9th at the Olympics, which is higher than 2/3's of the team that the USA fielded in London for weightlifting).

Wed, 08/22/2012 6:20pm

Janet Goldman '89 was a two-time Olympic speedskater, competing in Sarjevo 1984 and Calgary 1988. http://tech.mit.edu/V104/PDF/N13.pdf http://www.usspeedskating.org/athletes/alumni/olympians?field_alumni_year_value=1984

Thomas L. De Fazio

Wed, 08/22/2012 6:47pm

Johnny Marvin won a bronze medal (3rd) in the Finn in 1956.

Wed, 08/01/2012 3:19pm

Gary Piantedosi, alternate, straight four crew, '76 Montreal, http://www.rowingrigs.com/pubsite/index.php?option=com_content&view=article&id=2&Itemid=3 John G Everett, men's eight crew, '76 Montreal and '80 Call the boathouse for other alumnae Olympians.

Mon, 08/08/2016 3:24pm

Hello Robert, Apologies for the oversight. Accumulating a comprehensive list of MIT alumni who have competed in the Olympics is an inexact science! We will updated accordingly and your name to our ever-growing list. Thank you--Jay London

Mon, 08/06/2012 1:27pm

Thanks again! I will personally forward this information over to DAPER and ask them to add it to their master list. Any more names, please keep them coming!

Fri, 02/07/2014 6:51pm

Thank you Joe, Sorting Olympians by their college affiliation is a difficult process, especially during the 1980 games! Thanks again and we will updated Mark on our lists. Jay London

Fri, 02/07/2014 5:12pm

Mark J. T. Smith, '78, was on the 1980 U.S. Men's Fencing Team.

Thu, 02/27/2014 2:34pm

Hi Beaver, That's a great idea. We may add another designation on the full list of Olympians page that indicates that the athletes participated in the Olympics while attending MIT. Thanks again, Jay

Thu, 02/27/2014 2:28pm

Hello alum, Olympic alternates proved especially tricky to confirm. Since they did not compete, this is no record of their participation. In Van Lidth De Jeude's case, we relied on a variety of sources, most notably a 1979 article in Sports Illustrated and a 1981 article in People magazine, both of which mention his Olympic participation. Thank you for reading and please let me know if you have any additional Olympic alumni. Jay

Mon, 02/10/2014 3:47pm

Hi Mike, Thanks for your note. We do have Mr. Steinbrenner on our list but did not outright mention him in the story because it's been difficult to determine which events he participated in. Some articles reference his competition in the 220-yard hurdles but that event has never been contested in the Olympics. Also, there is no mention in any archived <em>Tech</em> articles or in Mr. Steinbrenner's <em>New York Times</em> obituary about his Olympic participation. The website www.sports-reference.com/olympics does not have him listed as a U.S. Olympian. However, there are multiple articles that mention his Olympic participation, including the MIT News article that you reference. For that reason, we have kept him on our list. (I will add his graduation year.) Hopefully, we will find out more information about Mr. Steinbrenner's participation. Olympic records, especially in the early days, are very spotty, and we're working to find out more information. Thank you for reading and please let me know if you come across any additional MIT alumni Olympians! Sincerely, Jay

Thu, 08/23/2012 4:22pm

Good luck Gwen! Please keep us updated! Thanks for letting us know. Jay

Wed, 08/22/2012 6:52pm

Hi Thomas, Thanks for the addition. DAPER has Johnny Marvin listed as competing in 1956, not as a medalist. We'll make sure that gets updated.

David Silberstein

Wed, 02/14/2018 1:42pm

(No subject)

I thought he qualified for the '80 games but US boycotted.

Jason London

Wed, 02/14/2018 2:54pm

Hello! The US did not participate in the 1980 games in Moscow, but still recognizes anyone who qualified for the games as an official Olympian. Thank you for reading! --Jay

ROBERT LEVIS

Mon, 08/08/2016 6:26am

Hey.... you forgot about me.. Robert Levis'64, son of Joseph Levis'26,the 3-time Olympian I represented Puerto Rico in the 1972 Olympics in Munich.

Robert Levis

Mon, 08/08/2016 4:25pm

No apologies are needed. Thank you for all the fine work you are doing in recognizing us MIT Olympians.

Fri, 02/07/2014 9:34pm

Hi David, The U.S. did not participate in the 1980 Summer Games but any athletes who qualified for the games were recognized and honored by the U.S. Olympic Committee. Jay London

Wed, 08/01/2012 3:34pm

Hello - Gary Piantedosi '76 and John Everett '76 are listed in the MIT Olympic History graph that we link to in the second paragraph. Thanks again! It was a tough list to collect, as there is no master database of MIT alumni Olympians.

Hajime Sano

Wed, 02/28/2018 5:15am

1980 US Olympic Boycott

Yes, Jimmy Carter withheld the US Olympic Team from the 1980 Moscow Games, in response to the Soviet invasion of Afghanistan. In return, the Soviets boycotted the 1984 Los Angeles Games.

Sat, 07/31/2021 9:20pm

Moscow Olympics

Yes, the US boycotted the 1980 Moscow Olympics due to the Soviet Union's involvement in Afghanistan. In return, the USSR boycotted the 1984 Los Angeles Olympics. Athletes selected for the 1980 Games are still considered Olympians. One example is rower Anita DeFrantz. (I think she rowed for Princeton.) DeFrantz would late go on to run LA '84, the non-profit that administers the 1984 Olympics' profits for Southern California youth sports. I had the pleasure of meeting her when I was President of the Southern California Speed Skating Association. DeFrantz would go on to become one of the first female IOC board members.

Hans Hoeflein

Mon, 01/05/2015 3:49pm

I think Nick Newman '56 Course XIII competed in sailing in 1956

Wed, 08/22/2012 7:59pm

Thank you, Mike. I have contacted DAPER and asked to add Janet to the list. Thanks again!

Sat, 02/22/2014 6:34pm

Perhaps as a footnote, we should add that Oscar Hedland, MIT Track and Field Coach in the 1950's, though not an alumnus, was an Olympic miler.....ran in the 1922 Olympics, or sometime about then. He was a fine coach. Joe Davis Class of '61, Hurdles and High Jump

Sat, 02/22/2014 3:55pm

Jay, didn't MIT's track coach in the 1950s, Oscar Hedlund, have a connection with the Olympics -- maybe a runner back in the early 1900's. Seems to me that was the rumor.

Sun, 08/05/2012 3:53pm

Sorry about missing the Olympic History web page. It's still missing Erland Van Lidth De Jeude, '77 who went as an alternate to WRESTLE at Montreal in '76. See: http://en.wikipedia.org/wiki/Erland_Van_Lidth_De_Jeude

Wed, 08/22/2012 7:36pm

Nancy Vespoli, MS ChemEng 1979, Rowing, 1980 Also, the myth in the MIT weight room at the time was that the bent bar in the corner was left behind by Chris Taylor, Wrestling 1976. However, I can't find any historical information stating that he attended MIT. Perhaps you can look through MIT records to see if he was in fact, most likely, a graduate student.

Igor Belakovskiy

Sat, 02/10/2018 1:06am

The photo you have for Steve Tucker is actually his team mate, Greg Ruckman, who is actually an alum of that other school in Cambridge. :)

Mon, 02/12/2018 2:17am

Hi! Steve had been misidentified in that photo. Thank you for the correction and thank you for reading! --Jay London

Sat, 02/10/2018 4:50am

I am guessing that "Andrew Weaver MArch " may be a typo, also I was a classmate of Liz Bradley and I am almost certain her PhD (here listed as '82) did not come a year before her bachelor's in '83!

Mon, 02/12/2018 2:12am

Hello! In MIT nomenclature, MArch (with a capital A) stands for master's of architecture. Andrew received his master's of architecture from MIT in 1986. --Jay London

Elizabeth Bradley

Sat, 02/10/2018 11:21pm

The lightcone

I think you meant 1992 (not 1982) for my Ph.D. Liz Bradley

Mon, 02/12/2018 2:19am

PhD '92

Hi Liz! Thank you--we've updated your PhD year. Thanks for reading! --Jay London

George Mitsuoka

Mon, 02/12/2018 12:49pm

Faculty/Staff in the Olympics?

In the late 80s, Rafael Nickel, Gold Medalist in Team Epee for Germany in 1984, was a researcher and assistant coach for the MIT Fencing Team

John Everett

Sat, 02/24/2018 11:59pm

Go Beavers!

Chester Riley was alternate for Tokyo Olympics in 1964 (not 1962) John Everett '76 is also PhD '91 Mark Smith was also on 1980 Fencing Thanks.

Tue, 02/27/2018 9:44pm

Hi! We've updated our story. Thanks for your updates and thanks for reading. -- Jay London

Wed, 02/28/2018 5:21am

Jordan Malone

Great article on MIT Olympians! I first met Jordan Malone '18 when he came to train with us (Southern California Speed Skating Association) around 2002. I am a skater, coach, and administrator with the SCSSA (1984-present). I also helped him navigate the MIT admissions process after he retired from international competition in 2014. I see there are mixed references to his sport as both short track speed skating and speed skating. His sport is in fact short track speed skating. I see a speed skating logo in the 2014 By The Numbers chart, but not short track speed skating. I don't know if the chart has been updated since then. If you are still tracking MIT Olympians, or have developed the MIT Olympians database you previously mentioned, please update to reflect his sport. Thank you! Hajime Sano '82

Nedzad Lomigora

Tue, 07/20/2021 6:38pm

Olympian alumnus not included in the list - Nedzad Lomigora

There is an Olympian alumnus that competed in Lillehammer Olympics in 1994 in Luge sport representing Bosnia and Herzegovina. Nedzad Lomigora 1999, SM - Master Of Science, 2M - Mech Eng - Sm/Eng See his profile here: https://olympics.com/en/athletes/nedzad-lomigora

Marko Slusarczuk

Sat, 07/31/2021 4:23pm

Steve Cucchiaro

You omitted Steve Cucchiaro '74 who was on the Olympic sailing team. https://spectrum.mit.edu/spring-2010/giving-back-6/

Mon, 08/02/2021 3:27pm

Contact MIT for more info!

Hi Steve, Please reach out to me at [email protected] and we can include your name in the list. Thank you!

Sat, 07/31/2021 6:02pm

Thanks so much Jay for compiling this list and to everyone for contributing! This is so cool!

Paulo Correia

Sun, 08/01/2021 11:53am

Nicole Freedman did not graduate from MIT in 1994. She transferred to Stanford after 2 years and graduated there. Source: me. I befriended Nicole when I got to Stanford for my Masters, we were on the cycling team together there, and one of my claims to fame is that over many dozens of town line sprints, Nicole only beat me once.

Fri, 12/01/2023 12:08am

Thanks for the great MIT Olympian article. A lot has been added in the last few years. One minor correction to note - Jordan Malone should be listed as a short track speed skater, not a speed skater. In Olympic parlance, speed skating refers to long track speed skating. They are related but separate sports. When Jordan came to LA to train with our coach Wilma Boomstra (around 2001), he found out I was an MIT alum and told me of his desire to attend MIT when he retired from short track. Fast forward to 2014, I showed him around MIT the summer right after he retired from competition. I like to think I had a small role in his attending and graduation from MIT. Thank you! Sincerely, Hajime Sano '82 Past President, Southern California Speed Skating Association

Fri, 12/01/2023 6:33pm

Thank you for pointing out this inconsistency, Hajime, we have fixed the incorrect reference under Facts and Figures to the correct "short track speed skating."

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7.4: Fluctuation-dissipation Theorem

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• Konstantin K. Likharev
• Stony Brook University

Similar questions may be asked about a more general situation, when the Hamiltonian \(\hat{H}_{s}\) of the system of interest \((s)\) , in the composite Hamiltonian (68), is not specified at all, but the interaction between that system and its environment still has a bilinear form similar to Eqs. (70) and (6.130): \[\hat{H}_{\text {int }}=-\hat{F}\{\lambda\} \hat{x},\] where \(x\) is some observable of our system \(s\) - say, its generalized coordinate or generalized momentum. It may look incredible that in this very general situation one still can make a very simple and powerful statement about the statistical properties of the generalized force \(F\) , under only two (interrelated) conditions - which are satisfied in a huge number of cases of interest:

(i) the coupling of system \(s\) of interest to its environment \(e\) is weak - in the sense that the perturbation theory (see Chapter 6 ) is applicable, and

(ii) the environment may be considered as staying in thermodynamic equilibrium, with a certain temperature \(T\) , regardless of the process in the system of interest. \({ }^{31}\)

This famous statement is called the fluctuation-dissipation theorem (FDT). \({ }^{32}\) Due to the importance of this fundamental result, let me derive it. \({ }^{33}\) Since by writing Eq. (68) we treat the whole system \((s+e)\) as a Hamiltonian one, we may use the Heisenberg equation (4.199) to write \[i \hbar \dot{\hat{F}}=[\hat{F}, \hat{H}]=\left[\hat{F}, \hat{H}_{e}\right],\] because, as was discussed in the last section, operator \(\hat{F}\{\lambda\}\) commutes with both \(\hat{H}_{s}\) and \(\hat{x}\) . Generally, very little may be done with this equation, because the time evolution of the environment’s Hamiltonian depends, in turn, on that of the force. This is where the perturbation theory becomes indispensable. Let us decompose the force operator into the following sum: \[\hat{F}\{\lambda\}=\langle\hat{F}\rangle+\hat{\widetilde{F}}(t), \text { with }\langle\hat{\widetilde{F}}(t)\rangle=0,\] where (here and on, until further notice) the sign \(\langle\ldots\rangle\) means the statistical averaging over the environment alone, i.e. over an ensemble with absolutely similar evolutions of the system \(s\) , but random states of its environment. \({ }^{34}\) From the point of view of the system \(s\) , the first term of the sum (still an operator!) describes the average response of the environment to the system dynamics (possibly, including such irreversible effects as friction), and has to be calculated with a proper account of their interaction - as we will do later in this section. On the other hand, the last term in Eq. (92) represents random fluctuations of the environment, which exist even in the absence of the system \(s\) . Hence, in the first non-zero approximation in the interaction strength, the fluctuation part may be calculated ignoring the interaction, i.e. treating the environment as being in thermodynamic equilibrium: \[i \hbar \dot{\tilde{F}}=\left[\hat{\widetilde{F}},\left.\hat{H}_{e}\right|_{\mathrm{eq}}\right] .\] Since in this approximation the environment’s Hamiltonian does not have an explicit dependence on time, the solution of this equation may be written by combining Eqs. (4.190) and (4.175): \[\hat{F}(t)=\exp \left\{+\left.\frac{i}{\hbar} \hat{H}_{e}\right|_{\mathrm{eq}} t\right\} \hat{F}(0) \exp \left\{-\left.\frac{i}{\hbar} \hat{H}_{e}\right|_{\mathrm{eq}} t\right\} .\] Let us use this relation to calculate the correlation function of the fluctuations \(F(t)\) , defined similarly to Eq. (80), but taking care of the order of the time arguments (very soon we will see why): \[\left\langle\widetilde{F}(t) \widetilde{F}\left(t^{\prime}\right)\right\rangle=\left\langle\exp \left\{+\frac{i}{\hbar} \hat{H}_{e} t\right\} \hat{F}(0) \exp \left\{-\frac{i}{\hbar} \hat{H}_{e} t\right\} \exp \left\{+\frac{i}{\hbar} \hat{H}_{e} t^{\prime}\right\} \hat{F}(0) \exp \left\{-\frac{i}{\hbar} \hat{H}_{e} t^{\prime}\right\}\right\rangle .\] (Here, for the notation brevity, the thermal equilibrium of the environment is just implied.) We may calculate this expectation value in any basis, and the best choice for it is evident: in the environment’s stationary-state basis, the density operator of the environment, its Hamiltonian, and hence the exponents in Eq. (95) are all represented by diagonal matrices. Using Eq. (5), the correlation function becomes \[\begin{aligned} \left\langle\widetilde{F}(t) \widetilde{F}\left(t^{\prime}\right)\right\rangle &=\operatorname{Tr}\left[\hat{w} \exp \left\{+\frac{i}{\hbar} \hat{H}_{e} t\right\} \hat{F}(0) \exp \left\{-\frac{i}{\hbar} \hat{H}_{e} t\right\} \exp \left\{+\frac{i}{\hbar} \hat{H}_{e} t^{\prime}\right\} \hat{F}(0) \exp \left\{-\frac{i}{\hbar} \hat{H}_{e} t^{\prime}\right\}\right] \\ & \equiv \sum_{n}\left[\hat{w} \exp \left\{+\frac{i}{\hbar} \hat{H}_{e} t\right\} \hat{F}(0) \exp \left\{-\frac{i}{\hbar} \hat{H}_{e} t\right\} \exp \left\{+\frac{i}{\hbar} \hat{H}_{e} t^{\prime}\right\} \hat{F}(0) \exp \left\{-\frac{i}{\hbar} \hat{H}_{e} t^{\prime}\right\}\right]_{n n} \\ &=\sum_{n, n^{\prime}} W_{n} \exp \left\{+\frac{i}{\hbar} E_{n} t\right\} \hat{F}_{n n^{\prime}} \exp \left\{-\frac{i}{\hbar} E_{n^{\prime}} t\right\} \exp \left\{+\frac{i}{\hbar} E_{n^{\prime}} t^{\prime}\right\} \hat{F}_{n^{\prime} n} \exp \left\{-\frac{i}{\hbar} E_{n} t^{\prime}\right\} \\ & \equiv \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \exp \left\{+\frac{i}{\hbar}\left(E_{n}-E_{n^{\prime}}\right)\left(t-t^{\prime}\right)\right\} \end{aligned}\] Here \(W_{n}\) are the Gibbs distribution probabilities given by Eq. (24), with the environment’s temperature \(T\) , and \(F_{n n^{\prime}} \equiv F_{n n}\) ( \((0)\) are the Schrödinger-picture matrix elements of the interaction force operator.

We see that though the correlator (96) is a function of the difference \(\tau \equiv t-t\) ’ only (as it should be for fluctuations in a macroscopically stationary system), it may depend on the order of its arguments. This is why let us mark this particular correlation function with the upper index "+", \[K_{F}^{+}(\tau) \equiv\left\langle\widetilde{F}(t) \widetilde{F}\left(t^{\prime}\right)\right\rangle=\sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \exp \left\{+\frac{i \widetilde{E} \tau}{\hbar}\right\}, \quad \text { where } \widetilde{E} \equiv E_{n}-E_{n^{\prime}}\] while its counterpart, with the swapped times \(t\) and \(t\) ’, with the upper index "-": \[K_{F}^{-}(\tau) \equiv K_{F}^{+}(-\tau)=\left\langle\widetilde{F}\left(t^{\prime}\right) \widetilde{F}(t)\right\rangle=\sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \exp \left\{-\frac{i \widetilde{E} \tau}{\hbar}\right\} .\] So, in contrast with classical processes, in quantum mechanics the correlation function of fluctuations \(\widetilde{F}\) is not necessarily time-symmetric: \[K_{F}^{+}(\tau)-K_{F}^{-}(\tau) \equiv K_{F}^{+}(\tau)-K_{F}^{+}(-\tau)=\left\langle\widetilde{F}(t) \widetilde{F}\left(t^{\prime}\right)-\widetilde{F}\left(t^{\prime}\right) \widetilde{F}(t)\right\rangle=2 i \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \sin \frac{\widetilde{E} \tau}{\hbar} \neq 0,\] so that \(\hat{F}(t)\) gives one more example of a Heisenberg-picture operator whose "values", taken in different moments of time, generally do not commute - see Footnote 49 in Chapter 4. (A good sanity check here is that at \(\tau=0\) , i.e. at \(t=t^{\prime}\) , the difference (99) between \(K_{F}^{+}\) and \(K_{F}^{-}\) vanishes.) Now let us return to the force operator’s decomposition (92), and calculate its first (average) component. To do that, let us write the formal solution of Eq. (91) as follows: \[\hat{F}(t)=\frac{1}{i \hbar} \int_{-\infty}^{t}\left[\hat{F}\left(t^{\prime}\right), \hat{H}_{e}\left(t^{\prime}\right)\right] d t^{\prime} .\] On the right-hand side of this relation, we still cannot treat the Hamiltonian of the environment as an unperturbed (equilibrium) one, even if the effect of our system \((s)\) on the environment is very weak, because this would give zero statistical average of the force \(F(t)\) . Hence, we should make one more step of our perturbative treatment, taking into account the effect of the force on the environment. To do this, let us use Eqs. (68) and (90) to write the (so far, exact) Heisenberg equation of motion for the environment’s Hamiltonian, \[i \hbar \dot{\hat{H}}_{e}=\left[\hat{H}_{e}, \hat{H}\right]=-\hat{x}\left[\hat{H}_{e}, \hat{F}\right],\] and its formal solution, similar to Eq. (100), but for time \(t\) ’ rather than \(t\) : \[\hat{H}_{e}\left(t^{\prime}\right)=-\frac{1}{i \hbar} \int_{-\infty}^{t^{\prime}} \hat{x}\left(t^{\prime \prime}\right)\left[\hat{H}_{e}\left(t^{\prime \prime}\right), \hat{F}\left(t^{\prime \prime}\right)\right] d t^{\prime \prime} .\] Plugging this equality into the right-hand side of Eq. (100), and averaging the result (again, over the environment only!), we get \[\langle\hat{F}(t)\rangle=\frac{1}{\hbar^{2}} \int_{-\infty}^{t} d t^{\prime} \int_{-\infty}^{t^{\prime}} d t^{\prime \prime} \hat{x}\left(t^{\prime \prime}\right)\left\langle\left[\hat{F}\left(t^{\prime}\right),\left[\hat{H}_{e}\left(t^{\prime \prime}\right), \hat{F}\left(t^{\prime \prime}\right)\right]\right]\right\rangle .\] This is still an exact result, but now it is ready for an approximate treatment, implemented by averaging in its right-hand side over the unperturbed (thermal-equilibrium) state of the environment. This may be done absolutely similarly to that in Eq. (96), at the last step using Eq. (94): \[\begin{aligned} &\left\langle\left[\hat{F}\left(t^{\prime}\right),\left[\hat{H}_{e}\left(t^{\prime \prime}\right), \hat{F}\left(t^{\prime \prime}\right)\right]\right\rfloor=\operatorname{Tr}\left\{\mathrm{w}\left[\mathrm{F}\left(t^{\prime}\right),\left[\mathrm{H}_{e} \mathrm{~F}\left(t^{\prime \prime}\right)\right]\right]\right\}\right. \\ &\equiv \operatorname{Tr}\left\{\mathrm{w}\left[\mathrm{F}\left(t^{\prime}\right) \mathrm{H}_{e} \mathrm{~F}\left(t^{\prime \prime}\right)-\mathrm{F}\left(t^{\prime}\right) \mathrm{F}\left(t^{\prime \prime}\right) \mathrm{H}_{e}-\mathrm{H}_{e} \mathrm{~F}\left(t^{\prime \prime}\right) \mathrm{F}\left(t^{\prime}\right)+\mathrm{F}\left(t^{\prime \prime}\right) \mathrm{H}_{e} \mathrm{~F}\left(t^{\prime}\right)\right]\right\} \\ &=\sum_{n, n^{\prime}} W_{n}\left[F_{n n^{\prime}}\left(t^{\prime}\right) E_{n^{\prime}} F_{n^{\prime} n}\left(t^{\prime \prime}\right)-F_{n n^{\prime}}\left(t^{\prime}\right) F_{n^{\prime} n}\left(t^{\prime \prime}\right) E_{n}-E_{n} F_{n n^{\prime}}\left(t^{\prime \prime}\right) F_{n^{\prime} n}\left(t^{\prime}\right)+F_{n n^{\prime}}\left(t^{\prime \prime}\right) E_{n^{\prime}} F_{n^{\prime} n}\left(t^{\prime \prime}\right)\right] \\ &\equiv-\sum_{n, n^{\prime}} W_{n} \widetilde{E}\left|F_{n n^{\prime}}\right|^{2}\left[\exp \left\{\frac{i \widetilde{E}\left(t^{\prime}-t^{\prime \prime}\right)}{\hbar}\right\}+\text { c.c. }\right] . \end{aligned}\] Now, if we try to integrate each term of this sum, as Eq. (103) seems to require, we will see that the lower-limit substitution (at \(t^{\prime}, t^{\prime \prime} \rightarrow-\infty\) ) is uncertain because the exponents oscillate without decay. This mathematical difficulty may be overcome by the following physical reasoning. As illustrated by the example considered in the previous section, coupling to a disordered environment makes the "memory horizon" of the system of our interest \((s)\) finite: its current state does not depend on its history beyond a certain time scale. \({ }^{35}\) As a result, the function under the integrals of Eq. (103), i.e. the sum (104), should self-average at a certain finite time. A simplistic technique for expressing this fact mathematically is just dropping the lower-limit substitution; this would give the correct result for Eq. (103). However, a better (mathematically more acceptable) trick is to first multiply the functions under the integrals by, respectively, \(\exp \left\{\varepsilon\left(t-t^{\prime}\right)\right\}\) and \(\exp \left\{\varepsilon\left(t^{\prime}-t^{\prime \prime}\right)\right\}\) , where \(\varepsilon\) is a very small positive constant, then carry out the integration, and after that follow the limit \(\varepsilon \rightarrow 0\) . The physical justification of this procedure may be provided by saying that the system’s behavior should not be affected if its interaction with the environment was not kept constant but rather turned on gradually - say, exponentially with an infinitesimal rate \(\varepsilon\) . With this modification, Eq. (103) becomes \[\langle\hat{F}(t)\rangle=-\frac{1}{\hbar^{2}} \sum_{n, n^{\prime}} W_{n} \widetilde{E}\left|F_{n n^{\prime}}\right|^{2} \lim _{\varepsilon \rightarrow 0} \int_{-\infty}^{t} d t^{\prime} \int_{-\infty}^{t^{\prime}} d t^{\prime \prime} \hat{x}\left(t^{\prime \prime}\right)\left[\exp \left\{\frac{i \widetilde{E}\left(t^{\prime}-t^{\prime \prime}\right)}{\hbar}+\varepsilon\left(t^{\prime \prime}-t\right)\right\}+\text { c.c. }\right] \text {. }\] This double integration is over the area shaded in Fig. 6, which makes it obvious that the order of integration may be changed to the opposite one as \[\int_{-\infty}^{t} d t^{\prime} \int_{-\infty}^{t^{\prime}} d t^{\prime \prime} \ldots=\int_{-\infty}^{t} d t^{\prime \prime} \int_{t^{\prime \prime}}^{t} d t^{\prime} \ldots=\int_{-\infty}^{t} d t^{\prime \prime} \int_{t^{\prime \prime}-t}^{0} d\left(t^{\prime}-t\right) \ldots \equiv \int_{-\infty}^{t} d t^{\prime \prime} \int_{0}^{\tau} d \tau^{\prime} \ldots,\] where \(\tau^{\prime} \equiv t-t^{\prime}\) , and \(\tau \equiv t-t^{\prime \prime}\) .

As a result, Eq. (105) may be rewritten as a single integral, \[\langle\hat{F}(t)\rangle=\int_{-\infty}^{t} G\left(t-t^{\prime \prime}\right) \hat{x}\left(t^{\prime \prime}\right) d t^{\prime \prime} \equiv \int_{0}^{\infty} G(\tau) \hat{x}(t-\tau) d \tau,\] whose kernel, \[\begin{aligned} G(\tau>0) & \equiv-\frac{1}{\hbar^{2}} \sum_{n, n^{\prime}} W_{n} \widetilde{E}\left|F_{n n^{\prime}}\right|^{2} \lim _{\varepsilon \rightarrow 0} \int_{0}^{\tau}\left[\exp \left\{\frac{i \widetilde{E}\left(\tau-\tau^{\prime}\right)}{\hbar}-\varepsilon \tau\right\}+\text { c.c. }\right] d \tau^{\prime} \\ &=\lim _{\varepsilon \rightarrow 0} \frac{2}{\hbar} \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \sin \frac{\widetilde{E} \tau}{\hbar} e^{-\varepsilon \tau} \equiv \frac{2}{\hbar} \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \sin \frac{\widetilde{E} \tau}{\hbar} \end{aligned}\] does not depend on the particular law of evolution of the system \((s)\) under study, i.e. provides a general characterization of its coupling to the environment.

In Eq. (107) we may readily recognize the most general form of the linear response of a system (in our case, the environment), taking into account the causality principle, where \(G(\tau)\) is the response function (also called the "temporal Green’s function") of the environment. Now comparing Eq. (108) with Eq. (99), we get a wonderfully simple universal relation, \[\langle[\hat{\tilde{F}}(\tau), \hat{\widetilde{F}}(0)]\rangle=i \hbar G(\tau) .\] that emphasizes once again the quantum nature of the correlation function’s time asymmetry. (This relation, called the Green-Kubo (or just "Kubo") formula after the works by Melville Green (1954) and Ryogo Kubo (1957), does not come up in the easier derivations of the FDT, mentioned in the beginning of this section.)

However, for us the relation between the function \(G(\tau)\) and the force’s anti-commutator, \[\left\langle\{\hat{\widetilde{F}}(t+\tau), \hat{\tilde{F}}(t)\} \equiv\langle\hat{\widetilde{F}}(t+\tau) \hat{\tilde{F}}(t)+\hat{\tilde{F}}(t) \hat{\widetilde{F}}(t+\tau)\rangle \equiv K_{F}^{+}(\tau)+K_{F}^{-}(\tau),\right.\] is much more important, because of the following reason. Eqs. (97)-(98) show that the so-called symmetrized correlation function, \[\begin{aligned} K_{F}(\tau) & \equiv \frac{K_{F}^{+}(\tau)+K_{F}^{-}(\tau)}{2}=\frac{1}{2}\langle\{\hat{\widetilde{F}}(\tau), \hat{\widetilde{F}}(0)\}\rangle=\lim _{\varepsilon \rightarrow 0} \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \cos \frac{\widetilde{E} \tau}{\hbar} e^{-2 \varepsilon|\tau|} \\ & \equiv \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \cos \frac{\widetilde{E} \tau}{\hbar} \end{aligned}\] which is an even function of the time difference \(\tau\) , looks very similar to the response function (108), "only" with another trigonometric function under the sum, and a constant front factor. \({ }^{36}\) This similarity may be used to obtain a direct algebraic relation between the Fourier images of these two functions of \(\tau\) . Indeed, the function (111) may be represented as the Fourier integral \({ }^{37}\) \[K_{F}(\tau)=\int_{-\infty}^{+\infty} S_{F}(\omega) e^{-i \omega \tau} d \omega=2 \int_{0}^{+\infty} S_{F}(\omega) \cos \omega \tau d \omega,\] with the reciprocal transform \[S_{F}(\omega)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} K_{F}(\tau) e^{i \omega \tau} d \tau=\frac{1}{\pi} \int_{0}^{+\infty} K_{F}(\tau) \cos \omega \tau d \tau,\] of the symmetrized spectral density of the variable \(F\) , defined as \[S_{F}(\omega) \delta\left(\omega-\omega^{\prime}\right) \equiv \frac{1}{2}\left\langle\hat{F}_{\omega} \hat{F}_{-\omega^{\prime}}+\hat{F}_{-\omega^{\prime}} \hat{F}_{\omega}\right\rangle \equiv \frac{1}{2}\left\langle\left\{\hat{F}_{\omega}, \hat{F}_{-\omega^{\prime}}\right\}\right\rangle,\] where the function \(\hat{F}_{\omega}\) (also a Heisenberg operator rather than a \(c\) -number!) is defined as \[\hat{F}_{\omega} \equiv \frac{1}{2 \pi} \int_{-\infty}^{+\infty} \hat{F}(t) e^{i \omega t} d t, \quad \text { so that } \hat{F}(t)=\int_{-\infty}^{+\infty} \hat{F}_{\omega} e^{-i \omega t} d \omega .\] The physical meaning of the function \(S_{F}(\omega)\) becomes clear if we write Eq. (112) for the particular case \(\tau=0\) : \[K_{F}(0) \equiv\left\langle\hat{\widetilde{F}}^{2}\right\rangle=\int_{-\infty}^{+\infty} S_{F}(\omega) d \omega=2 \int_{0}^{+\infty} S_{F}(\omega) d \omega\] This formula infers that if we pass the function \(F(t)\) through a linear filter cutting from its frequency spectrum a narrow band \(d \omega\) of physical (positive) frequencies, then the variance \(\left\langle F_{\mathrm{f}}^{2}\right\rangle\) of the filtered signal \(F_{\mathrm{f}}(t)\) would be equal to \(2 S_{F}(\omega) d \omega\) - hence the name "spectral density". 38

Let us use Eqs. (111) and (113) to calculate the spectral density of fluctuations \(\widetilde{F}(t)\) in our model, using the same \(\varepsilon\) -trick as at the deviation of Eq. (108), to quench the upper-limit substitution: \[\begin{aligned} S_{F}(\omega) &=\sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \frac{1}{2 \pi} \lim _{\varepsilon \rightarrow 0} \int_{-\infty}^{+\infty} \cos \frac{\widetilde{E} \tau}{\hbar} e^{-\varepsilon \mid \tau} \mid e^{i \omega \tau} d \tau \\ & \equiv \frac{1}{2 \pi} \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \lim _{\varepsilon \rightarrow 0} \int_{0}^{+\infty}\left[\exp \left\{\frac{i \widetilde{E} \tau}{\hbar}\right\}+\text { c.c. }\right] e^{-\varepsilon \tau} e^{i \omega \tau} d \tau \\ &=\frac{1}{2 \pi} \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \lim _{\varepsilon \rightarrow 0}\left[\frac{1}{i(\widetilde{E} / \hbar+\omega)-\varepsilon}+\frac{1}{i(-\widetilde{E} / \hbar+\omega)-\varepsilon}\right] . \end{aligned}\] Now it is a convenient time to recall that each of the two summations here is over the eigenenergies of the environment, whose spectrum is virtually continuous because of its large size, so that we may transform each sum into an integral - just as this was done in Sec. 6.6: \[\sum_{n} \ldots \rightarrow \int \ldots d n=\int \ldots \rho\left(E_{n}\right) d E_{n},\] where \(\rho(E) \equiv d n / d E\) is the environment’s density of states at a given energy. This transformation yields \[S_{F}(\omega)=\frac{1}{2 \pi} \lim _{\varepsilon \rightarrow 0} \int d E_{n} W\left(E_{n}\right) \rho\left(E_{n}\right) \int d E_{n^{\prime}} \rho\left(E_{n^{\prime}}\right)\left|F_{n n^{\prime}}\right|^{2}\left[\frac{1}{i(\widetilde{E} / \hbar-\omega)-\varepsilon}+\frac{1}{i(-\widetilde{E} / \hbar-\omega)-\varepsilon}\right] .\] Since the expression inside the square bracket depends only on a specific linear combination of two energies, namely on \(\widetilde{E} \equiv E_{n}-E_{n^{\prime}}\) , it is convenient to introduce also another, linearly-independent combination of the energies, for example, the average energy \(\bar{E} \equiv\left(E_{n}+E_{n^{\prime}}\right) / 2\) , so that the state energies may be represented as \[E_{n}=\bar{E}+\frac{\widetilde{E}}{2}, \quad E_{n^{\prime}}=\bar{E}-\frac{\widetilde{E}}{2} .\] With this notation, Eq. (119) becomes \[\begin{gathered} S_{F}(\omega)=-\frac{\hbar}{2 \pi} \lim _{\varepsilon \rightarrow 0} \int d \bar{E}\left[\int d \widetilde{E} W\left(\bar{E}+\frac{\widetilde{E}}{2}\right) \rho\left(\bar{E}+\frac{\widetilde{E}}{2}\right) \rho\left(\bar{E}-\frac{\widetilde{E}}{2}\right)\left|F_{n n^{\prime}}\right|^{2} \frac{1}{i(\widetilde{E}-\hbar \omega)-\hbar \varepsilon}\right. \\ \left.+\int d \widetilde{E} W\left(\bar{E}+\frac{\widetilde{E}}{2}\right) \rho\left(\bar{E}+\frac{\widetilde{E}}{2}\right) \rho\left(\bar{E}-\frac{\widetilde{E}}{2}\right)\left|F_{n n^{\prime}}\right|^{2} \frac{1}{i(-\widetilde{E}-\hbar \omega)-\hbar \varepsilon}\right] . \end{gathered}\] Due to the smallness of the parameter \(\hbar \varepsilon\) (which should be much smaller than all genuine energies of the problem, including \(k_{\mathrm{B}} T, \hbar \omega, E_{n}\) , and \(E_{n}\) ), each of the internal integrals in Eq. (121) is dominated by an infinitesimal vicinity of one point, \(\widetilde{E}_{\pm}=\pm \hbar \omega .\) In these vicinities, the state densities, the matrix elements, and the Gibbs probabilities do not change considerably, and may be taken out of the integral, which may be then worked out explicitly: 39 \[\begin{aligned} S_{F}(\omega) &=-\frac{\hbar}{2 \pi} \lim _{\varepsilon \rightarrow 0} \int d \bar{E} \rho_{+} \rho_{-}\left[W_{+}\left|F_{+}\right|^{2} \int_{-\infty}^{+\infty} \frac{d \widetilde{E}}{i(\widetilde{E}-\hbar \omega)-\hbar \varepsilon}+W_{-}\left|F_{-}\right|^{2} \int_{-\infty}^{+\infty} \frac{d \widetilde{E}}{i(-\widetilde{E}-\hbar \omega)-\hbar \varepsilon}\right] \\ & \equiv-\frac{\hbar}{2 \pi} \lim _{\varepsilon \rightarrow 0} \int d \bar{E} \rho_{+} \rho_{-}\left[W_{+}\left|F_{+}\right|^{2} \int_{-\infty}^{+\infty} \frac{-i(\widetilde{E}-\hbar \omega)-\hbar \varepsilon}{(\widetilde{E}-\hbar \omega)^{2}+(\hbar \varepsilon)^{2}} d \widetilde{E}+W_{-}\left|F_{-}\right|^{2} \int_{-\infty}^{+\infty} \frac{i(\widetilde{E}+\hbar \omega)-\hbar \varepsilon}{(\widetilde{E}+\hbar \omega)^{2}+(\hbar \varepsilon)^{2}} d \widetilde{E}\right] \\ &=\frac{\hbar}{2} \int \rho_{+} \rho_{-}\left[W_{+}\left|F_{+}\right|^{2}+W_{-}\left|F_{-}\right|^{2}\right] d \bar{E}, \end{aligned}\] where the indices \(\pm\) mark the functions’ values at the special points \(\widetilde{E}_{\pm}=\pm \hbar \omega\) , i.e. \(E_{n}=E_{n}^{\prime} \pm \hbar \omega\) . The physics of these points becomes simple if we interpret the state \(n\) , for which the equilibrium Gibbs distribution function equals \(W_{n}\) , as the initial state of the environment, and \(n\) ’ as its final state. Then the top-sign point corresponds to \(E_{n}{ }^{\prime}=E_{n}-\hbar \omega\) , i.e. to the result of emission of one energy quantum \(\hbar \omega\) of the "observation" frequency \(\omega\) by the environment to the system \(s\) of our interest, while the bottom-sign point \(E_{n^{\prime}}=E_{n}+\hbar \omega\) , corresponds to the absorption of such quantum by the environment. As Eq. (122) shows, both processes give similar, positive contributions into the force fluctuations.The situation is different for the Fourier image of the response function \(G(\tau),{ }^{40}\) \[\chi(\omega) \equiv \int_{0}^{+\infty} G(\tau) e^{i \omega \tau} d \tau,\] that is usually called either the generalized susceptibility or the response function - in our case, of the environment. Its physical meaning is that according to Eq. (107), the complex function \(\chi(\omega)=\chi^{\prime}(\omega)+\) \(i \chi^{\prime \prime}(\omega)\) relates the Fourier amplitudes of the generalized coordinate and the generalized force: \({ }^{41}\) \[\left\langle\hat{F}_{\omega}\right\rangle=\chi(\omega) \hat{x}_{\omega} .\] The physics of its imaginary part \(\chi\) " \((\omega)\) is especially clear. Indeed, if \(x_{\omega}\) represents a sinusoidal classical process, say \[x(t)=x_{0} \cos \omega t \equiv \frac{x_{0}}{2} e^{-i \omega t}+\frac{x_{0}}{2} e^{+i \omega t}, \text { i.e. } x_{\omega}=x_{-\omega}=\frac{x_{0}}{2},\] then, in accordance with the correspondence principle, Eq. (124) should hold for the \(c\) -number complex amplitudes \(F_{\omega}\) and \(x_{\omega}\) , enabling us to calculate the time dependence of the force as \[\begin{aligned} F(t) &=F_{\omega} e^{-i \omega t}+F_{-\omega} e^{+i \omega t}=\chi(\omega) x_{\omega} e^{-i \omega t}+\chi(-\omega) x_{-\omega} e^{+i \omega t}=\frac{x_{0}}{2}\left[\chi(\omega) e^{-i \omega t}+\chi^{*}(\omega) e^{+i \omega t}\right] \\ &=\frac{x_{0}}{2}\left[\left(\chi^{\prime}+i \chi^{\prime \prime}\right) e^{-i \omega t}+\left(\chi^{\prime}-i \chi^{\prime \prime}\right) e^{+i \omega t}\right] \equiv x_{0}\left[\chi^{\prime}(\omega) \cos \omega t+\chi^{\prime \prime}(\omega) \sin \omega t\right] \end{aligned}\] We see that \(\chi\) ’" \((\omega)\) weighs the force’s part (frequently called quadrature) that is \(\pi / 2\) -shifted from the coordinate \(x\) , i.e. is in phase with its velocity, and hence characterizes the time-average power flow from the system into its environment, i.e. the energy dissipation rate: 42 \[\overline{\mathscr{P}}=\overline{-F(t) \dot{x}(t)}=\overline{-x_{0}\left[\chi^{\prime}(\omega) \cos \omega t+\chi^{\prime \prime}(\omega) \sin \omega t\right]\left(-\omega x_{0} \sin \omega t\right)}=\frac{x_{0}^{2}}{2} \omega \chi^{\prime \prime}(\omega) .\] Let us calculate this function from Eqs. (108) and (123), just as we have done for the spectral density of fluctuations: \[\begin{aligned} \chi^{\prime \prime}(\omega) &=\operatorname{Im} \int_{0}^{+\infty} G(\tau) e^{i \omega \tau} d \tau=\frac{2}{\hbar} \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \lim _{\varepsilon \rightarrow 0} \operatorname{Im} \int_{0}^{+\infty} \frac{1}{2 i}\left(\exp \left\{i \frac{\widetilde{E} \tau}{\hbar}\right\}-\text { c.c. }\right) e^{i \omega \tau} e^{-\varepsilon \tau} d \tau \\ &=\sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \lim _{\varepsilon \rightarrow 0} \operatorname{Im}\left(\frac{1}{-\widetilde{E}-\hbar \omega-i \hbar \varepsilon}-\frac{1}{\widetilde{E}-\hbar \omega-i \hbar \varepsilon}\right) \\ & \equiv \sum_{n, n^{\prime}} W_{n}\left|F_{n n^{\prime}}\right|^{2} \lim _{\varepsilon \rightarrow 0}\left(\frac{\hbar \varepsilon}{(\widetilde{E}+\hbar \omega)^{2}+(\hbar \varepsilon)^{2}}-\frac{\hbar \varepsilon}{(\widetilde{E}-\hbar \omega)^{2}+(\hbar \varepsilon)^{2}}\right) \end{aligned}\] Making the transfer (118) from the double sum to the double integral, and then the integration variable transfer (120), we get \[\begin{aligned} \chi^{\prime \prime}(\omega)=\lim _{\varepsilon \rightarrow 0} \int d \bar{E}\left[\int_{-\infty}^{+\infty} W\left(\bar{E}+\frac{\widetilde{E}}{2}\right) \rho\left(\bar{E}+\frac{\widetilde{E}}{2}\right) \rho\left(\bar{E}-\frac{\widetilde{E}}{2}\right)\left|F_{n n^{\prime}}\right|^{2} \frac{\hbar \varepsilon}{(\widetilde{E}+\hbar \omega)^{2}+(\hbar \varepsilon)^{2}} d \widetilde{E}\right.\\ \left.-\int_{-\infty}^{+\infty} W\left(\bar{E}+\frac{\widetilde{E}}{2}\right) \rho\left(\bar{E}+\frac{\widetilde{E}}{2}\right) \rho\left(\bar{E}-\frac{\widetilde{E}}{2}\right)\left|F_{n n^{\prime}}\right|^{2} \frac{\hbar \varepsilon}{(\widetilde{E}-\hbar \omega)^{2}+(\hbar \varepsilon)^{2}} d \widetilde{E}\right] . \end{aligned}\] Now using the same argument about the smallness of parameter \(\varepsilon\) as above, we may take the spectral densities, the matrix elements of force, and the Gibbs probabilities out of the integrals, and work out the remaining integrals, getting a result very similar to Eq. (122): \[\chi^{\prime \prime}(\omega)=\pi \int \rho_{+} \rho_{-}\left[W_{-}\left|F_{-}\right|^{2}-W_{+}\left|F_{+}\right|^{2}\right] d \bar{E} .\] In order to relate these two results, it is sufficient to notice that according to Eq. (24), the Gibbs probabilities \(W_{\pm}\) are related by a coefficient depending on only the temperature \(T\) and observation frequency \(\omega\) : \[W_{\pm} \equiv W\left(\bar{E}+\frac{\widetilde{E}_{\pm}}{2}\right) \equiv W\left(\bar{E} \pm \frac{\hbar \omega}{2}\right)=\frac{1}{Z} \exp \left\{-\frac{\bar{E} \pm \hbar \omega / 2}{k_{\mathrm{B}} T}\right\}=W(\bar{E}) \exp \left\{\mp \frac{\hbar \omega}{2 k_{\mathrm{B}} T}\right\},\] so that both the spectral density (122) and the dissipative part (130) of the generalized susceptibility may be expressed via the same integral over the environment energies: \[\begin{aligned} &S_{F}(\omega)=\hbar \cosh \left(\frac{\hbar \omega}{2 k_{\mathrm{B}} T}\right) \int \rho_{+} \rho_{-} W(\bar{E})\left[\left|F_{+}\right|^{2}+\left|F_{-}\right|^{2}\right] d \bar{E}, \\ &\chi^{\prime \prime}(\omega)=2 \pi \sinh \left(\frac{\hbar \omega}{2 k_{\mathrm{B}} T}\right) \int \rho_{+} \rho_{-} W(\bar{E})\left[\left|F_{+}\right|^{2}+\left|F_{-}\right|^{2}\right] d \bar{E}, \end{aligned}\] and hence are universally related as \[S_{F}(\omega)=\frac{\hbar}{2 \pi} \chi^{\prime \prime}(\omega) \operatorname{coth} \frac{\hbar \omega}{2 k_{\mathrm{B}} T} .\] This is, finally, the much-celebrated Callen-Welton’s fluctuation-dissipation theorem (FDT). It reveals a fundamental, intimate relationship between these two effects of the environment ("no dissipation without fluctuation") - hence the name. A curious feature of the FDT is that Eq. (134) includes the same function of temperature as the average energy (26) of a quantum oscillator of frequency \(\omega\) , though, as the reader could witness, the notion of the oscillator was by no means used in its derivation. As will see in the next section, this fact leads to rather interesting consequences and even conceptual opportunities.

In the classical limit, \(\hbar \omega<<k_{\mathrm{B}} T\) , the FDT is reduced to \[S_{F}(\omega)=\frac{\hbar}{2 \pi} \chi^{\prime \prime}(\omega) \frac{2 k_{\mathrm{B}} T}{\hbar \omega}=\frac{k_{\mathrm{B}} T}{\pi} \frac{\operatorname{Im} \chi(\omega)}{\omega} .\] In most systems of interest, the last fraction is close to a finite (positive) constant within a substantial range of relatively low frequencies. Indeed, expanding the right-hand side of Eq. (123) into the Taylor series in small \(\omega\) , we get \[\chi(\omega)=\chi(0)+i \omega \eta+\ldots, \quad \text { with } \chi(0)=\int_{0}^{\infty} G(\tau) d \tau, \quad \text { and } \eta \equiv \int_{0}^{\infty} G(\tau) \tau d \tau .\] Since the temporal Green’s function \(G\) is real by definition, the Taylor expansion of \(\chi^{\prime \prime}(\omega) \equiv \operatorname{Im} \chi(\omega)\) at \(\omega=0\) starts with the linear term \(\omega \eta\) , where \(\eta\) is a certain real coefficient, and unless \(\eta=0\) , is dominated by this term at small \(\omega\) . The physical sense of the constant \(\eta\) becomes clear if we consider an environment that provides a force described by a simple, well-known kinematic friction law \[\langle\hat{F}\rangle=-\eta \dot{\hat{x}}, \quad \text { with } \eta \geq 0,\] where \(\eta\) is usually called the drag coefficient. For the Fourier images of coordinate and force, this gives the relation \(F_{\omega}=i \omega \eta x_{\omega}\) , so that according to Eq. (124), \[\chi(\omega)=i \omega \eta, \quad \text { i.e. } \frac{\chi^{\prime \prime}(\omega)}{\omega} \equiv \frac{\operatorname{Im} \chi(\omega)}{\omega}=\eta \geq 0\] With this approximation, and in the classical limit, the FDT (134) is reduced to the well-known Nyquist formula: 43 \[S_{F}(\omega)=\frac{k_{\mathrm{B}} T}{\pi} \eta, \quad \text { i.e. }\left\langle F_{\mathrm{f}}^{2}\right\rangle=4 k_{\mathrm{B}} T \eta d v\] According to Eq. (112), if such a constant spectral density \({ }^{44}\) persisted at all frequencies, it would correspond to a delta-correlated process \(F(t)\) , with \[K_{F}(\tau)=2 \pi S_{F}(0) \delta(\tau)=2 k_{\mathrm{B}} T \eta \delta(\tau)\]

• cf. Eqs. (82) and (83). Since in the classical limit the right-hand side of Eq. (109) is negligible, and the correlation function may be considered an even function of time, the symmetrized function under the integral in Eq. (113) may be rewritten just as \(\langle F(\tau) F(0)\rangle\) . In the limit of relatively low observation frequencies (in the sense that \(\omega\) is much smaller than not only the quantum frontier \(k_{\mathrm{B}} T / \hbar\) but also the frequency scale of the function \(\left.\chi^{\prime \prime}(\omega) / \omega\right)\) , Eq. (138) may be used to recast Eq. (135) in the form 45

\[\eta \equiv \lim _{\omega \rightarrow 0} \frac{\chi^{\prime \prime}(\omega)}{\omega}=\frac{1}{k_{\mathrm{B}} T} \int_{0}^{\infty}\langle F(\tau) F(0)\rangle d \tau\] To conclude this section, let me return for a minute to the questions formulated in our earlier discussion of dephasing in the two-level model. In that problem, the dephasing time scale is \(T_{2}=1 / 2 D_{\varphi}\) . Hence the classical approach to the dephasing, used in Sec. 3, is adequate if \(\hbar D_{\varphi} \ll k_{\mathrm{B}} T\) . Next, we may identify the operators \(\hat{f}\) and \(\hat{\sigma}_{z}\) participating in Eq. (70) with, respectively, \((-\hat{F})\) and \(\hat{x}\) participating in the general Eq. (90). Then the comparison of Eqs. (82), (89), and (140) yields \[\frac{1}{T_{2}} \equiv 2 D_{\varphi}=\frac{4 k_{\mathrm{B}} T}{\hbar^{2}} \eta\] so that, for the model described by Eq. (137) with a temperature-independent drag coefficient \(\eta\) , the rate of dephasing by a classical environment is proportional to its temperature.

\({ }^{31}\) The most frequent example of the violation of this condition is the environment’s overheating by the energy flow from system \(s\) . Let me leave it to the reader to estimate the overheating of a standard physical laboratory room by a typical dissipative quantum process - the emission of an optical photon by an atom. (Hint: it is extremely small.)

\({ }^{32}\) The FDT was first derived by Herbert Callen and Theodore Allen Welton in 1951, on the background of an earlier derivation of its classical limit by Harry Nyquist in 1928 .

\({ }^{33}\) The FDT may be proved in several ways that are shorter than the one given below - see, e.g., either the proof in SM Secs. \(5.5\) and \(5.6\) (based on H. Nyquist’s arguments), or the original paper by H. Callen and T. Welton, Phys. Rev. 83, 34 (1951) - wonderful in its clarity. The longer approach I will describe here, besides giving the important Green-Kubo formula (109) as a byproduct, is a very useful exercise in the operator manipulation and the perturbation theory in its integral form - different from the differential forms used in Chapter 6 . If the reader is not interested in this exercise, they may skip the derivation and jump straight to the result expressed by Eq. (134), which uses the notions defined by Eqs. (114) and (123).

\({ }_{34}\) For usual ("ergodic") environments, without intrinsic long-term memories, this statistical averaging over an ensemble of environments is equivalent to averaging over intermediate times - much longer than the correlation time \(\tau_{\mathrm{c}}\) of the environment, but still much shorter than the characteristic time of evolution of the system under analysis, such as the dephasing time \(T_{2}\) and the energy relaxation time \(T_{1}\) - both still to be calculated.

\({ }^{35}\) Actually, this is true for virtually any real physical system - in contrast to idealized models such as a dissipation-free oscillator that swings for ever and ever with the same amplitude and phase, thus "remembering" the initial conditions.

\({ }^{36}\) For the heroic reader who has suffered through the calculations up to this point: our conceptual work is done! What remains is just some simple math to bring the relation between Eqs. (108) and (111) to an explicit form.

\({ }^{37}\) Due to their practical importance, and certain mathematical issues of their justification for random functions, Eqs. (112)-(113) have their own grand name, the Wiener-Khinchin theorem, though the math rigor aside, they are just a straightforward corollary of the standard Fourier integral transform (115).

\({ }^{38}\) An alternative popular measure of the spectral density of a process \(F(t)\) is \(S_{F}(v) \equiv\left\langle F_{\mathrm{f}}^{2}\right\rangle / d v=4 \pi S_{F}(\omega)\) , where \(v\) \(=\omega / 2 \pi\) is the "cyclic" frequency (measured in \(\mathrm{Hz}\) ).

\({ }^{39}\) Using, e.g., MA Eq. (6.5a). (The imaginary parts of the integrals vanish, because the integration in infinite limits may be always re-centered to the finite points \(\pm \hbar \omega\) .) A math-enlightened reader may have noticed that the integrals might be taken without the introduction of small \(\varepsilon\) , using the Cauchy theorem - see MA Eq. (15.1).

\({ }^{40}\) The integration in Eq. (123) may be extended to the whole time axis, \(-\infty<\tau<+\infty\) , if we complement the definition (107) of the function \(G(\tau)\) for \(\tau>0\) with its definition as \(G(\tau)=0\) for \(\tau<0\) , in correspondence with the causality principle.

\({ }^{41}\) In order to prove this relation, it is sufficient to plug expression \(\hat{x}_{s}=\hat{x}_{\omega} e^{-i \omega t}\) , or any sum of such exponents, into Eqs. (107) and then use the definition (123). This (simple) exercise is highly recommended to the reader.

\({ }^{42}\) The sign minus in Eq. (127) is due to the fact that according to Eq. (90), \(F\) is the force exerted on our system ( \((s)\) by the environment, so that the force exerted by our system on the environment is \(-F\) . With this sign clarification, the expression \(\mathscr{P}=-F \dot{x}=-F v\) for the instant power flow is evident if \(x\) is the usual Cartesian coordinate of a 1D particle. However, according to analytical mechanics (see, e.g., CM Chapters 2 and 10), it is also valid for any {generalized coordinate, generalized force} pair which forms the interaction Hamiltonian (90).

\({ }^{43}\) Actually, the 1928 work by H. Nyquist was about the electronic noise in resistors, just discovered experimentally by his Bell Labs colleague John Bertrand Johnson. For an Ohmic resistor, as the dissipative "environment" of the electric circuit it is connected with, Eq. (137) is just the Ohm’s law, and may be recast as either \(\langle V\rangle=-R(d Q / d t)=R I\) , or \(\langle I\rangle=-G(d \Phi / d t)=G V\) . Thus for the voltage \(V\) across an open circuit, \(\eta\) corresponds to its resistance \(R\) , while for current \(I\) in a short circuit, to its conductance \(G=1 / R\) . In this case, the fluctuations described by Eq. (139) are referred to as the Johnson-Nyquist noise. (Because of this important application, any model leading to Eq. (138) is commonly referred to as the Ohmic dissipation, even if the physical nature of the variables \(x\) and \(F\) is quite different from voltage and current.)

\({ }^{44}\) A random process whose spectral density may be reasonably approximated by a constant is frequently called the white noise, because it is a random mixture of all possible sinusoidal components with equal weights, reminding the spectral composition of the natural white light.

\({ }^{45}\) Note that in some fields (especially in physical kinetics and chemical physics), this particular limit of the Nyquist formula is called the Green-Kubo (or just "Kubo") formula. However, in the view of the FDT development history (described above), it is much more reasonable to associate these names with Eq. (109) - as it is done in most fields of physics.

MIT Libraries logo MIT Libraries

MIT Specifications for Thesis Preparation

Approved November 2022 for use in the 2022-2023 academic year. Updated March 2023 to incorporate changes to MIT Policies and Procedures 13.1.3 Intellectual Property Not Owned by MIT .

View this page as an accessible PDF .

Table of Contents

• Thesis Preparation Checklist

Timeline for submission and publication

• Bachelor’s degree thesis
• Graduate degree thesis

Dual degree theses

Joint theses, what happens to your thesis, title selection, embedded links.

• Special circumstances

Signature page

Abstract page.

• Acknowledgments

Biographical notes

Table of contents, list of figures.

• List of tables
• List of supplemental material

Notes and bibliographic references

Open licensing, labeling copyright in your thesis, use of previously published material in your thesis, digital supplementary material, physical supplementary material, starting with accessible source files, file naming.

• How to submit thesis information to the MIT Libraries

Placing a temporary hold on your thesis

Changes to a thesis after submission, permission to reuse or republish from mit theses, general information.

This guide has been prepared by the MIT Libraries, as prescribed by the Committee on Graduate Programs and the Committee on Undergraduate Program, to assist students and faculty in the preparation of theses. The Institute is committed to the preservation of each student’s thesis because it is both a requirement for the MIT degree and a record of original research that contains information of lasting value.

In this guide, “department” refers to a graduate or undergraduate program within an academic unit, and “thesis” refers to the digital copy of the written thesis. The official thesis version of record, which is submitted to the MIT Libraries, is the digital copy of the written thesis that has been approved by the thesis committee and certified by the department in fulfillment of a student’s graduation requirement.

The requirements in this guide apply to all theses and have been specified both to facilitate the care and dissemination of the thesis and to assure the preservation of the final approved document. Individual departments may dictate more stringent requirements.

Before beginning your thesis research, remember that the final output of this research—your thesis document—should only include research findings that may be shared publicly, in adherence with MIT’s policy on Open Research and Free Interchange of Information . If you anticipate that your thesis will contain content that requires review by an external sponsor or agency, it is critical that you allow sufficient time for this review to take place prior to thesis submission. 

Questions not answered in this guide should be referred to the appropriate department officer or to the MIT Libraries ( [email protected] ).

• Final edited and complete thesis PDF is due to your department on the date specified in the Academic Calendar.
• Hold requests should be submitted to the Vice Chancellor for Undergraduate and Graduate Education or TLO concurrent with your thesis submission.
• Thesis information is due to the MIT Libraries before your date of graduation.
• Departments must transfer theses to the MIT Libraries within 30 days from the last day of class (end of term).
• One week later (30 days from the last day of classes + 7 days) or one week after the degree award date (whichever is later) the MIT Libraries may begin publishing theses in DSpace@MIT.
• If you have requested and received a temporary (up to 90-day) hold on the publication of your thesis from the Vice Chancellor, your thesis will be placed on hold as soon as it is received by the Libraries, and the 90-day hold will begin 30 days from the last day of class (end of term).
• If your thesis research is included in a disclosure to the TLO, the TLO may place your thesis on temporary hold with the Libraries, as appropriate.

Submitting your thesis document to your department

Your thesis document will be submitted to your department as a PDF, formatted and including the appropriate rights statement and sections as outlined in these specifications. Your department will provide more specific guidance on submitting your files for certification and acceptance.

Your department will provide information on submitting:

• A PDF/A-1  of your final thesis document (with no signatures)
• Signature page (if required by your department; your department will provide specific guidance)
• Original source files used to create the PDF of your thesis (optional, but encouraged)
• Supplementary materials  (optional and must be approved by your advisor and program)

Degree candidates must submit their thesis to the appropriate office of the department in which they are registered on the dates specified in the Academic Calendar. ( Academic Calendar | MIT Registrar ). September, February, and May/June are the only months in which degrees are awarded.

Bachelor’s degree theses

Graduate degree theses, submitting your thesis information to the libraries.

Information about your thesis must be submitted to the Libraries thesis submission and processing system  prior to your day of graduation. The information you provide must match the title page and abstract of your thesis . See How to submit thesis information to the MIT Libraries section for more details .

The academic department is required to submit the thesis to the MIT Libraries within one month after the last day of the term in which the thesis was submitted ( Faculty Regulation 2.72 ). The thesis document becomes part of the permanent archival collection. All thesis documents that have been approved will be transferred electronically to the MIT Libraries by a department representative via the MIT Libraries thesis submission and processing system .

The full-text PDF of each thesis is made publicly available in DSpace@MIT . A bibliographic record will appear in the MIT Libraries’ catalog, as well as the OCLC database WorldCat, which is accessible to libraries and individuals worldwide. Authors may also opt-in to having their thesis made available in the ProQuest Dissertations & Theses Global database.

Formatting specifications

Your work will be a more valuable research tool for other scholars if it can be located easily. Search engines use the words in the title, and sometimes other descriptive words, to locate works. Therefore,

• Be sure to select a title that is a meaningful description of the content of your manuscript; and
• Do: “The Effects of Ion Implantation and Annealing on the Properties of Titanium Silicide Films on Silicon Substrates”
• Do: “Radiative Decays on the J/Psi to Two Pseudoscalar Final States”

You may include clickable links to online resources within the thesis file. Make the link self-descriptive so that it can stand on its own and is natural language that fits within the surrounding writing of your paragraph. The full URL should be included as a footnote or bibliography citation (dependent on citation style).

• Sentence in thesis: Further information is available on the MIT Writing and Communications Center’s website . The full-text PDF of each thesis is made publicly available in DSpace@MIT .
• Footnote or Bibliography: follow the rules of your chosen citation style and include the full website URL, in this case http://libraries.mit.edu/mit-theses

Sections of your thesis

Required (all information should be on a single page)

The title page should contain the title, name of the author (this can be the author’s preferred name), previous degrees, the degree(s) to be awarded at MIT, the date the degree(s) will be conferred (May/June, September, or February only), copyright notice (and legend, if required), and appropriate names of thesis supervisor(s) and student’s home department or program officer.

The title page should have the following fields in the following order and centered (including spacing) :

Thesis title as submitted to registrar

Author’s preferred name

Previous degree information, if applicable

Submitted to the [department name] in partial fulfillment of the requirements for the degree(s) of

[degree name]

Massachusetts Institute of Technology

Month and year degree will be granted (May or June, September, February ONLY)

Copyright statement

This permission legend MUST follow: The author hereby grants to MIT a nonexclusive, worldwide, irrevocable, royalty-free license to exercise any and all rights under copyright, including to reproduce, preserve, distribute and publicly display copies of the thesis, or release the thesis under an open-access license.

[Insert 2 blank lines]

Note: The remaining fields are left aligned and not centered

Authored by: [Author name]

[Author’s department name] (align with the beginning of the author’s name from the previous line)

[Date thesis is to be presented to the department] (align with the beginning of the author’s name from the first line)

Certified by: [Advisor’s full name as it appears in the MIT catalog]

   [Advisor’s department as it appears in the MIT catalog] (align with the beginning of the advisor’s name from the previous line), Thesis supervisor

Accepted by: [name]

[title – line 1] (align with the beginning of the name from the previous line)

[title – line 2] (align with the beginning of the name from the first line)

Note: The name and title of this person varies in different degree programs and may vary each term; contact the departmental thesis administrator for specific information

• Students in joint graduate programs (such as Harvard-MIT Health Sciences and Technology and Woods Hole Oceanographic Institution) should list both their MIT thesis supervisor and the supervisor from the partner academic institution.
• The name and title of the department or the program officer varies in different degree programs and may vary each term. Contact the departmental graduate administrator for specific information.
• For candidates receiving two degrees, both degrees to be awarded should appear on the title page. For candidates in dual degree programs, all degrees and departments or programs should appear on the title page, and the names of both department heads/committee chairs are required. Whenever there are co-supervisors, both names should appear on the title page.

Here are some PDF examples of title pages:

• Bachelor’s Degree – using a Creative Commons license
• PhD candidate – using a Creative Commons license
• Master’s candidate – dual degrees
• Masters’ candidates – multiple authors
• Masters’ candidates – multiple authors with dual degrees and extra committee members
• Bachelor’s Degree – change of thesis supervisor

Title page: Special circumstances – change of thesis supervisor

If your supervisor has recently died or is no longer affiliated with the Institute:

• Both this person and your new supervisor should be listed on your title page
• Under the new supervisor’s name, state that they are approving the thesis on behalf of the previous supervisor
• An additional page should be added to the thesis, before the acknowledgments page, with an explanation about why a new supervisor is approving your thesis on behalf of your previous supervisor. You may also thank the new supervisor for acting in this capacity
• Review this PDF example of a title page with a change in supervisor

If your supervisor is external to the Institute (such as an industrial supervisor):

• You should acknowledge this individual on the Acknowledgements page as appropriate, but should not list this person on the thesis title page
• The full thesis committee and thesis readers can be acknowledged on the Acknowledgements page, but should not be included on the title page

Not Required

Please consult with your department to determine if they are requiring or requesting an additional signature page.

Each thesis must include an abstract of generally no more than 500 words single-spaced. The abstract should be thought of as a brief descriptive summary, not a lengthy introduction to the thesis. The abstract should immediately follow the title page.

The abstract page should have the following fields in the following order and centered (including spacing):

• Thesis title

Submitted to the [Department] on [date thesis will be submitted] in Partial Fulfillment of the Requirements for the Degree of [Name of degree to be received]

[Insert 1 blank line]

Single-spaced summary; approximately 500 words or less; try not to use formulas or special characters

Thesis supervisor: [Supervisor’s name]

Title: [Title of supervisor]

The Abstract page should include the same information as on the title page. With the thesis title, author name, and submitting statement above the abstract, the word “ABSTRACT” typed before the body of the text, and the thesis supervisor’s name and title below the abstract.

Acknowledgements

An acknowledgement page may be included and is the appropriate place to include information such as external supervisor (such as an industrial advisor) or a list of the full thesis committee and thesis readers. Please note that your thesis will be publicly available online at DSpace@MIT , which is regularly crawled and indexed by Google and other search-engine providers.

The thesis may contain a short biography of the candidate, including institutions attended and dates of attendance, degrees and honors, titles of publications, teaching and professional experience, and other matters that may be pertinent. Please note that your thesis will be publicly available online at DSpace@MIT , which is regularly crawled and indexed by Google and other search-engine providers.

List of Tables

List of supplemental material.

Whenever possible, notes should be placed at the bottom of the appropriate page or in the body of the text. Notes should conform to the style appropriate to the discipline. If notes appear at the bottom of the page, they should be single-spaced and included within the specified margins.

It may be appropriate to place bibliographic references either at the end of the chapter in which they occur or at the end of the thesis.

The style of quotations, footnotes, and bibliographic references may be prescribed by your department. If your department does not prescribe a style or specify a style manual, choose one and be consistent. Further information is available on the MIT Writing and Communications Center’s website .

Ownership of copyright

The Institute’s policy concerning ownership of thesis copyright is covered in Rules and Regulations of the Faculty, 2.73 and MIT Policies and Procedures 13.1.3 . Copyright covers the intellectual property in the words and images in the thesis. If the thesis also includes patentable subject matter, students should contact the Technology Licensing Office (TLO) prior to submission of their thesis.

Under these regulations, students retain the copyright to student theses.

The student must, as a condition of a degree award, grant to MIT a nonexclusive, worldwide, irrevocable, royalty-free license to exercise any and all rights under copyright, including to reproduce, preserve, distribute and publicly display copies of the thesis, or release the thesis under an open-access license. The MIT Libraries publish the thesis on DSpace@MIT , allowing open access to the research output of MIT.

You may also, optionally, apply a Creative Commons License to your thesis. The Creative Commons License allows you to grant permissions and provide guidance on how your work can be reused by others. For more information about CC: https://creativecommons.org/about/cclicenses/ . To determine which CC license is right for you, you can use the CC license chooser .

You must include an appropriate copyright notice on the title page of your thesis. This should include the following:

• the symbol “c” with a circle around it © and/or the word “copyright”
• the year of publication (the year in which the degree is to be awarded)
• the name of the copyright owner
• the words “All rights reserved” or your chosen Creative Commons license
• Also include the following statement below the ©“ The author hereby grants to MIT a nonexclusive, worldwide, irrevocable, royalty-free license to exercise any and all rights under copyright, including to reproduce, preserve, distribute and publicly display copies of the thesis, or release the thesis under an open-access license.”
• Also include the following statement below the © “The author hereby grants to MIT a nonexclusive, worldwide, irrevocable, royalty-free license to exercise any and all rights under copyright, including to reproduce, preserve, distribute and publicly display copies of the thesis, or release the thesis under an open-access license.”

You are responsible for obtaining permission, if necessary, to include previously published material in your thesis. This applies to most figures, images, and excerpts of text created and published by someone else; it may also apply to your own previous work. For figures and short excerpts from academic works, permission may already be available through the MIT Libraries (see here for additional information ). Students may also rely on fair use , as appropriate. For assistance with copyright questions about your thesis, you can contact [email protected] .

When including your own previously published material in your thesis, you may also need to obtain copyright clearance. If, for example, a student has already published part of the thesis as a journal article and, as a condition of publication, has assigned copyright to the journal’s publisher, the student’s rights are limited by what the publisher allows. More information about publisher policies on reuse in theses is available here.

Students can hold onto sufficient rights to reuse published articles (or excerpts of these) in their thesis if they are covered by MIT’s open access policy. Learn more about MIT’s open access policy and opt-in here . Contact [email protected] for more information.

When including your own previously published articles in your thesis, check with your department for specific requirements, and consider the following:

• Ensure you have any necessary copyright permissions to include previously published material in your thesis.
• Be sure to discuss copyright clearance and embargo options with your co-authors and your advisor well in advance of preparing your thesis for submission.
• Include citations of where portions of the thesis have been previously published.
• When an article included has multiple authors, clearly designate the role you had in the research and production of the published paper that you are including in your thesis.

Supplemental material and research data

Supplemental material that may be submitted with your thesis is the materials that are essential to understanding the research findings of your thesis, but impossible to incorporate or embed into a PDF. Materials submitted to the MIT Libraries may be provided as supplemental digital files or in some cases physical items. All supplementary materials must be approved for submission by your advisor. The MIT Libraries can help answer questions you may have about managing the supplementary material and other research materials associated with your research.

Contact [email protected] early in your thesis writing process to determine the best way to include supplemental materials with your thesis.

You may also have other research data and outputs related to your thesis research that are not considered supplemental material and should not be submitted with your thesis. Research materials include the facts, observations, images, computer program results, recordings, measurements, or experiences on which a research output—an argument, theory, test or hypothesis, or other output—is based. These may also be termed, “research data.” This term relates to data generated, collected, or used during research projects, and in some cases may include the research output itself. Research materials should be deposited in appropriate research data repositories and cited in your thesis . You may consult the MIT Libraries’ Data Management Services website for guidance or reach out to Data Management Services (DMS)( [email protected] ), who can help answer questions you may have about managing your thesis data and choosing suitable solutions for longer term storage and access.

• Supplementary information may be submitted with your thesis to your program after approval from your thesis advisor. 
• Supplemental material should be mentioned and summarized in the written document, for example, using a few key frames from a movie to create a figure.
• A list of supplementary information along with brief descriptions should be included in your thesis document. For digital files, the description should include information about the file types and any software and version needed to open and view the files.
• Issues regarding the format of non-traditional, supplemental content should be resolved with your advisor.
• Appendices and references are not considered supplementary information.
• If your research data has been submitted to a repository, it should not also be submitted with your thesis.
• Follow the required file-naming convention for supplementary files: authorLastName-kerb-degree-dept-year-type_supplemental.ext
• Captioning ( legally required ): text versions of the audio content, synchronized with the video: ways to get your video captioned
• Additional content, not required:
• For video, an audio description: a separate narrative audio track that describes important visual content, making it accessible to people who are unable to see the video
• Transcripts: should capture all the spoken audio, plus on-screen text and descriptions of key visual information that wouldn’t otherwise be accessible without seeing the video

For physical components that are integral to understanding the thesis document, and which cannot be meaningfully conveyed in a digital form, the author may submit the physical items to the MIT Libraries along with their thesis document. When photographs or a video of a physical item (such as a model) would be sufficient, the images should be included in the thesis document, and a video could be submitted as digital supplementary material.

An example of physical materials that would be approved for submission as part of the thesis would be photographs that cannot be shared digitally in our repository due to copyright restrictions. In this case, the photographs could be submitted as a physical volume that is referred to in the thesis document.

As with digital supplementary information and research materials, physical materials must be approved for submission by your advisor. Contact [email protected] early in your thesis writing process to determine if physical materials should accompany your thesis, and if so how to schedule a transfer of materials to the MIT Libraries.

Creating your thesis document/digital format

You are required to submit a PDF/A-1 formatted thesis document to your department. In addition, it is recommended that original files, or source files, (such a .doc or .tex) are submitted alongside the PDF/A-1 to better ensure long-term access to your thesis.

You should create accessible files that support the use of screen readers and make your document more easily readable by assistive technologies. This will expand who is able to access your thesis. By creating an accessible document from the beginning, there will be less work required to remediate the PDF that gets created. Most software offers a guide for creating documents that are accessible to screen readers. Review the guidelines provided by the MIT Libraries .

In general:

• Use styles and other layout features for headings, lists, tables, etc. If you don’t like the default styles associated with the headings, you can customize them.
• Avoid using blank lines to add visual spacing and instead increase the size of the spaces before and/or after the line.
• Avoid using text boxes.
• Embed URLs.
• Anchor images to text when inserting them into a doc.
• Add alt-text to any images or figures that convey meaning (including, math formulas).
• Use a sans serif font.
• Add basic embedded metadata, such as author, title, year of graduation, department, keywords etc. to your thesis via your original author tool.

Creating a PDF/A-1

PDF/A-1 (either a or b) is the more suitable format for long term preservation than a basic PDF. It ensures that the PDF format conforms to certain specifications which make it more likely to open and be viewable in the long term. It is best for static content that will not change in the future, as this is the most preservation-worthy version and does not allow for some complex elements that could corrupt or prevent the file from being viewable in the future. Guidelines on how to convert specific file types to PDF/A .

In general: (should we simplify these bullets)

• Convert to PDF/A directly from your original files (text, Word, InDesign, LaTeX, etc.). It is much easier and better to create valid PDF/A documents from your original files than from a regular PDF. Converting directly will ensure that fonts and hyperlinks are embedded in the document.
• Do not embed multimedia files (audio and video), scripts, executables, lab notebooks, etc. into your PDF. Still images are fine. The other formats mentioned may be able to be submitted as supplemental files.
• Do not password protect or encrypt your PDF file.
• Validate your PDF/A file before submitting it to your department.

All digital files must be named according to this scheme: authorLastName-kerb-degree-dept-year-type_other.ext

• Thesis PDF: macdonald-mssimon-mcp-dusp-2023-thesis.pdf
• Signature page: macdonald-mssimon-mcp-dusp-2023-sig.pdf
• Original source file: macdonald-mssimon-mcp-2023-source.docx
• Supplemental file: macdonald-mssimon-mcp-2023-supplmental_1.mov
• Second supplemental file: macdonald-mssimon-mcp-2023-supplmental_2.mov
• Read Me file about supplemental: macdonald-mssimon-mcp-2023-supplemental-readme.txt

How to submit thesis information to the MIT Libraries

Before your day of graduation, you should submit your thesis title page metadata to the MIT Libraries  prior to your day of graduation. The submission form requires Kerberos login.

Student submitted metadata allows for quicker Libraries processing times. It also provides a note field for you to let Libraries’ staff know about any metadata discrepancies.

The information you provide must match the title page and abstract of your thesis . Please have a copy of your completed thesis on hand to enter this information directly from your thesis. If any discrepancies are found during processing, Libraries’ staff will publish using the information on the approved thesis document. You will be asked to confirm or provide:

• Preferred name of author(s)as they appear on the title page of the thesis
• ORCID provides a persistent digital identifier that distinguishes you from every other researcher. The goal is to support the creation of a permanent, clear, and unambiguous record of scholarly communication by enabling reliable attribution of authors and contributors. Read ORCID FAQs to learn more
• Department(s)
• A license is optional, and very difficult to remove once published. The Creative Commons License allows you to grant permissions and provide guidance on how your work can be reused by others. Read more information about CC .
• Thesis supervisor(s)
• If you would like the full-text of your thesis to be made openly available in the ProQuest Dissertation & Theses Global database (PQDT), you can indicate that in the Libraries submission form.
• Open access inclusion in PQDT is at no cost to you, and increases the visibility and discoverability of your thesis. By opting in you are granting ProQuest a license to distribute your thesis in accordance with ProQuest’s policies. Further information can be found in the ProQuest Dissertations and Theses Author FAQ .
• Full-text theses and associated supplemental files will only be sent to ProQuest once any temporary holds have been lifted, and the thesis has been published in DSpace@MIT.
• Regardless of opting-in to inclusion in PQDT, the full text of your thesis will still be made openly available in DSpace@MIT . Doctoral Degrees: Regardless of opting-in the citation and abstract of your thesis will be included in PQDT.

Thesis research should be undertaken in light of MIT’s policy of open research and the free interchange of information . Openness requires that, as a general policy, thesis research should not be undertaken on campus when the results may not be published. From time to time, there may be a good reason for delaying the distribution of a thesis to obtain patent protection, or for reasons of privacy or security. To ensure that only those theses that meet certain criteria are withheld from distribution and that they are withheld for the minimum period, the Institute has established specific review procedures.

Written notification of patent holds and other restrictions must reach the MIT Libraries before the thesis in question is received by the MIT Libraries. Theses will not be available to the public prior to being published by the MIT Libraries. The Libraries may begin publishing theses in DSpace@MIT one month and one week from the last day of classes.

Thesis hold requests should be directed to the Technology Licensing Office (TLO) ( [email protected] ) when related to MIT-initiated patent applications (i.e., MIT holds intellectual property rights; patent application process via TLO). Requests for a thesis hold must be made jointly by the student and advisor directly to the MIT Technology Licensing Office as part of the technology disclosure process.

Thesis hold or restricted access requests should be directed to the Office of the Vice Chancellor ([email protected]) when related to:

• Student-initiated patents (student holds intellectual property rights as previously determined by TLO) [up to 90-day hold]
• Pursuit of business opportunities (student holds intellectual property rights as previously determined by TLO)[up to 90-day hold]
• Government restrictions [up to 90-day hold]
• Privacy and security [up to 90-day hold]
• Scholarly journal articles pending publication [up to 90-day hold]
• Book publication [up to 24-month hold]

In the unusual circumstance that a student wants to request a hold beyond the initial 90-day period, they should contact the Office of Vice President for Research , who may consult with the TLO and/or the Office of the Vice Chancellor, as appropriate to extend the hold. Such requests must be supported by evidence that explains the need for a longer period.

Find information about each type of publication hold, and to learn how to place a hold on your thesis

After publication

Your thesis will be published on DSpace@MIT . Theses are processed by the MIT Libraries and published in the order they are transferred by your department. The Libraries will begin publishing theses in DSpace@MIT one month and one week from the last day of classes.

All changes made to a thesis, after it has been submitted to the MIT Libraries by your department, must have approval from the Vice Chancellor or their designee. Thesis documents should be carefully reviewed prior to submission to ensure they do not contain misspellings or incorrect formatting. Change requests for these types of minor errors will not be approved.

There are two types of change requests that can be made:

• Errata: When the purpose is to correct significant errors in content, the author should create an errata sheet using the form and instructions (PDF)  and obtain approval first from both the thesis supervisor or program chair, before submitting for review by the Vice Chancellor.
• Substitution: If the purpose of the change is to excise classified, proprietary, or confidential information, the author should fill out the  application form (PDF) and have the request approved first by the thesis supervisor or program chair, before submitting for review by the Vice Chancellor.

Students and supervisors should vet thesis content carefully before submission to avoid these scenarios whenever possible.

You are always authorized to post electronic versions of your own thesis, in whole or in part, on a website, without asking permission. If you hold the copyright in the thesis, approving and/or denying requests for permission to use portions of the thesis in third-party publications is your responsibility.

MIT Libraries Thesis Team https://libguides.mit.edu/mit-thesis-faq [email protected] | https://thesis-submit.mit.edu/

Distinctive Collections Room 14N-118 | 617-253-5690 https://libraries.mit.edu/distinctive-collections/

Technology Licensing Office [email protected] | 617-253-6966 http://tlo.mit.edu/

Office of the General Counsel [email protected]  | 617-452-2082 http://ogc.mit.edu/

Office of Graduate Education Room 3-107 | 617-253-4680 http://oge.mit.edu/ [email protected]

MIT Libraries,  Scholarly Communications https://libraries.mit.edu/scholarly/ [email protected]

Office of  the Vice Chancellor Room 7-133 | 617-253-6056 http://ovc.mit.edu [email protected]

Office of the Vice President for Research Room 3-234 | 617-253-8177 [email protected]

MIT Writing and Communications Center Room E18-233 [email protected] | https://cmsw.mit.edu/writing-and-communication-center/

• Bibliography
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John Bertrand

John bertrand skippered australia ii to a 4-3 victory over liberty, wresting the america's cup from the new york yacht club, which had held it for 132 years. the great grandson of thomas pearkes, the english master engineer who prepared the challenging america's cup yachts for sir thomas lipton, john bertrand had a score to settle. he did it magnificently on september 26, 1983 when he won the seventh race against dennis conner and liberty. there had never been a seventh or even a sixth race in the history of america's cup racing. there was no question that australia ii was faster than liberty - faster than a challenger had ever been relative to a defender. australia ii was built with a secret winged keel that provided excellent windward ability while permitting very low wetted surface in a yacht of minimum rule size, the winning combination for conditions in newport. in the penultimate race dubbed "the race of the century," the lead changed several times until australia ii gained 1 minute and 21 seconds in the crucial sixth leg of the race in a dying southerly wind off newport. bertrand applied a near perfect cover during a 47 tack fight to the finish area and crossed the line 41 seconds ahead of liberty..

• Undergraduate
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• Knight Science Journalism
• Related Programs
• Undergraduate Theses

These students either “joint-majored”, “double/second-majored” or did a second degree in STS, along with a science/engineering field.

AKERA, ATSUSHI “A Social Technology: Ethnography of a Computer Terminal Room” June 1986 (Charlie Weiner, thesis advisor)

ARDHASSERIL, ROSHAN “Nuclear State: Pakistan, Domestic and International” June 2013 (Theodore Postol, thesis advisor)

BARRETT, BERTRAND H. “Theory and Design of an Educational Computing Environment” June 1985 (J.C.R. Licklider, thesis advisor)

BECERRA, JUANA C. “Herman Feshbach: What it Meant to be a Physicist in the Twentieth Century” June 2015 (David Kaiser, thesis advisor)

BELAND, CHRISTOPHER D. “Digital Technology and Copyright Law” February 2002 (David I. Kaiser, thesis advisor)

BESPOLKA, CARL G. “Green Energy Policy in the Federal Republic of Germany” May 1983 (Michael Geisler, thesis advisor)

BEST, WALDO T. “Subjective Confidence in Technology” May 1988 (Thomas Sheridan, thesis advisor) [double S.B. degrees in Humanities and Engineering]

BROWN, DAVID J. “A Framework for Analyzing Residential Electricity Consumption” May 1983 (Ted Greenwood, thesis advisor)

BRYAN, ERIC FAIN “Financing Invention” May 1988 (Robert Rines, thesis advisor)

BYFIELD, LAINI “Modern Medicine vs. Traditional Medicine” June 1999 (Joe Dumit and Hugh Gusterson, thesis advisors)

COWAN, THOMAS “Network Control in a Globalized World: How Visa and Swift’s Founding Structures Serve Their Stakeholders on the International Stage” June 2017 (William Deringer, thesis advisor)

CUNNINGHAM, KEVIN “Contemporary Computer Software and the Writing Process” February 1984 (James Paradis, thesis tutor)

DUBRANSKY, JULIAN “The politicization of science during the COVID-19 pandemic in the United States” June 2021 (John Durant, thesis advisor)

FRANCO, KATHERINE A. “The Idealists and the Pragmatists. A Comparative History of Free Software and Open Source Software” May 2005 (Rosalind Williams, thesis advisor)

GARFINKEL, SIMSON L. “The Context of Funding in the Sociological Research of Paul F. Lazarsfeld” June 1987 (Peter Buck, thesis tutor) [Simson graduated in 1987 with 3 separate S.B. degrees in Chemistry, Political Science, and Humanities]

GILLESPIE, JAMES JUDSON “Going Nowhere: Pittsburg’s Attempt to Build a Subway, 1910-1935” 1990 (Robert Fogelson, thesis advisor)

GLAVIN, MITCHELL “School Attendance for Children with Acquired Immune Deficiency Syndrome: An Example of AIDS Policy” June 1987 (Harvey Sapolsky, thesis advisor)

GLENHABER, MEHITABEL “‘Space Became Their Highway’: The L-5 Society and the closing of the Final Frontier” June 2019 (William Deringer, thesis advisor)

GORDON, EDWARD A. “The Impact of Internet Content Regulation on the Freedom of Expression Around the World” June 1999 (David Mindell, thesis advisor)

HANSON, ELIZABETH A. “Scientific Motherhood: American Childrearing, 1890-1915” June 1984 (Merritt Roe Smith, thesis advisor)

HE, YIRAN “Breakout: How Materials Start-Ups Separate from and Stay Connected to Academic Spaces” May 2020 (William Deringer, thesis advisor)

HEIM, STEVEN F. “Sustaining Vermont: Cooperatives in Vermont’s Economic Development” February 1997 (Deborah Fitzgerald and Alice Amsden, thesis advisors)

HONG, HYEONSIL June 1990 [S.B. in Humanities and Engineering]

HORO, UZUKI “Can MIT Tolerate Its Self-criticism? – a Case of David Noble” May 2023 ( John Durant and Robin Scheffler, thesis advisors)

HUANG, TERESA “Between the Real and the Virtual: Development of Complex Relationships and Communities in the Age of the Internet” June 1997 (Sherry Turkle, thesis advisor)

JONES, BRIANNA “Defining ‘Good Science’ in Today’s World: A Video Compilation of Perspectives and Advice for Incoming Graduate Students” June 2015 (Rosalind Williams, thesis advisor)

KEEGAN, BRIAN “Defending New Jerusalem: The Foundation and Transformation of MIT’s Program in Science, Technology, and Society” June 2006 (Rosalind Williams, thesis advisor)

LEE, JENNIFER JUNG-WUK “Engineering a Sanitary Environment: William Thompson Sedgwick and Public Health Work, 1884-1921” May 1994 (Evelynn Hammonds, thesis advisor)

LYNCH, ALISON June 1990 [S.B. in Humanities and Science]

MANOLIU, MIHAI “Synthesis and Transformation: Moving Beyond Doomsday” June 1984 (John R. Ross, thesis advisor)

MARTIN, MARISSA L. “Defining a New Science: Lessons from a Brief History of the Brain Sciences at MIT” May 2000 (Joe Dumit, thesis advisor)

McBATH, BRUCE COURTNEY 1981 [S.B. in Humanities and Science]

NICHOLLS, GINA-MONIQUE R. “The Offensive and Defensive Politics of Deploying Theater Ballistic Missile Defenses in East Asia” June 2000 (Theodore Postol, thesis advisor)

PRATHER, DARCY 1991 [double B.S. degrees in Humanities and Engineering]

RAHL, GARY M. “The Auditorium and the Space Station: The Death of the American Myth” June 1989 (Leon Trilling, thesis advisor)

REUSS, RONALD “Computer-Aided Reading” June 1986 (David Clark, thesis supervisor) [double B.S. degrees in Humanities and Engineering]

REZA, FAISAL “Human Cloning: Science, Ethics, Policy, Society” February 2003 (Hugh Gusterson, thesis advisor) [double S.B. degrees in Humanities and Science]

SAWICKI, ANDRES “The Paradox Theory in Attention Deficit Hyperactivity Disorder: From Research to Marketing” January 2003 (Joseph Dumit, thesis advisor) [double S.B. degrees in Humanities and Science]

SAYLOR, MICHAEL “A Machiavellian Interpretation of Political Dynamics” June 1987 (John Sterman, thesis advisor) [double S.B. degrees in Humanities and Engineering]

SHAH, SAMEER “Perception of Risk: Disaster Scenarios at Brookhaven” June 2003 (Hugh Gusterson, thesis advisor) [double S.B. degrees in Humanities and Science]

SHAH, VAIBHAVI “The Politics and Perceptibility of Breath During The COVID-19 Pandemic” February 2021 (Robin Scheffler, thesis advisor)

SHARIFI, JAMSHIED 40-minute original music composition in lieu of thesis May 1983 [S.B. in Humanities and Engineering]

SHAWCROSS, PAUL J. “The American Civil Space Program: Preparing for the Next Twenty-Five Years” February 1988 (Kosta Tsipis, thesis advisor)[double major in STS and Aero/Astro]

SKLAR, BRANDON “The Philosophical Interpretation of Quantum Mechanics” June 1982

SOLORZANO, RAMON “An Appropriate Technology: Movement Towards a Value-Laden Approach to Technology” September 1984 (Larry Bucciarelli, thesis advisor)

STICKGOLD-SARAH, JESSIE “Form and Usage: The Evolving Identity of the Computerized Medical Record” February 1997 (Deborah Fitzgerald, thesis advisor)

THOMPSON, ELIZABETH “Artificial Skin: Its Path to Adoption” February 1986 (John Sterman, thesis advisor)

WEIGEL, ANNALISA May 1995 [double major in STS and Aero/Astro]

WIENER, MATTHEW CHARLES “Attitudes Towards Computers in the Soviet Union, 1970-1986: An examination of popular-science writing” May 1987 (Paul Josephson, thesis advisor) [double S.B. in Humanities and Science]

XU, SHEILA ZHI “The Emergence of a Deaf Economy” June 2014 (Rosalind Williams, thesis advisor)

• Joint Major (21E/21S)
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• Concentration
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• Meet with us!

Undergraduate Officer

Professor Dwai Banerjee

E51-171 dwai@mit.edu

Academic Administrator

Karen Gardner

E51-163f 617-253-9759 kgardner@mit.edu

John Urschel

I am an assistant professor in the MIT Math department. My research is focused on matrix analysis and numerical analysis, with an emphasis on theoretical results and provable guarantees for practical problems.

I am also a Junior Fellow at the Harvard Society of Fellows (currently on leave). Previously, I was a member of the Institute for Advanced Study, under the supervision of Peter Sarnak . I completed my PhD in math at MIT in 2021, and had the pleasure of being advised by Michel Goemans .

• A New Upper Bound for the Growth Factor in Gauss. Elimin. with Complete Pivoting, Preprint [ PDF ]
• Nodal Decompositions of Symmetric Matrices, Int. Math. Res. Not., to appear [ PDF ]
• Representing the Special Linear Group with Block Unitriangular Matrices, PNAS Nexus (2023) [ PDF ]
• Maximum Spread of Graphs and Bipartite Graphs, Comm. AMS (2022) [ PDF ]
• Uniform Error Estimates for the Lanczos Method, SIMAX (2021) [ PDF ]
• On the Characterization and Uniqueness of Centroidal Voronoi Tessellations, SINUM (2017) [ PDF ]
• Learning Determinantal Point Processes with Moments and Cycles, ICML (2017) [ PDF ]

Below, you can find a brief description of my research, a full list of my publications, my current and past teaching, and some outreach programs I am involved in. Here is a (most likely outdated) CV .

• Discrete Trace Thms. and Energy Min. Spring Embeddings of Planar Graphs, LAA (2021). [ PDF ]
• Estimating the Matrix p -> q Norm, Preprint [ PDF ]
• Some New Results on the Maximum Growth Factor in Gaussian Elimination, SIMAX, to appear [ PDF ]

Instabilities in the Sun-Jupiter-Asteroid Three Body Problem u , with Joseph Galante. Celestial Mechanics and Dynamical Astronomy, 2013. [ PDF ]

Math 041: Trigonometry and Analytic Geometry, Penn State, Spring 2013