Mobolaji's Personal Webpage

Instructor: Mobolaji Williams mwilliams@physics.harvard.edu
TA: Jason T Necaise necaisej@mit.edu

Lec:	MWF 1:15-2:45pm	Room:	5-233
MW OH:	MWF 12:30-1:15pm	Room:	5-233
JN OH:	MF 6-7:30pm	Room:	Simmons 6th Floor Lounge

Course Information

Syllabus

This course will introduce the concepts and formalism at the foundations of statistical physics.

Statistical physics concerns the physics of systems with many degrees of freedom. In terms of content, by the end of the course, students should understand qualitative and quantitative definitions of entropy, the implications of the laws of thermodynamics, and why the Boltzmann distribution is important in modeling systems at finite temperature. In terms of skills, students should have increased their familiarity with mathematical methods in the physical science, learned how to write short programs to simulate random events, and become more adept at articulating their understanding of physics.

Announcements:

[Aug 2, 18] I'm leaving your final exam, challenge problem, and 5th pset with Fernando. You can pick it up from him tonight after the final banquet. [July 31, 18] I've graded your finals and included any points you received from the challenge problem. The mean was 75 with a standard deviation of 38. There is a large standard deviation in the scores mostly because students who completed any part of the challenge problem received a large score boost. Also, when grading I noticed there was a typo in the exam. In the statistical physics of permutations problem (problem 5), the argument of the exponential should have been "-βλj" instead of simply "-βλ". This mistake seemed to have affected your solutions for part (a) and (c), so I was very lenient in grading part (a) and I've eliminated part (c) from the exam, so the exam is now out of 125 points instead of out of 130. The solutions to the final are online. Also, I'll set something up to return your assignments, challenge problems, and final exams to you before you leave for the summer. Finally, I think you all did fairly well on the exam and in the class in general. The course was difficult, and in retrospect, I can think of a few ways it could have been made smoother for some students, but I hope the experience was not too traumatic for you all. If anything the most important lesson you should take from this course is how your study habits should change and improve when encountering difficult technical material. Good luck on the calculus exam tomorrow. [July 29, 18] I've returned your graded pset 4 to Fernando and you can pick it up from his room. Some of you requested one-on-one sessions. I won't be coming to MIT tomorrow, but we can still schedule skype/google hangout meetings. Anyone who is interested can let me know, and we can decide on a time that works for both of us. [July 28, 18] In the spin_model_simulation.nb code, the last line of the code for b, the quantity Savg[mu, H, kBT] (which represents the array of spins), should be replaced with Savgtheory[mu, H, kBT] (which represents the average of the array of spins). Specifically, you have to calculate the function you defined in the line of the code above for the given values of mu, H, and kBT. The code should have included this already, so I have updated the online version. Also, we will have our review session tomorrow from 1pm to 3pm in room 5-233. Hopefully, the class will be open; we will find a different classroom otherwise. [July 26, 18] A student alerted me to a typo in part (b) of the challenge problem. In this part, "Eq.(17)" should be replaced with "Fig.(3)". I've corrected the mistake on the website, so refreshing the page should show the corrected version. [July 24, 18] Please, bring your laptops tomorrow. The class activity will require them. Three things: - DNA dimerization - Correction of Gap in previous lecture - Reading for activity tomorrow DNA dimerization I posted the lecture notes (09) for the model of DNA dimerization. We did not have time to cover this model in class but with our previous work on the ideal gas model, Laplace's method, and Gamma functions the notes should be comprehensible. Still the model presented in the notes is the most mathematically sophisticated one we've seen so far, so you might have to work through it a few times to understand it. Lecture Notes: "Gap" in derivation. I posted the lecture notes 10 for yesterday's lecture. In the notes, I more completely explain the previous "gap in the derivation" of the final result. Short version: There is no actual "gap". Skip the next few lines if you're not interested in the theoretical explanation. Summarizing the notes, the result p^eq_j/p^eq_k = π_(k->j)/π_(j->k) does not actually follow directly from 0 = Σ_κ( p^eq_k π_(k->j) - p^eq_j π_(j->k) ) Rather, (1) implies (2). So although we began with (2) in the derivation, in moving to (1), we are essentially taking it as an additional assumption. This is something I never noticed before. I have seen (1) a dozen times and it is an important result of non-equilibrium statistical physics, but it turns out it is not as fundamental as it appears. Reading for activity tomorrow Read Section 2.2 from lecture notes 10. Even if it is incomprehensible, the first look will prepare you for the activity tomorrow. [July 22, 18] Two corrections to the current assignment. - For Eq.(4), I flipped the expressions on each side of the equality, so the Hint refers to the wrong term. You should actually begin on the left side and show that it leads to Eq(3). - For Homework 5, the reference to Stirling's approximation refers to what was derived in Problem 4. [July 21, 18] Tomorrow we will have assignment office hours at 1pm in the 8th floor tower C lounge with room number 837. I'll arrive at 12:30pm if anyone has any questions about the course material in general. Also, lecture notes for Friday's class have been updated and revised on the course website in order to complete the derivation of the relationship between < s > and xbar. This derivation closely resembles that used to complete the last parts of the challenge problem. [July 21, 18] This is just a reminder that as you are composing you solutions, you should include explanations of your solutions and not just mathematical calculations without context. Look at the solutions on the course website for a model of how a write up should look. Problem set 1 was technically the "grace problem set" for you to make the mistake once and not be excessively penalized. From problem set 4 on wards, if you only provide an answer or a derivation without an explanation, you will get at max 1 point for that part of the problem. [July 19, 18] In 3(d) of the the problem set, the reference should be to lecture notes 06 not lecture notes 07. The referenced equation is also something we derived in last Friday's lecture on the Boltzmann distribution and partition function. [July 17, 18] As I mentioned last Wednesday, tonight I am available to answer questions until the assignment is due. So feel free to send me any questions you have. [July 16, 18] A student alerted me to a typo in Eq.(6) of pset 4. The exponential should be exp(βμΗ) Namely, there should be a factor of H next to the μ. I corrected the online version and it should appear when you reload the document. [July 13, 18] A student asked a question about problem 1, and I think the answer will benefit everyone. Here is the question. <i>On problem 1, should I use the energy defined using the (-JN/2)m^2 definition (the one on the problem we did in class) when writing out the free energy or just keep the summation in the expression?</i> Neither! You should replace the summation over s_i with m (the average spin), but you should not get (-JN/2)m^2. The energy definition (-JN/2)m^2 is for the mean-field Ising model. The energy term for the mean-field Ising model is based on the assumption that the magnetic field is generated by the mean-spin of the entire system. For the problem set, we are not considering the mean-field Ising model. So, instead we just have an external magnetic field H, and each spin interacts with that external magnetic field. So the energy in the free energy should be linear in m rather than quadratic in m. [July 13, 18] Three things - Lecture Notes - Problem Set 4 - Midterm Evaluations Lecture Notes In lecture today, we covered the Boltzmann distribution and the partition function, but we did not go over an example. Please read Lecture Notes 06 "Boltzmann distribution and Partition Function" which works through an example of applying the Boltzmann distribution to a network of spins in an external magnetic field. Attached is the full problem worksheet from Wednesday. For Problem Set 4 - Problem 5 (b) i. You should have seven terms for this part. - Problem 6 is too difficult, so I am removing it from this assignment and adding it to Problem Set 5 as a challenge problem. Even as a challenge problem, I'll include some additional context so it's easier to solve in Problem Set 5. So feel free to ignore this problem 6 for this assignment. I created an updated version of the assignment with the problem's new status. Midterm Evaluations - Midterm evaluations will be this Sunday from 1-4:30pm in Rm 668a. We might have to move to a new location at 3pm when there are other office hours in the lounge. Each evaluation should only be about 10 mins long. Sign up here (https://doodle.com/poll/72nq9m56swmikdk7). Please sign up for a time slot by tomorrow afternoon otherwise I'll send you an email. [July 11, 18] A student asked a question about problem 1, and I think the answer will benefit everyone. Here is the question. On problem 1, should I use the energy defined using the (-JN/2)m^2 definition (the one on the problem we did in class) when writing out the free energy or just keep the summation in the expression? Neither! You should replace the summation over s_i with m (the average spin), but you should not get (-JN/2)m^2. The energy definition (-JN/2)m^2 is for the mean-field Ising model. The energy term for the mean-field Ising model is based on the assumption that the magnetic field is generated by the mean-spin of the entire system. For the problem set, we are not considering the mean-field Ising model. So, instead we just have an external magnetic field H, and each spin interacts with that external magnetic field. So the energy in the free energy should be linear in m rather than quadratic in m. [July 11, 18] A student alerted me to a typo in the problem set. For problem 5 (b). ii, the problem statement should read ii. Express each of the terms in i. as a factorial (Note: 0! = 1.), and rewrite the result of i. in terms of these factorials. The final expression should have three terms. Namely, ii. should refer to part i, and not part (b). I have corrected the website version. Thank you anonymous student! [July 11, 18] I posted the lecture notes for today's class (Lecture Notes 05). Section 6 of the lecture notes covers the mean-field Ising model in full. You will continue working through the model in section tomorrow, but it would be good if you review the relevant part of the notes beforehand so you have an idea of what the larger question is and why we solve it the way we do. Also, the course going forward will be working through many important problems/techniques in statistical physics. On the website, there is a short guide to working through these many derivations ("Derivations as Arguments"). The important point is to try to understand the larger question the derivation seeks to answer in addition to understanding the specific mathematics of the derivation. [July 11, 18] I posted the lecture notes for today's class (Lecture Notes 05). Section 6 of the lecture notes covers the mean-field Ising model in full. You will continue working through the model in section tomorrow, but it would be good if you review the relevant part of the notes beforehand so you have an idea of what the larger question is and why we solve it the way we do. Also, the course going forward will be working through many important problems/techniques in statistical physics. On the website, there is a short guide to working through these many derivations ("Derivations as Arguments"). The important point is to try to understand the larger question the derivation seeks to answer in addition to understanding the specific mathematics of the derivation. [July 11, 18] You might have noticed that the problem set solutions have been removed from the website. This is a temporary change. The solutions should be back up by this weekend, but let me know if you have any questions about a specific problem on the assignment. [July 8, 18] Just a reminder that we will have office hours today starting at 1pm in the Simmons 2nd floor lounge. We'll be in the lounge farthest from the main entrance to Simmons. [June 29, 18] Here are answers to the anonymous questions/suggestions of lecture today: <i>Give more real life examples of the equations and processes we have been doing so they're more easily conceptualized</i> Good suggestion! Using equations is often good for generality, but considering the most general case off the bat can be confusing. Hopefully, today's examples of the Poisson distribution were sufficiently concrete. Virtually all of our future work will be using variables instead of actual numbers (this is just the nature of theoretical physics), but hopefully we will have ample time to discuss the conceptual underpinnings as well. <i>I am confused about the combinatorics discussed today and the probability taught today. </i> and <i>Combinatorics </i> Understandable! It is a new topic for many of you. I'll actually try to put up solutions to the review document I gave out today so you can check your solution attempts. Also, what, specifically, is confusing? Details help pinpoint the source of confusion. <i>How do we know when to Taylor series and when not to? </i> The use of Taylor series in physics is more art than science. Whenever we are analytically computing something, and we find that we can go no further with standard techniques, we often apply the Taylor series to approximate complex functions with something similar. So we apply Taylor series according to the objectives of our calculation. We will be applying it many times throughout the course. <i>Discuss the meaning of continuous probability distributions</i> Good idea! There is a sense of the meaning of continuous distributions in the lecture notes and in the handout, but it is good to review this concept because we will see it again when we analyze gas systems. Hopefully, I'll be able to discuss continuous distributions more carefully during office hours on Sunday. [June 27, 18] Anticipating what we will be able to cover by next week, I have changed assignment #2. It is the same as the previous version except I replaced the final problem with one on probability distributions. If you have already looked at the assignment online, you might have to reload the page to get an updated version. Also, you might have noticed the link to the lecture note 02 is not live. This is intentional. I'm still editing these lecture notes, and will hopefully post them tonight. Don't fret if you were confused today. The lecture was fast (especially the last part), but I now have a better sense of where the class is and will modify the material accordingly. [June 26, 18] Another student pointed out a contradiction between the figure and a comment about the figure for problem 4. The figure is drawn with θ defined from the horizontal but I later state it is defined from the vertical. The second reference to θ should have it as defined from the horizontal. I corrected the problem set online. Also, don't worry if you already solved the problem defining θ from the vertical. The final answer will just be π/2 minus the result obtained from the alternative definition of θ, and you will still get full credit. In any case, I am more interested in seeing the correct approach to the problem rather than the correct answer. [June 25, 18] A student alerted me to an ambiguity in the problem statement for 4(a). When I write "total distance" traveled, I mean "total horizontal distance". Thank you anonymous student! Also, for this assignment and (all future assignments), feel free to email me questions about the assignment up until your lights out curfew. [June 24, 18] Welcome new students to the web page for Physics III. Throughout the summer, I'll post lecture notes, assignments, solutions, and supplementary material to this webpage. Hope you're ready for an intense and rewarding six weeks!

---

Lecture notes Lec 01, Mon June 25: Introductions, Structure of Physics Lec 02, Wed June 27: Change, Chance, and Counting Workshop, Fri June 29: Taylor Series, Probability, Combinatorics (Review) Lec 03, Mon July 2 and Fri July 6: Entropy from Information Lec 04, Mon July 9: Laws of Thermodynamics Lec 05, Wed July 11: Free Energy and Order Parameters Lec 06, Fri July 13: Boltzmann Distribution and Partition Function Lec 07, Mon July 16 and Wed July 18: Statistical Physics of the Ideal Gas Lec 08, Wed July 18 and Friday July 20: Laplace's Method and Mean-Field Ising Model Lec 09, (Supplement): Model of DNA Dimerization Lec 10, Mon July 23 and Wed July 25: Simulating Statistical Physics Systems Lec 11, (Supplement): Non-Equilibrium Statiatiscal Physics	Supplementary Notes Sup 01, Presenting Your Work Sup 02, Learning Derivations
Assingments Problem Set 1: Mathematics and Physics Review (Due Tues June 26) Problem Set 2: Probability and Counting (Due Tues July 3) Problem Set 3: Beginning Statistical Physics (Due Tues July 10) Problem Set 4: Free Energy and Partition Functions (Due Tues July 17) Problem Set 5: Statistical Physics, The Ideal Gas, and Simulations (Due Tues July 24) Problem Set 6: MCMC and Review (Due Mon July 30)	Solution sets Solution set #1 Solution set #2 Solution set #3 Solution set #4 Solution set #5 Challenge Problem Solution Solution set #6
Exams Final Exam Information Final Exam (July 31)	Exam Solutions Final Exam Solutions
Mathematica Code coin_flip_simulation.nb (Coin-Flip Assignment) coin_flip_simulation_soln.nb (Coin-Flip Solution) spin_model_simulation.nb (Spin Model Simulation Assignment) spin_model_simulation.nb (Spin Model Simulation Solution)	Starter problems Starter Problems Wed July 13 Workshop Wed July 25 Workshop

Collaboration Policy

Feel free to discuss problems with your classmates, the TA, or the instructor. Also, feel free to consult online resources. However, your final submission must represent your own reasoning and calculation only supplemented by personal resources. At the end of your assignment, list the people you collaborated with and the online resources you referenced.

Useful References

Stanford Statistical Physics by Susskind: A series of seven lectures on statistical physics. This is a more advanced course which ends up discussing quantum systems at the end.

Entropy, Order Parameters, and Complexity by Sethna: Online version of a modern textbook on statistical physics. Covers many unique applications of statistical physics in the problems. This course overlaps with chapters 1, 3, 5, and 6.

MIT 8.044 OCW Course site OpenCourseWare collection of materials for MIT's 8.044, the first statistical physics course in the undergraduate physics sequence.