HOW TO SELF LEARN STATISTICAL MECHANICS
Hola
Everyone!! I am writing this blog after a quite long time. This time I like to
discuss an overview of Statical Mechanics at beginners level. The first thing that pops up in our
mind when we hear thermodynamics or statistical physics is the tragic ending of
great physicist Ludwig Boltzmann. Also such sad stories of other contemporary
legends like Robert Mayer(after whom universal gas constant R is named), Paul Ehrenfest(after whom Ehrenfest’s Theorem in QM is
named), Ralph Fowler, Hermann von Helmholtz(after whom Helmholtz potential,
Helmholtz theorems are named), Williard Gibbs, Rudolf Diesel(founder of diesel
engine whose death is a mystery till the date); also had a similar ending
though for some different reasons. So, the daemon attached to the subject will
make you scary before you even start the subject. But except for Boltzmann and Ehrenfest; rest didn’t commit suicide as you might have read
in some pop-sci articles, instead, they had scumbled to other health ailments
or faced some difficult days at the later part of their life due to many
reasons which had nothing to do with their research career. But there are many other reasons which make
the subject quite different from other subjects.
Before learning stat mech it is important to know what makes
this subject unique regarding its historical significance and origin. This
qualitative understanding will let you appreciate the subject when you explore
different topics. Also, I will discuss prerequisites and resources. So, let’s
crank the engine!
We know that thermodynamics was formulated by the pioneers of
the 17th -19th century. This included formulation of Gas
laws, the concept of heat or calorie, different laws based on energy
conservation, and finally the idea of entropy and other thermodynamics
potentials. This was coupled with engineering innovations of different types of
engines and devices that helped in utilizing mechanical
advantage thus inducing the Industrial Revolution. But once the main theories
were formulated the major interest shifted to optimization of existing engines
and concepts like increasing efficiency, finding suitable cycles to perform the
certain task; because of which we have dozens of thermodynamics cycles, which
you might have learned if you had a course in Engineering Thermodynamics. But the theoretical aspects had a different story. The
main there was nothing certain about the microscopic behavior of a given
macroscopic phenomenon since there wasn’t any substantial proof for the
existence of atomic and subatomic particles. Amongst the concept of entropy had
something peculiar which revealed some of the secrets of nature. The other
concepts were intuitively pleasing, but we had no reason to accept the entropy
principle, but this ignited some fresh ideas about explaining
thermodynamics. Well, here I am not
exactly going by hierarchy I am naively trying to explain how things were
overall. So pardon me if I have messed up with the chronology.
So, it is in this period some of the greatest minds came forward with an idea to explain macroscopic phenomena in terms of the microscopic phenomenon. There were early efforts by Maxwell, Clausius to explain different properties of molecules or constituent atoms in an average or statistical manner using probabilistic distributions. Surprisingly there were some papers centuries-old proposed by other mathematicians and physicists but they had little or no influence directly except for mathematical concepts. This was followed by other prominent people including Boltzmann with the concept of trying to explain the system in terms of ensembles and entropy. (Here I am not using the exact words used in the original papers but I am just trying to summarize the main idea).
For this reason, you will find statistical mechanics have many ad-hoc assumptions and ansatz which is quite difficult to accept though accepted and explained theoretically and verified experimentally. So this method of viewing the subject from a different perspective makes it quite difficult at first since you have learned the same concept differently now you are trying to explain with some crazy concepts. Also, mathematical concepts of probability, asymptotic analysis, and other statistical tools all together make the subject highly counter-intuitive. At least this was the case for me when I first encountered the topics. So now let me mention how to approach the subject.
Major Prerequisites
I am not
going to repeat the basic mathematical concepts of Calculus, Linear Algebra
which I have already mentioned in the previous posts.
a. a. Mathematical
Prerequisites
1. 1.Probability and Statistics: Well the first thing you should know is probability and statistics. So the natural question would be at what level? Knowing probability at an advanced level will be extremely helpful. But I will mention what worked for me. It will be more than enough to cover all topics in any standard Mathematical Physics textbooks like; Riley-Hobson, Arfken-Weber, or Balki ‘s book on the same topic. I read the chapter on probability and statistics from Riley-Hobson only; which I found more than sufficient for a beginner's course on statistical mechanics. But the way these techniques are used in the subject is quite different. Concepts related to the central limit theorem, generating function, cumulant, different probability distributions will be useful. Also, statistical mechanics books discuss all the essential topics when they are covered.
2.Fourier Transforms: Well you don’t need to know F.T in detail, but you should know the physical meaning of these concepts and also about their relations with probability and statistics. Any book on Mathematical Methods will serve the purpose.
3.Asymptotic Analysis: Again this is a small topic which you should know. Some special tricks and theorems will be discussed in the statistical mechanics textbook which is sufficient. For all who looking to get a better idea you can read Chapter:6 Asymptotic Expansion of Integrals in the book “Advanced Mathematical Methods for Scientist and Engineers” by Bender and Orzag (title and chapter number may vary subject to change in each edition)
b. b.Physics Prerequisites
1.Thermodynamics: You should have a good understanding of
thermodynamical concepts. So brush up on old topics once. You will wonder that
both thermodynamics and statistical physics will give identical results which
will make you respect all legends who contributed to their attempt to look subject
from a different perspective.
2.Quantum Mechanics: Knowledge of quantum mechanics at least on
the elementary topics (more precisely up to chater11 of R.Shankar ) is
essential. Though you don’t encounter it at an early stage you need QM when you
learn Quantum statistical mechanics at a later stage. Particularly difference between distinguishable and indistinguishable particles, symmetries etc..
3.Classical Mechanics: Knowledge of advanced classical mechanics
topics like Hamiltonian Principle, Phase Space, Liouville Theorem, and Legendre-Transformation will be very useful.
Reference Materials
Now let us begin with major textbooks that will be useful
1.Thermodynamics and Statistical Mechanics by Greiner-Neise-Stocker: This is a part of Walter Greiner's Theoretical Physics books series. This is one of the less known(maybe?) fantastic books to understand the subject from an absolute beginner's level. The book starts with a discussion on major theoretical topics related to thermodynamics in pedagogical sequence starting from gas laws and basic definition of thermodynamics followed by entropy and thermodynamics potentials. The authors take enough time to explain the shortcomings in thermodynamics and all the unanswered questions. Part two of this book deals with statical mechanics. The author's way of introducing concepts of the ensemble, explaining Equi-Priori theorem, the ergodic hypothesis under the light of statistics and probability is highly commendable. An emphasis on the philosophy of the subject makes it very interesting. The author takes time in explaining each mathematical step in great mathematical details wherever required and this doesn’t make you feel “How the hell this result was obtained!!” To be frank this was the subject where I saw all 24 Greek letters in their capital and small forms together but this book helped to understand what the expression means instead of taking it for granted. I like to stress topics related to Quantum Statistical Mechanics are described exceptionally. Another big positive thing about this book is the solved example; each example teaches you some real application-level problems including topics in condensed matter physics.
2.Statistical Mechanics by Pathria and Beale: This is another standard book used for statistical mechanics. This book is also similar to Greiner but there are many more topics covered in the book which deal with advanced statistical methane topics in the later chapters. Also, there are more applicational topics in comparison to Grainer particularly for Quantum Statistical Mechanics(these topics are not covered in). I found the content and explanation are very similar (maybe you can check the Bibliography of both books )
3.Statistical Physics of Particle by Mehran Kardar: This is a concise book on statistical mechanics and the second volume of the book Statistical Physics of Fields deals with advanced level topics. This is more of a lecture note rather book because this is authored by MIT professor Prof.Mehran Kardar which is highly based on his class lectures and notes at MIT. (These lectures are readily available at MIT OCW ). So I like to rate this at a slightly higher level compared to previous ones since the descriptions feel quite inadequate and mathematical steps are not described in detail as in the previous books. Some derivations were not clear personally for me. This might be because the prerequisites and other topics have been specifically developed based on the MIT curriculum. You can also be following the video lectures on which the textbook is based. Which helps to cover the book easily. This book has concepts explained in a mathematically rigorous manner. There is a dedicated chapter on probability which is more than sufficient to learn statistical mechanics at the initial level. And the problems at the end are application-level problems that make concepts very clear. You can find a solution manual also for this but give it a try before giving up. There are nearly 10 problems at the end of each chapter. Topics on Kinetic Theory of gases, Cluster Expansion, Virial Coefficients, and applications of some condensed matter theory topics are also covered.
There are many other books like F.Reif ,Huang Kerson, Stephen Blundell, Landau-Lifshitz etc which are all great. But since I haven't read any of them, I am unable to review here. If you have read any of the above books share your experience in comment section.
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