Epilogue
We have introduce the subject of statistical mechanics in the 18 weeks semester. During the lecture, we introduce the basic framework of statistical mechanics (also, I smuggle some references about the related interesting research subjects ;) ). Let’s take an over view of what we have achieved from the logical point of view.
We ask: what is thermodynamics? We answer this question on a conceptual level.
We start our introduction by pointing out the phenomenological nature of thermodynamics. Because of that, thermodynamics is applied to various different fields.
We emphasis the point of view that thermodynamics is an effective theory of the many-body system.
At this point, we don’t know what is the theoretical framework for effective theories. But, let’s move on.
We have a warm up chapter about basic understanding of statistics and use it to explore some of the key ideas we mentioned previously. We study the random walks and picture what effective theory means.
We introduce the idea of scale invariance and the idea of universality class using the exact solvable random walk model.
From the microscopic model we build an effective macroscopic description of the physics, i.e. the diffusion equation. This is our first exposure of the example of effective theory.
We make a mathematical connection of the random walk model with the central limit theorem which is a mathematical application of the universality class.
The solution of the diffusion equation is connect to the central limit theorem.
We ask again : what is thermodynamics? We answer this question from the formal structure of thermodynamics. We formulate what is thermodynamics by picturing the formal framework of thermodynamics using the function entropy. With several postulated properties of entropy, one can recover the whole thermodynamics. This is just a mathematical concise way to describe thermodynamic. However, the physical picture is not the point we want to emphasis here. The key point of this approach is to emphasis the fundamental importance to have the microscopic interpretation of the core concept – entropy.
What is thermodynamics? The effective theory of statistical mechanics.
The great answer provided by Boltzmann herald the new era of statistical mechanics. Boltzmann’s postulate, \(S=k_B\ln(\vert\Gamma\vert)\), concisely encapsulate the implicit assumption of statistical independence and the connection between macroscopic and microscopic physics.
We reasoned the importance of phase space volume \(\vert \Gamma \vert\) via the Liouville’s theorem. And reasoned why Liouville’s theorem suggest a possible static statistical description of the many-body system.
From the Boltzmann’s postulate, we reason one of the very first ensemble–microcanonical ensemble and develop the corresponding ensembles under different constraints.
We learn how to apply the idea of statistical mechanics to non-interacting systems. We found several generic behavior for those non-interacting systems. Especially, the quantum effect of many-body systems leads to two distinct statistics: Fermi-Dirac and Bose-Einstein statistics. The distinct statistical behavior leads to dramatically different phenomena. We also briefly introduce the second quantization from the principle of indistinguishability in quantum mechanics.
We try to extend our understanding to interacting systems by studying the phases and phase transitions of the simplest non-trivial model–Ising model. We use physical reasoning, exact solution and mean field theory to explore the possible physics. What we have learned actually leads to bigger questions: how to systematically understand different phases of matter? Especially, we are using the simple Ising model which cannot be the full story of the possible theory behind phases and phase transitions. It turns out, the theory standing behind the study of phases and phase transitions is “the theory of theories”, the renormalization group framework. In the take-home project, I hope the astonishing similarity of the critical exponents triggers your curiosity that how come the two very distinct systems, with distinct theoretical descriptions, could have very similarly behavior under suitable conditions.
We answer most questions for non-interacting systems. However, we only discuss the effect of interacting systems using naive mean field theory and leave the phenomena and various approaches for interacting systems for statistical mechanics (II).
After walking you through the statistical mechanics (I) this semester, I hope the materials not only extend your tool box to think about many-body system (or complex systems) but also motivate you to explore emergent many-body effects beyond non-interacting limits. Our semester ends here and hope you have a wonderful 2023 summer.
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