Mathematical transformations and physics

[nonequilibrium]

In relation to thermodynamics or statistical mechanics, it is often briefly mentioned that the partition function is the Laplace transform of the microcanonical $\Omega$, or that the Helmholtz free energy is the Legendre transformation of entropy, etc. But in the courses I've had, it stayed to simply noting these facts.

I was just wondering, is there a deeper reason for these appearances of transformations in these contexts? More specifically: is there a more mathematical book on statistical mechanics that actually gives these matters sufficient attention (and their reason should then be that it leads to elegant physics, not just for the math of it, if you understand what I mean).

 

[Valerie Park]

The answer to your question is yes. There are several books that provide a more mathematical treatment of statistical mechanics and thermodynamics, including:1) Statistical Mechanics: Theory and Molecular Simulation by Mark Tuckerman 2) Principles of Statistical Mechanics by H.A. Kramers 3) Classical and Quantum Statistical Mechanics by George Uhlenbeck and George Wannier 4) Statistical Mechanics: A Set of Lectures by Richard Feynman 5) Introduction to Statistical Mechanics by David Chandler 6) Statistical Mechanics by Robert B. Laughlin 7) Statistical Mechanics: Algorithms and Computations by Werner Krauth 8) Statistical Mechanics: An Introduction by David J. Thouless 9) Statistical Mechanics by Pierre Gaspard 10) Introduction to Statistical Physics by Silvio R.A. Salinas. These books all provide a more in-depth look at the transformations and the mathematics behind them. They discuss in detail the concepts of entropy, Helmholtz free energy, partition functions, and their relationship to each other. They also provide examples and explanations of how these concepts can be used to solve various problems in statistical physics.