Thermodynamics and statistical mechanics#

Reference: [Pel11] Chap. 1 and Chap. 2, [Set21] Chap. 6.4

Summary

  • Thermodynamics use a small and finite set of variables to describe the macroscopic property of \(10^{23}\) degrees of freedom. Statistical mechanics provides a microscopic reasoning behind thermodynamics. It is achieved through several nontrivial approximations

    1. The deterministic description is replaced by the statistical description.

    2. The system is assumed to be homogeneous.

  • The central problem of thermodynamics is about the final equilibrium thermodynamic properties given the initial thermodynamic properties of the system.

  • Key concepts in thermodynamics

    1. Extensive and intensive variables

    2. the central problem of thermodynamics

    3. Lagendre transformation

  • Thermodynamics can be constructed by assuming some properties of the function “entropy”

    1. Entropy is the solution of the central problem of thermodynamics.

    2. Entropy is a function of extensive variables.

    3. Entropy is an extensive qunatity.

    4. Entropy is a concave function in the extensive variables.

    5. Entropy is a monotonically increasing function of the internal energy.

There are several ways to build the connection between thermodynamics and statistical mechanics. We will choose to formulate thermodynamics by defining the properties of the entropy function. The relation of this approach with the thermodynamic laws will be left for the readers to think about.

What?

The thermodynamic laws are

  1. Transitivity of equilibria: if system \(A\) and system \(C\) are in equilibrum. So is \(B\) and \(C\). Then, \(A\) and \(B\) are in equilibrium

  2. Conservation of energy: total energy of an isolated system including the heat energy is constant.

  3. Entropy always increses.

  4. Entropy at \(T=0\) is 0.

How they are related to the properties of entropy?

We will start with a simple example: the idea gas model to illustrate the logical jump between the deterministic description and the statistical description. After we have a rough idea what assumptions are made and what’s the meaning of statistical description. We will review the thermodyanmics circulating around the definition of entropy function. We would like to emphasis the fact that the idea of entropy in thermodynamics is an abstract concept without microscopic interpretation. The microscopic interpretation of entropy is the starting point of the formulation of statistical mechanics.

The purpose of this chapter is to bridge our understanding toward the statistical postulate which is the foundation of statistical mechanics. With the help of this chapter, we will discuss the fundamental postulates of statistical mechanics.