Introduction to Mathematical Physics/Statistical physics/Introduction

Statistical physics goal is to describe matter's properties at macroscopic scale from a microscopic description (atoms, molecules, etc\dots). The great number of particles constituting a macroscopic system justifies a probabilistic description of the system. Quantum mechanics (see chapter ---chapmq---) allows to describe part of systems made by a large number of particles: Schr\"odinger equation provides the various accessible states as well as their associated energy. Statistical physics allows to evaluate occupation probabilities $P_{l}$ of a quantum state $l$. It introduces fundamental concepts as temperature, heat\dots

Obtaining of probability $P_{l}$ is done using the statistical physics principle that states that physical systems tend to go to a state of ``maximum disorder [ph:physt:Reif64], [ph:physt:Diu89]. A disorder measure is given by the statistical entropy\index{entropy}

$S=-k_{B}\sum P_{l}\ln P_{l}$

The statistical physics principle can be enounced as:

Postulate:

At macroscopic equilibrium, statistical distribution of microscopic states is, among all distributions that verify external constraints imposed to the system, the distribution that makes statistical entropy maximum.

Remark:

This problem corresponds to the classical minimization (or maximization) problem presented at section chapmetvar) which can be treated by Lagrange multiplier method.