# 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 ${\displaystyle P_{l}}$ of a quantum state ${\displaystyle l}$. It introduces fundamental concepts as temperature, heat\dots

Obtaining of probability ${\displaystyle P_{l}}$ 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[1]\index{entropy}

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

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.

1. This formula is analog to the information entropy chosen by Shannon in his information theory [ph:physt:Shannon49].