# Statistical physics

Statistical physics is a branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, neurology, and even some social sciences, such as sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion.[1]

In particular, statistical mechanics develops the phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.

## Statistical mechanics

Statistical mechanics provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, therefore explaining thermodynamics as a natural result of statistics, classical mechanics, and quantum mechanics at the microscopic level. Because of this history, the statistical physics is often considered synonymous with statistical mechanics or statistical thermodynamics.[note 1]

One of the most important equations in Statistical mechanics (analogous to $F=ma$ in mechanics, or the Schrödinger equation in quantum mechanics ) is the definition of the partition function $Z$, which is essentially a weighted sum of all possible states $q$ available to a system.

$Z = \sum_q \mathrm{e}^{-\frac{E(q)}{k_BT}}$

where $k_B$ is the Boltzmann constant, $T$ is temperature and $E(q)$ is energy of state $q$. Furthermore, the probability of a given state, $q$, occurring is given by

$P(q) = \frac{ {\mathrm{e}^{-\frac{E(q)}{k_BT}}}}{Z}$

Here we see that very-high-energy states have little probability of occurring, a result that is consistent with intuition.

A statistical approach can work well in classical systems when the number of degrees of freedom (and so the number of variables) is so large that exact solution is not possible, or not really useful. Statistical mechanics can also describe work in non-linear dynamics, chaos theory, thermal physics, fluid dynamics (particularly at high Knudsen numbers), or plasma physics.

Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes the large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a Monte Carlo simulation to yield insight into the dynamics of a complex system.

## References

1. ^ Huang, Kerson. Introduction to Statistical Physics (2nd ed.). CRC Press. p. 15. ISBN 978-1-4200-7902-9.

## Books

Thermal and Statistical Physics (lecture notes, Web draft 2001) by Mallett M., Blumler P.

BASICS OF STATISTICAL PHYSICS: Second Edition by Harald J W Müller-Kirsten (University of Kaiserslautern, Germany)