Statistical physics

Statistical physics explains and quantitatively describes superconductivity, superfluidity, turbulence, collective phenomena in solids and plasma, and the structural features of liquid. It underlies the modern astrophysics. In solid state physics, statistical physics aids the study of liquid crystals, phase transitions, and critical phenomena. Many experimental studies of matter are entirely based on the statistical description of a system. These include the scattering of cold neutrons, X-ray, visible light, and more. Statistical physics also plays a role in materials science, nuclear physics, astrophysics, chemistry, biology and medicine (e.g. study of the spread of infectious diseases).

Statistical mechanics
Thermodynamics Kinetic theory
Particle statistics Spin–statistics theorem Identical particles Maxwell–Boltzmann Bose–Einstein Fermi–Dirac Parastatistics Anyonic statistics Braid statistics
Thermodynamic ensembles NVE Microcanonical NVT Canonical µVT Grand canonical NPH Isoenthalpic–isobaric NPT Isothermal–isobaric
Models Debye Einstein Ising Potts
Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
Scientists Maxwell Boltzmann Gibbs Einstein Ehrenfest von Neumann Tolman Debye Fermi Bose

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, statistical physics is often considered synonymous with statistical mechanics or statistical thermodynamics.[note 1]

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

${\displaystyle Z=\sum _{q}\mathrm {e} ^{-{\frac {E(q)}{k_{B}T}}}}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle T}$ is temperature and ${\displaystyle E(q)}$ is energy of state ${\displaystyle q}$. Furthermore, the probability of a given state, ${\displaystyle q}$, occurring is given by

${\displaystyle P(q)={\frac {\mathrm {e} ^{-{\frac {E(q)}{k_{B}T}}}}{Z}}}$

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 the 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 properties of a complex system. Monte Carlo methods are important in computational physics, physical chemistry, and related fields, and have diverse applications including medical physics, where they are used to model radiation transport for radiation dosimetry calculations.[2][3][4]

• Fundamentals of Statistical and Thermal Physics by Frederick Reif
• BASICS OF STATISTICAL PHYSICS: Second Edition

by Harald J W Müller-Kirsten (University of Kaiserslautern, Germany)

• Statistical Physics and other resources by Leo P Kadanoff
• Statistical Physics - Statics, Dynamics and Renormalization by Leo P Kadanoff
• History and outlook of statistical physics by Dieter Flamm