Statistical physics

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.

A significant contribution (at different times) in development of statistical physics was given by Satyendra Nath Bose, James Clerk Maxwell, Ludwig Boltzmann, J. Willard Gibbs, Marian Smoluchowski, Albert Einstein, Enrico Fermi, Richard Feynman, Lev Landau, Vladimir Fock, Werner Heisenberg, Nikolay Bogolyubov, Benjamin Widom, Lars Onsager, and others. Statistical physics is studied in the nuclear center at Los Alamos. Also, Pentagon has organized a large department for the study of turbulence at Princeton University. Work in this area is also being conducted by Saclay (Paris), Max Planck Institute, Netherlands Institute for Atomic and Molecular Physics and other research centers.

Statistical physics allowed us to explain and quantitatively describe superconductivity, superfluidity, turbulence, collective phenomena in solids and plasma, and the structural features of liquid. It underlies the modern astrophysics. It is statistical physics that helped us to create such intensively developing study of liquid crystals and to construct a theory of phase transition 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 plays a major role in Physics of Solid State Physics, Materials Science, Nuclear Physics, Astrophysics, Chemistry, Biology and Medicine (e.g. study of the spread of infectious diseases), Information Theory and Technique but also in those areas of technology owing to their development in the evolution of Modern Physics. It still has important applications in theoretical sciences such as Sociology and Linguistics and is useful for researchers in higher education, corporate governance, and industry.

• Statistical ensemble
• Statistical field theory
• Mean sojourn time
• Dynamics of Markovian particles
• Complex network
• Mathematical physics
• Combinatorics and physics
• Binomial distribution

• Fundamentals of Statistical and Thermal Physics by Frederick Reif

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)

• 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