# 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, neuroscience, and even some social sciences, such as sociology[1] and linguistics.[2] Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion.[3]

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

One of the most important equations in statistical mechanics (analogous [in what sense?] 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 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.

### Quantum statistical mechanics

Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system; this can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.

## Scientists and universities

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, Benjamin and Jeremy Chubb (also inventors of the titanium sublimation pump), 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.

## Achievements

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.

## Notes

1. ^ This article presents a broader sense of the definition of statistical physics.

## References

1. ^ Raducha, Tomasz; Gubiec, Tomasz (April 2017). "Coevolving complex networks in the model of social interactions". Physica A: Statistical Mechanics and its Applications. 471: 427–435. arXiv:1606.03130. doi:10.1016/j.physa.2016.12.079. ISSN 0378-4371.
2. ^ Raducha, Tomasz; Gubiec, Tomasz (2018-04-27). "Predicting language diversity with complex networks". PLOS ONE. 13 (4): e0196593. doi:10.1371/journal.pone.0196593. ISSN 1932-6203. PMC 5922521. PMID 29702699.
3. ^ Huang, Kerson (2009-09-21). Introduction to Statistical Physics (2nd ed.). CRC Press. p. 15. ISBN 978-1-4200-7902-9.