TheInfoList

Statistical physics is a branch of
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

that evolved from a foundation of
statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
, which uses methods of
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
and
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

, and particularly the
mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently
stochastic Stochastic () refers to the property of being well described by a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...
nature. Its applications include many problems in the fields of physics,
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

,
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

,
neuroscience Neuroscience is the of the . It is a science that combines , , , , , and to understand the fundamental and emergent properties of s, and s. The understanding of the biological basis of , , , , and has been described by as the "epic chal ...

. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics develops the phenomenological results of
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

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 classical mechanics, which is concerned with the motion of particles or objects when subjected to a force.

# Scope

Statistical physics explains and quantitatively describes
superconductivity Superconductivity is a set of physical properties observed in certain materials where electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current An electric current is a st ...

,
superfluidity File:Liquid helium Rollin film.jpg, The liquid helium is in the superfluid phase. A thin invisible film creeps up the inside wall of the bowl and down on the outside. A drop forms. It will fall off into the liquid helium below. This will repeat unt ...
,
turbulence In fluid dynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities o ...

, collective phenomena in
solid Solid is one of the four fundamental states of matter 4 (four) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is an ...

s and
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral) or heliotrope, a mineral aggregate * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood plasma ...
, and the structural features of
liquid A liquid is a nearly incompressible In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, ...

. It underlies the modern
astrophysics Astrophysics is a science that employs the methods and principles of physics in the study of astronomical objects and phenomena. Among the subjects studied are the Sun, other stars, galaxy, galaxies, extrasolar planets, the interstellar medium and ...
. In solid state physics, statistical physics aids the study of
liquid crystals Liquid crystals (LCs) are a state of matter which has properties between those of conventional liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flow ...
,
phase transition In chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in ...
s, and
critical phenomenaIn physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...
. Many experimental studies of matter are entirely based on the statistical description of a system. These include the scattering of cold
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...

s,
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Moti ...

,
visible light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nano ...
, 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

Statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and forc ...
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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

as a natural result of statistics, classical mechanics, and
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
at the microscopic level. Because of this history, statistical physics is often considered synonymous with statistical mechanics or
statistical thermodynamics In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throug ...
.This article presents a broader sense of the definition of statistical physics. One of the most important equations in statistical mechanics (akin to $F=ma$ in
Newtonian mechanics Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: ''Law 1''. A body continues ...
, or the
Schrödinger equation The Schrödinger equation is a linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a ma ...
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. : $Z = \sum_q \mathrm^$ where $k_B$ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), c ...
, $T$ is
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

and $E\left(q\right)$ is
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

of state $q$. Furthermore, the probability of a given state, $q$, occurring is given by : $P\left(q\right) = \frac$ 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 Degrees of Freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...
(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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
chaos theory Chaos theory is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields ...
,
thermal physics Example of a thermal column between the ground and a cumulus A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by vo ...
,
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
(particularly at high
Knudsen number The Knudsen number (Kn) is a dimensionless number In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quant ...
s), or
plasma physics Plasma () 1, where \nu_ is the electron gyrofrequency and \nu_ is the electron collision rate. It is often the case that the electrons are magnetized while the ions are not. Magnetized plasmas are ''anisotropic'', meaning that their properties ...
.

## Quantum statistical mechanics

Quantum statistical mechanics Quantum statistical mechanics is statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not as ...
is
statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
applied to quantum mechanical systems. In quantum mechanics, a
statistical ensemble In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
(probability distribution over possible
quantum state In quantum physics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is o ...
s) is described by a
density operator In quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking i ...
''S'', which is a non-negative,
self-adjointIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
,
trace-classIn mathematics, a trace-class operator is a compact operator for which a Trace (linear algebra), trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear op ...
operator of trace 1 on the
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''H'' describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by
quantum logic In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 1936 paper by Garrett Birkhoff and John von Neumann, w ...
.

# Monte Carlo method

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 Monte Carlo methods, or Monte Carlo experiments, are a broad class of computation Computation is any type of calculation A calculation is a deliberate process that transforms one or more inputs into one or more results. The term is used in a v ...
to yield insight into the properties of a
complex system A complex system is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by it ...
. Monte Carlo methods are important in
computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a Scientific theory, quantitative theory already exists. Historically, computational physics was the first application of modern comp ...
,
physical chemistry Physical chemistry is the study of macroscopic scale, macroscopic and particulate phenomena in chemistry, chemical systems in terms of the principles, practices, and concepts of physics such as Motion (physics), motion, energy, force, time, therm ...
, and related fields, and have diverse applications including
medical physics#REDIRECT Medical physics Medical physics (also called biomedical physics, medical biophysics, applied physics in medicine, physics applications in medical science, radiological physics or hospital radio-physics) is, in general, the application of p ...
, where they are used to model radiation transport for radiation dosimetry calculations.

* Combinatorics and physics *
Complex network In the context of network theory, a complex network is a Graph (discrete mathematics), graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattice graph, lattices or random graphs but o ...
* Dynamics of Markovian particles *
Mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
* Mean sojourn time *
Statistical field theoryIn theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contra ...