Particle statistics is a particular description of multiple

particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...

s in

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

. A key prerequisite concept is that of a

statistical ensemble In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...

(an idealization comprising the

state space In computer science, a state space is a discrete space representing the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial ...

of possible states of a system, each labeled with a probability) that emphasizes properties of a large system as a whole at the expense of knowledge about parameters of separate particles. When an ensemble describes a system of particles with similar properties, their number is called the

particle number In thermodynamics, the particle number (symbol ) of a thermodynamic system is the number of constituent particles in that system. The particle number is a fundamental thermodynamic property which is conjugate to the chemical potential. Unlike m ...

and usually denoted by ''N''.

Classical statistics

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

, all particles ( fundamental and

composite particle This is a list of known and hypothesized microscopic particles in particle physics, condensed matter physics and cosmology. Standard Model elementary particles Elementary particles are particles with no measurable internal structure; that is, ...

s, atoms, molecules, electrons, etc.) in the system are considered distinguishable. This means that individual particles in a system can be tracked. As a consequence, switching the positions of any pair of particles in the system leads to a different configuration of the system. Furthermore, there is no restriction on placing more than one particle in any given state accessible to the system. These characteristics of classical positions are called

Maxwell–Boltzmann statistics In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density ...

Quantum statistics

The fundamental feature of

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

that distinguishes it from classical mechanics is that particles of a particular type are indistinguishable from one another. This means that in an ensemble of similar particles, interchanging any two particles does not lead to a new configuration of the system. In the language of quantum mechanics this means that the

wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...

of the system is invariant up to a phase with respect to the interchange of the constituent particles. In the case of a system consisting of particles of different kinds (for example, electrons and protons), the wave function of the system is invariant up to a phase separately for both assemblies of particles. The applicable definition of a particle does not require it to be elementary or even "microscopic", but it requires that all its

degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...

(or ''internal states'') that are relevant to the physical problem considered shall be known. All quantum particles, such as

lepton In particle physics, a lepton is an elementary particle of half-integer spin (Spin (physics), spin ) that does not undergo strong interactions. Two main classes of leptons exist: electric charge, charged leptons (also known as the electron-li ...

s and

baryon In particle physics, a baryon is a type of composite particle, composite subatomic particle that contains an odd number of valence quarks, conventionally three. proton, Protons and neutron, neutrons are examples of baryons; because baryons are ...

s, in the universe have three translational motion degrees of freedom (represented with the wave function) and one discrete degree of freedom, known as spin. Progressively more "complex" particles obtain progressively more internal freedoms (such as various

quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...

s in an

atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...

), and, when the number of internal states that "identical" particles in an ensemble can occupy dwarfs their count (the particle number), then effects of quantum statistics become negligible. That's why quantum statistics is useful when one considers, say, helium liquid or

ammonia Ammonia is an inorganic chemical compound of nitrogen and hydrogen with the chemical formula, formula . A Binary compounds of hydrogen, stable binary hydride and the simplest pnictogen hydride, ammonia is a colourless gas with a distinctive pu ...

gas (its

molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...

s have a large, but conceivable number of internal states), but is useless applied to systems constructed of

macromolecule A macromolecule is a "molecule of high relative molecular mass, the structure of which essentially comprises the multiple repetition of units derived, actually or conceptually, from molecules of low relative molecular mass." Polymers are physi ...

s. While this difference between classical and quantum descriptions of systems is fundamental to all of quantum statistics, quantum particles are divided into two further classes on the basis of the

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

of the system. The spin–statistics theorem binds two particular kinds of combinatorial symmetry with two particular kinds of spin symmetry, namely

bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-integer ...

and fermions.

Classical statistics

Quantum statistics

See also