World Library  
Flag as Inappropriate
Email this Article

Particle statistics

Article Id: WHEBN0003524476
Reproduction Date:

Title: Particle statistics  
Author: World Heritage Encyclopedia
Language: English
Subject: Statistical mechanics, Bose–Einstein statistics, Fermi–Dirac statistics, Isoenthalpic–isobaric ensemble, Maxwell–Boltzmann statistics
Collection: Particle Statistics, Statistical Mechanics
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Particle statistics

Particle statistics is a particular description of multiple particles in statistical mechanics. Its core concept is a statistical ensemble that emphasizes properties of a large system as a whole at the expense of knowledge about parameters of separate particles. When an ensemble consists of particles with similar properties, their number is called the particle number and usually denoted by N.

Contents

  • Classical statistics 1
  • Quantum statistics 2
    • Bose–Einstein statistics 2.1
    • Fermi–Dirac statistics 2.2

Classical statistics

In classical mechanics all the particles (fundamental and composite particles, atoms, molecules, electrons, etc.) in the system are considered distinguishable. This means that one can label and track each individual particle in a system. As a consequence, changing the position of any two particles in the system leads to a completely different configuration of the entire system. Furthermore there is no restriction on placing more than one particle in any given state accessible to the system. Classical statistics is called Maxwell–Boltzmann statistics (or M–B statistics).

Quantum statistics

Quantum occupancy nomograms.

The fundamental feature of quantum mechanics that distinguishes it from classical mechanics is that particles of a particular type are indistinguishable from one another. This means that in an assembly consisting of similar particles, interchanging any two particles does not lead to a new configuration of the system (in the language of quantum mechanics: the wave function 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 (or internal states) that are relevant to the physical problem considered shall be known. All quantum particles, such as leptons and baryons, 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 numbers in an atom), 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 gas (its molecules have a large, but conceivable number of internal states), but is useless applied to systems constructed of macromolecules.

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

Bose–Einstein statistics

In Bose–Einstein statistics (B–E Statistics) interchanging any two particles of the system leaves the resultant system in a symmetric state. That is, the wave function of the system before interchanging equals the wave function of the system after interchanging.

It is important to emphasize that the wave function of the system has not changed itself. This has very important consequences on the state of the system: There is no restriction to the number of particles that can be placed in a single state (accessible to the system). It is found that the particles that obey Bose–Einstein statistics are the ones which have integer spins, which are therefore called bosons (named after Bose). Examples of bosons include photons and helium-4 (both atoms and nuclei). One type of system obeying B–E statistics is the Bose–Einstein condensate where all particles of the assembly exist in the same state.

Fermi–Dirac statistics

In Fermi–Dirac statistics (F–D statistics) interchanging any two particles of the system leaves the resultant system in an antisymmetric state. That is, the wave function of the system before interchanging is the wave function of the system after interchanging, with an overall minus sign.

Again, the wave function of the system itself does not change. The consequence of the negative sign on the Fermi–Dirac statistics can be understood in the following way:

Suppose that the particles that are interchanged belong to the same state. Since the particles are considered indistinguishable from one another then changing the coordinates of the particles should not have any change on the system's wave function (because by our assumptions the particles are in the same state). Therefore, the wave function before interchanging similar states equals the wave function after interchanging similar states.

Combining (or adding, literally speaking) the above statement with the fundamental asymmetry of the Fermi–Dirac system leads us to conclude that the wave function of the system before interchanging equals zero.

This shows that in Fermi–Dirac statistics, more than one particle cannot occupy a single state accessible to the system. This is called Pauli's exclusion principle.

It is found that particles with half-integral spin (or fermions) obey the Fermi–Dirac statistics. This includes electrons, protons, helium-3 (both atoms and nuclei) etc.

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.