PlanetPhysics/Probability Distribution Functions in Physics

\newcommand{\sqdiagram}[9]{Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

This is a contributed topic on probability distribution functions and their

applications in physics, mostly in spectroscopy, quantum mechanics, statistical mechanics and the theory of extended QFT operator algebras (extended symmetry, quantum groupoids with Haar measure and quantum algebroids).

Probability Distribution Functions in PhysicsEdit

Physical ExamplesEdit

{\mathbf Fermi-Dirac distribution}

This is a widely used probability distribution function (pdf) applicable to all fermion particles in quantum statistical mechanics, and is defined as:

where denotes the energy of the fermion system and is the chemical potential of the fermion system at an absolute temperature T.

A classical example of a continuous probability distribution function on is the Gaussian distribution , or normal distribution where is a parameter related to the width of the distribution (measured for example at half-heigth).

In high-resolution spectroscopy, however, similar but much narrower continuous distribution functions called Lorentzians are more common; for example, high-resolution NMR absorption spectra of neat liquids consist of such Lorentzians whereas rigid solids exhibit often only Gaussian peaks resulting from both the overlap as well as the marked broadening of Lorentzians.

General definitions of probability distribution functionsEdit

One needs to introduce first a Borel space Failed to parse (unknown function "\borel"): {\displaystyle \borel} , then consider a measure space Failed to parse (unknown function "\borel"): {\displaystyle S_M:= (\Omega, \borel, \mu)} , and finally define a real function that is measurable `almost everywhere' on its domain and is also normalized to unity. Thus, consider Failed to parse (unknown function "\borel"): {\displaystyle (\Omega, \borel, \mu)} to be a measure space . A probability distribution function (pdf) on (the domain) is a function such that:

  1. is -measurable
  2. is nonnegative -measurable-almost everywhere.
  3. satisfies the equation

Thus, a probability distribution function induces a probability measure on the measure space Failed to parse (unknown function "\borel"): {\displaystyle (\Omega, \borel)} , given by for all Failed to parse (unknown function "\borel"): {\displaystyle x \in \borel} . The measure is called the associated probability measure of . and are different measures although both have the same underlying measurable space Failed to parse (unknown function "\borel"): {\displaystyle S_M := (\Omega, \borel)} .

The discrete distribution (dpdf)

Consider a countable set with a counting measure imposed on , such that , is the cardinality of , for any subset . A discrete probability distribution function (\mathbf dpdf) on can be then defined as a nonnegative function satisfying the equation

A simple example of a is any Poisson distribution on Failed to parse (unknown function "\naturals"): {\displaystyle \naturals} (for any real number ), given by the formula for any Failed to parse (unknown function "\naturals"): {\displaystyle i \in \naturals} .

Taking any probability (or measure) space defined by the triplet Failed to parse (unknown function "\borel"): {\displaystyle (\Omega, \borel, \mu)} and a random variable , one can construct a distribution function on by defining The resulting function is called the distribution of on

The continuous distribution (cpdf)

Consider a measure space specified as the triplet Failed to parse (unknown function "\borel"): {\displaystyle (\reals, \borel_\lambda, \lambda)} , that is, the set of real numbers equipped with a Lebesgue measure . Then, one can define a continuous probability distribution function (cpdf ) is simply a measurable, nonnegative almost everywhere function such that

The associated measure has a [RadonNikodymTheorem Radon--Nikodym derivative] with respect to equal to :

One defines the cummulative distribution function, or {\mathbf cdf ,} of by the formula for all

All SourcesEdit

[1][2]

ReferencesEdit

  1. B. Aniszczyk. 1991. A rigid Borel space., Proceed. AMS. , 113 (4):1013-1015., available online.
  2. A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in Math.}, Springer-Verlag, Berlin, {\mathbf 725}: 19-14.