PlanetPhysics/Probability Distribution Functions in Physics

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This is a contributed topic on probability distribution functions and their

applications in physics, mostly in spectroscopy, [[../QuantumParadox/|quantum mechanics]], [[../ThermodynamicLaws/|statistical mechanics]] and the theory of extended [[../HotFusion/|QFT]] [[../Groupoid/|operator algebras]] (extended symmetry, [[../QuantumGroupoids/|quantum groupoids]] with [[../HigherDimensionalQuantumAlgebroid/|Haar measure]] and [[../LongRangeCoupling/|quantum algebroids]]).

Probability Distribution Functions in Physics

Physical Examples

{\mathbf [[../FermiDiracDistribution/|Fermi-Dirac distribution]]}

This is a widely used probability distribution function (pdf) applicable to all [[../QuarkAntiquarkPair/|fermion]] [[../Particle/|particles]] in [[../QuantumStatisticalTheories/|quantum statistical mechanics]], and is defined as:

$f_{D-F}(ϵ)=\frac{1}{1+exp(\frac{ϵ-\mu }{kT})},$

where $ϵ$ denotes the [[../CosmologicalConstant/|energy]] of the fermion [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] and $\mu$ is the chemical potential of the fermion system at an [[../ThermodynamicLaws/|absolute temperature]] T.

A classical example of a continuous probability distribution function on $ℝ$ is the Gaussian distribution , or normal distribution $f(x):=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-(x-m{)}^2/2{\sigma }^2},$ where ${\sigma }^2$ is a [[../Parameter/|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 [[../Bijective/|functions]] called Lorentzians are more common; for example, high-resolution $1^{}H$ [[../SpectralImaging/|NMR]] [[../FluorescenceCrossCorrelationSpectroscopy/|absorption]] spectra of neat liquids consist of such Lorentzians whereas rigid [[../CoIntersections/|solids]] exhibit often only Gaussian peaks resulting from both the overlap as well as the marked broadening of Lorentzians.

General definitions of probability distribution functions

One needs to introduce first a [[../BorelSpace/|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 [[../Bijective/|domain]] $\mathrm{\Omega }$ and is also normalized to unity. Thus, consider Failed to parse (unknown function "\borel"): {\displaystyle (\Omega, \borel, \mu)} to be a measure space $S_M$. A probability distribution function (pdf) on (the domain) $\mathrm{\Omega }$ is a function $f_p:\mathrm{\Omega }⟶ℝ$ such that:

1. $f_p$ is $\mu$-measurable
2. $f_p$ is nonnegative $\mu$-measurable-almost everywhere.
3. $f_p$ satisfies the equation

$\underset{\mathrm{\Omega }}{\int }{f}_p(x)d\mu =1.$

Thus, a probability distribution function $f_p$ induces a probability measure $M_P$ on the measure space Failed to parse (unknown function "\borel"): {\displaystyle (\Omega, \borel)} , given by $M_P(X):=\underset{X}{\int }{f}_p(x)d\mu =\underset{\mathrm{\Omega }}{\int }{1}_X{f}_p(x)d\mu ,$ for all Failed to parse (unknown function "\borel"): {\displaystyle x \in \borel} . The measure $M_P$ is called the associated probability measure of $f_p$. $M_P$ and $\mu$ are different measures although both have the same underlying [[../InvariantBorelSet/|measurable space]] Failed to parse (unknown function "\borel"): {\displaystyle S_M := (\Omega, \borel)} .

The discrete distribution (dpdf)

Consider a countable set $I$ with a counting measure imposed on $I$, such that $\mu (A):=|A|$, is the cardinality of $A$, for any subset $A\subset I$. A discrete probability distribution function (\mathbf dpdf) $f_d$ on $I$ can be then defined as a nonnegative function $f_d:I⟶ℝ$ satisfying the equation $\sum _{i\in I}{f}_d(i)=1.$

A simple example of a $dpdf$ is any Poisson distribution $P_r$ on Failed to parse (unknown function "\naturals"): {\displaystyle \naturals} (for any real number $r$), given by the [[../Formula/|formula]] $P_r(i):=e^{-r}\frac{r^i}{i!},$ for any Failed to parse (unknown function "\naturals"): {\displaystyle i \in \naturals} .

Taking any probability (or measure) space $S_M$ defined by the triplet Failed to parse (unknown function "\borel"): {\displaystyle (\Omega, \borel, \mu)} and a random variable $X:\mathrm{\Omega }⟶I$, one can construct a distribution function on $I$ by defining $f(i):=\mu (\{X=i\}).$ The resulting $\mathrm{\Delta }$ function is called the distribution of $X$ on $I.$

The continuous distribution (cpdf)

Consider a measure space $S_M$ 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 ) $f_c:ℝ⟶ℝ$ is simply a measurable, nonnegative almost everywhere function such that $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_c(x)dx=1.$

The associated measure has a [RadonNikodymTheorem Radon--Nikodym derivative] with respect to $\lambda$ equal to $f_c$: $\frac{dP}{d\lambda }=f_c.$

One defines the cummulative distribution function, or {\mathbf cdf ,} $F$ of $f_c$ by the formula $F(x):=P(\{X\le x\})=\underset{-\mathrm{\infty }}{\overset{x}{\int }}f(t)dt,$ for all $x\in ℝ.$