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, 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 Physics

Physical Examples

{\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:

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

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

A classical example of a continuous probability distribution function on $\mathbb {R}$ 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 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 $^{1}H$ 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 functions

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 $\Omega$ 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) $\Omega$ is a function $f_{p}:\Omega \longrightarrow \mathbb {R}$ such that:

1. $f_{p}$ is $\mu$-measurable
2. $f_{p}$ is nonnegative $\mu$-measurable-almost everywhere.
3. $f_{p}$ satisfies the equation

$\int _{\Omega }f_{p}(x)\ d\mu =1.$

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 $M_{P}(X):=\int _{X}f_{p}(x)\ d\mu =\int _{\Omega }1_{X}f_{p}(x)\ d\mu ,$ 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 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 $f_{d}:I\longrightarrow \mathbb {R}$ 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 $P_{r}(i):=e^{-r}{\frac {r^{i}}{i!}},$ 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 $X:\Omega \longrightarrow I$, one can construct a distribution function on $I$ by defining $f(i):=\mu (\{X=i\}).$ The resulting $\Delta$ 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 ) $f_{c}:\mathbb {R} \longrightarrow \mathbb {R}$ is simply a measurable, nonnegative almost everywhere function such that $\int _{-\infty }^{\infty }f_{c}(x)\ dx=1.$

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 $F(x):=P(\{X\leq x\})=\int _{-\infty }^{x}f(t)\ dt,$ for all $x\in \mathbb {R} .$