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).

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 $$\reals$$ 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 $$^1H$$ 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 $$\borel$$, then consider a measure space $$S_M:= (\Omega, \borel, \mu)$$, and finally define a real function that is measurable `almost everywhere' on its domain $$\Omega$$ and is also normalized to unity. Thus, consider $$(\Omega, \borel, \mu)$$ to be a measure space $$S_M$$. A probability distribution function (pdf) on (the domain) $$\Omega$$ is a function $$f_p: \Omega \longrightarrow \reals$$ such that:

$$ \int_{\Omega} f_p(x)\ d\mu = 1. $$
 * 1) $$f_p$$ is $$\mu$$-measurable
 * 2) $$f_p$$ is nonnegative $$\mu$$-measurable-almost everywhere.
 * 3) $$f_p$$ satisfies the equation

Thus, a probability distribution function $$f_p$$ induces a probability measure $$M_P$$ on the measure space $$(\Omega, \borel)$$, given by $$M_P(X) := \int_X f_p(x)\ d\mu = \int_{\Omega} 1_X f_p(x)\ d\mu,$$ for all $$x \in \borel$$. 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 $$S_M := (\Omega, \borel)$$.

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 $$f_d : I \longrightarrow \reals$$ satisfying the equation $$\sum_{i \in I} f_d(i) = 1.$$

A simple example of a $$dpdf$$ is any Poisson distribution $$P_r$$ on $$\naturals$$ (for any real number $$r$$), given by the formula $$ P_r(i) := e^{-r} \frac{r^i}{i!}, $$ for any $$i \in \naturals$$.

Taking any probability (or measure) space $$S_M$$ defined by the triplet $$(\Omega, \borel, \mu)$$ 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 $$(\reals, \borel_\lambda, \lambda)$$, that is, the set of real numbers equipped with a Lebesgue measure. Then, one can define a continuous probability distribution function (cpdf ) $$f_c : \reals \longrightarrow \reals$$ 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 $$x \in \reals.$$