probability distribution functions in physicsProbabilityDistributionFunctionsInPhysics2009-04-22 12:45:272009-04-23 23:42:44Topicprobability distribution functionprobability measureassociated probability measurePoisson distributiondpdfdiscrete probability distribution functiondistribution of $X$ on $I$cumulative distribution functioncdfRadon--Nikodym derivativeGaussian distributionnormal distributionchemical potentialquantum groupoids with Haar measureRadon measureFermi-Dirac distribution function%%Planet physics followed by special preamble
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\def\D{\mathsf{D}}This is a contributed topic on probability distribution functions and their
applications in physics (mostly in quantum mechanics, statistical mechanics
and the theory of extended QFT operator algebras, extended symmetry, quantum groupoids with Haar measure and quantum algebroids).
\section{Probability Distribution Functions in Physics}
One needs to introduce first a Borel space $\borel$, then a measure space $(\Omega, \borel, \mu)$, and finally define a real function that is measurable `almost everywhere' on its domain $\Omega$ and is also normalized to unity.
Let $(\Omega, \borel, \mu)$ be a measure space. A {\em probability distribution function} on (the domain) $\Omega$ is a function $f: \Omega \longrightarrow \reals$ such that:
\begin{enumerate}
\item $f$ is $\mu$-measurable
\item $f$ is nonnegative $\mu$-almost everywhere.
\item $f$ satisfies the equation
$$
\int_{\Omega} f(x)\ d\mu = 1
$$
\end{enumerate}
Thus, a probability distribution function $f$ induces a {\em probability measure} $P$ on the measure space $(\Omega, \borel)$, given by
$$
P(X) := \int_X f(x)\ d\mu = \int_{\Omega} 1_X f(x)\ d\mu,
$$
for all $X \in \borel$. The measure $P$ is called the {\em associated probability measure} of $f$. Note that $P$ and $\mu$ are different measures, \PMlinkescapetext{even} though they both share the same underlying measurable space $(\Omega, \borel)$.
\section{Examples}
\subsection{Fermi-Dirac Distribution}
This is a widely used probability distribution function 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 {\em chemical potential} of the fermion system at an absolute temperature T.
\subsection{The Discrete Case: 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 {\em discrete probability distribution function (\bf 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 is the Poisson distribution $P_r$ on $\naturals$ (for any real number $r$), which is given by
$$
P_r(i) := e^{-r} \frac{r^i}{i!}
$$
for any $i \in \naturals$.
Given any probability (or measure) space $(\Omega, \borel, \mu)$ and any random variable $X: \Omega \longrightarrow I$, one costructs a distribution function on $I$ by taking $f(i) := \mu(\{X = i\})$. The resulting function $\Delta$ is called the {\em distribution of $X$ on $I$}.
\subsection{The Continuous Case: cpdf}
Let $(\Omega, \borel, \mu)$ equals $(\reals, \borel_\lambda, \lambda)$ be the set of real numbers equipped with a Lebesgue measure. Then a continuous probability distribution function ({\em 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 \PMlinkexternal{Radon--Nikodym derivative}{RadonNikodymTheorem} with respect to $\lambda$ equal to $f_c$:
$$
\frac{dP}{d\lambda} = f_c.
$$
One defines the {\em cummulative distribution function, or {\bf cdf},} $F$ of $f_c$ by the formula
$$
F(x) := P(\{X \leq x\}) = \int_{-\infty}^x f(t)\ dt,
$$
for all $x \in \reals$.
\begin{example}
A classical example of a continuous probability distribution function on $\reals$ is the {\em Gaussian distribution}, or {\em 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).
\end{example}
In high-resolution spectroscopy, however, similar but much narrower continuous distributions called {\em 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.