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).
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.
where
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.
Thus, a probability distribution function
induces a probability measure
on the measure space
, given by
for all
Consider a countable set
with a counting measure imposed on
, such that
, is the cardinality of
, for any subset
. A discrete probability distribution function (dpdf)
on
can be then defined as a nonnegative function
satisfying the equation
A simple example of a
is any Poisson distribution
on
(for any real number
), given by the formula
for any
Taking any probability (or measure) space
defined by the triplet
and a random variable
, one can construct a distribution function on
by defining
The resulting
Consider a measure space
specified as the triplet
, 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 Radon-Nikodym derivative
with respect to
equal to
:
for all
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