probability distribution functions in physics

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).

0.1 Probability Distribution Functions in Physics

0.1.1 Physical Examples

Example 0.1. 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:

1 fD−F (𝜖) = ---------𝜖−μ-, 1 + exp (kT )

where 𝜖 denotes the energy of the fermion system and μ is the chemical potential of the fermion system at an absolute temperature T.

Example 0.2. A classical example of a continuous probability distribution function on ℝ is the Gaussian distribution, or normal distribution

1 2 2 f (x) := -√----e−(x−m) ∕2σ , σ 2π

where σ² 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.

0.2 General definitions of probability distribution functions

Definition 0.1. One needs to introduce first a Borel space 𝔅, then consider a measure space S_M := (Ω,𝔅,μ), and finally define a real function that is measurable ‘almost everywhere’ on its domain Ω and is also normalized to unity. Thus, consider (Ω,𝔅,μ) to be a measure space S_M. A probability distribution function (pdf) on (the domain) Ω is a function f_p : Ω→ℝ such that:

1.: f_p is μ-measurable
2.: f_p is nonnegative μ-measurable-almost everywhere.
3.: f_p satisfies the equation $∫ f (x ) dμ = 1. Ω p$

Thus, a probability distribution function f_p induces a probability measure M_P on the measure space (Ω,𝔅), given by

∫ ∫ MP (X ) := fp(x) dμ = 1X fp(x) d μ, X Ω

for all x ∈ 𝔅. The measure M_P is called the associated probability measure of f_p. M_P and μ are different measures although both have the same underlying measurable space S_M := (Ω,𝔅).

Definition 0.2. The discrete distribution (dpdf)

Consider a countable set I with a counting measure imposed on I, such that μ(A) := |A|, is the cardinality of A, for any subset A ⊂ I. A discrete probability distribution function (dpdf) f_d on I can be then defined as a nonnegative function f_d : I→ℝ satisfying the equation

∑ fd(i) = 1. i∈I

A simple example of a dpdf is any Poisson distribution P_r on ℕ (for any real number r), given by the formula

i Pr (i) := e−rr-, i!

for any i ∈ ℕ.

Taking any probability (or measure) space S_M defined by the triplet (Ω,𝔅,μ) and a random variable X : Ω→I, one can construct a distribution function on I by defining

f(i) := μ ({X = i}).

The resulting Δ function is called the distribution of X on I.

Definition 0.3. The continuous distribution (cpdf)

Consider a measure space S_M specified as the triplet (ℝ,𝔅_λ,λ), 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

∫ ∞ fc(x ) dx = 1. −∞

The associated measure has a Radon–Nikodym derivative with respect to λ equal to f_c:

dP --- = fc. dλ

Definition 0.4. One defines the cummulative distribution function, or cdf, F of f_c by the formula

∫ x F (x) := P ({X ≤ x} ) = f(t) dt, −∞

for all x ∈ ℝ.

References

[1] B. Aniszczyk. 1991. A rigid Borel space., Proceed. AMS., 113 (4):1013-1015., available online.

[2] A. Connes.1979. Sur la théorie noncommutative de l’ integration, Lecture Notes in Math., Springer-Verlag, Berlin, 725: 19-14.

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