Note on exponential families of distributions

TL;DR

This paper demonstrates that any probability distribution can be expressed in exponential form, implying broad applicability in representing equilibrium states of classical and quantum systems in grand canonical form.

Contribution

It establishes a general representation theorem for probability distributions in exponential form, extending to classical and quantum equilibrium distributions.

Findings

Any probability distribution can be represented exponentially.

Equilibrium distributions in classical and quantum systems are expressible in grand canonical form.

The result unifies the representation of diverse physical systems.

Abstract

We show that an arbitrary probability distribution can be represented in exponential form. In physical contexts, this implies that the equilibrium distribution of any classical or quantum dynamical system is expressible in grand canonical form.