The microscopic meaning of grand potential resulting from combinatorial approach to a general system of particles

TL;DR

This paper introduces a novel combinatorial approach to interacting fluids, revealing a microscopic interpretation of the grand potential as an exponential generating function for cluster states, providing new insights into phase transitions.

Contribution

It presents a new combinatorial method using Bell polynomials to interpret the grand potential microscopically, linking it to cluster internal states and phase transition mechanisms.

Findings

Derived exact probability formula for particle cluster configurations.

Revealed grand potential as a generating function for cluster states.

Provided an approximate density of states expression.

Abstract

We present a completely new approach to the problem of interacting fluids, which we believe may provide important insights into microscopic mechanisms that lead to the occurrence of phase transitions. The approach exploits enumerative properties and combinatorial meaning of Bell polynomials. We derive the exact formula for probability of a general system of N particles at temperature T to consist of k weakly coupled clusters of various sizes. We also show that the grand potential of the system may be considered as the exponential generating function for the number of internal states (thermodynamic probability) of these clusters. The microscopic interpretation of the grand potential is novel and surprising, especially if one recalls that until now the only thermodynamic meaning of this free energy was known. We also derive an approximated expression for the density of states.