The Fokker-Planck operator at a continuous phase transition

TL;DR

This paper analyzes the spectral properties of the Fokker-Planck operator in systems undergoing continuous phase transitions, showing eigenvalues accumulate at zero as system size grows, regardless of critical temperature definitions.

Contribution

It demonstrates that eigenvalues of the Fokker-Planck operator accumulate at zero at the transition point in the thermodynamic limit, independent of finite-size critical temperature choices.

Findings

Eigenvalues accumulate at zero as system size increases

Accumulation is unaffected by critical temperature definitions

Results hold below and above the upper critical dimension

Abstract

I consider a physical system described by a continuous field theory and enclosed in a large but finite cubical box with periodic boundary conditions. The system is assumed to undergo a continuous phase transition at some critical point. The \phi^4 theory that is a continuous version of the Ising model is such a system but there are many other examples corresponding to higher spin, higher symmetry etc. The eigenfunctions of the corresponding Fokker-Planck operator can be chosen, of course, to be eigenfunctions of the momentum operator. It is shown that the eigenvalues of the FP operator, corresponding to each eigenvalue q of the momentum operator, evaluated at a transition point of the finite system, accumulate at zero, when the size of the system tends to infinity. There are many reasonable ways of defining a critical temperature of a finite system, that tends to the critical temperature of the infinite system as the size of the system tends to infinity. The accumulation of eigenvalues is neither affected by the specific choice of critical temperature of the finite system nor by whether the system is below or above its upper critical dimension.