Microcanonical solution of the mean-field $\phi^4$ model: comparison with time averages at finite size

TL;DR

This paper provides an exact microcanonical solution for the mean-field $$ model, comparing finite-size time averages with the thermodynamic limit, highlighting finite size effects and convergence behavior.

Contribution

It introduces two methods for solving the mean-field $$ model microcanonically and compares finite-size simulation results with the thermodynamic limit.

Findings

Finite size effects scale as N^{-1}

Time averages converge to N limit values as N increases

Finite N time averages are influenced by slow dynamics

Abstract

We solve the mean-field $\phi^4$ model in an external magnetic field in the microcanonical ensemble using two different methods. The first one is based on Rugh's microcanonical formalism and leads to express macroscopic observables, such as temperature, specific heat, magnetization and susceptibility, as time averages of convenient functions of the phase-space. The approach is applicable for any finite number of particles $N$. The second method uses large deviation techniques and allows us to derive explicit expressions for microcanonical entropy and for macroscopic observables in the $N \to\infty$ limit. Assuming ergodicity, we evaluate time averages in molecular dynamics simulations and, using Rugh's approach, we determine the value of macroscopic observables at finite $N$. These averages are affected by a slow time evolution, often observed in systems with long-range interactions. We then show how the finite $N$ time averages of macroscopic observables converge to their corresponding $N\to\infty$ values as $N$ is increased. As expected, finite size effects scale as $N^{-1}$.