Microcanonical solution of the mean-field φ4 model: Comparison with time averages at finite size

We study the mean-field φ4 model in an external magnetic field in the microcanonical ensemble using two different methods. The first one is based on Rughʹs 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→∞ limit. Assuming ergodicity, we evaluate time averages in molecular dynamics simulations and, using Rughʹs 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→∞ values as N is increased. As expected, finite size effects scale as N-1.