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

Abstract

We solve the mean-field ${\varphi }^4$ 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 $N^{-1}$.

arXiv comment18 pages, 1 figure

References in corpus (11)

• Negative Specific Heat in Astronomy, Physics and Chemistry
• Inequivalence of ensembles in a system with long range interactions
• Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model
• Large deviation techniques applied to systems with long-range interactions
• Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model
• Complete Analysis of Phase Transitions and Ensemble Equivalence for the Curie-Weiss-Potts Model
• A Geometric, Dynamical Approach to Thermodynamics
• The mean-field phi4-model: entropy, analyticity, and configuration space topology
• Clustering and ensembles inequivalence in the phi-4 and phi-6 mean-field Hamiltonian models
• Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model
• On the inequivalence of statistical ensembles

Cited by in corpus (7)

• The large deviation approach to statistical mechanics
• Statistical mechanics and dynamics of solvable models with long-range interactions
• Phase transitions and configuration space topology
• Negative magnetic susceptibility and nonequivalent ensembles for the mean-field ${\varphi }^4$ spin model
• On the thermodynamics of classical micro-canonical systems
• Stability of inhomogeneous states in mean-field models with a local potential
• Quantum fluctuations inhibit symmetry breaking in the HMF model