Slow relaxation in microcanonical warming of a Ising lattice

TL;DR

This study investigates the slow relaxation dynamics of a semi-infinite Ising lattice under microcanonical warming, revealing power-law decay of correlations and complex temperature-dependent behavior, with faster equilibration in finite systems.

Contribution

It demonstrates the slow, power-law relaxation of correlations in an infinite Ising lattice under microcanonical dynamics and shows faster equilibration in finite systems, highlighting the dynamics at different temperatures.

Findings

Correlations decay to equilibrium via power laws, especially at infinite temperature.

Finite systems reach equilibrium more rapidly than infinite ones.

The relaxation exponents depend non-trivially on temperature.

Abstract

We study the warming process of a semi-infinite cylindrical Ising lattice initially ordered and coupled at the boundary to a heat reservoir. The adoption of a proper microcanonical dynamics allows a detailed study of the time evolution of the system. As expected, thermal propagation displays a diffusive character and the spatial correlations decay exponentially in the direction orthogonal to the heat flow. However, we show that the approach to equilibrium presents an unexpected slow behavior. In particular, when the thermostat is at infinite temperature, correlations decay to their asymptotic values by a power law. This can be rephrased in terms of a correlation length vanishing logarithmically with time. At finite temperature, the approach to equilibrium is also a power law, but the exponents depend on the temperature in a non-trivial way. This complex behavior could be explained in terms of two dynamical regimes characterizing finite and infinite temperatures, respectively. When finite sizes are considered, we evidence the emergence of a much more rapid equilibration, and this confirms that the microcanonical dynamics can be successfully applied on finite structures. Indeed, the slowness exhibited by correlations in approaching the asymptotic values are expected to be related to the presence of an unsteady heat flow in an infinite system.