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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

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Slow relaxation in long-range interacting systems with stochastic dynamics.

Shamik Gupta¹, David Mukamel

¹Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel.

Physical Review Letters

|September 28, 2010

Summary

Quasistationary states in Hamiltonian systems are disrupted by stochastic processes. These long-lived nonequilibrium states become crossover phenomena, not long-term behaviors, when energy-conserving stochasticity is introduced.

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Area of Science:

Statistical mechanics
Complex systems dynamics

Background:

Quasistationary states are long-lived nonequilibrium states in deterministic systems with long-range interactions.
These states relax to equilibrium over time scales that scale algebraically with system size.
The Hamiltonian mean-field model is a paradigmatic example exhibiting quasistationarity.

Purpose of the Study:

To investigate the impact of nondeterministic, energy-conserving stochastic processes on quasistationary states.
To determine if quasistationarity persists under the influence of stochastic dynamics.
To analyze the robustness of long-lived nonequilibrium states in generalized models.

Main Methods:

Generalization of the Hamiltonian mean-field model to include energy-conserving stochastic processes.
Analysis using the Boltzmann equation.
Application of a scaling approach.
Conducting numerical simulations.

Main Results:

The system relaxes to equilibrium on time scales that do not diverge algebraically with system size when stochastic processes are present.
Quasistationarity is observed only as a crossover phenomenon.
The duration of the crossover phenomenon is dependent on the strength of the stochastic process.

Conclusions:

Stochastic processes fundamentally alter the nature of quasistationary states.
Quasistationarity is not a robust long-term behavior in the presence of energy-conserving stochasticity.
The introduction of stochasticity leads to a finite relaxation time to equilibrium, overriding algebraic divergence.