Abstract
In this chapter, we explore the facet of noise occurring in a finite-size system due to sampling of initial conditions and its interplay with that introduced through interaction with the environment. Within the ambit of a model long-range interacting system of classical Heisenberg spins, we pursue our study focusing specifically on the dynamical relaxation of the system to Boltzmann-Gibbs equilibrium. Considering the deterministic spin precessional dynamics, we reveal the full range of quasistationary behavior observed during the relaxation, whereby the system gets trapped in nonequilibrium states for times that diverge with the system size, and consequently exhibits a slow relaxation to equilibrium. By contrast, the corresponding stochastic dynamics, modeling the interaction of the system with the environment and constructed in the spirit of the stochastic Landau-Lifshitz-Gilbert equation, shows a fast relaxation to equilibrium on a size-independent timescale. In particular, no signature of quasistationarity is observed in this case when the noise is strong enough. We also observe similar fast relaxation in the Glauber Monte Carlo dynamics of the model. Our findings on the spin dynamics thus establish that the quasistationarity observed in deterministic dynamics is washed away by fluctuations induced through contact with the environment.
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Notes
- 1.
Let us remark that stochasticity in the sense of unpredictability (due to sensitive dependence on the initial conditions) may also result from chaos in fully-deterministic dynamical systems even when the dynamics is low-dimensional.
- 2.
In this chapter, the Roman and the Greek indices are used to label the spins and the spin components, respectively.
- 3.
In this regard, one may refer to the phenomenon of Jeans instability in a prototypical long-range system, the gravitational systems [9].
- 4.
One may note that for the quadratic model (\(n=1\)) with \(\epsilon > \epsilon _c\), the quantity \(\langle \cos ^2\theta \rangle \) is strictly a constant for infinite N. For finite N, it however shows fluctuations about this constant value.
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Das, D., Gupta, S. (2023). Relaxation Dynamics of Mean-Field Classical Spin Systems. In: Facets of Noise. Fundamental Theories of Physics, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-031-45312-0_8
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