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Register a new userViolent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams
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A classical long-range-interacting $N$-particle system relaxes to thermal equilibrium on time scales growing with $N$; in the limit $Nto infty$ such a relaxation time diverges. However, a completely non-collisional relaxation process, known as violent relaxation, takes place on a much shorter time scale independent of $N$ and brings the system towards a non-thermal quasi-stationary state. A finite system will eventually reach thermal equilibrium, while an infinite system will remain trapped in the quasi-stationary state forever. For times smaller than the relaxation time the distribution function of the system obeys the collisionless Boltzmann equation, also known as the Vlasov equation. The Vlasov dynamics is invariant under time reversal so that it does not naturally describe a relaxational dynamics. However, as time grows the dynamics affects smaller and smaller scales in phase space, so that observables not depending upon small-scale details appear as relaxed after a short time. Herewith we present an approximation scheme able to describe violent relaxation in a one-dimensional toy-model, the Hamiltonian Mean Field (HMF). The approach described here generalizes the one proposed in G. Giachetti and L. Casetti, J. Stat. Mech.: Theory Exp. 2019, 043201 (2019), that was limited to cold initial conditions, to generic initial conditions, allowing us to to predict non-equilibrium phase diagrams that turn out to be in good agreement with those obtained from the numerical integration of the Vlasov equation.
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In $N$-body systems with long-range interactions mean-field effects dominate over binary interactions (collisions), so that relaxation to thermal equilibrium occurs on time scales that grow with $N$, diverging in the $Ntoinfty$ limit. However, a faster and non-collisional relaxation process, referred to as violent relaxation, sets in when starting from generic initial conditions: collective oscillations (referred to as virial oscillations) develop and damp out on timescales not depending on the systems size. After the damping of such oscillations the system is found in a quasi-stationary state that survives virtually forever when the system is very large. During violent relaxation the distribution function obeys the collisionless Boltzmann (or Vlasov) equation, that, being invariant under time reversal, does not naturally describe a relaxation process. Indeed, the dynamics is moved to smaller and smaller scales in phase space as time goes on, so that observables that do not depend on small-scale details appear as relaxed after a short time. We propose an approximation scheme to describe collisionless relaxation, based on the introduction of moments of the distribution function, and apply it to the Hamiltonian Mean Field (HMF) model. To the leading order, virial oscillations are equivalent to the motion of a particle in a one-dimensional potential. Inserting higher-order contributions in an effective way, inspired by the Caldeira-Leggett model of quantum dissipation, we derive a dissipative equation describing the damping of the oscillations, including a renormalization of the effective potential and yielding predictions for collective properties of the system after the damping in very good agreement with numerical simulations. Here we restrict ourselves to cold initial conditions; generic initial conditions will be considered in a forthcoming paper.
We consider a modification of the well studied Hamiltonian Mean-Field model by introducing a hard-core point-like repulsive interaction and propose a numerical integration scheme to integrate numerically its dynamics. Our results show that the outcome of the initial violent relaxation is altered, and also that the phase-diagram is modified with a critical temperature at a higher value than in the non-collisional counterpart.
We survey the portfolio of computational strategies available for tackling the generic problems of phase behavior - free-energy-estimation and coexistence-curve mapping.
We study the dynamics of a classical disordered macroscopic model completely isolated from the environment reproducing, in a classical setting, the quantum quench protocol. We show that, depending on the pre and post quench parameters the system approaches equilibrium, succeeding to act as a bath on itself, or remains out of equilibrium, in two different ways. In one of the latter, the system stays confined in a metastable state in which it undergoes stationary dynamics characterised by a single temperature. In the other, the system ages and its dynamics are characterised by two temperatures associated to observations made at short and long time differences (high and low frequencies). The parameter dependence of the asymptotic states is rationalised in terms of a dynamic phase diagram with one equilibrium and two out of equilibrium phases. Aspects of pre-thermalisation are observed and discussed. Similarities and differences with the dynamics of the dissipative model are also explained.
We review generalized Fluctuation-Dissipation Relations which are valid under general conditions even in ``non-standard systems, e.g. out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperperturbed dynamics. In these relations, typically one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in non-standard cases, including driven granular media, systems with a multiscale structure, active matter and systems showing anomalous diffusion.
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