Nonequilibrium Phase Transitions in Systems with Long-Range Interactions

Abstract

We introduce a generalized Hamiltonian mean field model-an XY model with both linear and quadratic coupling between spins and explicit Hamiltonian dynamics. In addition to the usual paramagnetic and ferromagnetic phases, this model also possesses a nematic phase. The generalized Hamiltonian mean field model can be solved explicitly using Boltzmann-Gibbs statistical mechanics, in both canonical and microcanonical ensembles. However, when the resulting microcanonical phase diagram is compared with the one obtained using molecular dynamics simulations, it is found that the two are very different. We will present a dynamical theory which allows us to explicitly calculate the phase diagram obtained using molecular dynamics simulations without any adjustable parameters. The model illustrates the fundamental role played by dynamics as well the inadequacy of Boltzmann-Gibbs statistics for systems with long-range forces in the thermodynamic limit.

Landau-like theory for universality of critical exponents in quasistationary states of isolated mean-field systems

“…The idea is to use the fact that the bracket · A can be removed for any function ψ(H A ) as ψ(H A ) A = ψ(H A ), since the bracket represents the average over an iso-J A curve while H A is constant along the curve. Keeping this fact in mind and denoting the order of external force as O(H ext ) = O(h), we expand f A = F I (H I ) A as (15) where the asymptotic Hamiltonian H A is expanded as (16) small. We note that the last term of the right-hand-side of (15) is not of O(h 2 ) at the critical point, but this change does not affect the following discussions since the term will be omitted.…”

“…This model (6) includes the HMF model [14] by setting K = 1 and V 1 = 1, and the generalized HMF model [16] by K = 2, V 1 = and V 2 = 1 − . The corresponding single body effective Hamiltonian is…”

Dynamics, thermodynamics, and phase transitions of classical spins interacting through the magnetic field

“…6 we show the short time dynamics of relaxation of SCISM to the ferromagnetic state. Similar to the coarsening dynamics of systems with short-range interactions, the relaxation dynamics is algebraic, very different from the exponential relaxation to a ferromagnetic QSS observed in the HMF model [16]. This suggests that the ferromagnetic state to which the system evolves is a true equilibrium and not a ferromagnetic QSS.…”

Quenches across the self-organization transition in multimode cavities

“…When the fields are sufficiently weak, then Q = Q = 0 1 2 , the density is homogeneous and there is no structural order. We call this phase paramagnetic, borrowing the notation of the generalized Hamiltonian mean-field model (GHMF) [21][22][23] to which this model can be mapped. The possible ordered phases in the steady state are illustrated in figure 1(b) and take one of four sets of values.…”