Mean-field Equations Of Motion Research Articles

Applicability of Mean-Field Theory for Time-Dependent Open Quantum Systems with Infinite-Range Interactions.

Understanding quantum many-body systems with long-range or infinite-range interactions is of relevance across a broad set of physical disciplines, including quantum optics, nuclear magnetic resonance, and nuclear physics. From a theoretical viewpoint, these systems are appealing since they can be efficiently studied with numerics, and in the thermodynamic limit are expected to be governed by mean-field equations of motion. Over the past years the capabilities to experimentally create long-range interacting systems have dramatically improved permitting their control in space and time. This allows us to induce and explore a plethora of nonequilibrium dynamical phases, including time crystals and even chaotic regimes. However, establishing the emergence of these phases from numerical simulations turns out to be surprisingly challenging. This difficulty led to the assertion that mean-field theory may not be applicable to time-dependent infinite-range interacting systems. Here, we rigorously prove that mean-field theory in fact exactly captures their dynamics, in the thermodynamic limit. We further provide bounds for finite-size effects and their dependence on the evolution time.

Mean-field theory for driven domain walls in disordered environments

We study the mean field equation of motion for driven domain walls in random media. We discuss the two cases of an external constant as well as an oscillating driving force. Our main focus lies on the critical dynamics close to the depinning transition, which we study by analytical and numerical methods. We find power-law scaling for the velocity as well as the hysteresis loop area.

Exact dynamics of multicomponent Bose-Einstein condensates in optical lattices in one, two, and three dimensions

Numerous exact solutions to the nonlinear mean-field equations of motion are constructed for multicomponent Bose-Einstein condensates on one-, two-, and three-dimensional optical lattices. We find both stationary and nonstationary solutions, which are given in closed form. Among these solutions are a vortex-antivortex array on the square optical lattice and modes in which two or more components slosh back and forth between neighboring potential wells. We obtain a variety of solutions for multicomponent condensates on the simple cubic lattice, including a solution in which one condensate is at rest and the other flows in a complex three-dimensional array of intersecting vortex lines. A number of physically important solutions are stable for a range of parameter values, as we show by direct numerical integration of the equations of motion.

New perspectives on multiple-copy, mean-field molecular dynamics methods

Mean-field molecular dynamics (MD) techniques are designed to improve phase-space sampling in MD simulations. Reviewed here are theoretical and practical contributions from our group, and the ideas our contributions are based upon, beginning with the original time-dependent Hartree technique, locally enhanced sampling (LES), and a new, purely classical derivation of multiple-copy, mean-field equations of motion. This view is used to provide new insights into approximations inherent in LES, as well as the basis for a recently proposed method, ensembles eXtracted from atomic coordinate transformations (the EXACT approximation).

Mean-field and Monte Carlo studies of the magnetization-reversal transition in the Ising model

Detailed mean field and Monte Carlo studies of the dynamic magnetization-reversal transition in the Ising model in its ordered phase under a competing external magnetic field of finite duration have been presented here. Approximate analytical treatment of the mean field equations of motion shows the existence of diverging length and time scales across this dynamic transition phase boundary. These are also supported by numerical solutions of the complete mean field equations of motion and the Monte Carlo study of the system evolving under Glauber dynamics in both two and three dimensions. Classical nucleation theory predicts different mechanisms of domain growth in two regimes marked by the strength of the external field, and the nature of the Monte Carlo phase boundary can be comprehended satisfactorily using the theory. The order of the transition changes from a continuous to a discontinuous one as one crosses over from coalescence regime (stronger field) to nucleation regime (weaker field). Finite size scaling theory can be applied in the coalescence regime, where the best fit estimates of the critical exponents are obtained for two and three dimensions.

Length and time scale divergences at the magnetization-reversal transition in the Ising model.

The divergences of both the length and time scales, at the magnetization-reversal transition in the Ising model under a pulsed field, have been studied in the linearized limit of the mean field theory. Both the length and time scales are shown to diverge at the transition point and it has been checked that the nature of the time scale divergence agrees well with the result obtained from the numerical solution of the mean field equation of motion. Similar growths in length and time scales are also observed, as one approaches the transition point, using Monte Carlo simulations. However, these are not of the same nature as the mean field case. Nucleation theory provides a qualitative argument that explains the nature of the time scale growth. To study the nature of growth of the characteristic length scale, we have looked at the cluster size distribution of the reversed spin domains and have defined a pseudocorrelation length that has been observed to grow at the phase boundary of the transition.

A numerical study of an expanding plasma of quarks in a chiral model

We numerically solve the transport equations for a quark gas described by the the Nambu-Jona-Lasinio model. The mean field equations of motion, which consist of the Vlasov equation for the density and the gap equation for the mean field, are discussed, and energy and momentum conservation are proven. Numerical solutions of the partial differential equations are obtained by applying finite difference methods. For an expanding fireball the light quark mass evolves from small values initially to the value of 350 MeV. This leads to a depletion of the high energy part of the quark spectrum and an enhancement at low momenta. When collisions are included one obtains an equation of the Boltzmann type, where the transition amplitudes depend on the properties of the medium. These equations are given for flavor SU(3), i.e. including strangeness. They are solved numerically in the relaxation time approximation and the time evolution of various observables is given. Medium effects in the relaxation times do not significantly influence the shape of the spectra. The mass of the strange quark changes little during the expansion. The strangeness yield and the slope temperatures of the final spectra are studied as a function of the size of the initial fireball.

Spin-reversal transition in Ising model under pulsed field

In this communication we report the existence of a dynamic “spin-reversal” transition in an Ising system perturbed by a pulsed external magnetic field. The transition is achieved by tuning the strength ( h p ) and/or the duration ( Δt) of the pulse which is applied in a direction opposite to the existing order. We have studied this transition in the kinetic Ising model in two-dimensions using Monte-Carlo technique, and solved numerically, the mean-field equation of motion. The transition is essentially dynamic in nature and it takes the system from one ordered equilibrium phase to another by means of the growth of opposite spin domains (in the kinetic Ising case) induced during the period when the pulsed field is applied.

Nonequilibrium phase transitions in a simple three-state lattice gas

We investigate the dynamics of a three-state stochastic lattice gas consisting of holes and two oppositely “charged” species of particles, under the influence of an “electric” field at zero total charge. Interacting only through an excluded-volume constraint, particles exchange with holes and, on a slower time scale, with each other. Using a combination of Monte Carlo simulations and meanfield equations of motion, we study a set of suitably defined order parameters, their histograms and fluctuations, as well as the current through the system. With increasing particle density and drive, the system first orders into a charge-segregated state, and then disorders again near complete filling. The transition is first order at low densities and turns second order at higher ones. The finite-size and aspect-ratio dependence of characteristic quantities is discussed at the mean-field level.

A mean-field equation of motion for the dynamic Ising model

A mean-field type of approximation is used to derive two differential equations, one approximately representing the average behavior of the Ising model with Glauber (spin-flip) stochastic dynamics, and the other doing the same for Kawasaki (spin-exchange) dynamics. The proposed new equations are compared with the Cahn-Allen and Cahn-Hilliard equations representing the same systems and with information about the exact behavior of the microscopic models.

On the Mean Field Equations of Motion in the Theories with Unstable Vacuum

The problem of finding the mean field equations of motion in the theories with unstable vacuum is considered. The one- and two-loop quantum corrections to classical equations of motion are obtained. It is shown that these quantum corrections are real and causal.