Equilibrium and nonequilibrium properties of systems with long-range interactions

We briefly review some equilibrium and nonequilibrium properties of systems with long-range interactions. Such systems, which are characterized by a potential that weakly decays at large distances, have striking properties at equilibrium, like negative specific heat in the microcanonical ensemble, temperature jumps at first order phase transitions, broken ergodicity. Here, we mainly restrict our analysis to mean-field models, where particles globally interact with the same strength. We show that relaxation to equilibrium proceeds through quasi-stationary states whose duration increases with system size. We propose a theoretical explanation, based on Lynden-Bell’s entropy, of this intriguing relaxation process. This allows to address problems related to nonequilibrium using an extension of standard equilibrium statistical mechanics. We discuss in some detail the example of the dynamics of the free electron laser, where the existence and features of quasi-stationary states is likely to be tested experimentally in the future. We conclude with some perspectives to study open problems and to find applications of these ideas to dipolar media.     

Nano- and Pico-Scale Transport Phenomena in Fluids

A new theoretical approach to describe pre-hydrodynamic stages of evolution in nonequilibrium fluids is presented. The local density, velocity, and temperature fields are expressed as integrals over Green's functions that depend on initial-state ensemble averages of dynamical quantities. For systems in which the initial states are nonuniform in only one spatial direction, explicit expressions for the Green's functions are derived in terms of initial-state ensemble averages of moments of particle displacements and products of particle velocities and particle displacements.     

Phase transitions in four-dimensional AdS black holes with a nonlinear electrodynamics source

In this work we consider black hole solutions to Einstein's theory coupled to a nonlinear power-law electromagnetic field with a fixed exponent value. We study the extended phase space thermodynamics in canonical and grand canonical ensembles, where the varying cosmological constant plays the role of an effective thermodynamic pressure. We examine thermodynamical phase transitions in such black holes and find that both first- and second-order phase transitions can occur in the canonical ensemble while, for the grand canonical ensemble, Hawking–Page and second-order phase transitions are allowed.     

Nonequilibrium model for the contact process in an ensemble of constant particle number.

We introduce and analyze numerically a nonequilibrium model with a conserved dynamics which is a realization of the contact process in an ensemble of constant particle number. The model possesses just one process in which particles jump around landing only on empty sites next to an existing particle. Particles are not allowed to land on a vacant site surrounded by empty sites. In contrast with the ordinary contact process, the present model does not have an absorbing state. In spite of lacking an absorbing state, the model displays properties that, in the thermodynamic limit, are identical to those of the ordinary contact process.     

Yang-Lee theory for a nonequilibrium phase transition

To analyze phase transitions in a nonequilibrium system, we study its grand canonical partition function as a function of complex fugacity. Real and positive roots of the partition function mark phase transitions. This behavior, first found by Yang and Lee under general conditions for equilibrium systems, can also be applied to nonequilibrium phase transitions. We consider a one-dimensional diffusion model with periodic boundary conditions. Depending on the diffusion rates, we find real and positive roots and can distinguish two regions of analyticity, which can be identified with two different phases. In a region of the parameter space, both of these phases coexist. The condensation point can be computed with high accuracy.     

Classification of Phase Transitions and Ensemble Inequivalence, in Systems with Long Range Interactions

Systems with long range interactions in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors. We propose a classification of microcanonical phase transitions, of their link to canonical ones, and of the possible situations of ensemble inequivalence. We discuss previously observed phase transitions and inequivalence in self-gravitating, two-dimensional fluid dynamics and non-neutral plasmas. We note a number of generic situations that have not yet been observed in such systems.     

Thermostats: Analysis and application

Gaussian isokinetic and isoenergetic deterministic thermostats are reviewed in the correct historical context with their later justification using Gauss' principle of least constraint. The Nose-Hoover thermostat for simulating the canonical ensemble is also developed. For some model systems the Lyapunov exponents satisfy the conjugate pairing rule and a Hamiltonian formulation is obtained. We prove the conjugate pairing rule for nonequilibrium systems where the force is derivable from a potential. The generalized symplectic structure and Hamiltonian formulation is discussed. The application of such thermostats to the Lorentz gas is considered in some detail. The periodic orbit expansion methods are used to calculate averages and to categorize the generic transitions in the structure of the attractor. We prove that the conductivity in the nonequilibrium Lorentz gas is non-negative. (c) 1998 American Institute of Physics.     

Pattern selection processes and noise induced pattern-forming transitions in periodic systems with transient dynamics

We apply the phase field crystal method for nonequilibrium patterning to stochastic systems with an external source in which transient dynamics is essential. Considering a prototype model for a one-component periodic system subjected to external influence kind of irradiation we study properties of pattern selection processes and external noise induced pattern-forming transitions. These processes are examined by means of the structure function dynamics analysis. Nonequilibrium pattern-forming transitions are analyzed numerically.     

Asymmetrically coupled directed percolation systems

We introduce a dynamical model of coupled directed percolation systems with two particle species. The two species A and B are coupled asymmetrically in that A particles branch B particles, whereas B particles prey on A particles. This model may describe epidemic spreading controlled by reactive immunization agents. We study nonequilibrium phase transitions with attention focused on the multicritical point where both species undergo the absorbing phase transition simultaneously. In one dimension, we find that the inhibitory coupling from B to A is irrelevant and the model belongs to the unidirectionally coupled directed percolation class. On the contrary, a mean-field analysis predicts that the inhibitory coupling is relevant and a new universality appears with a variable dynamic exponent. Numerical simulations on small-world networks confirm our predictions.     

Limits of the equivalence of time and ensemble averages in shear flows

In equilibrium systems, time and ensemble averages of physical quantities are equivalent due to ergodic exploration of phase space. In driven systems, it is unknown if a similar equivalence of time and ensemble averages exists. We explore effective limits of such convergence in a sheared bubble raft using averages of the bubble velocities. In independent experiments, averaging over time leads to well-converged velocity profiles. However, the time averages from independent experiments result in distinct velocity averages. Ensemble averages are approximated by randomly selecting bubble velocities from independent experiments. Increasingly better approximations of ensemble averages converge toward a unique velocity profile. Therefore, the experiments establish that in practical realizations of nonequilibrium systems, temporal averaging and ensemble averaging can yield convergent (stationary) but distinct distributions.     

Nonequilibrium quantum phase transitions in the Dicke model

We establish a set of nonequilibrium quantum phase transitions in the Dicke model by considering a monochromatic nonadiabatic modulation of the atom-field coupling. For weak driving the system exhibits a set of sidebands which allow the circumvention of the no-go theorem which otherwise forbids the occurrence of superradiant phase transitions. At strong driving we show that the system exhibits a rich multistable structure and exhibits both first- and second-order nonequilibrium quantum phase transitions.     

On the problem of the metastable region at morphological instability

The study deals with numerical analysis of the morphological stability of a growing round particle with respect to harmonic perturbations of an arbitrary amplitude. Various growth regimes (from diffusion to kinetic-limited) are considered. It is found that the critical size of the particle stability decreases as the perturbation amplitude increases and tends to the value, which was determined analytically elsewhere using the maximum entropy production principle. This result is a crucial argument in support of the hypothesis that the entropy production can be used for analysis of a nonequilibrium phase transitions similarly to thermodynamic potentials in the case of equilibrium phase transitions.     

Geometric phase contribution to quantum nonequilibrium many-body dynamics

We study the influence of geometry of quantum systems underlying space of states on its quantum many-body dynamics. We observe an interplay between dynamical and topological ingredients of quantum nonequilibrium dynamics revealed by the geometrical structure of the quantum space of states. As a primary example we use the anisotropic XY ring in a transverse magnetic field with an additional time-dependent flux. In particular, if the flux insertion is slow, nonadiabatic transitions in the dynamics are dominated by the dynamical phase. In the opposite limit geometric phase strongly affects transition probabilities. This interplay can lead to a nonequilibrium phase transition between these two regimes. We also analyze the effect of geometric phase on defect generation during crossing a quantum-critical point.     

Evidence for Fisher renormalization in the compressible phi4 model

We present novel Fourier Monte Carlo simulations of a compressible phi4-model on a simple-cubic lattice with linear-quadratic coupling of order parameter and strain, focusing on the detection of fluctuation-induced first-order transitions and deviations from standard critical behavior. The former is indeed observed in the constant stress ensemble and for auxetic systems at constant strain, while for regular isotropic systems at constant strain, we find strong evidence for Fisher-renormalized critical behavior and are led to predict the existence of a tricritical point.     

Entropy production and time asymmetry in nonequilibrium fluctuations

The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems: namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann's constant.     

Signatures for “phase” transitions in finite systems

The existence of “phase” transitions in finite nuclear systems has been predicted in several models. However fluctuations in the relevant order parameters are generally large enough to prevent their use as signatures for the transitions. In present work both the specific heat and the square of the fluctuations in the internal energy over temperature squared have been calculated in a realistic calculation in²⁴Mg. These are equivalent in an exact canonical ensemble calculation for fixed particle number but differ in the thermal mean field approach. Both clearly indicate the presence of “phase” transitions, the first being more useful for the higher lying second order shape transitions, the second more appropriate for the lower lying first order transitions.     

Memory effects in the two-level model for glasses

We study an ensemble of two-level systems interacting with a thermal bath. This is a well-known model for glasses. The origin of memory effects in this model is a quasistationary but nonequilibrium state of a single two-level system, which is realized due to a finite-rate cooling and slow thermally activated relaxation. We show that single-particle memory effects, such as negativity of the specific heat under reheating, vanish for a sufficiently disordered ensemble. In contrast, a disordered ensemble displays a collective memory effect [similar to the Kovacs effect], where nonequilibrium features of the ensemble are monitored via a macroscopic observable. An experimental realization of the effect can be used to further assess the consistency of the model.     

Universality class of absorbing phase transitions with a conserved field

We investigate the critical behavior of systems exhibiting a continuous absorbing phase transition in the presence of a conserved field coupled to the order parameter. The results obtained point out the existence of a new universality class of nonequilibrium phase transitions that characterizes a vast set of systems including conserved threshold transfer processes and stochastic sandpile models.