Localization of eigenvectors in random graphs

We consider spatial organization of point defects in the generalized model of defects formation in elastic medium by taking into account defects production by irradiation influence and stochastic contribution for defects dynamics satisfying the fluctuation dissipation relation. We have found that depending on initial conditions and control parameters reduced to defects generation rate caused by irradiation, temperature and the stochastic source intensity different stationary structures of defects can be organized during the system evolution. Studying phase transitions between phases characterized by low- and high defect densities in stochastic system we have shown that such phenomena are described by mechanisms inherent in entropy-driven phase transitions. Stationary patterns are studied by amplitude analysis of unstable slow modes.     

A general framework for non-equilibrium phenomena: the master equation and its formal consequences

We consider non-equilibrium systems defined by a state space, and by a stochastic dynamics and its stationary state. The dynamics need not satisfy detailed balance. In this abstract framework we do the following: (1) define and analyze “relative entropy”, (2) study dissipation in the relaxation to the stationary state, as well as the extra dissipation to maintain the system in its stationary state against some detailed balance dynamics, (3) extend the fluctuation-dissipation theorem and the Onsager relations, and (4) give a formula for the stationary state in terms of a summation over trees.     

Current Large Deviations in a Driven Dissipative Model

We consider lattice gas diffusive dynamics with creation-annihilation in the bulk and maintained out of equilibrium by two reservoirs at the boundaries. This stochastic particle system can be viewed as a toy model for granular gases where the energy is injected at the boundary and dissipated in the bulk. The large deviation functional for the particle currents flowing through the system is computed and some physical consequences are discussed: the mechanism for local current fluctuations, dynamical phase transitions, the fluctuation-relation.     

Adiabatic Elimination in Quantum Stochastic Models

We consider a physical system with a coupling to bosonic reservoirs via a quantum stochastic differential equation. We study the limit of this model as the coupling strength tends to infinity. We show that in this limit the solution to the quantum stochastic differential equation converges strongly to the solution of a limit quantum stochastic differential equation. In the limiting dynamics the excited states are removed and the ground states couple directly to the reservoirs.     

Domino mechanisms in photoinduced phase transitions

Photoinduced structural phase transitions via excited electronic states are discussed theoretically using a one-dimensional model composed of localized electrons and lattices under the adiabatic or diabatic approximation. We show that the global structural change by photoexcitation only at a site is possible, and we clarify conditions for the occurrence of such phenomena. Spatiotemporal dynamics of nonequilibrium first-order phase transitions is also investigated in detail in terms of photoinduced nucleations and domino processes of the domain boundaries (domain walls), which are in striking contrast to the mean-field dynamics. In the adiabatic regime, after the spontaneous emission of a photon, an initial local structural change (i) remains locally, (ii) induces cooperatively a global structural change, or (iii) disappears and returns to the initial phase. Dynamical features of the case (ii) are characterized by the deterministic (semichaotic) domino process; domain walls between the two phases move determinis-tically at a constant velocity (with changing speed) without further spontaneous emissions in the case of strong (weak) dissipation. In the diabatic regime, similar three types of structural change exist. The domain-wall dynamics is described as the stochastic domino process, which is accompanied by the successive radiative transitions. A new theoretical treatment is also proposed to study crossover between the adiabatic and diabatic regimes.     

Kinetic Theory of Jet Dynamics in the Stochastic Barotropic and 2D Navier-Stokes Equations

We discuss the dynamics of zonal (or unidirectional) jets for barotropic flows forced by Gaussian stochastic fields with white in time correlation functions. This problem contains the stochastic dynamics of 2D Navier-Stokes equation as a special case. We consider the limit of weak forces and dissipation, when there is a time scale separation between the inertial time scale (fast) and the spin-up or spin-down time (large) needed to reach an average energy balance. In this limit, we show that an adiabatic reduction (or stochastic averaging) of the dynamics can be performed. We then obtain a kinetic equation that describes the slow evolution of zonal jets over a very long time scale, where the effect of non-zonal turbulence has been integrated out. The main theoretical difficulty, achieved in this work, is to analyze the stationary distribution of a Lyapunov equation that describes quasi-Gaussian fluctuations around each zonal jet, in the inertial limit. This is necessary to prove that there is no ultraviolet divergence at leading order, in such a way that the asymptotic expansion is self-consistent. We obtain at leading order a Fokker–Planck equation, associated to a stochastic kinetic equation, that describes the slow jet dynamics. Its deterministic part is related to well known phenomenological theories (for instance Stochastic Structural Stability Theory) and to quasi-linear approximations, whereas the stochastic part allows to go beyond the computation of the most probable zonal jet. We argue that the effect of the stochastic part may be of huge importance when, as for instance in the proximity of phase transitions, more than one attractor of the dynamics is present.     

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.     

The dynamic phase transition in the spin-1/2 Ising system within the path probability method

We study the dynamic phase transitions and present the dynamic phase diagrams of the spin-1/2 Ising system under the presence of a time-varying (sinusoidal) external magnetic field within the path probability method (PPM) of Kikuchi and we observe that the PPM gives exactly the same result as with the Glauber-type stochastic dynamics based on the mean-field theory (DMFT). We also investigate the influence of the rate constant on the dynamic phase diagrams in detail and five new and interesting dynamic phase diagrams are found. We notice that the derivation of the dynamic equations by using the PPM is more clear and easier than within the DMFT and the Glauber-type stochastic dynamics based on the effective-field theory (DEFT). The advantages and disadvantages of the PPM over the DMFT and DEFT are also discussed.     

Stochastic bifurcation for a tumor–immune system with symmetric Lévy noise

In this paper, we investigate stochastic bifurcation for a tumor–immune system in the presence of a symmetric non-Gaussian Lévy noise. Stationary probability density functions will be numerically obtained to define stochastic bifurcation via the criteria of its qualitative change, and bifurcation diagram at parameter plane is presented to illustrate the bifurcation analysis versus noise intensity and stability index. The effects of both noise intensity and stability index on the average tumor population are also analyzed by simulation calculation. We find that stochastic dynamics induced by Gaussian and non-Gaussian Lévy noises are quite different.     

On the Generalized Langevin Equation for a Rouse Bead in a Nonequilibrium Bath

We present the reduced dynamics of a bead in a Rouse chain which is submerged in a bath containing a driving agent that renders it out-of-equilibrium. We first review the generalized Langevin equation of the middle bead in an equilibrated bath. Thereafter, we introduce two driving forces. Firstly, we add a constant force that is applied to the first bead of the chain. We investigate how the generalized Langevin equation changes due to this perturbation for which the system evolves towards a steady state after some time. Secondly, we consider the case of stochastic active forces which will drive the system to a nonequilibrium state. Including these active forces results in an extra contribution to the second fluctuation–dissipation relation. The form of this active contribution is analysed for the specific case of Gaussian, exponentially correlated active forces. We also discuss the resulting rich dynamics of the middle bead in which various regimes of normal diffusion, subdiffusion and superdiffusion can be present.     

Entanglement evolution in multipartite cavity-reservoir systems under local unitary operations

We analyze the entanglement evolution of two cavity photons being affected by the dissipation of two individual reservoirs. Under an arbitrary local unitary operation on the initial state, it is shown that there is only one parameter which changes the entanglement dynamics. For the bipartite subsystems, we show that the entanglement of the cavity photons is correlated with that of the reservoirs, although the local operation can delay the time at which the photon entanglement disappears and advance the time at which the reservoir entanglement appears. Furthermore, via a new defined four-qubit entanglement measure and two three-qubit entanglement measures, we study the multipartite entanglement evolution in the composite system, which allows us to analyze quantitatively both bipartite and multipartite entanglement within a unified framework. In addition, we also discuss the entanglement evolution with an arbitrary initial state.     

The Response of Glassy Systems to Random Perturbations: A Bridge Between Equilibrium and Off-Equilibrium

We discuss the response of aging systems with short-range interactions to a class of random perturbations. Although these systems are out of equilibrium, the limit value of the free energy at long times is equal to the equilibrium free energy. By exploiting this fact, we define a new order parameter function, and we relate it to the ratio between response and fluctuation, which is in principle measurable in an aging experiment. For a class of systems possessing stochastic stability, we show that this new order parameter function is intimately related to the static order parameter function, describing the distribution of overlaps between clustering states. The same method is applied to investigate the geometrical organization of pure states. We show that the ultrametric organization in the dynamics implies static ultrametricity, and we relate these properties to static separability, i.e., the property that the measure of the overlap between pure states is essentially unique. Our results, especially relevant for spin glasses, pave the way to an experimental determination of the order parameter function.     

Stochastic model for the dynamics of a spin-oscillator coupled system

We construct a stochastic model for the dynamics of a one-dimensional system consisting of bilinearly coupled harmonic oscillators and spins. The spin dynamics is defined as a Glauber model where the spins are effectively coupled through their interaction with the oscillators. To maintain internal thermal equilibrium in the composite system, which does not exhibit Onsager symmetry, we introduce a phenomenological retarded friction in the oscillator equation of motion and relate it to the spin correlation function through a fluctuation-dissipation theorem. The oscillator susceptibility is derived and the behavior of its poles as functions of wavevector and temperature is studied. The results are compared to those obtained by other authors who have studied similar systems, using irreversible thermodynamics. In contrast to ours, these treatments do not give an explicit result for the wavevector dependence of the poles.     

Duality for Stochastic Models of Transport

We study three classes of continuous time Markov processes (inclusion process, exclusion process, independent walkers) and a family of interacting diffusions (Brownian energy process). For each model we define a boundary driven process which is obtained by placing the system in contact with proper reservoirs, working at different particle densities or different temperatures. We show that all the models are exactly solvable by duality, using a dual process with absorbing boundaries. The solution does also apply to the so-called thermalization limit in which particles or energy is instantaneously redistributed among sites. The results shows that duality is a versatile tool for analyzing stochastic models of transport, while the analysis in the literature has been so far limited to particular instances. Long-range correlations naturally emerge as a result of the interaction of dual particles at the microscopic level and the explicit computations of covariances match, in the scaling limit, the predictions of the macroscopic fluctuation theory.     

On the Stochastic Dynamics of Disordered Spin Models

In this article we discuss several aspects of the stochastic dynamics of spin models. The paper has two independent parts. Firstly, we explore a few properties of the multi-point correlations and responses of generic systems evolving in equilibrium with a thermal bath. We propose a fluctuation principle that allows us to derive fluctuation–dissipation relations for many-time correlations and linear responses. We also speculate on how these features will be modified in systems evolving slowly out of equilibrium, such as finite-dimensional or dilute spin-glasses. Secondly, we present a formalism that allows one to derive a series of approximated equations that determine the dynamics of disordered spin models on random (hyper) graphs.     

Lattice gas models in contact with stochastic reservoirs: Local equilibrium and relaxation to the steady state

Extending the results of a previous work, we consider a class of discrete lattice gas models in a finite interval whose bulk dynamics consists of stochastic exchanges which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. We establish here the local equilibrium structure of the stationary measures for these models. Further, we prove as a law of large numbers that the time-dependent empirical density field converges to a deterministic limit process which is the solution of the initial-boundary value problem for a nonlinear diffusion equation.Supported in part by NSF Grants DMR89-18903 and INT85-21407. G.E. and H.S. also supported by the Deutsche Forschungsgemeinschaft     

Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models

We consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior, which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. The unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non-reversible. We study the structure of the stationary measure in the hydrodynamic limit, as the microscopic lattice size goes to infinity. In particular, we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by Fick's law.Dedicated to Res Jost and Arthur WightmanSupported in part by NSF Grants DMR 89-18903 and INT 8521407. H.S. also supported by the Deutsche Forschungsgemeinschaft     

Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators. Received: 12 March 2001 / Accepted: 5 August 2001     

Relaxation to Stationary Nonequilibrium States in Stochastic Ginzburg–Landau Models

We propose an approach to investigate properties of the time relaxation to stationary nonequilibrium states of correlation functions of stochastic Ginzburg–Landau models with noise (temperature of the reservoirs in contact with the system) changing in space. The formalism relates the stochastic expectations to correlation functions of an imaginary time field theory, and it allows us to study the nonlinear dynamics in terms of a field theory given by a perturbation of a Gaussian measure related to the (easier) linear dynamical problem. To show the usefulness of the formalism, we argue that a perturbative analysis within the integral representation is enough to give us the time relaxation rates of the correlations in some situations.