Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems

We study the stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. In the course of this transition diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed. In order to understand the details and mechanisms of these noise-induced dynamics we consider the thermodynamic limit N-->infinity of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good qualitative agreement with dynamical scenarios observed in the stochastic simulations of large ensembles.     

A generalized Boltzmann equation for nuclear dynamics

We derive a generalized Boltzmann equation from an extended time-dependent mean-field theory, which is self-consistent and incorporates two-body collisions due to the residual interaction. Obtained from a systematic reduction of a more general, but complicated, many-body theory, this kinetic equation retains many of the quantum effects of the many-body system. Due to proper treatment of quantum causality, its collision integral contains terms which are associated with the principal-value parts of propagators in a quantum-mechanically correct memory kernel. Thus, it properly generalizes the Uehling-Uhlenbeck equation and provides a quantum kinetic theory for nuclear dynamics in both low- and intermediate-energy regions. The new features in this Boltzmann equation are investigated in a nuclear-matter model with a simple effective interaction. We solve for the small-amplitude solutions corresponding to the response of the system to an arbitrary weak external field. The results are contrasted with the collisionless limit and Uehling-Uhlenbeck limit and conclusions are drawn about the dynamical effects of the two-body collisions on the quasiparticles.     

Relaxation of vacuum energy in <Emphasis Type="Italic">q</Emphasis>-theory

The q-theory formalism aims to describe the thermodynamics and dynamics of the deep quantum vacuum. The thermodynamics leads to an exact cancellation of the quantum-field zero-point-energies in equilibrium, which partly solves the main cosmological constant problem. But, with reversible dynamics, the spatially flat Friedmann–Robertson–Walker universe asymptotically approaches the Minkowski vacuum only if the Big Bang already started out in an initial equilibrium state. Here, we extend q-theory by introducing dissipation from irreversible processes. Neglecting the possible instability of a de-Sitter vacuum, we obtain different scenarios with either a de-Sitter asymptote or collapse to a final singularity. The Minkowski asymptote still requires fine-tuning of the initial conditions. This suggests that, within the q-theory approach, the decay of the de-Sitter vacuum is a necessary condition for the dynamical solution of the cosmological constant problem.     

Kinetic description of vacuum particle production in collisions of ultrarelativistic nuclei

The dynamics of partons that emerge as the result of quantum tunneling in a spatially uniform time-dependent field is studied under conditions prevalent in ultrarelativistic heavy-ion collisions. A self-consistent set of coupled equations that consists of the renormalized Maxwell equation and the Vlasov kinetic equation that involves a source and which is derived on a dynamical basis is solved numerically. The time dependence of the distributions of internal fields and currents for bosons and fermions is investigated within this back-reaction mechanism, and their momentum spectra are constructed. Clear evidence that oscillations in the time dependence of parton distributions in phase-space cells are of a stochastic character is obtained, and a significant irregularity in the momentum distribution on large time scales is found. If the influence of the back reaction is disregarded, these effects disappear completely, the oscillations becoming regular. A possible thermalization scenario for such a quasiparticle plasma is considered in the relaxation-time approximation. A locally equilibrium state is described within the two-component thermodynamics of particles and antiparticles. The possibility of introducing temperature under conditions of a strong vacuum polarization is discussed.     

Effects of random-crystal-field on the kinetic Blume Capel model

The effects of random crystal field on the stationary states of the kinetic spin-1 Blume Capel model are studied using the Glauber-type stochastic dynamics under the time-dependent oscillating field assumption. Our investigation, based on the equilibrium ground state phase diagram, revealed many interesting phenomena. The known phases, in the equilibrium case, are obtained for high field and are represented by limit cycles. The phase diagram of the pure pure kinetic Ising spin- and spin-1 Blume Capel models are deduced as particular cases. First-order, second-order transition lines, dynamical critical and dynamical double critical end points are also obtained.     

Stochastic dynamics: A review of stochastic calculus of variations

We present the main results of a variational calculus for Markovian stochastic processes which allows us to characterize the dynamics of probabilistic systems by extremal properties for some functionals of processes. They generalize, by construction, the main variational formulations of classical dynamics. This framework is used for the dynamical analysis of Nelson's stochastic mechanics, an approach to quantum mechanics in which the concept of trajectory for particles still makes sense. The semiclassical limit is formulated in terms of the second variation of the starting functional. We also use the proposed stochastic calculus of variations in the context of statistical mechanics of systems far from equilibrium, namely, to solve the Onsager-Machlup problem.On leave from Département de Physique Théorique, Université de Genève, CH-I2II, Genève 4, Switzerland.     

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.     

Zeno dynamics for open quantum systems

In this paper, we formulate limit Zeno dynamics of general open systems as the adiabatic elimination of fast components. We are able to exploit previous work on adiabatic elimination of quantum stochastic models to give explicitly the conditions under which open Zeno dynamics will exist. The open systems formulation is further developed as a framework for Zeno master equations, and Zeno filtering (that is, quantum trajectories based on a limit Zeno dynamical model). We discuss several models from the point of view of quantum control. For the case of linear quantum stochastic systems, we present a condition for stability of the asymptotic Zeno dynamics.     

An Inverse Problem in Quantum Statistical Physics

We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density n(x)? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential.     

Non-Markovian quantum jumps

Open quantum systems that interact with structured reservoirs exhibit non-Markovian dynamics. We present a quantum jump method for treating the dynamics of such systems. This approach is a generalization of the standard Monte Carlo wave function (MCWF) method for Markovian dynamics. The MCWF method identifies decay rates with jump probabilities and fails for non-Markovian systems where the time-dependent rates become temporarily negative. Our non-Markovian quantum jump approach circumvents this problem and provides an efficient unraveling of the ensemble dynamics.     

A General Metric for the Similarity of Both Stochastic and Deterministic System Dynamics

Many problems in the study of dynamical systems—including identification of effective order, detection of nonlinearity or chaos, and change detection—can be reframed in terms of assessing the similarity between dynamical systems or between a given dynamical system and a reference. We introduce a general metric of dynamical similarity that is well posed for both stochastic and deterministic systems and is informative of the aforementioned dynamical features even when only partial information about the system is available. We describe methods for estimating this metric in a range of scenarios that differ in respect to contol over the systems under study, the deterministic or stochastic nature of the underlying dynamics, and whether or not a fully informative set of variables is available. Through numerical simulation, we demonstrate the sensitivity of the proposed metric to a range of dynamical properties, its utility in mapping the dynamical properties of parameter space for a given model, and its power for detecting structural changes through time series data.     

Controllable coupling and quantum correlation dynamics of two double quantum dots coupled via a transmission line resonator

We propose a theoretical scheme to generate a controllable and switchable coupling between two double-quantum-dot (DQD) spin qubits by using a transmission line resonator (TLR) as a bus system. We study dynamical behaviors of quantum correlations described by entanglement correlation (EC) and discord correlation (DC) between two DQD spin qubits when the two spin qubits and the TLR are initially prepared in X-type quantum states and a coherent state, respectively. We demonstrate that in the EC death regions there exist DC stationary states in which the stable DC amplification or degradation can be generated during the dynamical evolution. It is shown that these DC stationary states can be controlled by initial-state parameters, the coupling, and detuning between qubits and the TLR. We reveal the full synchronization and anti-synchronization phenomena in the EC and DC time evolution, and show that the EC and DC synchronization and anti-synchronization depends on the initial-state parameters of the two DQD spin qubits. It is shown that the initial quantum correlation may be suppressed completely when the evolution time approaches to the infinity in the presence of dissipation. These results shed new light on dynamics of quantum correlations.