Asymptotic Stationary State Research Articles

Transition from ergodic to many-body localization regimes in open quantum systems in terms of the neural-network ansatz

The purpose of our work is to investigate asymptotic stationary states of an open disordered many-body quantum model which is characterized by an ergodic — many-body localization (MBL) phase transition. To find these states, we use the neural-network ansatz, a new method of modeling complex many-body quantum states discussed in the recent literature. Our main result is that that the ergodic phase — MBL transition is detectable in the performance of the neural network that is trained to reproduce the asymptotic states of the model. While the network is able to reproduce, with a relatively high accuracy, ergodic states, it fails to do so when the model system enter the MBL phase. We conclude that MBL features of the model translate into the cost function landscape which becomes corrugated and acquires many local minima.

Nonlinear amplification of ocean waves in straits

Using the exact Hasselmann equation, we study wind-driven deep-water ocean waves in a strait with the wind directed orthogonally to the shore. The strait has “dissipative” shores with no reflection from the shorelines. We show that the evolution of wave turbulence can be divided into two different regimes in time. During the first regime, waves propagate with the wind, and the wind-driven sea can be described by self-similar solutions of the Hasselmann equation. The second regime starts after a sufficiently significant accumulation of wave energy at the downwind boundary. From this instant, an ensemble of waves propagating against the wind starts to form. Moreover, waves orthogonal to the wind arise and propagate along the strait. The wave system eventually reaches an asymptotic stationary state in which two types of wave motion coexist: an ensemble of self-similar waves propagating with the wind and quasimonochromatic waves propagating almost orthogonally to the wind direction and tending to slant against the wind at the angle of 15° with respect to the shore of turbulence origination. These “secondary waves” arise only as a result of an intensive nonlinear wave interaction. The total wave energy exceeds its expected value approximately by a factor of two compared with the energy calculated in the absence of shores. We expect that this amplification increases substantially in the presence of reflective shores. We propose calling this “secondary” laser-like mechanism “nonlinear ocean wave amplification” (abbreviated NOWA).

Heisenberg-Langevin versus quantum master equation

The quantum master equation is an important tool in the study of quantum open systems. It is often derived under a set of approximations, chief among them the Born (factorization) and Markov (neglect of memory effects) approximations. In this article we study the paradigmatic model of quantum Brownian motion of an harmonic oscillator coupled to a bath of oscillators with a Drude-Ohmic spectral density. We obtain analytically the \emph{exact} solution of the Heisenberg-Langevin equations, with which we study correlation functions in the asymptotic stationary state. We compare the \emph{exact} correlation functions to those obtained in the asymptotic long time limit with the quantum master equation in the Born approximation \emph{with and without} the Markov approximation. In the latter case we implement a systematic derivative expansion that yields the \emph{exact} asymptotic limit under the factorization approximation \emph{only}. We find discrepancies that could be significant when the bandwidth of the bath $\Lambda$ is much larger than the typical scales of the system. We study the \emph{exact} interaction energy as a \emph{proxy} for the correlations missed by the Born approximation and find that its dependence on $\Lambda$ is similar to the \emph{discrepancy} between the exact solution and that of the quantum master equation in the Born approximation. We quantify the regime of validity of the quantum master equation in the Born approximation with or without the Markov approximation in terms of the system's relaxation rate $\gamma$, its \emph{unrenormalized} natural frequency $\Omega$ and $\Lambda$: $\gamma/\Omega \ll 1$ and \emph{also} $\gamma \Lambda/\Omega^2 \ll 1$. The reliability of the Born approximation is discussed within the context of recent experimental settings and more general environments.

Kinematic wave models of lane-drop bottlenecks

Lane-drop bottlenecks lead to both capacity reduction and capacity drop, but there are no systematic studies on traffic dynamics, especially the stationary traffic patterns when the upstream link is congested, at such critical bottlenecks within the framework of kinematic wave theory. In this paper, we study three kinematic wave models for both discontinuous and continuous lane-drop bottlenecks, where the number of lanes drops either suddenly or gradually. In the first two models, we apply the multilane LWR model (Lighthill and Whitham, 1955; Richards, 1956) for a discontinuous and continuous lane-drop bottleneck, respectively; in the third model, we assume that the standing wave is instantaneous inside a continuous lane-drop bottleneck. For the first model, we apply the kinematic wave theory developed in Jin et al. (2009) to solve the Riemann problem. In particular, we present an optimization formulation for the entropy condition, solve the stationary states and kinematic waves on both upstream and downstream links, and define instantaneous discontinuous standing waves connecting the upstream and downstream links’ stationary states. For the second model, we define a generalized Riemann problem and show that the asymptotic stationary states and kinematic waves on both links are the same as those in the first model, but the asymptotic standing wave is continuous, comprising of stationary states inside the whole lane-drop zone. Assuming standing waves to be instantaneous as in the first model but continuous as in the second model, we present a new model of a continuous lane-drop bottleneck and a new theoretical framework to solve the generalized Riemann problem. Finally we develop three corresponding Cell Transmission Models and verify theoretical results, especially the presence and structure of continuous standing waves inside a continuous lane-drop zone, with numerical examples. We find that vehicles’ acceleration rates across the standing waves can be infinite in the first model and unrealistically large in the other two and that there is no capacity drop in all of the models. The theoretical framework of the third model forms a basis for a follow-up study, (Jin, 2017), in which we introduce bounded acceleration as an additional constraint on the instantaneous standing waves and present a behavioral model of capacity drop.

Transient dynamics in trial-offer markets with social influence: Trade-offs between appeal and quality.

We study a trial-offer market where consumers may purchase one of two competing products. Consumer preferences are affected by the products quality, their appeal, and their popularity. While the asymptotic convergence or stationary states of these, and related dynamical systems, has been vastly studied, the literature regarding the transitory dynamics remains surprisingly sparse. To fill this gap, we derive a system of Ordinary Differential Equations, which is solved exactly to gain insight into the roles played by product qualities and appeals in the market behavior. We observe a logarithmic tradeoff between quality and appeal for medium and long-term marketing strategies: The expected market shares remain constant if a decrease in quality is followed by an exponential increase in the product appeal. However, for short time horizons, the trade-off is linear. Finally, we study the variability of the dynamics through Monte Carlo simulations and discover that low appeals may result in high levels of variability. The model results suggest effective marketing strategies for short and long time horizons and emphasize the significance of advertising early in the market life to increase sales and predictability.

Coherence of mechanical oscillators mediated by coupling to different baths

We study the non-equilibrium dynamics of two coupled mechanical oscillators with general linear couplings to two uncorrelated thermal baths at temperatures $T_1$ and $T_2$, respectively. We obtain the complete solution of the Heisenberg-Langevin equations, which reveal a coherent mixing among the normal modes of the oscillators as a consequence of their off-diagonal couplings to the baths. Unique renormalization aspects resulting from this mixing are discussed. Diagonal and off-diagonal (coherence) correlation functions are obtained analytically in the case of strictly Ohmic baths with different couplings in the strong and weak coupling regimes. An asymptotic non-equilibrium stationary state emerges for which we obtain the complete expressions for the correlations and coherence. Remarkably the coherence survives in the high temperature, classical limit for $T_1 \neq T_2$. In the case of vanishing detuning between the oscillator normal modes both coupling to one and the same bath the coherence retains memory of the initial conditions at long time. A perturbative expansion of the early time evolution reveals that the emergence of coherence is a consequence of the entanglement between the normal modes of the oscillators \emph{mediated} by their couplings to the baths. This \emph{suggests} the survival of entanglement in the high temperature limit for different temperatures of the baths which is essentially a consequence of the non-equilibrium nature of the asymptotic stationary state. An out of equilibrium setup with small detuning and large $|T_1- T_2|$ produces non-vanishing steady-state coherence and entanglement in the high temperature limit of the baths.

Integrable turbulence generated from modulational instability of cnoidal waves

We study numerically the nonlinear stage of the modulational instability (MI) of cnoidal waves in the framework of the focusing one-dimensional nonlinear Schrödinger (NLS) equation. Cnoidal waves are exact periodic solutions of the NLS equation which can be represented as the lattices of overlapping solitons. The MI of these lattices leads to the development of ‘integrable turbulence’ (Zakharov 2009 Stud. Appl. Math. 122 219–34). We study the major characteristics of turbulence for the dn-branch of cnoidal waves and demonstrate how these characteristics depend on the degree of ‘overlapping’ between the solitons within the cnoidal wave.Integrable turbulence, which develops from the MI of the dn-branch of cnoidal waves, asymptotically approaches its stationary state in an oscillatory way. During this process, kinetic and potential energies oscillate around their asymptotic values. The amplitudes of these oscillations decay with time as , , the phases contain nonlinear phase shift decaying as t−1/2, and the frequency of the oscillations is equal to the double maximal growth rate of the MI, . In the asymptotic stationary state, the ratio of potential to kinetic energy is equal to −2. The asymptotic PDF of the wave intensity is close to the exponential distribution for cnoidal waves with strong overlapping, and is significantly non-exponential for cnoidal waves with weak overlapping of the solitons. In the latter case, the dynamics of the system reduces to two-soliton collisions, which occur at an exponentially small rate and provide an up to two-fold increase in amplitude compared with the original cnoidal wave. For all cnoidal waves of the dn-branch, the rogue waves at the time of their maximal elevation have a quasi-rational profile similar to that of the Peregrine solution.

Lévy flights and nonhomogenous memory effects: Relaxation to a stationary state.

The non-Markovian stochastic dynamics involving Lévy flights and a potential in the form of a harmonic and nonlinear oscillator is discussed. The subordination technique is applied and the memory effects, which are nonhomogeneous, are taken into account by a position-dependent subordinator. In the nonlinear case, the asymptotic stationary states are found. The relaxation pattern to the stationary state is derived for the quadratic potential: the density decays like a linear combination of the Mittag-Leffler functions. It is demonstrated that in the latter case the density distribution satisfies a fractional Fokker-Planck equation. The densities for the nonlinear oscillator reveal a complex picture, qualitatively dependent on the potential strength, and the relaxation pattern is exponential at large time.

Quantum walk, entanglement and thermodynamic laws

We consider a special dynamics of a quantum walk (QW) on a line. The walker, initially localized at the origin of the line with arbitrary chirality, evolves to an asymptotic stationary state. In this stationary state a measurement is performed and the state resulting from this measurement is used to start a second QW evolution to achieve a second asymptotic stationary state. In previous works, we developed the thermodynamics associated with the entanglement between the coin and position degrees of freedom in the QW. Here we study the application of the first and second laws of thermodynamics to the process between the two stationary states mentioned above. We show that: (i) the entropy change has upper and lower bounds that are obtained analytically as functions of the initial conditions. (ii) the energy change is associated to a heat-transfer process.

The ideal versus the reality: topology and turbulence in the current density in tokamak

Models for the current density profile in magnetically confined plasma have been proposed starting from first principles, basically in analogy with the self-organization of the vorticity in the two-dimensional Euler fluid. The principle of maximum entropy has led to the derivation of the Liouville equation, under some criticism regarding the possibility of a statistical approach. We identify a class of asymptotic stationary states for the coupled fields of current-density and vorticity, using the Backlund transform. For this class of asymptotic states the maxima of the vorticity and current density must coincide. We also discuss a possible relevance of the field-theoretical solutions sphaleron for magnetic reconnection in low-collision regimes.

Transparent control of an exactly solvable two-level system via combined modulations

Analytically exact solutions of a driven two-level system are extremely important in many areas of physics [E. Barnes and S. Das Sarma, Phys. Rev. Lett. 109, 060401 (2012)]. Here we present a simple method of synchronous and asynchronous combined modulations to generate such exact solutions, and apply several of the simplest solutions to perform the coherent control of quantum states. The interesting physical phenomena, such as the arbitrary population transfers between the different and/or asymptotic stationary states and the controlled coherent population oscillation, are achieved analytically and numerically. Simplification of the exact solutions and the modulated fields renders the control strategies more transparent and the qubit manipulation more convenient in experiments.

Optimal mutation rates in dynamic environments: The Eigen model

We consider the Eigen quasispecies model with a dynamic environment. For an environment with sharp-peak fitness in which the most-fit sequence moves by k spin-flips each period T we find an asymptotic stationary state in which the quasispecies population changes regularly according to the regular environmental change. From this stationary state we estimate the maximum and the minimum mutation rates for a quasispecies to survive under the changing environment and calculate the optimum mutation rate that maximizes the population growth. Interestingly we find that the optimum mutation rate in the Eigen model is lower than that in the Crow-Kimura model, and at their optimum mutation rates the corresponding mean fitness in the eigenmodel is lower than that in the Crow-Kimura model, suggesting that the mutation process which occurs in parallel to the replication process as in the Crow-Kimura model gives an adaptive advantage under changing environment.

The geologic evolution of Mars: Episodicity of resurfacing events and ages from cratering analysis of image data and correlation with radiometric ages of Martian meteorites

We have investigated 12 prominent regions on Mars in detail with respect to their geologic evolution through time on the basis of detailed geologic mapping exercises and determining age relationships through crater counting techniques using imagery. New data in combination with previously obtained data have been analyzed by way of a refined method of cratering age extraction that also gives fine details of periods of resurfacing. We have found that there has been volcanic and fluvial/glacial geologic activity on the Martian surface at all times from > 4 Ga ago until today. This activity shows episodic pulses in intensity of both volcanic and fluvial/glacial processes at ~ 3.8–3.3 Ga, 2.0–1.8 Ga, 1.6 to 1.2 Ga, ~ 800 to 300 m.y., ~ 200 m.y., and ~ 100 m.y., and a possible weaker phase around ~ 2.5–2.2 Ga ago. In between these episodes, there was relative quiescence of volcanic and/or fluvial/glacial activity. The episodes we find on the Martian surface in the crater frequency analyses of HRSC, MOC and THEMIS data coincide with some age groups of the Martian meteorites (~ 1.3 Ga, ~ 600–300 m.y., ~ 170 m.y.). It appears that the surface activity expressions and their episodicity are related to the interior evolution of the planet when convection in the asymptotic stationary state changes from the so-called stagnant-lid regime to an episodic behavior. Similarities in episodic behavior are found for the other terrestrial planets: Venus, the Earth's moon, and the Earth itself suggesting a common general relationship in the evolutionary tracks.

Asymptotic traffic dynamics arising in diverge–merge networks with two intermediate links

Basic road network components, such as merging and diverging junctions, contribute much to formation and propagation of traffic congestion. However, in contrast to extensive studies of traffic dynamics on a road link, little is understood for those on a road network. In this study, we provide a comprehensive analysis of asymptotic traffic dynamics arising in diverge–merge networks with two intermediate links with first-order kinematic wave models. In addition, traffic demand at the origin and traffic demand at the destination are constant and maximum, and the proportion of vehicles choosing an intermediate link is constant. First, we examine all possible asymptotic stationary states and the sufficient and necessary conditions for their existence. With an example, we find that non-stationary states can include damped (DPO) and persistent periodic oscillations (PPO), and present an equivalent dynamical system that can be used to distinguish these two types of oscillations. This study can be a springboard for better understanding of traffic dynamics, especially formation and propagation of traffic congestion, on general road networks.

Normal and anomalous solitons in the theory of dynamical Cooper pairing

We obtain multisoliton solutions of the time-dependent Bogoliubov--de Gennes equations or, equivalently, Gorkov equations that describe the dynamics of a fermionic condensate in the dissipationless regime. There are two kinds of solitons---normal and anomalous. At large times, normal multisolitons asymptote to unstable stationary states of the BCS Hamiltonian with zero order parameter (normal states), while the anomalous ones tend to eigenstates characterized by a nonzero anomalous average. Under certain circumstances, multisoliton solutions break up into sums of single solitons. In the linear analysis near the stationary states, solitons correspond to unstable modes. Generally, they are nonlinear extensions of these modes, so that a stationary state with $k$ unstable modes gives rise to a $k$-soliton solution. We relate parameters of the multisolitons to those of the asymptotic stationary state, which determines the conditions necessary for exciting solitons. We further argue that the dynamics in many physical situations is multisoliton.

Temporal variations in the convective style of planetary mantles

Investigations of mantle convection with temperature- and strain-rate-dependent viscosity have shown the existence of fundamentally different convective styles: By varying e.g. the Rayleigh number, the viscosity contrast or the strain-rate dependency of viscosity, the planform of convection in the asymptotic stationary state changes from the so-called stagnant lid regime to an episodic behaviour and further to a state characterised by a permanently mobilised surface. Our studies suggest that this transition may not only be induced by a change of parameters but also occurs temporally for fixed parameters. We have in fact observed convective systems in the stagnant lid regime that show isolated events of surface mobilisation occurring out of a thermally equilibrated state. We use a 3D numerical mantle convection model to investigate mantle convection and surface dynamics as a coupled fluid dynamical system. Our studies focus on the existence of a transitional regime in which temporal variations between the stagnant lid and the episodic regime occur. We were able to deduce a mobilisation criterion that describes the stability of the stagnant surface, thus allowing for a quantitative analysis of the transition to a (temporarily) mobilised surface. This criterion is also suitable to predict the occurrence of surface mobilisation events.

The characteristic function method applied to molecular dynamics of inelastic granular gases

In this work we study the dissipative dynamics of a system of smooth hard spheres with a constant restitution coefficient and small number of particles in the homogeneous cooling regime. We focus on the velocity distribution from simulations in the Molecular Dynamics approach. The main goal of this paper is to present a methodology based on the application of the characteristic function technique and the so-called W function, introduced by Lévy to measure the distance of distributions from the Gaussian. We use this methodology to (i) characterize asymptotic stationary states independently of initial conditions; (ii) study the multiple dependence of these stationary states on the number of particles, density regimes and boundary conditions for a fixed restitution coefficient; (iii) discuss the existence of a limit state in the thermodynamic limit; (iv) address the open problem of the convergence of a Sonine expansion in the highly dissipative regime and (v) measure the overpopulated high energy tails. Moreover, we investigate in what sense the theoretical results related to the Enskog–Boltzmann equation can be reproduced.

Generation of Werner states and preservation of entanglement in a noisy environment

We study the influence of noisy environment on the evolution of two-atomic system in the presence of collective damping. Generation of Werner states as asymptotic stationary states of evolution is described. We also show that for some initial states the amount of entanglement is preserved during the evolution.

Testing the Edwards hypothesis in spin systems under tapping dynamics

The Edwards hypothesis of ergodicity of blocked configurations for gently tapped granular materials is tested for abstract models of spin systems on random graphs and spin chains with kinetic constraints. The tapping dynamics is modeled by considering two distinct mechanisms of energy injection: thermal and random tapping. We find that ergodicity depends upon the tapping procedure (i.e. the way the blocked configurations are dynamically accessed): for thermal tapping ergodicity is a good approximation, while it fails to describe the asymptotic stationary state reached by the random tapping dynamics.

Different strategies of evolution in a predator–prey system

We propose a lattice model of evolution of a predator–prey system. Both species in order to survive must eat. The prey eats the grass growing in unlimited quantities on most of the cells of the lattice. On the remaining cells the prey find cover against predators, but not food. We compare four active strategies—the prey move in the direction of grassy cells, they move away from the predators, the predators move in the direction of the prey and both species move in the direction of their respective food resources. These strategies are compared with random wandering of both species. We show that the character of the asymptotic stationary state depends on the initial concentrations of predators and prey, density of the shelter for the prey and on the strategy adopted. The predators are more vulnerable to the proper choice of the strategy. The strategies which lead in most cases to the extinction of predators and prey are those in which the predators are searching the prey.