Asymptotic Steady State Research Articles

Supertransient Chaos in a Single and Coupled Liénard Systems.

We report the appearance of supertransient chaos in a single and two-coupled Liénard system with the influence of external periodic force. The existence of transient dynamics in a model is significantly long before it settles into the asymptotic steady state of periodic dynamics understood as supertransient chaos. The two diffusively coupled forced Liénard systems exhibit extremely long transient dynamics when their frequencies of the external forcing are slightly mismatched. Additionally, the coupled system signifies supertransient hyperchaotic dynamics for a specific set of system parameters. This study involves different numerical characterizations, statistical analysis, and hardware implementation using an analog electronic circuit.

Distinguishing Between Long-Transient and Asymptotic States in a Biological Aggregation Model

Aggregations are emergent features common to many biological systems. Mathematical models to understand their emergence are consequently widespread, with the aggregation–diffusion equation being a prime example. Here we study the aggregation–diffusion equation with linear diffusion in one spatial dimension. This equation is known to support solutions that involve both single and multiple aggregations. However, numerical evidence suggests that the latter, which we term ‘multi-peaked solutions’ may often be long-transient solutions rather than asymptotic steady states. We develop a novel technique for distinguishing between long transients and asymptotic steady states via an energy minimisation approach. The technique involves first approximating our study equation using a limiting process and a moment closure procedure. We then analyse local minimum energy states of this approximate system, hypothesising that these will correspond to asymptotic patterns in the aggregation–diffusion equation. Finally, we verify our hypotheses through numerical investigation, showing that our approximate analytic technique gives good predictions as to whether a state is asymptotic or transient. Overall, we find that almost all twin-peaked, and by extension multi-peaked, solutions are transient, except for some very special cases. We demonstrate numerically that these transients can be arbitrarily long-lived, depending on the parameters of the system.

Asymptotic steady state of a dissipative long-range interacting spin system at finite temperature

Many of the recently developed quantum systems that are considered promising candidates for the underlying technology of quantum computing enjoy long-range interaction, which is even tunable in some of them. Most of these systems can be described using the Heisenberg spin model to represent their interaction with the same system or a different one in hybrid structures. We consider here a finite two-dimensional spin system with a varying spin-spin long-range interaction under the effect of an external uniform magnetic field. We investigate the dynamics of the system and its asymptotic behavior at different degrees of anisotropy and interaction range under coupling to a thermal dissipative environment, starting from different initial states. We show that the system spin state and bipartite entanglement evolve in time to reach asymptotic values that are enhanced significantly by the degree of anisotropy and interaction range, whereas the thermal dissipative environment degrades the entanglement asymptotic equilibrium value as the temperature increases. Interestingly, while the robustness of the spin system against the environment’s thermal dissipative effects increases with the interaction range and degree of anisotropy, the entanglement between the far sites on the lattice shows its highest resilience in the partially anisotropic system, which might be attributed to a critical behavior taking place in the system. While the system’s early dynamics varies significantly depending on the initial state, the asymptotic equilibrium value is found to be completely independent.

Self-consistent soliton evolution in single-two-mode fiber lasers.

Ultrafast few-mode fiber lasers have received increasing attention from basic research to practical applications due to their unique pulse performance and intriguing nonlinear dynamics. Here, we experimentally and numerically reveal the formation and evolution behaviors of a soliton in a mode-locked fiber laser composed of two-mode and single-mode fibers. The LP11 pulse walks away from the LP01 pulse in the two-mode fiber due to modal dispersion and then transforms into an auxiliary LP01 pulse after entering the single-mode fiber. After re-entering the two-mode fiber, the LP01 pulse excites the LP11 pulse via mode coupling; therefore, the LP11 pulse also consists of dominant and auxiliary pulses. Such a soliton fiber laser converges to an asymptotic steady state with unlocked spatial modes arising from the interplay between the strong modal dispersion and weak mode coupling.

A resistive MHD model and simulation on plasma flow evolution in the presence of resonant magnetic perturbation in a tokamak

Nonaxisymmetric magnetic fields such as the intrinsic error field and the externally applied resonant magnetic perturbation (RMP) in a tokamak are known to influence the plasma momentum transport and flow evolution through plasma response, which itself strongly depends on the plasma flow as well. The nonlinear interaction between plasma response and flow has been previously modeled in the conventional error field theory with the “no-slip” condition, which has been recently extended to allow the “free-slip” condition. In this work, we further target this specific process and numerically simulate the nonlinear plasma response and flow evolution in the presence of a single-helicity RMP in a circular-shaped model tokamak configuration, based on the full resistive MHD model in the initial-value code NIMROD. Time evolution of the parallel (to k) flow or “slip frequency” profile and its asymptotic steady state obtained from the NIMROD simulations are compared with both conventional and extended nonlinear response theories. Here, k is the wave vector of the propagating island. Good agreement with the extended theory with free-slip condition has been achieved for the parallel flow profile evolution in response to RMP in all resistive regimes, whereas the difference from the conventional theory with the no-slip condition tends to diminish as the plasma resistivity approaches zero.

On the dynamic optimality of yardstick regulation

This paper proposes a generalization of Shleifer’s (RAND J Econ 16:319–327, 1985) model of yardstick competition to a dynamic framework. In a differential game setting, we show that the yardstick mechanism effectively replicates the first-best solution if players adopt open-loop behaviour rules and are symmetric at the initial time; in the absence of initial symmetry, the social efficiency is reached only in the asymptotic steady state. On the contrary, if players adopt Markovian behaviour rules, then the yardstick pricing rule cannot achieve the first-best solution along the equilibrium path of any Markov Perfect Nash Equilibrium.

Microstructural complexity and dimensional changes in heavily irradiated zirconium

Using atomistic simulations based on the creation-relaxation algorithm, we explore the evolution of microstructure in irradiated zirconium over a broad range of radiation exposure. In agreement with experimental observations, we find that at relatively low temperatures, microstructure evolves towards an asymptotic dynamic steady state forming at doses close to 1 dpa. Simulations show the spontaneous formation of $a$-type interstitial dislocation loops, gradually transforming into an $a$-type extended dislocation network and giving rise to macroscopic anisotropic dimensional changes in a textured material. A fully developed $a$-type interstitial/vacancy and $c$-type vacancy dislocation microstructure corresponds to the highest degree of irradiation growth anisotropy and vanishingly small volumetric swelling of the material.

Subdiffusive dynamics and critical quantum correlations in a disorder-free localized Kitaev honeycomb model out of equilibrium

Disorder-free localization has recently emerged as a mechanism for ergodicity breaking in homogeneous lattice gauge theories. In this work we show that this mechanism can lead to unconventional states of quantum matter as the absence of thermalization lifts constraints imposed by equilibrium statistical physics. We study a Kitaev honeycomb model in a skew magnetic field subject to a quantum quench from a fully polarized initial product state and observe nonergodic dynamics as a consequence of disorder-free localization. We find that the system exhibits a subballistic power-law entanglement growth and quantum correlation spreading, which is otherwise typically associated with thermalizing systems. In the asymptotic steady state the Kitaev model develops volume-law entanglement and power-law decaying dimer quantum correlations even at a finite energy density. Our work sheds light onto the potential for disorder-free localized lattice gauge theories to realize quantum states in two dimensions with properties beyond what is possible in an equilibrium context.

Time-dependent homogeneous states of binary granular suspensions

The time evolution of a homogeneous bidisperse granular suspension is studied in the context of the Enskog kinetic equation. The influence of the surrounding viscous gas on the solid particles is modeled via a deterministic viscous drag force plus a stochastic Langevin-like term. It is found first that, regardless of the initial conditions, the system reaches (after a transient period lasting a few collisions per particle) a universal unsteady hydrodynamic regime where the distribution function of each species not only depends on the dimensionless velocity (as in the homogeneous cooling state) but also on the instantaneous temperature scaled with respect to the background temperature. To confirm this result, theoretical predictions for the time-dependent partial temperatures are compared against direct simulation Monte Carlo (DSMC) results; the comparison shows an excellent agreement confirming the applicability of hydrodynamics in granular suspensions. Also, in the transient regime, the so-called Mpemba-like effect (namely, when an initially hotter sample cools sooner than the colder one) is analyzed for inelastic collisions. The theoretical analysis of the Mpemba effect is performed for initial states close to and far away from the asymptotic steady state. In both cases, good agreement is found again between theory and DSMC results. As a complement to the previous studies, we determine in this paper the dependence of the steady values of the dynamic properties of the suspension on the parameter space of the system. More specifically, we focus our attention on the temperature ratio T1/T2 and the fourth degree cumulants c1 and c2 (measuring the departure of the velocity distributions f1 and f2 from their Maxwellian forms). While our approximate theoretical expression for T1/T2 agrees very well with computer simulations, some discrepancies are found for the cumulants. Finally, a linear stability analysis of the steady state solution is also carried out showing that the steady state is always linearly stable.

Thermal Robustness of Entanglement in a Dissipative Two-Dimensional Spin System in an Inhomogeneous Magnetic Field.

Recently new novel magnetic phases were shown to exist in the asymptotic steady states of spin systems coupled to dissipative environments at zero temperature. Tuning the different system parameters led to quantum phase transitions among those states. We study, here, a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian Lindblad environment at finite temperature. We show how applying an inhomogeneous magnetic field to the system at different degrees of anisotropy may significantly affect the spin states, and the entanglement properties and distribution among the spins in the asymptotic steady state of the system. In particular, applying an inhomogeneous field with an inward (growing) gradient toward the central spin is found to considerably enhance the nearest neighbor entanglement and its robustness against the thermal dissipative decay effect in the completely anisotropic (Ising) system, whereas the beyond nearest neighbor ones vanish entirely. The spins of the system in this case reach different steady states depending on their positions in the lattice. However, the inhomogeneity of the field shows no effect on the entanglement in the completely isotropic (XXX) system, which vanishes asymptotically under any system configuration and the spins relax to a separable (disentangled) steady state with all the spins reaching a common spin state. Interestingly, applying the same field to a partially anisotropic (XYZ) system does not just enhance the nearest neighbor entanglements and their thermal robustness but all the long-range ones as well, while the spins relax asymptotically to very distinguished spin states, which is a sign of a critical behavior taking place at this combination of system anisotropy and field inhomogeneity.

Design approach to stabilizing controller in state space for non-driven micromechanical gyro

The gyro is a new silicon micromechanical gyro, which has no a driving structure but only a sensing structure. The non-driven gyro is installed on a rotating carrier and utilizes the spinning of the carrier to obtain an angular momentum. When the carrier produces a transverse rotation, the sensing mass of the gyro is acted by Coriolis force to sense the input angular velocity. In applications, we found that the shock causes the Si pendulum to go out of steady state, and to fail to work. So, the stability is a key problem. In this paper, we put forward a feasible way to overcome this problem. The method uses modeling in state space to construct a state feedback controller. We firstly introduce the working mechanism of the gyro and obtain the dynamic equation. Then we use Riccati equation to design a steady controller that has a ‘sandwich’ structure. For the stabilizing controller, we calculate the anti-torsion stillness coefficient and analyze the damping. Finally, some shock tests had been done. The theoretical simulation and experimental test shown that the designed stabilizing controller makes the gyro reach an asymptotic steady state in a short time.

Pre-asymptotic dynamics of the infinite size Neumann (p = 2 spherical) model

In this contribution we further study the classical disordered p = 2 spherical model with Hamiltonian dynamics, or in integrable systems terms, the Neumann model, in the infinite size limit. We summarise the asymptotic results that some of us presented in a recent publication, and we deepen the analysis of the pre-asymptotic dynamics. We also discuss the possible description of the asymptotic steady state with a generalised Gibbs ensemble.

Strong vorticity fluctuations and antiferromagnetic correlations in axisymmetric fluid equilibria

The macroscale structure and microscale fluctuation statistics of late-time asymptotic steady state flows in cylindrical geometries is studied using the methods of equilibrium statistical mechanics. The axisymmetric assumption permits an effective two-dimensional description in terms of the (toroidal) flow field $\sigma$ about the cylinder axis and the vorticity field $\xi$ that generates mixing within the (poloidal) planes of fixed azimuth. As for a number of other 2D fluid systems, extending the classic 2D Euler equation, the flow is constrained by an infinite number of conservation laws, beyond the usual kinetic energy and angular momentum. All must be accounted for in a consistent equilibrium description. It is shown that the most directly observable impact of the conservation laws is on $\sigma$, which displays interesting large-scale radius-dependent flow structure. However, unlike in some previous treatments, we find that the thermodynamic temperature is always positive. As a consequence, except for an infinitesimal boundary layer that maintains the correct (conserved) value of the overall poloidal circulation, the impact on $\xi$ resides in the statistics of the strongly fluctuating, fine-scale mixing, where it is sensitive to `antiferromagnetic' microscale correlations that help maintain the analogue of local charge neutrality. The poloidal flow is macroscopically featureless, displaying no large scale circulating jet- or eddy-like features (which typically emerge as negative temperature states in analogous Euler and quasigeostrophic equilibria).

Micro-dynamics of neutral flow induced dusty plasma flow

We present a detailed experimental study of gas flow induced motion of dust particles in a DC glow discharge plasma. The characteristics of the dust dynamics are investigated as a function of the differential gas flow rate, the background neutral pressure, the dust particle size, and the neutral species of the gas. The experiments have been carried out in the table top Dusty Plasma Experimental (DPEx) device in which a plasma is created between a disk shaped anode and a grounded cathode in a Π-shaped Pyrex glass tube. The asymptotic steady state flow velocity of the injected micron sized dust particles is found to increase with an increase in neutral flow velocity and decrease with an increase in the background pressure. Furthermore, this velocity is seen to be independent of the size of the dust particles, but decreases with an increase in the mass of the background gas. A simple theoretical model, based on estimates of the various forces acting on the dust particles, is used to elucidate the role of neutrals in the flow dynamics of the dust particles. Our experiments thus provide a detailed microscopic understanding of some of the past phenomenological observations of dust flows in the DPEx device and can prove useful in future experimental implementations of dust flow experiments.

Statistical scaling laws for chemical oscillators

Universality classes are defined for the nonlinear, oscillatory Belousov–Zhabotinsky (BZ) reaction. We examine the decay to asymptotic steady state for supercritical Hopf bifurcation by considering a phenomenological approach supported by numerical simulations and confirmed by a analytical description. The formalism is general and it is expected to be universal for systems exhibiting Hopf bifurcations.

Generic pure quantum states as steady states of quasi-local dissipative dynamics

We investigate whether a generic pure state on a multipartite quantum system can be the unique asymptotic steady state of locality-constrained purely dissipative Markovian dynamics. In the tripartite setting, we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one, solely depending on the subsystems’ dimension. A complete analytical characterization is given when the central subsystem is a qubit. In the N-partite case, we provide conditions on the subsystems’ size and the nature of the locality constraint, under which random pure states cannot be quasi-locally stabilized generically. Also, allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic, our results offer insights into the extent to which random pure states may arise as unique ground states of frustration-free parent Hamiltonians. We further argue that, to a high probability, pure quantum states sampled from a t-design enjoy the same stabilizability properties of Haar-random ones as long as suitable dimension constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a connection between the tasks of quasi-local state stabilization and unique state reconstruction from local tomographic information, and provide a constructive procedure for determining a generic N-partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties, improving over existing results.

Dynamical quantum phase transitions in systems with continuous symmetry breaking

Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three dimensional O(N) model following a quantum quench from a symmetry breaking initial state. The O(N) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero-crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O(N) model are general and apply to generic systems with continuous symmetry breaking.

Two-Species Migration and Clustering in Two-Dimensional Domains

We extend two-species models of individual aggregation or clustering to two-dimensional spatial domains, allowing for more realistic movement of the populations compared with one spatial dimension. We assume that the domain is bounded and that there is no flux into or out of the domain. The motion of the species is along fitness gradients which allow the species to seek out a resource. In the case of competition, species which exploit the resource alone will disperse while avoiding one another. In the case where one of the species is a predator or generalist predator which exploits the other species, that species will tend to move toward the prey species, while the prey will tend to avoid the predator. We focus on three primary types of interspecies interactions: competition, generalist predator–prey, and predator–prey. We discuss the existence and stability of uniform steady states. While transient behaviors including clustering and colony formation occur, our stability results and numerical evidence lead us to believe that the long-time behavior of these models is dominated by spatially homogeneous steady states when the spatial domain is convex. Motivated by this, we investigate heterogeneous resources and hazards and demonstrate how the advective dispersal of species in these environments leads to asymptotic steady states that retain spatial aggregation or clustering in regions of resource abundance and away from hazards or regions or resource scarcity.

Quench-induced entanglement and relaxation dynamics in Luttinger liquids

We investigate the time evolution towards the asymptotic steady state of a one dimensional interacting system after a quantum quench. We show that at finite time the latter induces entanglement between right- and left- moving density excitations, encoded in their cross-correlators, which vanishes in the long-time limit. This behavior results in a universal time-decay in system spectral properties $ \propto t^{-2} $, in addition to non-universal power-law contributions typical of Luttinger liquids. Importantly, we argue that the presence of quench-induced entanglement clearly emerges in transport properties, such as charge and energy currents injected in the system from a biased probe, and determines their long-time dynamics. In particular, energy fractionalization phenomenon turns out to be a promising platform to observe the universal power-law decay $ \propto t^{-2} $ induced by entanglement and represents a novel way to study the corresponding relaxation mechanism.

Structure Preserving Schemes for Nonlinear Fokker–Planck Equations and Applications

In this paper we focus on the construction of numerical schemes for nonlinear Fokker-Planck equations that preserve the structural properties, like non negativity of the solution, entropy dissipation and large time behavior. The methods here developed are second order accurate, they do not require any restriction on the mesh size and are capable to capture the asymptotic steady states with arbitrary accuracy. These properties are essential for a correct description of the underlying physical problem. Applications of the schemes to several nonlinear Fokker-Planck equations with nonlocal terms describing emerging collective behavior in socio-economic and life sciences are presented.