Large Time Limit Research Articles

Charge transfer statistics and qubit dynamics at the tunneling Fermi-edge singularity

Tunneling of spinless electrons from a single-channel emitter into an empty collector through an interacting resonant level of the quantum dot (QD) is studied, when all Coulomb screening of charge variations on the dot is realized by the emitter channel and the system is mapped onto an exactly solvable model of a dissipative qubit. In this model we describe the qubit density matrix evolution with a generalized Lindblad equation, which permit us to count the tunneling electrons and therefore relate the qubit dynamics to the charge transfer statistics. In particular, the coefficients of its generating function equal to the time dependent probabilities to have the fixed number of electrons tunneled into the collector are expressed through the parameters of a non-Hermitian Hamiltonian evolution of the qubit pure states in between the successive electron tunnelings. From the long time asymptotics of the generating function we calculate Fano factors of the second and third order (skewness) and establish their relation to the extra average and cumulant, respectively, of the charge accumulated in the transient process of the empty QD evolution beyond their linear time dependence. It explains the origin of the sub and super Poisson shot noise in this system and shows that the super Poisson signals existence of a non-monotonous oscillating transient current and the qubit coherent dynamics. The mechanism is illustrated with particular examples of the generating functions, one of which coincides in the large time limit with the $1/3$ fractional Poissonian realized without the real fractional charge tunneling.

Assessing the structure of classical molecular optimal control landscapes

We consider laser-driven optimal control landscape of a molecule from a classical mechanical perspective. The goal of optimal control in the present work is to steer the molecule from an initial state to a target state, denoted by two distinct points in phase space. Thus, a particular control objective is given as the difference between the final achieved phase space point and the target. The corresponding control landscape is defined as the latter control objective as a functional of the control field. While previous examination of the landscape critical points (i.e., a suboptimal point on the landscape where there is a zero gradient) has shown that the landscape topology is generally trap-free, the structure of the landscape away from these critical points is not well understood. We explore the landscape structure by examining an underlying metric defined as the ratio R of the gradient-based optimization path length of the control field evolution to the Euclidean distance between a given initial control field and the resultant optimal control field, where the latter field corresponds to a point at the top of the landscape. We analyze the path length-to-distance ratio R analytically for a linear forced harmonic oscillator and numerically for a nonlinear forced Morse oscillator. For the linear forced harmonic oscillator, we find that R⩽2 and reaches its minimum value of 1 (i.e., corresponding to “a straight shot” through control space) in the large target time limit, as well as at special finite target times. The ratio R is similarly small for Morse oscillator simulations when following a steepest-ascent path to the top of the landscape, implying that the landscape is quite smooth and devoid of gnarled features. This conclusion is exemplified for a path discovered with R≃1.0 where simply following the initial gradient direction takes the climb very close to the top of the landscape. These findings are consistent with a variety of previous like simulations examining R in quantum control scenarios.

Open XXZ chain and boundary modes at zero temperature

We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.

Driving mechanisms of ratchet flow in thin liquid films under tangential two-frequency forcing

In a recent paper, we demonstrated the emergence of ratchet flows in thin liquid films subjected to tangential two-frequency vibrations [E. Sterman-Cohen, M. Bestehorn, and A. Oron, “Ratchet flow of thin liquid films induced by a two-frequency tangential forcing,” Phys. Fluids 30, 022101 (2018)], and asymmetric forcing was found to be a sole driving mechanism for these ratchet flows. In this paper, we consider other two-frequency excitations and reveal an additional driving mechanism of an emerging ratchet flow when the acceleration imparted by forcing is symmetric with respect to a certain moment of time within the forcing period (this type of forcing referred to as “symmetric forcing”). This driving mechanism exhibits an intricate interaction between forcing, capillarity, and gravity. We find that in contradistinction with the case of asymmetric forcing where the flow intensity reaches a constant value in the large-time limit, in the case of symmetric forcing the flow intensity exhibits oscillatory variation in time. We also discuss the flow intensity variation of the emerging ratchet flows with the fundamental wavenumber of the disturbance.

Large time behavior of solutions to a chemotaxis model with porous medium diffusion

In this paper, we study the large time behavior of a chemotaxis model with nonlinear diffusion and consumption{ut=Δum−∇⋅(u∇v)+μu(1−u),vt−Δv=−vu, where m>1. In a previous paper [5], we have proved the existence and uniform boundedness of global weak solutions for any nonnegative initial data and any m>1. In this work, we show that the weak solutions strongly converge to (1,0) in the large time limit.

Transition to Frozen Antiferromagnetic Pattern in Delayed Logistic Map

We study nonlinearly coupled logistic map with feedback in the limit of large delay time ([Formula: see text]). In the [Formula: see text]’s, the mapping of delayed dynamical systems to the spatially extended system was pursued in the context of optical systems by Arecchi and coworkers. We map the delayed system to a coupled map lattice of size [Formula: see text] with a sequential update and strictly one-sided coupling. Apart from aiding visualization, this formal equivalence helps us identify “phases” in a delayed system. One of the easily identified phases is that of frozen antiferromagnetic phase which is abundant in phase space. As coupling parameter ([Formula: see text]) increases, the system goes from periodic cascades along the coarse-grain region to chaos. We use the number of domain walls as an order parameter for locating this transition. The domain walls undergo random walk with drift and annihilate each other. At critical parameter value, we observe the power-law decay of this parameter. The decay exponents fall within the directed Ising universality class in general. We also find the escape time distribution, i.e. time required for reaching frozen antiferromagnetic regime. It diverges as it approaches the critical point. The divergence of average laminar length shows features indicating crisis induced intermittency.

Nonequilibrium dynamics of noninteracting fermions in a trap

We consider the real-time dynamics of N noninteracting fermions in d = 1. They evolve in a trapping potential V(x), starting from the equilibrium state in a potential V0(x). We study the time evolution of the Wigner function W(x, p, t) in the phase space (x, p), and the associated kernel which encodes all correlation functions. At t = 0 the Wigner function for large N is uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf over a scale eN being described by a universal scaling function related to the Airy function. We obtain exact solutions for the Wigner function, the density, and the correlations in the case of harmonic and inverse square potentials, for several V0(x). In the large-N limit, near the edges where the density vanishes, we obtain limiting kernels (of the Airy or Bessel types) that retain the form found in equilibrium, up to a time-dependent rescaling. For nonharmonic traps the evolution of the Fermi volume is more complex. Nevertheless we show that, for intermediate times, the Fermi surf is still described by the same equilibrium scaling function, with a nontrivial time- and space-dependent width which we compute analytically. We discuss the multi-time correlations and obtain their explicit scaling forms valid near the edge for the harmonic oscillator. Finally, we address the large-time limit where relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the “classical” regime . Using the diagonal ensemble we compute the Wigner function in the quantum case (large N, fixed ℏ ) and show that it agrees with the GGE. We also obtain the higher order (nonlocal) correlations in the diagonal ensemble.

Large-time asymptotics of a public goods game model with diffusion

We consider a spatially inhomogeneous public goods game model with diffusion. By utilising a generalised Hamiltonian structure of the model we study the existence of global classical solutions as well as the large time behaviour: first, the asymptotic convergence of the PDE to the corresponding ODE system is proven. This result entails also the periodic behaviour of PDE solutions in the large time limit. Secondly, a shadow system approximation is considered and the convergence of the PDE to the shadow system in the associated fast-diffusion limit is shown. Finally, the asymptotic convergence of the shadow to the ODE system is proven.

A continued fraction based approach for the Two-photon Quantum Rabi Model

We study the Two Photon Quantum Rabi Model by way of its spectral functions and survival probabilities. This approach allows numerical precision with large truncation numbers, and thus exploration of the spectral collapse. We provide independent checks and calibration of the numerical results by studying an exactly solvable case and comparing the essential qualitative structure of the spectral functions. We stress that the large time limit of the survival probability provides us with an indicator of spectral collapse, and propose a technique for the detection of this signal in the current and upcoming quantum simulations of the model.

Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions

This work is concerned with a prototypical model for the spatio-temporal evolution of a forager–exploiter system, consisting of two species which simultaneously consume a common nutrient, and which interact through a taxis-type mechanism according to which individuals from the exploiter subpopulation move upward density gradients of the forager subgroup. Specifically, the model [Formula: see text] for the population densities [Formula: see text] and [Formula: see text] of foragers and exploiters, as well as the nutrient concentration [Formula: see text], is considered in smoothly bounded domains [Formula: see text], [Formula: see text]. It is first shown that under an explicit condition linking the sizes of the resource production rate [Formula: see text] and of the initial nutrient concentration, an associated Neumann-type initial-boundary value problem admits a global solution within an appropriate generalized concept. The second of the main results asserts stabilization of these solutions toward spatially homogeneous equilibria in the large time limit, provided that [Formula: see text] satisfies a mild assumption on temporal decay. To the best of our knowledge, these are the first rigorous analytical results addressing taxis-type cross-diffusion mechanisms coupled in a cascade-like manner as in (⋆).

Multipoint distribution of periodic TASEP

The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multitime and multilocation distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large-time limit in the so-called relaxation time scale.

Diffusive transport in highly corrugated channels

Diffusion in waveguides with spatially modulated profiles is an important topic in modern electromagnetics and optics. Wave dynamics in the high-frequency asymptotics are governed by classical ray dynamics which can be characterised by looking at the diffusion of particles throughout the channel. We study the transport of particles (rays) in a channel with a sinusoidal profile at different values of the corrugation amplitude. We find that below a certain corrugation level the transport is ballistic, beyond this threshold, a diffusion-like behaviour emerges in the asymptotic limit of large times. In this regime particle transport slows down due to the trapping mechanism in the corrugated regions of the channel. We use the analogy with correlated random walks to discuss the observed transport regimes.

Effective driven dynamics for one-dimensional conditioned Langevin processes in the weak-noise limit

In this work we focus on fluctuations of time-integrated observables for a particle diffusing in a one-dimensional periodic potential in the weak-noise asymptotics. Our interest goes to rare trajectories presenting an atypical value of the observable, that we study through a biased dynamics in a large-deviation framework. We determine explicitly the effective probability-conserving dynamics which makes rare trajectories of the original dynamics become typical trajectories of the effective one. Our approach makes use of a weak-noise path-integral description in which the action is minimised by the rare trajectories of interest. For ‘current-type’ additive observables, we find criteria for the emergence of a propagative trajectory minimising the action for large enough deviations, revealing the existence of a dynamical phase transition at a fluctuating level, whose singular behaviour is between first and second order. In addition, we provide a new method to determine the scaled cumulant generating function of the observable without having to optimise the action. It allows one to show that the weak-noise and the large-time limits commute in this problem. Finally, we show how the biased dynamics can be mapped in practice to an explicit effective driven dynamics, which takes the form of a driven Langevin dynamics in an effective potential. The non-trivial shape of this effective potential is key to understand the link between the dynamical phase transition in the large deviations of current and the standard depinning transition of a particle in a tilted potential.

Dissipative solution to the Ericksen–Leslie system equipped with the Oseen–Frank energy

We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state.

Evolution of dispersal in spatial population models with multiple timescales.

We study the evolutionary stability of dispersal strategies, including but not limited to those that can produce ideal free population distributions (that is, distributions where all individuals have equal fitness and there is no net movement of individuals at equilibrium). The environment is assumed to be variable in space but constant in time. We assume that there is a separation of times scales, so that dispersal occurs on a fast timescale, evolution occurs on a slow timescale, and population dynamics and interactions occur on an intermediate timescale. Starting with advection-diffusion models for dispersal without population dynamics, we use the large time limits of profiles for population distributions together with the distribution of resources in the environment to calculate growth and interaction coefficients in logistic and Lotka-Volterra ordinary differential equations describing population dynamics. We then use a pairwise invasibility analysis approach motivated by adaptive dynamics to study the evolutionary and/or convergence stability of strategies determined by various assumptions about the advection and diffusion terms in the original advection-diffusion dispersal models. Among other results we find that those strategies which can produce an ideal free distribution are evolutionarily stable.

Entropy production and asymptotic factorization via thermalization: A collisional model approach

The Markovian evolution of an open quantum system is characterized by a positive entropy production, while the global entropy gets redistributed between the system and the environment degrees of freedom. Starting from these premises, we analyze the entropy variation of an open quantum system in terms of two distinct relations: the Clausius inequality, that provides an intrinsic bound for the entropy variation in terms of the heat absorbed by the system, and an extrinsic inequality, which instead relates the former to the corresponding entropy increment of the environment. By modeling the thermalization process with a Markovian collisional model, we compare and discuss the two bounds, showing that the latter is asymptotically saturated in the limit of large interaction time. In this regime not only the reduced density matrix of the system reaches an equilibrium configuration, but it also factorizes from the environment degrees of freedom. This last result is proven analytically when the system-bath coupling is sufficiently strong and through numerical analysis in the weak-coupling regime.

Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica

We consider the fluctuations in the stochastic growth of a one-dimensional interface of height described by the Kardar–Parisi–Zhang (KPZ) universality class. We study the joint probability distribution function (JPDF) of the interface heights at two-times t1 and , with droplet initial conditions at t = 0. In the large-time limit, this JPDF is expected to become a universal function of the time ratio , and of the (properly scaled) heights h(x,t1) and h(x,t2). Using the replica Bethe ansatz method for the KPZ equation, in De Nardis and Le Doussal (2017 J. Stat. Mech. 053212) we obtained a formula for the JPDF in the (partial) tail regime where h(x,t1) is large and positive, subsequently found to be in excellent agreement with the experimental and numerical data (De Nardis et al 2017 Phys. Rev. Lett. 118 125701). Here we show that our results are in perfect agreement with Johansson’s recent rigorous expression of the full JPDF (Johansson 2018 in preparation (arXiv:1802.00729)), thereby confirming the validity of our methods.

Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals

We study a nonlinear system of differential equations describing the evolution of a competitive two-species chemotaxis system with two chemicals in a bounded domain. The system consists of four PDEs, two equations of parabolic type describing the evolution of the competitive species and two elliptic equations modeling the distribution of the chemicals. By introducing global competitive/cooperative factors, we obtain, for different ranges of parameters, that any positive and bounded solution converges to a spatially homogeneous state. The proofs rely on the comparison principle for spatially homogeneous sub- and super-solutions. The existence and uniqueness of global classical solution are proved under assumptions on the initial data and appropriate conditions on the parameters of the system. Such solution stabilizes to spatially homogeneous equilibria in the large time limit, given coexistence or extinction of solutions for a range of parameters.

Rényi entropies of generic thermodynamic macrostates in integrable systems

We study the behaviour of Rényi entropies in a generic thermodynamic macrostate of an integrable model. In the standard quench action approach to quench dynamics, the Rényi entropies may be derived from the overlaps of the initial state with Bethe eigenstates. These overlaps fix the driving term in the thermodynamic Bethe ansatz (TBA) formalism. We show that this driving term can be also reconstructed starting from the macrostate’s particle densities. We then compute explicitly the stationary Rényi entropies after the quench from the dimer and the tilted Néel state in XXZ spin chains. For the former state we employ the overlap TBA approach, while for the latter we reconstruct the driving terms from the macrostate. We discuss in full detail the limits that can be analytically handled and we use numerical simulations to check our results against the large time limit of the entanglement entropies.

Kinetics of deposition in the diffusion-controlled limit.

The adsorption of particles diffusing in a half space bounded by the substrate and irreversibly sticking to the substrate upon contacts is investigated. We show that when absorbing particles are planar disks diffusing in the three-dimensional half space, the coverage approaches its saturated "jamming" value as t^{-1} in the large time limit (generally as t^{-1/(d-1)} when the substrate is d dimensional and d>1, and as e^{-t/ln(t)} when d=1). We also analyze the asymptotic behavior when particles are spherical and when particles are planar aligned squares.