Large Time Limit Research Articles

A convergent genus expansion for the plateau

We conjecture a formula for the spectral form factor of a double-scaled matrix integral in the limit of large time, large density of states, and fixed temperature. The formula has a genus expansion with a nonzero radius of convergence. To understand the origin of this series, we compare to the semiclassical theory of “encounters” in periodic orbits. In Jackiw-Teitelboim (JT) gravity, encounters correspond to portions of the moduli space integral that mutually cancel (in the orientable case) but individually grow at low energies. At genus one we show how the full moduli space integral resolves the low energy region and gives a finite nonzero answer.

Nonlinear Vlasov and Fokker-Planck Dynamics in Confined Systems with Drag: A Numerical Study

Abstract We explore numerically the behavior of a one-dimensional many-body system consisting of particles that interact through short range repulsive forces, and are also under the effects of drag forces, and of an external confining potential. The statistical dynamics of systems of this kind exhibits interesting links with the thermostatistical formalism based on the Sq non-additive entropies. In the regime of overdamped motion, these systems admit an effective description in terms of a non-linear Fokker-Planck equation. When the overdamped condition is relaxed, and inertial effects are explicitly taken into account, the system can be described by a Vlasov-like effective mean field dynamics. The Vlasov-like description of this type of systems has been recently investigated in the literature from an analytical point of view. In the present contribution we explore the behaviour of these system numerically, through direct molecular dynamics simulations. We consider examples of systems with four different short range repulsive forces, with a dependence on distance given by exponential, Heaviside, Lorentzian, and Bessel functions. The results of our numerical simulations are fully consistent with the predictions derived from the Vlasov-like mean field description. In particular, we verify that, in the asymptotic limit of large times, the system evolves towards a state exhibiting a spatial distribution of particles that coincides with the stationary solution of an appropriate nonlinear Fokker-Planck equation. This limit spatial distribution has the form of a q-Gaussian.

Dynamic fluctuations of current and mass in nonequilibrium mass transport processes

We study steady-state dynamic fluctuations of current and mass in several variants of random average processes on a ring of L sites. These processes violate detailed balance in the bulk and have nontrivial spatial structures: their steady states are not described by the Boltzmann–Gibbs distribution and can have nonzero spatial correlations. Using a microscopic approach, we exactly calculate the second cumulants, or the variance, ⟨Qi2(T)⟩c and ⟨Qsub2(l,T)⟩c , of cumulative (time-integrated) currents up to time T across the ith bond and across a subsystem of size l (summed over bonds in the subsystem), respectively. We also calculate the (two-point) dynamic correlation function of the subsystem mass. In particular, we show that, for large L≫1 , the second cumulant ⟨Qi2(T)⟩c of the cumulative current up to time T across the ith bond grows linearly as ⟨Qi2⟩c∼T for initial times T∼O(1) , subdiffusively as ⟨Qi2⟩c∼T1/2 for intermediate times 1≪T≪L2 , and then again linearly as ⟨Qi2⟩c∼T for long times T≫L2 . The scaled cumulant liml→∞,T→∞⟨Qsub2(l,T)⟩c/2lT of current across the subsystem of size l and up to time T converges to the density-dependent particle mobility χ(ρ) when the large subsystem-size limit is taken first, followed by the large-time limit; when the limits are reversed, it simply vanishes. Remarkably, regardless of the dynamical rules, the scaled current cumulant D⟨Qi2(T)⟩c/2χL≡W(y) as a function of scaled time y=DT/L2 can be expressed in terms of a universal scaling function W(y) , where D is the bulk-diffusion coefficient; interestingly, the intermediate-time subdiffusive and long-time diffusive growths can be connected through a single scaling function W(y) . The power spectra for current and mass are also exactly characterized by the respective scaling functions. Furthermore, we provide a microscopic derivation of equilibrium-like Green–Kubo and Einstein relations that connect the steady-state current fluctuations to an ‘operational’ mobility (i.e. the response to an external force field) and mass fluctuation, respectively.

Large deviations in statistics of the local time and occupation time for a run and tumble particle.

We investigate the statistics of the local time T=∫_{0}^{T}δ(x(t))dt that a run and tumble particle (RTP) x(t) in one dimension spends at the origin, with or without an external drift. By relating the local time to the number of times the RTP crosses the origin, we find that the local time distribution P(T) satisfies the large deviation principle P(T)∼e^{-TI(T/T)} in the large observation time limit T→∞. Remarkably, we find that in the presence of drift the rate function I(ρ) is nonanalytic: we interpret its singularity as a dynamical phase transition of first order. We then extend these results by studying the statistics of the amount of time R that the RTP spends inside a finite interval (i.e., the occupation time), with qualitatively similar results. In particular, this yields the long-time decay rate of the probability P(R=T) that the particle does not exit the interval up to time T. We find that the conditional end-point distribution exhibits an interesting change of behavior from unimodal to bimodal as a function of the size of the interval. To study the occupation time statistics, we extend the Donsker-Varadhan large-deviation formalism to the case of RTPs, for general dynamical observables and possibly in the presence of an external potential.

Unorientable topological gravity and orthogonal random matrix universality

The duality of Jackiw-Teitelboim (JT) gravity and a double scaled matrix integral has led to studies of the canonical spectral form factor (SFF) in the so called τ−scaled limit of large times, t → ∞, and fixed temperature, in order to demonstrate agreement with universal random matrix theory (RMT). Though this has been established for the unitary case, extensions to other symmetry classes requires the inclusion of unorientable manifolds in the sum over geometries, necessary to address time reversal invariance, and regularization of the corresponding prime geometrical objects, the Weil-Petersson (WP) volumes. We report here how universal signatures of quantum chaos, witnessed by the fidelity to the Gaussian orthogonal ensemble, emerge for the low-energy limit of unorientable JT gravity, i.e. the unorientable Airy model/topological gravity. To this end, we implement the loop equations for the corresponding dual (double-scaled) matrix model and find the generic form of the unorientable Airy WP volumes, supported by calculations using unorientable Kontsevich graphs. In an apparent violation of the gravity/chaos duality, the τ−scaled SFF on the gravity side acquires both logarithmic and power law contributions in t, not manifestly present on the RMT side. We show the expressions can be made to agree by means of bootstrapping-like relations hidden in the asymptotic expansions of generalized hypergeometric functions. Thus, we are able to establish strong evidence of the quantum chaotic nature of unorientable topological gravity.

On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it

Abstract We propose iterative algorithms to solve adversarial training problems in a variety of supervised learning settings of interest. Our algorithms, which can be interpreted as suitable ascent-descent dynamics in Wasserstein spaces, take the form of a system of interacting particles. These interacting particle dynamics are shown to converge toward appropriate mean-field limit equations in certain large number of particles regimes. In turn, we prove that, under certain regularity assumptions, these mean-field equations converge, in the large time limit, toward approximate Nash equilibria of the original adversarial learning problems. We present results for non-convex non-concave settings, as well as for non-convex concave ones. Numerical experiments illustrate our results.

Stabilization in a chemotaxis-consumption model involving Robin-type boundary conditions

The chemotaxis-consumption model{ut=∇⋅((u+1)α∇u)−∇⋅(u(u+1)β−1∇v),x∈Ω,t>00=Δv−uv,x∈Ω,t>0 is considered in a smooth bounded domain Ω⊂Rn under the boundary conditions (u+1)α∂νu=u(u+1)β−1∂νv and ∂νv=(γ−v)g on ∂Ω, with parameters α,β∈R, γ>0 and the nonnegative function g∈C1+ω(Ω¯) for some ω∈(0,1). If β−α≤1, it is proved that there exists a global bounded classical solution (u,v) for suitably regular initial data u0. Furthermore, for the given mass m=∫Ωu∞>0, the corresponding stationary system admits a unique classical solution (u∞,v∞), appearing as a large-time limit in the sense that if γ>0 is small enough, then‖u(⋅,t)−u∞‖L∞(Ω)+‖v(⋅,t)−v∞‖L∞(Ω)→0 as t→∞, with ∫Ωu∞=∫Ωu0=m>0.

Global boundedness and asymptotic behavior in a double haptotaxis model for oncolytic virotherapy

This work studies a haptotactic cross-diffusion system modeling oncolytic virotherapy{ut=Δu−∇⋅(u∇v)−ρuz,vt=−(u+w)v−κv,wt=DwΔw−ξw∇⋅(w∇v)−w+ρuz,zt=DzΔz−z−ρuz+βw, in a smooth bounded domain Ω⊂R2, with parameters Dw>0, Dz>0, ξw≥0, ρ≥0, κ>0 and β≥0. This system describes the interaction among uninfected and infected cancer cells, extracellular matrix and oncolytic viruses. It is rigorously proved that an associated no-flux initial-boundary value problem has a unique global classical solution which is bounded in (L∞(Ω))4. Moreover, it is shown that this bounded solution can approach a spatially constant equilibrium in the large time limit under the additional assumption that 0≤β<1.

The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions

The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions

Extremely persistent dense active fluids.

We study the dynamics of dense three-dimensional systems of active particles for large persistence times τp at constant average self-propulsion force f. These systems are fluid counterparts of previously investigated extremely persistent systems, which in the large persistence time limit relax only on the time scale of τp. We find that many dynamic properties of the systems we study, such as the mean-squared velocity, the self-intermediate scattering function, and the shear-stress correlation function, become τp-independent in the large persistence time limit. In addition, the large τp limits of many dynamic properties, such as the mean-square velocity and the relaxation times of the scattering function, and the shear-stress correlation function, depend on f as power laws with non-trivial exponents. We conjecture that these systems constitute a new class of extremely persistent active systems.

Asymptotic growth and decay of two-dimensional symmetric plasmas

We study the large time behavior of classical solutions to the two-dimensional Vlasov-Poisson (VP) and relativistic Vlasov-Poisson (RVP) systems launched by radially-symmetric initial data with compact support. In particular, we prove that particle positions and momenta grow unbounded as $ t \to \infty $ and obtain sharp rates on the maximal values of these quantities on the support of the distribution function for each system. Furthermore, we establish nearly sharp rates of decay for the associated electric field, as well as upper and lower bounds on the decay rate of the charge density in the large time limit. We prove that, unlike (VP) in higher dimensions, smooth solutions do not scatter to their free-streaming profiles as $ t \to \infty $ because nonlinear, long-range field interactions dominate the behavior of characteristics due to the exchange of energy from the potential to the kinetic term. In this way, the system may 'forget' any previous configuration of particles.

Large deviations for trajectory observables of diffusion processes in dimension d > 1 in the double limit of large time and small diffusion coefficient

For diffusion processes in dimension d > 1, the statistics of trajectory observables over the time-window [0,T] can be studied via the Feynman–Kac deformations of the Fokker–Planck generator, which can be interpreted as Euclidean non-Hermitian electromagnetic quantum Hamiltonians. It is interesting to compare the four regimes corresponding to the time T, either finite or large, and to the diffusion coefficient D, either finite or small. (1) For finite T and finite D, one needs to consider the full time-dependent quantum problem that involves the full spectrum of the Hamiltonian. (2) For large time T→+∞ and finite D, one only needs to consider the ground-state properties of the quantum Hamiltonian to obtain the generating function of rescaled cumulants and to construct the corresponding canonical conditioned processes. (3) For finite T and D → 0, one only needs to consider the dominant classical trajectory and its action satisfying the Hamilton–Jacobi equation, as in the semi-classical Wentzel–Kramers–Brillouin (WKB) approximation of quantum mechanics. (4) In the double limit T→+∞ and D → 0, the simplifications in the large deviations in TD of trajectory observables can be analyzed via the two orders of limits, i.e. either from the limit D → 0 of the ground-state properties of the quantum Hamiltonians of (2), or from the limit of long classical trajectories T→+∞ in the semi-classical WKB approximation of (3). This general framework is illustrated in dimension d = 2 with rotational invariance.

Geometric versus Entropic Gaussian Correlations in an Open Quantum System of Two Bosonic Modes

Different types of geometric and entropic quantum correlation quantifiers are studied for a system composed of two resonant bosonic modes embedded in a thermal bath. The description of the evolution of the correlation measures is formulated in the framework of the theory of open systems, based on completely positive quantum dynamical semigroups, by using both a geometric and entropic quantification of total nonclassical correlations of Gaussian states. We consider the special case when the initial squeezed thermal state of the system preserves its form in time. We show that time evolution of the measures strongly depends on the parameters characterising the initial state of the system (squeezing parameter and average thermal photon numbers of the two modes) and of the thermal environment (temperature of the thermal bath and dissipation rate). In the limit of large times all the considered measures asymptotically tend to zero value, corresponding to an asymptotic bimodal uncorrelated product state. We make a comparison between the behaviour of the evolution in time of the Gaussian geometric quantum correlations and Gaussian entropic quantum correlations.

Symmetry-resolved entanglement in fermionic systems with dissipation

We investigate symmetry-resolved entanglement in out-of-equilibrium fermionic systems subject to gain and loss dissipation, which preserves the block-diagonal structure of the reduced density matrix. We derive a hydrodynamic description of the dynamics of several entanglement-related quantities, such as the symmetry-resolved von Neumann entropy and the charge-imbalance-resolved fermionic negativity. We show that all these quantities admit a hydrodynamic description in terms of entangled quasiparticles. While the entropy is dominated by dissipative processes, the resolved negativity is sensitive to the presence of entangled quasiparticles, and it shows the typical ‘rise and fall’ dynamics. Our results hold in the weak-dissipative hydrodynamic limit of large intervals, long times and weak dissipation rates.

Can Dirac-type singularities in Keller-Segel systems be ruled out by power-type singular sensitivities?

This paper deals with the chemotaxis system with power-type singular sensitivities: ut=Δu−χ∇⋅(uvα∇v) and vt=Δv−v+u in a bounded domain Ω⊂R2, with χ>0 and α>0. It is notable that, whenever α=0, this system reduces to the minimal Keller-Segel model; the persistent Dirac-type singularities are known to occur at least in radial parabolic-elliptic setting. In this work, our first result shows that for any χ>0, if eitherα≥14, orα∈(0,14) and ‖u0‖L1 is small appropriately, then the corresponding initial-boundary value problem possesses a global generalized solution (u,v) with the property that u belongs to L1(Ω×(0,T)) for any T>0, which further implies that the general sensitivities 1vα, with α≥14, can rule out the emergence of Dirac-type singularities. Our second result indicates that, if α∈(0,12) and ‖u0‖L1 is small properly, then such global generalized solution becomes bounded and smooth at least eventually, and approaches the spatial equilibria in the large time limit.

On the Kolmogorov Distance for the Maximum Likelihood Estimator in the Explosive Ornstein-Uhlenbeck Process

The paper estimates the Kolmogorov distance between the distribution of the normalized maximum likelihood estimator of the positive drift parameter in the nonergodic Ornstein-Uhlenbeck process and the standard Cauchy distribution and shows exponential error rate for large time limit.

Logarithmic negativity in out-of-equilibrium open free-fermion chains: An exactly solvable case

We derive the quasiparticle picture for the fermionic logarithmic negativity in a tight-binding chain subject to gain and loss dissipation. We focus on the dynamics after the quantum quench from the fermionic Néel state. We consider the negativity between both adjacent and disjoint intervals embedded in an infinite chain. Our result holds in the standard hydrodynamic limit of large subsystems and long times, with their ratio fixed. Additionally, we consider the weakly-dissipative limit, in which the dissipation rates are inversely proportional to the size of the intervals. We show that the negativity is proportional to the number of entangled pairs of quasiparticles that are shared between the two intervals, as is the case for the mutual information. Crucially, in contrast with the unitary case, the negativity content of quasiparticles is not given by the Rényi entropy with Rényi index 1/21/2, and it is in general not easily related to thermodynamic quantities.

Interacting, running and tumbling: The active Dyson Brownian motion

We introduce and study a model in one dimension of N run-and-tumble particles (RTP) which repel each other logarithmically in the presence of an external quadratic potential. This is an “active” version of the well-known Dyson Brownian motion (DBM) where the particles are subjected to a telegraphic noise, with two possible states ± with velocity ±v 0. We study analytically and numerically two different versions of this model. In model I a particle only interacts with particles in the same state, while in model II all the particles interact with each other. In the large time limit, both models converge to a steady state where the stationary density has a finite support. For finite N, the stationary density exhibits singularities, which disappear when . In that limit, for model I, using a Dean-Kawasaki approach, we show that the stationary density of + (respectively −) particles deviates from the DBM Wigner semi-circular shape, and vanishes with an exponent 3/2 at one of the edges. In model II, the Dean-Kawasaki approach fails but we obtain strong evidence that the density in the large N limit (still) retains a Wigner semi-circular shape.

Leftward, rightward, and complete exit-time distributions of jump processes.

First-passage properties of continuous stochastic processes confined in a one-dimensional interval are well described. However, for jump processes (discrete random walks), the characterization of the corresponding observables remains elusive, despite their relevance in various contexts. Here we derive exact asymptotic expressions for the leftward, rightward, and complete exit-time distributions from the interval [0,x] for symmetric jump processes starting from x_{0}=0, in the large x and large time limit. We show that both the leftward probability F_{[under 0]̲,x}(n) to exit through 0 at step n and rightward probability F_{0,[under x]̲}(n) to exit through x at step n exhibit a universal behavior dictated by the large-distance decay of the jump distribution parametrized by the Levy exponent μ. In particular, we exhaustively describe the n≪(x/a_{μ})^{μ} and n≫(x/a_{μ})^{μ} limits and obtain explicit results in both regimes. Our results finally provide exact asymptotics for exit-time distributions of jump processes in regimes where continuous limits do not apply.

Scattering by source-type flows in disordered media

Scattering through a natural porous formation (by far the most ubiquitous example of disordered medium) represents a formidable tool to identify effective flow and transport properties. In particular, we are interested here in the scattering of a passive scalar as determined by a steady velocity field, which is generated by a line of singularity. The velocity undergoes erratic spatial variations, and concurrently, the evolution of the scattering is conveniently described within a stochastic framework that regards the conductivity of the hosting medium as a stationary, Gaussian, random field. Unlike the similar problem in uniform (in the mean) flow fields, the problem at stake results much more complex. Central to the present study is the fluctuation of the driving field that is computed in closed (analytical) form as a large time limit of the same quantity in the unsteady-state flow regime. The structure of the second-order moment Xrr, quantifying the scattering along the radial direction, is explained by the rapid change of the distance along which the velocities of two fluid particles become uncorrelated. Moreover, two approximate, analytical expressions are shown to be quite accurate in reproducing full simulations of Xrr. Finally, the same problem is encountered in other fields, belonging both to classical and to quantum physics. As such, our results lend themselves to being used within a context much wider than that exploited in the present study.