Large Time Limit Research Articles

Stationary time correlations for fermions after a quench in the presence of an impurity

We consider the quench dynamics of non-interacting fermions in one dimension in the presence of a finite-size impurity at the origin. This impurity is characterized by general momentum-dependent reflection and transmission coefficients which are changed from to at time t = 0. The initial state is at equilibrium with such that the system is cut in two independent halves with , , respectively, to the right and to the left of the impurity. We obtain the exact large time limit of the multi-time correlations. These correlations become time translationally invariant, and are non-zero in two different regimes: i) for where the system reaches a non-equilibrium steady state (NESS), ii) for , i.e., the ray regime. For a repulsive impurity these correlations are independent of , , while in the presence of bound states they oscillate and memory effects persist. We show that these nontrivial relaxational properties can be retrieved in a simple manner from the large time behaviour of the single particle wave functions.

Spectral form factor in the τ-scaling limit

We study the spectral form factor (SFF) of general topological gravity in the limit of large time and fixed temperature. It has been observed recently that in this limit, called the tau-scaling limit, the genus expansion of the SFF can be summed up and the late-time behavior of the SFF such as the ramp-plateau transition can be studied analytically. In this paper we develop a technique for the systematic computation of the higher order corrections to the SFF in the strict tau-scaling limit. We obtain the first five corrections in a closed form for the general background of topological gravity. As concrete examples, we present the results for the Airy case and Jackiw-Teitelboim gravity. We find that the above higher order corrections are the Fourier transforms of the corrections to the sine-kernel approximation of the Christoffel-Darboux kernel in the dual double-scaled matrix integral, which naturally explains their structure. Along the way we also develop a technique for the systematic computation of the corrections to the sine-kernel formula, which have not been fully explored in the literature before.

Establishing the accuracy of position-specific carbon isotope analysis of propane by GC-pyrolysis-GC-IRMS.

Position-specific (PS) δ13 C values of propane have proven their ability to provide valuable information on the evolution history of natural gases. Two major approaches to measure PS δ13 C values of propane are isotopic 13 C nuclear magnetic resonance (NMR) and gas chromatography-pyrolysis-gas chromatography-isotope ratio mass spectrometry (GC-Py-GC-IRMS). Measurement accuracy of the isotopic 13 C NMR has been verified, but the requirements of large sample size and long experimental time limit its applications. GC-Py-GC-IRMS is a more versatile method with a small sample size, but its accuracy has not been demonstrated. We measured the PS δ13 C values of propane from nine natural gases using both 13 C NMR and GC-Py-GC-IRMS, then evaluated the accuracy of the GC-Py-GC-IRMS method. The results show that large carbon isotope fractionations occurred for both terminal and central carbons within propane during pyrolysis. The isotope fractionations during the pyrolysis are reproducible at optimum conditions, but vary between the two GC-Py-GC-IRMS systems tested, affected by experimental conditions (e.g., pyrolysis temperature, flow rate, and reactor conditions). It is necessary to evaluate and calibrate each GC-Py-GC-IRMS system using propane gases with accurately determined PS δ13 C values. This study also highlights a need for PS isotope standards for propane and other molecules (e.g., butane and acetic acid).

Burgers’ equation in the complex plane

Burgers’ equation is a well-studied model in applied mathematics with connections to the Navier–Stokes equations in one spatial direction and traffic flow, for example. Following on from previous work, we analyse solutions to Burgers’ equation in the complex plane, concentrating on the dynamics of the complex singularities and their relationship to the solution on the real line. For an initial condition with a simple pole in each of the upper- and lower-half planes, we apply formal asymptotics in the small- and large-time limits in order to characterise the initial and later motion of the singularities. The small-time limit highlights how infinitely many singularities are born at t=0 and how they orientate themselves to lie increasingly close to anti-Stokes lines in the far field of the inner problem. This inner problem also reveals whether or not the closest singularity to the real axis moves toward the axis or away. For intermediate times, we use the exact solution, apply method of steepest descents, and implement the AAA approximation to track the complex singularities. Connections are made between the motion of the closest singularity to the real axis and the steepness of the solution on the real line. While Burgers’ equation is integrable (and has an exact solution), we deliberately apply a mix of techniques in our analysis in an attempt to develop methodology that can be applied to other nonlinear partial differential equations that do not.

Entanglement negativity in a fermionic chain with dissipative defects: exact results

We investigate the dynamics of the fermionic logarithmic negativity in a free-fermion chain with a localized loss, which acts as a dissipative impurity. The chain is initially prepared in a generic Fermi sea. In the standard hydrodynamic limit of large subsystems and long times, with their ratio fixed, the negativity between two subsystems is described by a simple formula, which depends only on the effective absorption coefficient of the impurity. The negativity grows linearly at short times, then saturating to a volume-law scaling. Physically, this reflects the continuous production with time of entangling pairs of excitations at the impurity site. Interestingly, the negativity is not the same as the Rényi mutual information with Rényi index , in contrast with the case of unitary dynamics. This reflects the interplay between dissipative and unitary processes. The negativity content of the entangling pairs is obtained in terms of an effective two-state mixed density matrix for the subsystems. Criticality in the initial Fermi sea is reflected in the presence of logarithmic corrections. The prefactor of the logarithmic scaling depends on the loss rate, suggesting a nontrivial interplay between dissipation and criticality.

Global existence and stabilization in a two-dimensional chemotaxis-Navier-Stokes system with consumption and production of chemosignals

We are concerned with the initial boundary value problems of chemotaxis-Navier-Stokes systems involving double chemosignals: one is an attractant consumed by the cells themselves and the other is an attractant or a repellent produced by the cells themselves. This work reveals that for any properly regular initial data, if the initial total cells mass is less than some threshold, then the corresponding attraction-atraction Navier-Stokes system admits a unique global classical solution, which, under extra two smallness assumptions on the initial data, is globally bounded and converges at exponential rate to the spatial equilibrium in the large time limit. As byproducts, the global classical solvability and stabilization of the Keller-Segel Navier-Stokes system and the uniform boundedness and large time behavior of the global classical solution of the associated attraction-repulsion Navier-Stokes system are simultaneously achieved.

Small-signal solutions to a nonlocal cross-diffusion model for interaction of scroungers with rapidly diffusing foragers

As a simplified version of a three-component taxis cascade model accounting for different migration strategies of two population groups in search of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional bounded convex domain with smooth boundary. For any given conveniently regular and biologically meaningful initial data, smallness conditions on the prescribed resource growth and on the initial nutrient signal concentration are identified which ensure the global existence of a global classical solution to the corresponding no-flux initial-boundary value problem. Moreover, under additional assumptions on the food production source these solutions are shown to be bounded, and to stabilize toward semi-trivial equilibria in the large time limit, respectively.

Duality relations in single-file diffusion

Single-file transport, which corresponds to the diffusion of particles that cannot overtake each other in narrow channels, is an important topic in out-of-equilibrium statistical physics. Various microscopic models of single-file systems have been considered, such as the simple exclusion process, which has reached the status of a paradigmatic model. Several different models of single-file diffusion have been shown to be related by a duality relation, which holds either microscopically or only in the hydrodynamic limit of large time and large distances. Here, we show that, within the framework of fluctuating hydrodynamics, these relations are not specific to these models and that, in the hydrodynamic limit, every single-file system can be mapped onto a dual single-file system, which we characterise. This general duality relation allows us to obtain new results for different models, by exploiting the solutions that are available for their dual model.

Qualitative behavior of solutions for a chemotaxis-haptotaxis system with gradient-dependent flux-limitation

This paper deals with the coupled chemotaxis-haptotaxis model with gradient-dependent flux-limitation where is a bounded domain with smooth boundary, χ, ξ and μ are positive parameters. Under the condition that then for any and all reasonably smooth initial data , the above system possesses a globally classical solution, which is uniformly bounded in time. Under the conditions that it asserts exponential decay of w in the large time limit, whereas both u and v persist in some certain sense.

Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the ‘aging effect’ as a hallmark of non-Markovianity where the discrete-time version of the ‘aging renewal process’ comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a t2-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (‘narrow’ with finite mean and variance) the walk exhibits normal diffusion and for ‘broad’ waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

Experimental verification of arcsine laws in mesoscopic nonequilibrium systems.

A large number of processes in the mesoscopic world occur out of equilibrium, where the time evolution of a system becomes immensely important since it is driven principally by dissipative effects. Nonequilibrium steady states (NESS) represent a crucial category in such systems, where relaxation timescales are comparable to the operational timescales. In this study, we employ a model NESS stochastic system, which is comprised of a colloidal microparticle optically trapped in a viscous fluid, externally driven by a temporally correlated noise, and show that time-integrated observables such as the entropic current, the work done on the system or the work dissipated by it, follow the three Lévy arcsine laws [A.C. Barato et al., Phys. Rev. Lett. 121, 090601 (2018)0031-900710.1103/PhysRevLett.121.090601], in the large time limit. We discover that cumulative distributions converge faster to arcsine distributions when it is near equilibrium and the rate of entropy production is small, because in that case the entropic current has weaker temporal autocorrelation. We study this phenomenon by changing the strength of the added noise as well as by perturbing our system with a flow field produced by a microbubble at close proximity to the trapped particle. We confirm our experimental findings with theoretical simulations of the systems. Our work provides an interesting insight into the NESS statistics of the meso-regime, where stochastic fluctuations play a pivotal role.

Quench dynamics of noninteracting fermions with a delta impurity

We study the out-of-equilibrium dynamics of noninteracting fermions in one dimension and in continuum space, in the presence of a delta impurity potential at the origin whose strength g is varied at time t = 0. The system is prepared in its ground state with g = g 0 = +∞, with two different densities and Fermi wave-vectors k L and k R on the two half-spaces x > 0 and x < 0 respectively. It then evolves for t > 0 as an isolated system, with a finite impurity strength g. We compute exactly the time dependent density and current. For a fixed position x and in the large time limit t → ∞, the system reaches a non-equilibrium stationary state (NESS). We obtain analytically the correlation kernel, density, particle current, and energy current in the NESS, and characterize their relaxation, which is algebraic in time. In particular, in the NESS, we show that, away from the impurity, the particle density displays oscillations which are the non-equilibrium analog of the Friedel oscillations. In the regime of ‘rays’, x/t = ξ fixed with x, t → ∞, we compute the same quantities and observe the emergence of two light cones, associated to the Fermi velocities k L and k R in the initial state. Interestingly, we find non trivial quantum correlations between two opposite rays with velocities ξ and −ξ which we compute explicitly. We extend to a continuum setting and to a correlated initial state the analytical methods developed in a recent work of Ljubotina, Sotiriadis and Prosen, in the context of a discrete fermionic chain with an impurity. We also generalize our results to an initial state at finite temperature, recovering, via explicit calculations, some predictions of conformal field theory in the low energy limit.

Domains of attraction of invariant distributions of the infinite Atlas model

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model does not have a unique stationary distribution and in fact for every a≥0, πa:=⨂ i=1∞Exp(2+ia) is a stationary measure for the gap process. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure πa if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to πa weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of πa for each a≥0. The cases a=0 and a>0 are qualitatively different as is seen from the analysis and the sufficient conditions that we provide. Proofs are based on the analysis of synchronous couplings, namely, couplings of the ranked particle systems started from different initial configurations, but driven using the same set of Brownian motions.

Point fields of last passage percolation and coalescing fractional Brownian motions.

We consider large-scale point fields which naturally appear in the context of the Kardar-Parisi-Zhang (KPZ) phenomenon. Such point fields are geometrical objects formed by points of mass concentration, and by shocks separating the sources of these points. We introduce similarly defined point fields for processes of coalescing fractional Brownian motions (cfBMs). The case of the Hurst index 2/3 is of particular interest for us since, in this case, the power law of the density decay is the same as that in the KPZ phenomenon. In this paper, we present strong numerical evidence that statistical properties of points fields in these two different settings are very similar. We also discuss theoretical arguments in support of the conjecture that they are exactly the same in the large-time limit. This would indicate that two objects may, in fact, belong to the same universality class.

Out-of-equilibrium dynamics arising from slow round-trip variations of Hamiltonian parameters across quantum and classical critical points

We address the out-of-equilibrium dynamics of many-body systems subject to slow time-dependent round-trip protocols across quantum and classical (thermal) phase transitions. We consider protocols where one relevant parameter w is slowly changed across its critical point wc = 0, linearly in time with a large time scale ts, from wi < 0 to wf > 0 and then back to wi < 0, thus entailing multiple passages through the critical point. Analogously to the one-way Kibble-Zurek protocols across a critical point, round-trip protocols develop dynamic scaling behaviors at both classical and quantum transitions, put forward within renormalization-group frameworks. The scaling scenario is analyzed within some paradigmatic models undergoing quantum and classical transitions belonging to the two-dimensional Ising universality class, such as one-dimensional quantum Ising models and fermionic wires, and two-dimensional classical Ising models (supplemented with a purely relaxational dynamics). While the dynamic scaling frameworks are similar for classical and quantum systems, substantial differences emerge due to the different nature of their dynamics, which is purely relaxational for classical systems (implying thermalization in the large-time limit at fixed model parameters), and unitary in the case of quantum systems. In particular, when the critical point separates two gapped (short-ranged) phases and the extreme value wf > 0 is kept fixed in the large-ts limit of the round-trip protocol, we observe hysteresis-like scenarios in classical systems, while quantum systems do not apparently develop a sufficiently robust scaling limit along the return way, due to the presence of rapidly oscillating relative phases among the relevant quantum states.

First-passage Brownian functionals with stochastic resetting

We study the statistical properties of first-passage time functionals of a one dimensional Brownian motion in the presence of stochastic resetting. A first-passage functional is defined as where t f is the first-passage time of a reset Brownian process x(τ), i.e., the first time the process crosses zero. In here, the particle is reset to x R > 0 at a constant rate r starting from x 0 > 0 and we focus on the following functionals: (i) local time , (ii) residence time , and (iii) functionals of the form with n > −2. For first two functionals, we analytically derive the exact expressions for the moments and distributions. Interestingly, the residence time moments reach minima at some optimal resetting rates. A similar phenomena is also observed for the moments of the functional A n . Finally, we show that the distribution of A n for large A n decays exponentially as for all values of n and the corresponding decay length a n is also estimated. In particular, exact distribution for the first passage time under resetting (which corresponds to the n = 0 case) is derived and shown to be exponential at large time limit in accordance with the generic observation. This behavioural drift from the underlying process can be understood as a ramification due to the resetting mechanism which curtails the undesired long Brownian first passage trajectories and leads to an accelerated completion. We confirm our results to high precision by numerical simulations.

Critical quantum metrology with fully-connected models: from Heisenberg to Kibble–Zurek scaling

Phase transitions represent a compelling tool for classical and quantum sensing applications. It has been demonstrated that quantum sensors can in principle saturate the Heisenberg scaling, the ultimate precision bound allowed by quantum mechanics, in the limit of large probe number and long measurement time. Due to the critical slowing down, the protocol duration time is of utmost relevance in critical quantum metrology. However, how the long-time limit is reached remains in general an open question. So far, only two dichotomic approaches have been considered, based on either static or dynamical properties of critical quantum systems. Here, we provide a comprehensive analysis of the scaling of the quantum Fisher information for different families of protocols that create a continuous connection between static and dynamical approaches. In particular, we consider fully-connected models, a broad class of quantum critical systems of high experimental relevance. Our analysis unveils the existence of universal precision-scaling regimes. These regimes remain valid even for finite-time protocols and finite-size systems. We also frame these results in a general theoretical perspective, by deriving a precision bound for arbitrary time-dependent quadratic Hamiltonians.

Dynamics of charge-imbalance-resolved entanglement negativity after a quench in a free-fermion model

The presence of a global internal symmetry in a quantum many-body system is reflected in the fact that the entanglement between its subparts is endowed with an internal structure, namely it can be decomposed as a sum of contributions associated to each symmetry sector. The symmetry resolution of entanglement measures provides a formidable tool to probe the out-of-equilibrium dynamics of quantum systems. Here, we study the time evolution of charge-imbalance-resolved negativity after a global quench in the context of free-fermion systems, complementing former works for the symmetry-resolved entanglement entropy. We find that the charge-imbalance-resolved logarithmic negativity shows an effective equipartition in the scaling limit of large times and system size, with a perfect equipartition for early and infinite times. We also derive and conjecture a formula for the dynamics of the charged Rényi logarithmic negativities. We argue that our results can be understood in the framework of the quasiparticle picture for the entanglement dynamics, and provide a conjecture that we expect to be valid for generic integrable models.

Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar

A no-flux initial-boundary value problem for the doubly degenrate parabolic system ut=∇·(uv∇u)+ℓuv,vt=Δv-uv,(⋆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} u_t = \ abla \\cdot \\big ( uv\ abla u\\big ) + \\ell uv, \\\\ v_t = \\Delta v - uv, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\\end{document}is considered in a smoothly bounded convex domain Omega subset mathbb {R}^n, with nge 1 and ell ge 0. The first of the main results asserts that for nonnegative initial data (u_0,v_0)in (L^infty (Omega ))^2 with u_0not equiv 0, v_0not equiv 0 and sqrt{v_0}in W^{1,2}(Omega ), there exists a global weak solution (u, v) which, inter alia, belongs to C^0(overline{Omega }times (0,infty )) times C^{2,1}(overline{Omega }times (0,infty )) and satisfies sup _{t>0} Vert u(cdot ,t)Vert _{L^p(Omega )}<infty for all pin [1,p_0) with p_0:=frac{n}{(n-2)_+}. It is next seen that for each of these solutions one can find u_infty in bigcap _{pin [1,p_0)} L^p(Omega ) such that, within an appropriate topological setting, (u(cdot ,t),v(cdot ,t)) approaches the equilibrium (u_infty ,0) in the large time limit. Finally, in the case nle 5 a result ensuring a certain stability property of any member in the uncountably large family of steady states (u_0,0), with arbitrary and suitably regular u_0:Omega rightarrow [0,infty ), is derived. This provides some rigorous evidence for the appropriateness of (star ) to model the emergence of a strikingly large variety of stable structures observed in experiments on bacterial motion in nutrient-poor environments. Essential parts of the analysis rely on the use of an apparently novel class of functional inequalities to suitably cope with the doubly degenerate diffusion mechanism in (star ).

The case of the biased quenched trap model in two dimensions with diverging mean dwell times

We investigate the biased quenched trap model on top of a two-dimensional lattice in the case of diverging expected dwell times. By utilizing the double-subordination approach and calculating the return probability in 2D, we explicitly obtain the disorder averaged probability density function of the particle’s position as a function of time (for any given bias) in the limit of large times (t → ∞). The first and second moments are calculated, and a formula for a general μth moment is found. The behavior of the first moment, i.e. ⟨x(t)⟩, presents non-linear response both in time and in the applied external force F 0. While the non-linearity in time occurs for any measurement time t, the non-linearity in F 0 is expected only when where α = T/T g, for temperatures T < T g. We support our analytic results by comparison to numerical simulations.