Large Time Limit Research Articles

Dynamics of nonisothermal two-thin-fluid-layer systems subjected to harmonic tangential forcing under Rayleigh–Taylor instability conditions

The stability of a nonisothermal system consisting of two superimposed fluid layers: a thin liquid film layer and a gas layer sandwiched between differentially heated horizontal solid plates in the gravity field, is investigated. The system is assumed to be subjected to the Rayleigh–Taylor instability (RTI) with the Marangoni effect that either enhances the RTI or opposes it and to the tangential harmonic vibration of the upper substrate. A set of reduced evolution equations is derived based on the weighted-residual integral boundary layer approach, and the investigation is carried out in the framework of this set. The base state of the system represents a time-periodic flow, and its linear stability analysis is carried out using the Floquet theory in the large-time limit. The nonlinear dynamics of the system is investigated numerically in the case of either a static or vibrating substrate. Among the possible outcomes of the nonlinear dynamics, there is the emergence of ruptured states of the liquid film with rupture taking place at either the upper or lower substrate and also the emergence of saturated continuous flows of the liquid film. We also find that the nonlinear dynamics of the system is consistent with the results of the linear stability analysis in terms of enhancement or attenuation of interfacial distortion.

The confined primal integral: a measure to benchmark heuristic MINLP solvers against global MINLP solvers

It is a challenging task to fairly compare local solvers and heuristics against each other and against global solvers. How does one weigh a faster termination time against a better quality of the found solution? In this paper, we introduce the confined primal integral, a new performance measure that rewards a balance of speed and solution quality. It emphasizes the early part of the solution process by using an exponential decay. Thereby, it avoids that the order of solvers can be inverted by choosing an arbitrarily large time limit. We provide a closed analytic formula to compute the confined primal integral a posteriori and an incremental update formula to compute it during the run of an algorithm. For the latter, we show that we can drop one of the main assumptions of the primal integral, namely the knowledge of a fixed reference solution to compare against. Furthermore, we prove that the confined primal integral is a transitive measure when comparing local solves with different final solution values. Finally, we present a computational experiment where we compare a local MINLP solver that uses certain classes of cutting planes against a solver that does not. Both versions show very different tendencies w.r.t. average running time and solution quality, and we use the confined primal integral to argue which of the two is the preferred setting.

Dampening effect of logistic source in a two-dimensional haptotaxis system with nonlinear zero-order interaction

This paper deals with the oncolytic virotherapy model{ut=Δu−∇⋅(u∇v)−uz+μu(1−u),vt=−(u+w)v,wt=DwΔw−w+uz,zt=DzΔz−z−uz+βw,(⋆) in a bounded domain Ω⊂R2 with smooth boundary, where μ,Dw,Dz and β are prescribed positive parameters. For any given suitably regular initial data, the global existence of classical solution to the corresponding homogeneous Neumann initial-boundary problem for a more general model allowing μ=0 was previously verified in Tao and Winkler (2020) [15]. This work further shows that whenever μ>0, the above-mentioned global classical solution to (⋆) is uniformly bounded; and moreover, if β<1, then the solution (u,v,w,z) stabilizes to the constant equilibrium (1,0,0,0) in the topology Lp(Ω)×(L∞(Ω))3 with any p>1 in a large time limit.

Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

We revisit the Lieb–Liniger model for n bosons in one dimension with attractive delta interaction in a half-space with diagonal boundary conditions. This model is integrable for the arbitrary value of , the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as b is decreased from the hard-wall case , with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar–Parisi–Zhang equation for the growth of a one-dimensional interface of height , on the half-space with boundary condition and droplet initial condition at the wall. We obtain explicit expressions, valid at all time t and arbitrary b, for the integer exponential (one-point) moments of the KPZ height field . From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for where the PDF is given by the GSE Tracy–Widom distribution (ii) bound for , where the PDF is a Gaussian. At the critical point , the PDF is given by the GOE Tracy–Widom distribution.

Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread

In this paper, we consider the following system{ut=Δu−χ∇⋅(u∇ω),vt=dvΔv−ξ∇⋅(v∇ω),mt=dmΔm+u−m,ωt=−(γ1u+m)ω, in two dimensional space with zero-flux boundary conditions. This model was proposed by Franssen et al. [7] to characterize the invasion and metastatic spread of cancer cells. We first establish the global existence of uniformly bounded global strong solutions. Then using the decay of ECM and the positivity of MDE, we further improve the regularity of obtained solutions, and achieve the uniform boundedness of solutions in the classical sense. Subsequently, we also prove the uniqueness of solutions. After that, we turn our attention to the large time behavior of solutions, and show that the global classical solution strongly converges to a semi-trivial steady state in the large time limit.

Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains

We consider an integrable system of two one-dimensional fermionic chains connected by a link. The hopping constant at the link can be different from that in the bulk. Starting from an initial state in which the left chain is populated while the right is empty, we present time-dependent full counting statistics and the Loschmidt echo in terms of Fredholm determinants. Using this exact representation, we compute the above quantities as well as the current through the link, the shot noise and the entanglement entropy in the large time limit. We find that the physics is strongly affected by the value of the hopping constant at the link. If it is smaller than the hopping constant in the bulk, then a local steady state is established at the link, while in the opposite case all physical quantities studied experience persistent oscillations. In the latter case the frequency of the oscillations is determined by the energy of the bound state and, for the Loschmidt echo, by the bias of chemical potentials.

The Linear Boltzmann Equation in Column Experiments of Porous Media

The use of the linear Boltzmann equation is proposed for transport in porous media in a column. By column experiments, we show that the breakthrough curve is reproduced by the linear Boltzmann equation. The advection-diffusion equation is derived from the linear Boltzmann equation in the asymptotic limit of large propagation distance and long time.

Totally asymmetric simple exclusion process with resetting

We study the one-dimensional totally asymmetric simple exclusion process (TASEP) with open boundaries having the additional dynamical feature of stochastic resetting to the initial, empty state. The system evolves according to the TASEP dynamics with particles entering the input side with rate and leaving the other side with rate . The system is brought back to its initial state of an empty lattice at random intervals . These intervals are drawn from probability distributions, for which we consider two possibilities—a power law with , and an exponential distribution . We use approximate expressions for the time evolution of density on the lattice for a normal TASEP to calculate the reset-averaged density as a function of time. We find that in the limit of large time, the system achieves a steady state when , while for we see a time-dependent scaling function. The large time behaviour of the density distribution shows a power law decay at the input boundary in all the phases while it shows a non-monotonic behaviour in the high-density phase of the TASEP. One sees this monotonic behaviour also for the exponential resetting, with the system always achieving a steady state in the limit of long times. We also perform numerical simulations, results from which show good agreement with our analytic expressions.

The transition rates of an isotropic quantum charged oscillator in the presence of external fields

In this work, explicit expressions for the transition rates of an isotropic quantum charged harmonic oscillator in the vicinity of a perfectly conducting half-space under the influence of an external classical source are obtained. In the absence of external sources, it is shown that the decay rate of an initially exited state of the oscillator is a periodic function in terms of the normalized distance to the plate. The modified transition rates in the presence of external classical sources are obtained in the large-time limit indicating a contribution proportional to the squared module of the Fourier transform of the external source. In the absence of the conducting plate and external sources, the results are in agreement with the free space case. The problem is generalized to the case of a real conducting half-space.

Large deviations for sticky Brownian motions

We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.

Small-Mass Solutions in the Two-Dimensional Keller--Segel System Coupled to the Navier--Stokes Equations

The fully parabolic Keller--Segel system is coupled to the incompressible Navier--Stokes equations through transport and buoyancy. It is shown that when posed with no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded planar domains and along with appropriate assumptions on regularity of the initial data, under a smallness condition exclusively involving the total initial population mass $m$ an associated initial-boundary value problem admits a globally defined generalized solution; in particular, this hypothesis is fully explicit and independent of the initial size of further solution components. Moreover, the obtained solution is seen to enjoy a certain temporally averaged boundedness property which, inter alia, rules out any finite-time collapse into persistent Dirac-type measures, as well as convergence to such singular profiles in the large time limit.

Heat Conservation and Fluctuations Between Quantum Reservoirs in the Two-Time Measurement Picture

This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans–Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in Andrieux et al. (in New J Phys 11(4):043014, 2009). We particularly fix some proofs of the latter article where the ultraviolet condition was not mentioned. We detail how these two symmetries lead respectively to fluctuation relations and a statistical refinement of heat conservation for isolated thermodynamic quantum systems. As in Andrieux et al. (in New J Phys 11(4):043014, 2009), we recover the fluctuation–dissipation theorem in the linear response theory, short of Green–Kubo relations. We illustrate the general theory on a number of canonical models.

Clustering of floating tracers in weakly divergent velocity fields.

This work deals with buoyant tracers floating at the ocean surface, where the geostrophic velocity component is two dimensional and rotational (nondivergent) and the ageostrophic component can contain rotational and potential (divergent) contributions that are comparable in size. We consider a random kinematic flow model and study the process of clustering, that is, aggregation of the floating tracer in localized spatial patches. In the long-time limit and in the cases of strongly and weakly divergent flows, the existing analytical theory predicts the process of exponential clustering, which is the emergence of spatial singularities containing all the available tracer. Here we confirm this analytical prediction, in numerical model solutions spanning different combinations of rotational and potential surface velocity components, and report that exponential clustering persists even in weakly divergent flows, however at significantly slower rates. For a wide range of parameters, we analyze not only the exponential clustering but also the other type of tracer aggregation, referred to as fragmentation clustering, as well as the coarse-graining effects on clustering. For the presented analysis we consider ensembles of Lagrangian particles and introduce and apply the statistical topography methodology.

Scaling behavior of the stationary states arising from dissipation at continuous quantum transitions

We study the critical behavior of the nonequilibrium dynamics and of the steady states emerging from the competition between coherent and dissipative dynamics close to quantum phase transitions. The latter is induced by the coupling of the system with a Markovian bath, such that the evolution of the system's density matrix can be effectively described by a Lindblad master equation. We devise general scaling behaviors for the out-of-equilibrium evolution and the stationary states emerging in the large-time limit for generic initial conditions in terms of the parameters of the Hamiltonian providing the coherent driving and those associated with the dissipative interactions with the environment. Our framework is supported by numerical results for the dynamics of a one-dimensional lattice fermion gas undergoing a quantum Ising transition in the presence of dissipative mechanisms which include local pumping and decay of particles.

Ensemble and effective dispersion in three-dimensional isotropic fractal media

We determine the time-dependent behavior of the dispersion coefficient for transport in formations with isotropic log-conductivity fields showing fractal behavior.We consider two different dispersion coefficients for point-like injection: (1) the ensemble dispersion coefficients, defined as half the rate of change of the second central moments of the ensemble-averaged concentration distribution and (2) the effective dispersion, which is half the rate of change of the expected second central moments. Our results show, that the two longitudinal macrodispersion coefficients steadily grow with time and remain different at all times in a fully fractal regime, indicating that no Fickian transport regime is ever reached. The resulting effective longitudinal transport model is consequently a fractional advection–dispersion equation. In the semi-fractal regime, a Gaussian transport regime is reached eventually. However, compared to the case of a classic non-fractal regime, the transient non-Gaussian regime lasts much longer. In the transverse direction, the two dispersion coefficients approach the same large-time limit also in fractal media highlighting the fundamental difference between longitudinal and transverse dispersion.

Coupled Kähler–Ricci solitons on toric Fano manifolds

We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled K\"ahler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for existence of torus-invariant solutions to a system of soliton type equations on toric Fano manifolds. Some of these solutions provide natural candidates for the large time limits of a certain geometric flow generalizing the K\"ahler-Ricci flow.

A Detailed Fluctuation Theorem for Heat Fluxes in Harmonic Networks Out of Thermal Equilibrium

We continue the investigation, started in [J. Stat. Phys. 166, 926-1015 (2017)], of a network of harmonic oscillators driven out of thermal equilibrium by heat reservoirs. We study the statistics of the fluctuations of the heat fluxes flowing between the network and the reservoirs in the nonequilibrium steady state and in the large time limit. We prove a large deviation principle for these fluctuations and derive the fluctuation relation satisfied by the associated rate function.

Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food

This work deals with a taxis cascade model for food consumption in two populations, namely foragers directly orienting their movement upward the gradients of food concentration and exploiters taking a parasitic strategy in search of food via tracking higher forager densities. As a consequence, the dynamics of both populations are adapted to the space distribution of food which is dynamically modified in time and space by the two populations. This model extends the classical one-species chemotaxis-consumption systems by additionally accounting for a second taxis mechanism coupled to the first in a consecutive manner. It is rigorously proved that for all suitably regular initial data, an associated Neumann-type initial-boundary value problem for the spatially one-dimensional version of this model possesses a globally defined bounded classical solution. Moreover, it is asserted that the considered two populations will approach spatially homogeneous distributions in the large time limit, provided that either the total population number of foragers or that of exploiters is appropriately small.

Finite-size and finite-time effects in large deviation functions near dynamical symmetry breaking transitions

We introduce and study a class of particle hopping models consisting of a single box coupled to a pair of reservoirs. Despite being zero-dimensional, in the limit of large particle number and long observation time, the current and activity large deviation functions of the models can exhibit symmetry-breaking dynamical phase transitions. We characterize exactly the critical properties of these transitions, showing them to be direct analogues of previously studied phase transitions in extended systems. The simplicity of the model allows us to study features of dynamical phase transitions which are not readily accessible for extended systems. In particular, we quantify finite-size and finite-time scaling exponents using both numerical and theoretical arguments. Importantly, we identify an analogue of critical slowing near symmetry breaking transitions and suggest how this can be used in the numerical studies of large deviations. All of our results are also expected to hold for extended systems.

Quantum information scrambling after a quantum quench

How quantum information is scrambled in the global degrees of freedom of nonequilibrium many-body systems is a key question to understand local thermalization. A consequence of scrambling is that in the scaling limit the mutual information between two intervals vanishes at all times, i.e., it does not exhibit a peak at intermediate times. Here we investigate the mutual information scrambling after a quantum quench in both integrable and nonintegrable one-dimensional systems. We study the mutual information between two intervals of finite length as a function of their distance. In integrable systems, the mutual information exhibits an algebraic decay with the distance between the intervals, signaling weak scrambling. This behavior may be qualitatively understood within the quasiparticle picture for the entanglement spreading. In the scaling limit of large intervals, times, and distances between the intervals, with their ratios fixed, this predicts a decay exponent equal to $1/2$. Away from the scaling limit, the power-law behavior persists, but with a larger (and model-dependent) exponent. For nonintegrable models, a much faster decay is observed, which can be attributed to the finite lifetime of the quasiparticles: unsurprisingly, nonintegrable models are better scramblers.