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1.
Phys Rev Lett ; 128(13): 130602, 2022 Apr 01.
Artigo em Inglês | MEDLINE | ID: mdl-35426706

RESUMO

We determine the full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti lattice gas model at long times by uncovering and exploiting complete integrability of the underlying equations of the macroscopic fluctuation theory. These equations are closely related to the derivative nonlinear Schrödinger equation (DNLS), and we solve them by the Zakharov-Shabat inverse scattering method (ISM) adapted by D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 798 (1978)JMAPAQ0022-248810.1063/1.523737 for the DNLS. We obtain explicit results for the exact large deviation function of the transferred heat for an initially localized heat pulse, where we uncover a nontrivial symmetry relation.

2.
Phys Rev E ; 105(3-1): 034106, 2022 Mar.
Artigo em Inglês | MEDLINE | ID: mdl-35428118

RESUMO

We evaluate the mean escape time of overdamped particles over potential barriers in short-correlated quenched Gaussian disorder potentials in one dimension at low temperature. The thermally activated escape is very sensitive to the form of the tail of the potential barrier probability distribution. We evaluate this tail by using the optimal fluctuation method. For monotone decreasing autocovariances, we reproduce the tail obtained by Lopatin and Vinokur (2001). However, for nonmonotonic autocovariances of the disorder potential which exhibit negative autocorrelations, we show that the tail is higher. This leads to an exponential increase of the mean escape time. The transition between the two regimes has the character of a first-order transition.

3.
Phys Rev E ; 104(5-1): 054131, 2021 Nov.
Artigo em Inglês | MEDLINE | ID: mdl-34942750

RESUMO

The "Brownian bees" model describes an ensemble of N independent branching Brownian particles. When a particle branches into two particles, the particle farthest from the origin is eliminated so as to keep the number of particles constant. In the limit of N→∞, the spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation. At long times the particle density approaches a spherically symmetric steady-state solution with a compact support. Here, we study fluctuations of the "swarm of bees" due to the random character of the branching Brownian motion in the limit of large but finite N. We consider a one-dimensional setting and focus on two fluctuating quantities: the swarm center of mass X(t) and the swarm radius ℓ(t). Linearizing a pertinent Langevin equation around the deterministic steady-state solution, we calculate the two-time covariances of X(t) and ℓ(t). The variance of X(t) directly follows from the covariance of X(t), and it scales as 1/N as to be expected from the law of large numbers. The variance of ℓ(t) behaves differently: It exhibits an anomalous scaling (1/N)lnN. This anomaly appears because all spatial scales, including a narrow region near the edges of the swarm where only a few particles are present, give a significant contribution to the variance. We argue that the variance of ℓ(t) can be obtained from the covariance of ℓ(t) by introducing a cutoff at the microscopic time 1/N where the continuum Langevin description breaks down. Our theoretical predictions are in good agreement with Monte Carlo simulations of the microscopic model. Generalizations to higher dimensions are briefly discussed.

4.
Phys Rev E ; 104(5-1): 054125, 2021 Nov.
Artigo em Inglês | MEDLINE | ID: mdl-34942795

RESUMO

Consider the short-time probability distribution P(H,t) of the one-point interface height difference h(x=0,τ=t)-h(x=0,τ=0)=H of the stationary interface h(x,τ) described by the Kardar-Parisi-Zhang (KPZ) equation. It was previously shown that the optimal path, the most probable history of the interface h(x,τ) which dominates the upper tail of P(H,t), is described by any of two ramplike structures of h(x,τ) traveling either to the left, or to the right. These two solutions emerge, at a critical value of H, via a spontaneous breaking of the mirror symmetry x↔-x of the optimal path, and this symmetry breaking is responsible for a second-order dynamical phase transition in the system. We simulate the interface configurations numerically by employing a large-deviation Monte Carlo sampling algorithm in conjunction with the mapping between the KPZ interface and the directed polymer in a random potential at high temperature. This allows us to observe the optimal paths, which determine each of the two tails of P(H,t), down to probability densities as small as 10^{-500}. At short times we observe mirror-symmetry-broken traveling optimal paths for the upper tail, and a single mirror-symmetric path for the lower tail, in good quantitative agreement with analytical predictions. At long times, even at moderate values of H, where the optimal fluctuation method is not supposed to apply, we still observe two well-defined dominating paths. Each of them violates the mirror symmetry x↔-x and is a mirror image of the other.

5.
Phys Rev E ; 103(3-1): 032140, 2021 Mar.
Artigo em Inglês | MEDLINE | ID: mdl-33862785

RESUMO

The "Brownian bees" model describes a system of N-independent branching Brownian particles. At each branching event the particle farthest from the origin is removed so that the number of particles remains constant at all times. Berestycki et al. [arXiv:2006.06486] proved that at N→∞ the coarse-grained spatial density of this particle system lives in a spherically symmetric domain and is described by the solution of a free boundary problem for a deterministic reaction-diffusion equation. Furthermore, they showed [arXiv:2005.09384] that, at long times, this solution approaches a unique spherically symmetric steady state with compact support: a sphere whose radius ℓ_{0} depends on the spatial dimension d. Here we study fluctuations in this system in the limit of large N due to the stochastic character of the branching Brownian motion, and we focus on persistent fluctuations of the swarm size. We evaluate the probability density P(ℓ,N,T) that the maximum distance of a particle from the origin remains smaller than a specified value ℓ<ℓ_{0} or larger than a specified value ℓ>ℓ_{0} on a time interval 0

6.
Phys Rev E ; 102(2-1): 022137, 2020 Aug.
Artigo em Inglês | MEDLINE | ID: mdl-32942446

RESUMO

The empirical velocity of a reaction-diffusion front, propagating into an unstable state, fluctuates because of the shot noises of the reactions and diffusion. Under certain conditions these fluctuations can be described as a diffusion process in the reference frame moving with the average velocity of the front. Here we address pushed fronts, where the front velocity in the deterministic limit is affected by higher-order reactions and is therefore larger than the linear spread velocity. For a subclass of these fronts-strongly pushed fronts-the effective diffusion constant D_{f}∼1/N of the front can be calculated, in the leading order, via a perturbation theory in 1/N≪1, where N≫1 is the typical number of particles in the transition region. This perturbation theory, however, overestimates the contribution of a few fast particles in the leading edge of the front. We suggest a more consistent calculation by introducing a spatial integration cutoff at a distance beyond which the average number of particles is of order 1. This leads to a nonperturbative correction to D_{f} which even becomes dominant close to the transition point between the strongly and weakly pushed fronts. At the transition point we obtain a logarithmic correction to the 1/N scaling of D_{f}. We also uncover another, and quite surprising, effect of the fast particles in the leading edge of the front. Because of these particles, the position fluctuations of the front can be described as a diffusion process only on very long time intervals with a duration Δt≫τ_{N}, where τ_{N} scales as N. At intermediate times the position fluctuations of the front are anomalously large and nondiffusive. Our extensive Monte Carlo simulations of a particular reacting lattice gas model support these conclusions.

7.
Phys Rev E ; 102(2-1): 022113, 2020 Aug.
Artigo em Inglês | MEDLINE | ID: mdl-32942466

RESUMO

We study the position distribution of a single active Brownian particle (ABP) on the plane. We show that this distribution has a compact support, the boundary of which is an expanding circle. We focus on a short-time regime and employ the optimal fluctuation method to study large deviations of the particle position coordinates x and y. We determine the optimal paths of the ABP, conditioned on reaching specified values of x and y, and the large deviation functions of the marginal distributions of x and of y. These marginal distributions match continuously with "near tails" of the x and y distributions of typical fluctuations, studied earlier. We also calculate the large deviation function of the joint x and y distribution P(x,y,t) in a vicinity of a special "zero-noise" point, and show that lnP(x,y,t) has a nontrivial self-similar structure as a function of x, y, and t. The joint distribution vanishes extremely fast at the expanding circle, exhibiting an essential singularity there. This singularity is inherited by the marginal x- and y-distributions. We argue that this fingerprint of the short-time dynamics remains there at all times.

8.
Phys Rev E ; 100(4-1): 042135, 2019 Oct.
Artigo em Inglês | MEDLINE | ID: mdl-31771031

RESUMO

Employing the optimal fluctuation method, we study the large deviation function of long-time averages (1/T)∫_{-T/2}^{T/2}x^{n}(t)dt,n=1,2,⋯, of centered stationary Gaussian processes. These processes are correlated and, in general, non-Markovian. We show that the anomalous scaling with time of the large-deviation function, recently observed for n>2 for the particular case of the Ornstein-Uhlenbeck process, holds for a whole class of stationary Gaussian processes.

9.
Phys Rev E ; 99(5-1): 052102, 2019 May.
Artigo em Inglês | MEDLINE | ID: mdl-31212513

RESUMO

The time that a diffusing particle spends in a certain region of space is known as the occupation time, or the residence time. Recently, the joint occupation-time statistics of an ensemble of noninteracting particles was addressed using the single-particle statistics. Here we employ the macroscopic fluctuation theory (MFT) to study the occupation-time statistics of many interacting particles. We find that interactions can significantly change the statistics and, in some models, even cause a singularity of the large-deviation function describing these statistics. This singularity can be interpreted as a dynamical phase transition. We also point out to a close relation between the MFT description of the occupation-time statistics of noninteracting particles and the level 2 large deviation formalism which describes the occupation-time statistics of a single particle.

10.
Phys Rev E ; 99(5-1): 052105, 2019 May.
Artigo em Inglês | MEDLINE | ID: mdl-31212556

RESUMO

We study large deviations of the time-averaged size of stochastic populations described by a continuous-time Markov jump process. When the expected population size N in the steady state is large, the large deviation function (LDF) of the time-averaged population size can be evaluated by using a Wentzel-Kramers-Brillouin (WKB) method, applied directly to the master equation for the Markov process. For a class of models that we identify, the direct WKB method predicts a giant disparity between the probabilities of observing an unusually small and an unusually large values of the time-averaged population size. The disparity results from a qualitative change in the "optimal" trajectory of the underlying classical mechanics problem. The direct WKB method also predicts, in the limit of N→∞, a singularity of the LDF, which can be interpreted as a second-order dynamical phase transition. The transition is smoothed at finite N, but the giant disparity remains. The smoothing effect is captured by the van-Kampen system size expansion of the exact master equation near the attracting fixed point of the underlying deterministic model. We describe the giant disparity at finite N by developing a different variant of WKB method, which is applied in conjunction with the Donsker-Varadhan large-deviation formalism and involves subleading-order calculations in 1/N.

11.
Phys Rev E ; 99(4-1): 042132, 2019 Apr.
Artigo em Inglês | MEDLINE | ID: mdl-31108640

RESUMO

We consider a stochastic interface h(x,t), described by the 1+1 Kardar-Parisi-Zhang (KPZ) equation on the half line x≥0 with the reflecting boundary at x=0. The interface is initially flat, h(x,t=0)=0. We focus on the short-time probability distribution P(H,L,t) of the height H of the interface at point x=L. Using the optimal fluctuation method, we determine the (Gaussian) body of the distribution and the strongly asymmetric non-Gaussian tails. We find that the slower-decaying tail scales as -sqrt[t]lnP≃|H|^{3/2}f_{-}(L/sqrt[|H|t]) and calculate the function f_{-} analytically. Remarkably, this tail exhibits a first-order dynamical phase transition at a critical value of L, L_{c}=0.60223⋯sqrt[|H|t]. The transition results from a competition between two different fluctuation paths of the system. The faster-decaying tail scales as -sqrt[t]lnP≃|H|^{5/2}f_{+}(L/sqrt[|H|t]). We evaluate the function f_{+} using a specially developed numerical method which involves solving a nonlinear second-order elliptic equation in Lagrangian coordinates. The faster-decaying tail also involves a sharp transition which occurs at a critical value L_{c}≃2sqrt[2|H|t]/π. This transition is similar to the one recently found for the KPZ equation on a ring, and we believe that it has the same fractional order, 5/2. It is smoothed, however, by small diffusion effects.

13.
Phys Rev E ; 97(5-1): 052110, 2018 May.
Artigo em Inglês | MEDLINE | ID: mdl-29906837

RESUMO

We determine the exact short-time distribution -lnP_{f}(H,t)=S_{f}(H)/sqrt[t] of the one-point height H=h(x=0,t) of an evolving 1+1 Kardar-Parisi-Zhang (KPZ) interface for flat initial condition. This is achieved by combining (i) the optimal fluctuation method, (ii) a time-reversal symmetry of the KPZ equation in 1+1 dimension, and (iii) the recently determined exact short-time height distribution -lnP_{st}(H,t)=S_{st}(H)/sqrt[t] for stationary initial condition. In studying the large-deviation function S_{st}(H) of the latter, one encounters two branches: an analytic and a nonanalytic. The analytic branch is nonphysical beyond a critical value of H where a second-order dynamical phase transition occurs. Here we show that, remarkably, it is the analytic branch of S_{st}(H) which determines the large-deviation function S_{f}(H) of the flat interface via a simple mapping S_{f}(H)=2^{-3/2}S_{st}(2H).

14.
Phys Rev E ; 97(4-1): 042130, 2018 Apr.
Artigo em Inglês | MEDLINE | ID: mdl-29758703

RESUMO

We study the short-time distribution P(H,L,t) of the two-point two-time height difference H=h(L,t)-h(0,0) of a stationary Kardar-Parisi-Zhang interface in 1+1 dimension. Employing the optimal-fluctuation method, we develop an effective Landau theory for the second-order dynamical phase transition found previously for L=0 at a critical value H=H_{c}. We show that |H| and L play the roles of inverse temperature and external magnetic field, respectively. In particular, we find a first-order dynamical phase transition when L changes sign, at supercritical H. We also determine analytically P(H,L,t) in several limits away from the second-order transition. Typical fluctuations of H are Gaussian, but the distribution tails are highly asymmetric. The tails -lnP∼|H|^{3/2}/sqrt[t] and -lnP∼|H|^{5/2}/sqrt[t], previously found for L=0, are enhanced for L≠0. At very large |L| the whole height-difference distribution P(H,L,t) is time-independent and Gaussian in H, -lnP∼|H|^{2}/|L|, describing the probability of creating a ramplike height profile at t=0.

15.
Phys Rev Lett ; 120(12): 120601, 2018 Mar 23.
Artigo em Inglês | MEDLINE | ID: mdl-29694078

RESUMO

The narrow escape problem deals with the calculation of the mean escape time (MET) of a Brownian particle from a bounded domain through a small hole on the domain's boundary. Here we develop a formalism which allows us to evaluate the nonescape probability of a gas of diffusing particles that may interact with each other. In some cases the nonescape probability allows us to evaluate the MET of the first particle. The formalism is based on the fluctuating hydrodynamics and the recently developed macroscopic fluctuation theory. We also uncover an unexpected connection between the narrow escape of interacting particles and thermal runaway in chemical reactors.

16.
Phys Rev E ; 95(6-1): 062124, 2017 Jun.
Artigo em Inglês | MEDLINE | ID: mdl-28709351

RESUMO

We study fluctuations of particle absorption by a three-dimensional domain with multiple absorbing patches. The domain is in contact with a gas of interacting diffusing particles. This problem is motivated by living cell sensing via multiple receptors distributed over the cell surface. Employing the macroscopic fluctuation theory, we calculate the covariance matrix of the particle absorption by different patches, extending previous works which addressed fluctuations of a single current. We find a condition when the sign of correlations between different patches is fully determined by the transport coefficients of the gas and is independent of the problem's geometry. We show that the fluctuating particle flux field typically develops vorticity. We establish a simple connection between the statistics of particle absorption by all the patches combined and the statistics of current in a nonequilibrium steady state in one dimension. We also discuss connections between the absorption statistics and (i) statistics of electric currents in multiterminal diffusive conductors and (ii) statistics of wave transmission through disordered media with multiple absorbers.

17.
Phys Rev E ; 95(1-1): 012134, 2017 Jan.
Artigo em Inglês | MEDLINE | ID: mdl-28208441

RESUMO

Height fluctuations of growing surfaces can be characterized by the probability distribution of height in a spatial point at a finite time. Recently there has been spectacular progress in the studies of this quantity for the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. Here we notice that, at or above a critical dimension, the finite-time one-point height distribution is ill defined in a broad class of linear surface growth models unless the model is regularized at small scales. The regularization via a system-dependent small-scale cutoff leads to a partial loss of universality. As a possible alternative, we introduce a local average height. For the linear models, the probability density of this quantity is well defined in any dimension. The weak-noise theory for these models yields the "optimal path" of the interface conditioned on a nonequilibrium fluctuation of the local average height. As an illustration, we consider the conserved Edwards-Wilkinson (EW) equation, where, without regularization, the finite-time one-point height distribution is ill defined in all physical dimensions. We also determine the optimal path of the interface in a closely related problem of the finite-time height-difference distribution for the nonconserved EW equation in 1+1 dimension. Finally, we discuss a UV catastrophe in the finite-time one-point distribution of height in the (nonregularized) KPZ equation in 2+1 dimensions.

18.
Phys Rev E ; 94(3-1): 032108, 2016 Sep.
Artigo em Inglês | MEDLINE | ID: mdl-27739726

RESUMO

We study the probability distribution P(H,t,L) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension when starting from a parabolic interface, h(x,t=0)=x^{2}/L. The limits of L→∞ and L→0 have been recently solved exactly for any t>0. Here we address the early-time behavior of P(H,t,L) for general L. We employ the weak-noise theory-a variant of WKB approximation-which yields the optimal history of the interface, conditioned on reaching the given height H at the origin at time t. We find that at small HP(H,t,L) is Gaussian, but its tails are non-Gaussian and highly asymmetric. In the leading order and in a proper moving frame, the tails behave as -lnP=f_{+}|H|^{5/2}/t^{1/2} and f_{-}|H|^{3/2}/t^{1/2}. The factor f_{+}(L,t) monotonically increases as a function of L, interpolating between time-independent values at L=0 and L=∞ that were previously known. The factor f_{-} is independent of L and t, signaling universality of this tail for a whole class of deterministic initial conditions.

19.
Phys Rev E ; 94(3-1): 032133, 2016 Sep.
Artigo em Inglês | MEDLINE | ID: mdl-27739741

RESUMO

We study the short-time behavior of the probability distribution P(H,t) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension. The process starts from a stationary interface: h(x,t=0) is given by a realization of two-sided Brownian motion constrained by h(0,0)=0. We find a singularity of the large deviation function of H at a critical value H=H_{c}. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry x↔-x of optimal paths h(x,t) predicted by the weak-noise theory of the KPZ equation. At |H|≫|H_{c}| the corresponding tail of P(H) scales as -lnP∼|H|^{3/2}/t^{1/2} and agrees, at any t>0, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of P scales as -lnP∼|H|^{5/2}/t^{1/2} and coincides with the corresponding tail for the sharp-wedge initial condition.

20.
Phys Rev E ; 93(5-2): 059901, 2016 May.
Artigo em Inglês | MEDLINE | ID: mdl-27301010

RESUMO

This corrects the article DOI: 10.1103/PhysRevE.93.020102.

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