arXiv subjects

Baruch Meerson

Publications and source records attributed to Baruch Meerson.

At least 37 records · Page 2Linked to original sources

arXiv2024

Inverse Scattering Method Solves the Problem of Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

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 Kaup and Newell (1978) 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.

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arXiv2024

Complete Integrability of the Problem of Full Statistics of Nonstationary Mass Transfer in the Simple Inclusion Process

The Simple Inclusion Process (SIP) interpolates between two well-known lattice gas models: the independent random walkers and the Kipnis-Marchioro-Presutti model. Here we study large deviations of nonstationary mass transfer in the SIP at long times in one dimension. We suppose that $N\gg 1$ particles start from a single lattice site at the origin, and we are interested in the probability $\mathcal{P}(M,N,T)$ of observing $M$ particles, $0\leq M\leq N$, to the right of the origin at a specified time $T\gg 1$. At large times, the corresponding probability distribution has a large-deviation behavior, $-\ln \mathcal{P}(M,N,T) \simeq \sqrt{T} s(M/N,N/\sqrt{T})$. We determine the rate function $s$ exactly by uncovering and utilizing complete integrability, by the inverse scattering method, of the underlying equations of the macroscopic fluctuation theory. We also analyze different asymptotic limits of the rate function $s$.

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arXiv2024

Macroscopic fluctuation theory of local time in lattice gases

The local time in an ensemble of particles measures the amount of time the particles spend in the vicinity of a given point in space. Here we study fluctuations of the empirical time average $R= T^{-1}\int_{0}^{T}ρ\left(x=0,t\right)\,dt$ of the density $ρ\left(x=0,t\right)$ at the origin (so that $R$ is the local time spent at the origin, rescaled by $T$) for an initially uniform one-dimensional diffusive lattice gas. We consider both the quenched and annealed initial conditions and employ the Macroscopic Fluctuation Theory (MFT). For a gas of non-interacting random walkers (RWs) the MFT yields exact large-deviation functions of $R$, which are closely related to the ones recently obtained by Burenev \textit{et al.} (2023) using microscopic calculations for non-interacting Brownian particles. Our MFT calculations, however, additionally yield the most likely history of the gas density $ρ(x,t)$ conditioned on a given value of $R$. Furthermore, we calculate the variance of the local time fluctuations for arbitrary particle- or energy-conserving diffusive lattice gases. Better known examples of such systems include the simple symmetric exclusion process, the Kipnis-Marchioro-Presutti model and the symmetric zero-range process. Our results for the non-interacting RWs can be readily extended to a step-like initial condition for the density.

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arXiv2024

Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices

We evaluate, in the large-$N$ limit, the complete probability distribution $\mathcal{P}(A,m)$ of the values $A$ of the sum $\sum_{i=1}^{N} |λ_i|^m$, where $λ_i$ ($i=1,2,\dots, N$) are the eigenvalues of a Gaussian random matrix, and $m$ is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang-Landau algorithm, we found that, in the limit of $N\to \infty$, the rate function of $\mathcal{P}(A,m)$ exhibits phase transitions of different characters. The phase diagram of the system on the $(A,m)$ plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for $m<2$ where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for $m>2$ where the maximum or minimum eigenvalue ``evaporates" from the rest of eigenvalues and dominates the statistics of $A$. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order $p=1+1/|m-1|$, where $0 2$ occur at the ground state of the Coulomb gas which corresponds to the Wigner's semicircular distribution.

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arXiv2023

Short-time large deviations of the spatially averaged height of a KPZ interface on a ring

Using the optimal fluctuation method, we evaluate the short-time probability distribution $P (\bar{H}, L, t=T)$ of the spatially averaged height $\bar{H} = (1/L) \int_0^L h(x, t=T) \, dx$ of a one-dimensional interface $h(x, t)$ governed by the Kardar-Parisi-Zhang equation $$ \partial_th=ν\partial_x^2h+\fracλ{2} \left(\partial_xh\right)^2+\sqrt{D}ξ(x,t) $$ on a ring of length $L$. The process starts from a flat interface, $h(x,t=0)=0$. Both at $λ\bar{H} < 0$, and at sufficiently small positive $λ\bar{H}$ the optimal (that is, the least-action) path $h(x,t)$ of the interface, conditioned on $\bar{H}$, is uniform in space, and the distribution $P (\bar{H}, L, T)$ is Gaussian. However, at sufficiently large $λ\bar{H} > 0$ the spatially uniform solution becomes sub-optimal and gives way to non-uniform optimal paths. We study them, and the resulting non-Gaussian distribution $P (\bar{H}, L, T)$, analytically and numerically. The loss of optimality of the uniform solution occurs via a dynamical phase transition of either first, or second order, depending on the rescaled system size $\ell = L/\sqrt{νT}$, at a critical value $\bar{H}=\bar{H}_{\text{c}}(\ell)$. At large but finite $\ell$ the transition is of first order. Remarkably, it becomes an "accidental" second-order transition in the limit of $\ell \to \infty$, where a large-deviation behavior $-\ln P (\bar{H}, L, T) \simeq (L/T) f(\bar{H})$ (in the units $λ=ν=D=1$) is observed. At small $\ell$ the transition is of second order, while at $\ell =O(1)$ transitions of both types occur.

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arXiv2023

Geometrical optics of large deviations of Brownian motion in inhomogeneous media

Geometrical optics provides an instructive insight into Brownian motion, ``pushed" into a large-deviations regime by imposed constraints. Here we extend geometrical optics of Brownian motion by accounting for diffusion inhomogeneity in space. We consider three simple model problems of Brownian motion on the line or in the plane in situations where the diffusivity of the Brownian particle depends on one spatial coordinate. One of our results describes ``Brownian refraction": an analog of refraction of light passing through a boundary between two media with different refraction indices.

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arXiv2023

Geometrical optics of first-passage functionals of random acceleration

Random acceleration is a fundamental stochastic process encountered in many applications. In the one-dimensional version of the process a particle is randomly accelerated according to the Langevin equation $\ddot{x}(t) = \sqrt{2D} ξ(t)$, where $x(t)$ is the particle's coordinate, $ξ(t)$ is Gaussian white noise with zero mean, and $D$ is the particle velocity diffusion constant. Here we evaluate the $A\to 0$ tail of the distribution $P_n(A|L)$ of the functional $I[x(t)]=\int_0^{T} x^n(t) dt=A$, where $T$ is the first-passage time of the particle from a specified point $x=L$ to the origin, and $n\geq 0$. We employ the optimal fluctuation method akin to geometrical optics. Its crucial element is determination of the optimal path -- the most probable realization of the random acceleration process $x(t)$, conditioned on specified $A\to 0$, $n$ and $L$. This realization dominates the probability distribution $P_n(A|L)$. We show that the $A\to 0$ tail of this distribution has a universal essential singularity, $P_n(A\to 0|L) \sim \exp\left(-\frac{α_n L^{3n+2}}{DA^3}\right)$, where $α_n$ is an $n$-dependent number which we calculate analytically for $n=0,1$ and $2$ and numerically for other $n$. For $n=0$ our result agrees with the asymptotic of the previously found first-passage time distribution.

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arXiv2023

Negative autocorrelations of disorder strongly suppress thermally activated particle motion in short-correlated quenched Gaussian disorder potentials

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 \emph{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 we show that the tail changes. It is much higher when the disorder potential exhibits negative autocorrelations, and it is much lower when the autocovariance is nonmonotonic but everywhere positive. This leads to an \emph{exponential} increase or decrease, respectively, of the mean escape time.

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arXiv2023

Large deviations of the interface height in the Golubović-Bruinsma model of stochastic growth

We study large deviations of the one-point height distribution, $\mathcal{P}(H,T)$, of a stochastic interface, governed by the Golubović-Bruinsma equation $$ \partial_{t}h=-ν\partial_{x}^{4}h+\fracλ{2}\left(\partial_{x}h\right)^{2}+\sqrt{D}\,ξ(x,t)\,, $$ where $h(x,t)$ is the interface height at point $x$ and time $t$, and $ξ(x,t)$ is the Gaussian white noise. The interface is initially flat, and $H$ is defined by the relation $h(x=0,t=T)=H$. Using the optimal fluctuation method (OFM), we focus on the short-time limit. Here the typical fluctuations of $H$ are Gaussian, and we evaluate the strongly asymmetric and non-Gaussian tails of $\mathcal{P}(H,T)$. We show that the upper tail scales as $-\ln \mathcal{P}(H,T) \sim H^{11/6}/T^{5/6}$. The lower tail, which scales as $-\ln \mathcal{P}(H,T) \sim H^{5/2}/T^{1/2}$, coincides with its counterpart for the Kardar-Parisi-Zhang equation, and we uncover a simple physical mechanism behind this universality. Finally, we verify our asymptotic results for the tails, and compute the large deviation function of $H$, numerically.

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arXiv2022

Path integrals for fractional Brownian motion and fractional Gaussian noise

The Wiener's path integral plays a central role in the studies of Brownian motion. Here we derive exact path-integral representations for the more general \emph{fractional} Brownian motion (fBm) and for its time derivative process -- the fractional Gaussian noise (fGn). These paradigmatic non-Markovian stochastic processes, introduced by Kolmogorov, Mandelbrot and van Ness, found numerous applications across the disciplines, ranging from anomalous diffusion in cellular environments to mathematical finance. Still, their exact path-integral representations were previously unknown. Our formalism exploits the Gaussianity of the fBm and fGn, relies on theory of singular integral equations and overcomes some technical difficulties by representing the action functional for the fBm in terms of the fGn for the sub-diffusive fBm, and in terms of the derivative of the fGn for the super-diffusive fBm. We also extend the formalism to include external forcing. The exact and explicit path-integral representations open new inroads into the studies of the fBm and fGn.

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arXiv2022

Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

We investigate non-stationary heat transfer in the Kipnis-Marchioro-Presutti (KMP) lattice gas model at long times in one dimension when starting from a localized heat distribution. At large scales this initial condition can be described as a delta-function, $u(x,t=0)=W δ(x)$. We characterize the process by the heat, transferred to the right of a specified point $x=X$ by time $T$, $$ J=\int_X^\infty u(x,t=T)\,dx\,, $$ and study the full probability distribution $\mathcal{P}(J,X,T)$. The particular case of $X=0$ has been recently solved [Bettelheim \textit{et al}. Phys. Rev. Lett. \textbf{128}, 130602 (2022)]. At fixed $J$, the distribution $\mathcal{P}$ as a function of $X$ and $T$ has the same long-time dynamical scaling properties as the position of a tracer in a single-file diffusion. Here we evaluate $\mathcal{P}(J,X,T)$ by exploiting the recently uncovered complete integrability of the equations of the macroscopic fluctuation theory (MFT) for the KMP model and using the Zakharov-Shabat inverse scattering method. We also discuss asymptotics of $\mathcal{P}(J,X,T)$ which we extract from the exact solution, and also obtain by applying two different perturbation methods directly to the MFT equations.

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arXiv2022

Competition in a system of Brownian particles: Encouraging achievers

We introduce and study analytically and numerically a simple model of inter-agent competition, where underachievement is strongly discouraged. We consider $N\gg 1$ particles performing independent Brownian motions on the line. Two particles are selected at random and at random times, and the particle closest to the origin is reset to it. We show that, in the limit of $N\to \infty$, the dynamics of the coarse-grained particle density field can be described by a nonlocal hydrodynamic theory which was encountered in a study of the spatial extent of epidemics in a critical regime. The hydrodynamic theory predicts relaxation of the system toward a stationary density profile of the "swarm" of particles, which exhibits a power-law decay at large distances. An interesting feature of this relaxation is a non-stationary "halo" around the stationary solution, which continues to expand in a self-similar manner. The expansion is ultimately arrested by finite-$N$ effects at a distance of order $\sqrt{N}$ from the origin, which gives an estimate of the average radius of the swarm. The hydrodynamic theory does not capture the behavior of the particle farthest from the origin -- the current leader. We suggest a simple scenario for typical fluctuations of the leader's distance from the origin and show that the mean distance continues to grow indefinitely as $\sqrt{t}$. Finally, we extend the inter-agent competition from $n=2$ to an arbitrary number $n$ of competing Brownian particles ($n\ll N$). Our analytical predictions are supported by Monte-Carlo simulations.

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arXiv2022

Fluctuations and first-passage properties of systems of Brownian particles with reset

We study stationary fluctuations in two models involving $N$ Brownian particles undergoing stochastic resetting to the origin in 1d. We start with the basic reset model where the particles reset independently (model A). Then we introduce nonlocal interparticle correlations by postulating that only the particle farthest from the origin is reset (model B). At long times both models approach nonequilibrium steady states. In the limit of $N\to \infty$, the steady-state particle density in model A has an infinite support, whereas in model B it has a compact support. A finite system radius, which scales at large $N$ as $\ln N$, appears in model A when $N$ is finite. In both models we study stationary fluctuations of the center of mass of the system and of the system's radius due to the random character of the Brownian motion and of the resetting events. In model A we determine exact distributions of these two quantities. The variance of the center of mass for both models scales as $1/N$. The variance of the radius is independent of $N$ in model A and exhibits an unusual scaling $(\ln N)/N$ in model B. The latter scaling is intimately related to the $1/f$ noise in the radius autocorrelations. Finally, we evaluate the mean first-passage time (MFPT) to a distant target in model A, model B, and the BBM. For model A we obtain an exact asymptotic expression for the MFPT which scales as $1/N$. For model B, and for the "Brownian bees" model, we propose a sharp upper bound for the MFPT. The bound assumes an ``evaporation" scenario, where the first passage requires multiple attempts of a single particle, which breaks away from the rest of the particles, to reach the target. The resulting MFPT for model B and the Brownian bees model scales exponentially with $\sqrt{N}$. We verify this bound by performing highly efficient weighted-ensemble simulations of the first passage in model B.

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arXiv2021

Fluctuations of a swarm of Brownian bees

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 a constant number of particles. In the limit of $N\to \infty$, 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 $\ell(t)$. Linearizing a pertinent Langevin equation around the deterministic steady state solution, we calculate the two-time covariances of $X(t)$ and $\ell(t)$. 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 $\ell(t)$ behaves differently: it exhibits an anomalous scaling $\ln N/N$. 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 $\ell(t)$ can be obtained from the covariance of $\ell(t)$ 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.

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arXiv2021

Observing symmetry-broken optimal paths of stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media

Consider the short-time probability distribution $\mathcal{P}(H,t)$ 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 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 $\mathcal{P}(H,t)$ -- is described by any of \emph{two} ramp-like 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 $x\leftrightarrow -x$ 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 $\mathcal{P}(H,t)$, 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 \emph{not} supposed to apply, we still observe two well-defined dominating paths. Each of them violates the mirror symmetry $x\leftrightarrow -x$ and is a mirror image of the other.

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arXiv2021

Persistent fluctuations of the swarm size of Brownian bees

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. (2020) proved that, at $N\to \infty$, 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. Further, they showed that, at long times, this solution approaches a unique spherically symmetric steady state with compact support: a sphere which radius $\ell_0$ 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 $\mathcal{P}(\ell,N,T)$ that the maximum distance of a particle from the origin remains smaller than a specified value $\ell<\ell_0$, or larger than a specified value $\ell>\ell_0$, on a time interval $0<t<T$, where $T$ is very large. We argue that $\mathcal{P}(\ell,N,T)$ exhibits the large-deviation form $-\ln \mathcal{P} \simeq N T R_d(\ell)$. For all $d$ we obtain asymptotics of the rate function $R_d(\ell)$ in the regimes $\ell \ll \ell_0$, $\ell\gg \ell_0$ and $|\ell-\ell_0|\ll \ell_0$. For $d=1$ the whole rate function can be calculated analytically. We obtain these results by determining the optimal (most probable) density profile of the swarm, conditioned on the specified $\ell$, and by arguing that this density profile is spherically symmetric with its center at the origin.

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arXiv2021

Statistics of first-passage Brownian functionals

We study the distribution of first-passage functionals ${\cal A}= \int_0^{t_f} x^n(t)\, dt$, where $x(t)$ is a Brownian motion (with or without drift) with diffusion constant $D$, starting at $x_0>0$, and $t_f$ is the first-passage time to the origin. In the driftless case, we compute exactly, for all $n>-2$, the probability density $P_n(A|x_0)=\text{Prob}.(\mathcal{A}=A)$. This probability density has an essential singular tail as $A\to 0$ and a power-law tail $\sim A^{-(n+3)/(n+2)}$ as $A\to \infty$. The former is reproduced by the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process for small $A$. For the case with a drift toward the origin, where no exact solution is known for general $n>-1$, the OFM predicts the distribution tails. For $A\to 0$ it predicts the same essential singular tail as in the driftless case. For $A\to \infty$ it predicts a stretched exponential tail $-\ln P_n(A|x_0)\sim A^{1/(n+1)}$ for all $n>0$. In the limit of large Péclet number $\text{Pe}= μx_0/(2D)\gg 1$, where $μ$ is the drift velocity, the OFM predicts a large-deviation scaling for all $A$: $-\ln P_n(A|x_0)\simeq\text{Pe}\, Φ_n\left(z= A/\bar{A}\right)$, where $\bar{A}=x_0^{n+1}/{μ(n+1)}$ is the mean value of $\mathcal{A}$. We compute the rate function $Φ_n(z)$ analytically for all $n>-1$. For $n>0$ $Φ_n(z)$ is analytic for all $z$, but for $-1 0$ with the same $|μ|$.

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arXiv2020

Velocity fluctuations of stochastic reaction fronts propagating into an unstable state: strongly pushed fronts

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 $D_f\sim 1/N$ of the front can be calculated, in the leading order, via a perturbation theory in $1/N \ll 1$, where $N\gg 1$ 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 non-perturbative 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 $Δt \gg τ_N$, where $τ_N$ scales as $N$. At intermediate times the position fluctuations of the front are anomalously large and non-diffusive. Our extensive Monte-Carlo simulations of a particular reacting lattice gas model support these conclusions.

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