arXiv subjects

Baruch Meerson

Publications and source records attributed to Baruch Meerson.

At least 73 records · Page 4Linked to original sources

arXiv2018

Nonequilibrium Steady State of a Weakly-Driven Kardar-Parisi-Zhang Equation

We consider an infinite interface in $d>2$ dimensions, governed by the Kardar-Parisi-Zhang (KPZ) equation with a weak Gaussian noise which is delta-correlated in time and has short-range spatial correlations. We study the probability distribution of the interface height $H$ at a point of the substrate, when the interface is initially flat. We show that, in a stark contrast with the KPZ equation in $d<2$, this distribution approaches a non-equilibrium steady state. The time of relaxation toward this state scales as the diffusion time over the correlation length of the noise. We study the steady-state distribution $\mathcal{P}(H)$ using the optimal-fluctuation method. The typical, small fluctuations of height are Gaussian. For these fluctuations the activation path of the system coincides with the time-reversed relaxation path, and the variance of $\mathcal{P}(H)$ can be found from a minimization of the (nonlocal) equilibrium free energy of the interface. In contrast, the tails of $\mathcal{P}(H)$ are nonequilibrium, non-Gaussian and strongly asymmetric. To determine them we calculate, analytically and numerically, the activation paths of the system, which are different from the time-reversed relaxation paths. We show that the slower-decaying tail of $\mathcal{P}(H)$ scales as $-\ln \mathcal{P}(H) \propto |H|$, while the faster-decaying tail scales as $-\ln \mathcal{P}(H) \propto |H|^3$. The slower-decaying tail has important implications for the statistics of directed polymers in random potential.

cond-mat.stat-mech↗

arXiv2018

Narrow escape of interacting diffusing particles

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 that allows us to evaluate the \emph{non-escape probability} of a gas of diffusing particles that may interact with each other. In some cases the non-escape 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.

cond-mat.stat-mech↗

arXiv2018

Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation

We use the optimal fluctuation method to evaluate the short-time probability distribution $\mathcal{P}\left(H,L,t\right)$ of height at a single point, $H=h\left(x=0,t\right)$, of the evolving Kardar-Parisi-Zhang (KPZ) interface $h\left(x,t\right)$ on a ring of length $2L$. The process starts from a flat interface. At short times typical (small) height fluctuations are unaffected by the KPZ nonlinearity and belong to the Edwards-Wilkinson universality class. The nonlinearity, however, strongly affects the (asymmetric) tails of $\mathcal{P}(H)$. At large $L/\sqrt{t}$ the faster-decaying tail has a double structure: it is $L$-independent, $-\ln\mathcal{P}\sim\left|H\right|^{5/2}/t^{1/2}$, at intermediately large $|H|$, and $L$-dependent, $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, at very large $|H|$. The transition between these two regimes is sharp and, in the large $L/\sqrt{t}$ limit, behaves as a fractional-order phase transition. The transition point $H=H_{c}^{+}$ depends on $L/\sqrt{t}$. At small $L/\sqrt{t}$, the double structure of the faster tail disappears, and only the very large-$H$ tail, $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, is observed. The slower-decaying tail does not show any $L$-dependence at large $L/\sqrt{t}$, where it coincides with the slower tail of the GOE Tracy-Widom distribution. At small $L/\sqrt{t}$ this tail also has a double structure. The transition between the two regimes occurs at a value of height $H=H_{c}^{-}$ which depends on $L/\sqrt{t}$. At $L/\sqrt{t} \to 0$ the transition behaves as a mean-field-like second-order phase transition. At $|H|<|H_c^{-}|$ the slower tail behaves as $-\ln\mathcal{P}\sim \left|H\right|^{2}L/t$, whereas at $|H|>|H_c^{-}|$ it coincides with the slower tail of the GOE Tracy-Widom distribution.

cond-mat.stat-mech↗

arXiv2017

Height distribution tails in the Kardar-Parisi-Zhang equation with Brownian initial conditions

For stationary interface growth, governed by the Kardar-Parisi-Zhang (KPZ) equation in 1 + 1 dimensions, typical fluctuations of the interface height at long times are described by the Baik-Rains distribution. Recently Chhita et al. [1] used the totally asymmetric simple exclusion process (TASEP) to study the height fluctuations in systems of the KPZ universality class for Brownian interfaces with arbitrary diffusion constant. They showed that there is a one-parameter family of long-time distributions, parametrized by the diffusion constant of the initial random height profile. They also computed these distributions numerically by using Monte Carlo (MC) simulations. Here we address this problem analytically and focus on the distribution tails at short times. We determine the (stretched exponential) tails of the height distribution by applying the Optimal Fluctuation Method (OFM) to the KPZ equation. We argue that, by analogy with other initial conditions, the "slow" tail holds at arbitrary times and therefore provides a proper asymptotic to the family of long-time distributions studied in Ref. [1]. We verify this hypothesis by performing large-scale MC simulations of a TASEP with a parallel-update rule. The "fast" tail, predicted by the OFM, is also expected to hold at arbitrary times, at sufficiently large heights.

cond-mat.stat-mech↗

arXiv2017

WKB theory of large deviations in stochastic populations

Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and people. Large deviations in stochastic population dynamics - such as those determining population extinction, fixation or switching between different states - are presently in a focus of attention of statistical physicists. We review recent progress in applying different variants of dissipative WKB approximation (after Wentzel, Kramers and Brillouin) to this class of problems. The WKB approximation allows one to evaluate the mean time and/or probability of population extinction, fixation and switches resulting from either intrinsic (demographic) noise, or a combination of the demographic noise and environmental variations, deterministic or random. We mostly cover well-mixed populations, single and multiple, but also briefly consider populations on heterogeneous networks and spatial populations. The spatial setting also allows one to study large fluctuations of the speed of biological invasions. Finally, we briefly discuss possible directions of future work.

cond-mat.stat-mech↗

arXiv2017

Fluctuations of absorption of interacting diffusing particles by multiple absorbers

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 non-equilibrium steady state in one dimension. We also discuss connections between the absorption statistics and (i) statistics of electric currents in multi-terminal diffusive conductors and (ii) statistics of wave transmission through disordered media with multiple absorbers.

cond-mat.stat-mech↗

arXiv2017

Large deviations of surface height in the $1+1$-dimensional Kardar-Parisi-Zhang equation: exact long-time results for $λH<0$

We study atypically large fluctuations of height $H$ in the 1+1-dimensional Kardar-Parisi-Zhang (KPZ) equation at long times $t$, when starting from a "droplet" initial condition. We derive exact large deviation function of height for $λH<0$, where $λ$ is the nonlinearity coefficient of the KPZ equation. This large deviation function describes a crossover from the Tracy-Widom distribution tail at small $|H|/t$, which scales as $|H|^3/t$, to a different tail at large $|H|/t$, which scales as $|H|^{5/2}/t^{1/2}$. The latter tail exists at all times $t>0$. It was previously obtained in the framework of the optimal fluctuation method. It was also obtained at short times from exact representation of the complete height statistics. The crossover between the two tails, at long times, occurs at $|H|\sim t$ as previously conjectured. Our analytical findings are supported by numerical evaluations using exact representation of the complete height statistics.

cond-mat.stat-mech↗

arXiv2016

Local average height distribution of fluctuating interfaces

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 \emph{local average height}. For the linear models the probability density of this quantity is well-defined in any dimension. The weak-noise theory (WNT) for these models yields the "optimal path" of the interface conditioned on a non-equilibrium 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 \emph{height-difference} distribution for the non-conserved EW equation in $1+1$ dimension. Finally, we discuss a UV catastrophe in the finite-time one-point distribution of height in the (non-regularized) KPZ equation in $2+1$ dimensions.

cond-mat.stat-mech↗

arXiv2016

Colonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure

We study the dynamics of colonization of a territory by a stochastic population at low immigration pressure. We assume a sufficiently strong Allee effect that introduces, in deterministic theory, a large critical population size for colonization. At low immigration rates, the average pre-colonization population size is small thus invalidating the WKB approximation to the master equation. We circumvent this difficulty by deriving an exact zero-flux solution of the master equation and matching it with an approximate non-zero-flux solution of the pertinent Fokker-Planck equation in a small region around the critical population size. This procedure provides an accurate evaluation of the quasi-stationary probability distribution of population sizes in the pre-colonization state, and of the mean time to colonization, for a wide range of immigration rates. At sufficiently high immigration rates our results agree with WKB results obtained previously. At low immigration rates the results can be very different.

q-bio.PE↗

arXiv2016

Dynamical phase transition in large deviation statistics of the Kardar-Parisi-Zhang equation

We study the short-time behavior of the probability distribution $\mathcal{P}(H,t)$ 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 $x \leftrightarrow -x$ of optimal paths $h(x,t)$ predicted by the weak-noise theory of the KPZ equation. At $|H|\gg |H_c|$ the corresponding tail of $\mathcal{P}(H)$ scales as $-\ln \mathcal{P} \sim |H|^{3/2}/t^{1/2}$ 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 $\mathcal{P}$ scales as $-\ln \mathcal{P} \sim |H|^{5/2}/t^{1/2}$ and coincides with the corresponding tail for the sharp-wedge initial condition.

cond-mat.stat-mech↗

arXiv2016

Short-time height distribution in 1d KPZ equation: starting from a parabola

We study the probability distribution $\mathcal{P}(H,t,L)$ 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 $L\to\infty$ and $L\to 0$ have been recently solved exactly for any $t>0$. Here we address the early-time behavior of $\mathcal{P}(H,t,L)$ 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 $H$ $\mathcal{P}(H,t,L)$ 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 $-\ln \mathcal{P}= f_{+}|H|^{5/2}/t^{1/2}$ 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 $L=\infty$ that were previously known. The factor $f_{-}$ is independent of $L$ and $t$, signalling universality of this tail for a whole class of deterministic initial conditions.

cond-mat.stat-mech↗

arXiv2016

Statistics of large currents in the Kipnis-Marchioro-Presutti model in a ring geometry

We use the macroscopic fluctuation theory to determine the statistics of large currents in the Kipnis-Marchioro-Presutti (KMP) model in a ring geometry. About 10 years ago this simple setting was instrumental in identifying a breakdown of the additivity principle in a class of lattice gases at currents exceeding a critical value. Building on earlier work, we assume that, for supercritical currents, the optimal density profile, conditioned on the given current, has the form of a traveling wave (TW). For the KMP model we find this TW analytically, in terms of elliptic functions, for any supercritical current $I$. Using this TW solution, we evaluate, up to a pre-exponential factor, the probability distribution $P(I)$. We obtain simple asymptotics of the TW and of $P(I)$ for currents close to the critical current, and for currents much larger than the critical current. In the latter case we show that $-\ln P (I) \sim I\ln I$, whereas the optimal density profile acquires a soliton-like shape. Our analytic results are in a very good agreement with Monte-Carlo simulations and numerical solutions of Hurtado and Garrido (2011).

cond-mat.stat-mech↗

arXiv2016

Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion

Consider a one-dimensional branching Brownian motion, and rescale the coordinate and time so that the rates of branching and diffusion are both equal to $1$. If $X_1(t)$ is the position of the rightmost particle of the branching Brownian motion at time $t$, the empirical velocity $c$ of this rightmost particle is defined as $c=X_1(t)/t$. Using the Fisher-KPP equation, we evaluate the probability distribution ${\mathcal P(c,t)}$ of this empirical velocity $c$ in the long time $t$ limit for $c > 2$. It was already known that, for a single seed particle, ${\mathcal P(c,t)} \sim \exp \,[-(c^2/4-1)t]$ up to a prefactor that can depend on $c$ and $t$. Here we show how to determine this prefactor. The result can be easily generalized to the case of multiple seed particles and to branching random walks associated to other traveling wave equations.

cond-mat.stat-mech↗

arXiv2016

Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation

Using the weak-noise theory, we evaluate the probability distribution $\mathcal{P}(H,t)$ of large deviations of height $H$ of the evolving surface height $h(x,t)$ in the Kardar-Parisi-Zhang (KPZ) equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height $H$ at time $t$. We argue that the tails of $\mathcal{P}$ behave, at arbitrary time $t>0$, and in a proper moving frame, as $-\ln \mathcal{P}\sim |H|^{5/2}$ and $\sim |H|^{3/2}$. The $3/2$ tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously observed at long times.

cond-mat.stat-mech↗

arXiv2015

Extinction of oscillating populations

Established populations often exhibit oscillations in their sizes. If a population is isolated, intrinsic stochasticity of elemental processes can ultimately bring it to extinction. Here we study extinction of oscillating populations in a stochastic version of the Rosenzweig-MacArthur predator-prey model. To this end we extend a WKB approximation (after Wentzel, Kramers and Brillouin) of solving the master equation to the case of extinction from a limit cycle in the space of population sizes. We evaluate the extinction rates and find the most probable paths to extinction by applying Floquet theory to the dynamics of an effective WKB Hamiltonian. We show that the entropic barriers to extinction change in a non-analytic way as the system passes through the Hopf bifurcation. We also study the subleading pre-exponential factors of the WKB approximation.

cond-mat.stat-mech↗

arXiv2015

Survival of interacting diffusing particles inside a domain with absorbing boundary

Suppose that a $d$-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density $n_0$. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability ${\mathcal P}$ that no particles are absorbed during a long time $T$. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time $T$. As a result, ${\mathcal P}$ decays exponentially with $T$ for a whole class of interacting diffusive gases in any dimension. For $d=1$ the stationary gas density profile and ${\mathcal P}$ can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that $-\ln {\mathcal P}\simeq D_0TL^{d-2} \,s(n_0)$, where $D_0$ is the gas diffusivity, and $L$ is the linear size of the system. We calculate the rescaled action $s(n_0)$ for $d=1$, for rectangular domains in $d=2$, and for spherical domains. Near close packing of the SSEP $s(n_0)$ can be found analytically for domains of any shape and in any dimension.

cond-mat.stat-mech↗

arXiv2015

Macroscopic fluctuation theory and first-passage properties of surface diffusion

We investigate non-equilibrium fluctuations of a solid surface governed by the stochastic Mullins-Herring equation with conserved noise. This equation describes surface diffusion of adatoms accompanied by their exchange between the surface and the bulk of the solid, when desorption of adatoms is negligible. Previous works dealt with dynamic scaling behavior of the fluctuating interface. Here we determine the probability that the interface first reaches a large given height at a specified time. We also find the optimal time history of the interface conditional on this non-equilibrium fluctuation. We obtain these results by developing a macroscopic fluctuation theory of surface diffusion.

cond-mat.stat-mech↗

arXiv2015

First Invader Dynamics in Diffusion-Controlled Absorption

We investigate the average time for the earliest particle to hit a spherical absorber when a homogeneous gas of freely diffusing particles with density $ρ$ and diffusivity $D$ is prepared in a deterministic state and is initially separated by a minimum distance $\ell$ from this absorber. In the high-density limit, this first absorption time scales as $\frac{\ell^2}{D}\frac{1}{\lnρ\ell}$ in one dimension; we also obtain the first absorption time in three dimensions. In one dimension, we determine the probability that the $k^{\rm th}$-closest particle is the first one to hit the absorber. At large $k$, this probability decays as $k^{1/3}\exp(-Ak^{2/3})$, with $A= 1.93299\ldots$ analytically calculable. As a corollary, the characteristic hitting time $T_k$ for the $k^{\rm th}$-closest particle scales as $k^{4/3}$; this corresponds to superdiffusive but still subballistic motion.

cond-mat.stat-mech↗