arXiv subjects

Arkady Vilenkin

Publications and source records attributed to Arkady Vilenkin.

17 recordsLinked to original sources

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.

cond-mat.stat-mech↗

arXiv2022

Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees

The ``Brownian bees'' model describes an ensemble of $N=$~const independent branching Brownian particles. The conservation of $N$ is provided by a modified branching process. When a particle branches into two particles, the particle which is farthest from the origin is eliminated simultaneously. The spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation in the limit of $N \gg 1$. At long times, the particle density approaches a spherically symmetric steady state solution with a compact support of radius $\bar{\ell}_0$. However, at finite $N$, the radius of this support, $L$, fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni {\em et al}., Phys. Rev. E. {\bf104}, 054131 (2021)]. It is proportional to $N^{-1}\ln N$ at $N\to\infty$. We investigate here the tails of the probability density function (PDF), $P(L)$, of the swarm radius, when the absolute value of the radius fluctuation $ΔL=L-\bar{\ell}_0$ is sufficiently larger than the typical fluctuations' scale determined by the variance. For negative deviations the PDF can be obtained in the framework of the optimal fluctuation method (OFM). This part of the PDF displays the scaling behavior: $\ln P\propto - N ΔL^2\, \ln^{-1}(ΔL^{-2})$, demonstrating a logarithmic anomaly at small negative $ΔL$. For the opposite sign of the fluctuation, $ΔL > 0$, the PDF can be obtained with an approximation of a single particle, running away. We find that $\ln P \propto -N^{1/2}ΔL$. We consider in this paper only the case, when $|ΔL|$ is much less than the typical radius of the swarm at $N\gg 1$.

cond-mat.stat-mech↗

arXiv2019

Time-averaged height distribution of the Kardar-Parisi-Zhang interface

We study the complete probability distribution $\mathcal{P}\left(\bar{H},t\right)$ of the time-averaged height $\bar{H}=(1/t)\int_0^t h(x=0,t')\,dt'$ at point $x=0$ of an evolving 1+1 dimensional Kardar-Parisi-Zhang (KPZ) interface $h\left(x,t\right)$. We focus on short times and flat initial condition and employ the optimal fluctuation method to determine the variance and the third cumulant of the distribution, as well as the asymmetric stretched-exponential tails. The tails scale as $-\ln\mathcal{P}\sim\left|\bar{H}\right|^{3/2} \! /\sqrt{t}$ and $-\ln\mathcal{P}\sim\left|\bar{H}\right|^{5/2} \! /\sqrt{t}$, similarly to the previously determined tails of the one-point KPZ height statistics at specified time $t'=t$. The optimal interface histories, dominating these tails, are markedly different. Remarkably, the optimal history, $h\left(x=0,t\right)$, of the interface height at $x=0$ is a non-monotonic function of time: the maximum (or minimum) interface height is achieved at an intermediate time. We also address a more general problem of determining the probability density of observing a given height history of the KPZ interface at point $x=0$.

cond-mat.stat-mech↗

arXiv2018

Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line

Consider a stochastic interface $h(x,t)$, described by the $1+1$ Kardar-Parisi-Zhang (KPZ) equation on the half-line $x\geq 0$. The interface is initially flat, $h(x,t=0)=0$, and driven by a Neumann boundary condition $\partial_x h(x=0,t)=A$ and by the noise. We study the short-time probability distribution $\mathcal{P}\left(H,A,t\right)$ of the one-point height $H=h(x=0,t)$. Using the optimal fluctuation method, we show that $-\ln \mathcal{P}\left(H,A,t\right)$ scales as $t^{-1/2} s \left(H,A t^{1/2}\right)$. For small and moderate $|A|$ this more general scaling reduces to the familiar simple scaling $-\ln \mathcal{P}\left(H,A,t\right)\simeq t^{-1/2} s(H)$, where $s$ is independent of $A$ and time and equal to one half of the corresponding large-deviation function for the full-line problem. For large $|A|$ we uncover two asymptotic regimes. At very short time the simple scaling is restored, whereas at intermediate times the scaling remains more general and $A$-dependent. The distribution tails, however, always exhibit the simple scaling in the leading order.

cond-mat.stat-mech↗

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↗

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

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↗

arXiv2014

Survival of a static target in a gas of diffusing particles with exclusion

Let a lattice gas of constant density, described by the symmetric simple exclusion process, be brought in contact with a "target": a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory (MFT), we evaluate the probability ${\mathcal P}(T)$ that no gas particle hits the target until a long but finite time $T$. We also find the most likely gas density history conditional on the non-hitting. The results depend on the dimension of space $d$ and on the rescaled parameter $\ell=R/\sqrt{D_0T}$, where $D_0$ is the gas diffusivity. For small $\ell$ and $d>2$, ${\mathcal P}(T)$ is determined by an exact stationary solution of the MFT equations that we find. For large $\ell$, and for any $\ell$ in one dimension, the relevant MFT solutions are non-stationary. In this case $\ln {\mathcal P}(T)$ scales differently with relevant parameters, and it also depends on whether the initial condition is random or deterministic. The latter effects also occur if the lattice gas is composed of non-interacting random walkers. Finally, we extend the formalism to a whole class of diffusive gases of interacting particles.

cond-mat.stat-mech↗

arXiv2014

Extreme Fluctuations of Current in the Symmetric Simple Exclusion Process: a Non-Stationary Setting

We use the macroscopic fluctuation theory (MFT) to evaluate the probability distribution P of extreme values of integrated current J at a specified time t=T in the symmetric simple exclusion process (SSEP) on an infinite line. As shown recently [Phys. Rev. E 89, 010101(R) (2014)], the SSEP belongs to the elliptic universality class. Here, for very large currents, the diffusion terms of the MFT equations can be neglected compared with the terms coming from the shot noise. Using the hodograph transformation and an additional change of variables, we reduce the "inviscid" MFT equations to Laplace's equation in an extended space. This opens the way to an exact solution. Here we solve the extreme-current problem for a flat deterministic initial density profile with an arbitrary density 0<n<1. The solution yields the most probable density history of the system conditional on the extreme current, and leads to a super-Gaussian extreme-current statistics, - ln P = F(n) J^3/T, in agreement with Derrida and Gerschenfeld [J. Stat. Phys. 137, 978 (2009)]. We calculate the function F(n) analytically. It is symmetric with respect to the half-filling density n=1/2, diverges as n approached 0 or 1, and exhibits a singularity F(n) |n-1/2| at the half-filling density n=1/2.

cond-mat.stat-mech↗

arXiv2012

Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts

The position of an invasion front, propagating into an unstable state, fluctuates because of the shot noise coming from the discreteness of reacting particles and stochastic character of the reactions and diffusion. A recent macroscopic theory [Meerson and Sasorov, Phys. Rev. E 84, 030101(R) (2011)] yields the probability of observing, during a long time, an unusually slow front. The theory is formulated as an effective classical Hamiltonian field theory which operates with the density field and the conjugate "momentum" field. Further, the theory assumes that the most probable density field history of an unusually slow front represents, up to small corrections, a traveling front solution of the Hamilton equations. Here we verify this assumption by solving the Hamilton equations numerically for models belonging to the directed percolation universality class.

cond-mat.stat-mech↗

arXiv2008

A nonlinear theory of non-stationary low Mach number channel flows of freely cooling nearly elastic granular gases

We use hydrodynamics to investigate non-stationary channel flows of freely cooling dilute granular gases. We focus on the regime where the sound travel time through the channel is much shorter than the characteristic cooling time of the gas. As a result, the gas pressure rapidly becomes almost homogeneous, while the typical Mach number of the flow drops well below unity. Eliminating the acoustic modes, we reduce the hydrodynamic equations to a single nonlinear and nonlocal equation of a reaction-diffusion type in Lagrangian coordinates. This equation describes a broad class of channel flows and, in particular, can follow the development of the clustering instability from a weakly perturbed homogeneous cooling state to strongly nonlinear states. If the heat diffusion is neglected, the reduced equation is exactly soluble, and the solution develops a finite-time density blowup. The heat diffusion, however, becomes important near the attempted singularity. It arrests the density blowup and brings about novel inhomogeneous cooling states (ICSs) of the gas, where the pressure continues to decay with time, while the density profile becomes time-independent. Both the density profile of an ICS, and the characteristic relaxation time towards it are determined by a single dimensionless parameter that describes the relative role of the inelastic energy loss and heat diffusion. At large values of this parameter, the intermediate cooling dynamics proceeds as a competition between low-density regions of the gas. This competition resembles Ostwald ripening: only one hole survives at the end.

cond-mat.soft↗

arXiv2006

Self-similar asymptotics for a class of Hele-Shaw flows driven solely by surface tension

We investigate the dynamics of relaxation, by surface tension, of a family of curved interfaces between an inviscid and viscous fluids in a Hele-Shaw cell. At t=0 the interface is assumed to be of the form |y|=A x^m, where A>0, m \geq 0, and x>0. The case of 0 1 corresponds to a cusp, whereas m=1 corresponds to a wedge. The inviscid fluid tip retreats in the process of relaxation, forming a lobe which size grows with time. Combining analytical and numerical methods we find that, for any m, the relaxation dynamics exhibit self-similar behavior. For m\neq 1 this behavior arises as an intermediate asymptotics: at late times for 0\leq m<1, and at early times for m>1. In both cases the retreat distance and the lobe size exhibit power law behaviors in time with different dynamic exponents, uniquely determined by the value of m. In the special case of m=1 (the wedge) the similarity is exact and holds for the whole interface at all times t>0, while the two dynamic exponents merge to become 1/3. Surprisingly, when m\neq 1, the interface shape, rescaled to the local maximum elevation of the interface, turns out to be universal (that is, independent of m) in the similarity region. Even more remarkably, the same rescaled interface shape emerges in the case of m=1 in the limit of zero wedge angle.

physics.flu-dyn↗

arXiv2006

Self-similar relaxation dynamics of a fluid wedge in a Hele-Shaw cell

Let the interface between two immiscible fluids in a Hele-Shaw cell have, at t=0, a wedge shape. As a wedge is scale-free, the fluid relaxation dynamics are self-similar. We find the dynamic exponent of this self-similar flow and show that the interface shape is given by the solution of an unusual inverse problem of potential theory. We solve this inverse problem analytically for an almost flat wedge, and numerically otherwise. The wedge solution is useful for analysis of pinch-off singularities.

physics.flu-dyn↗

arXiv2005

Scaling and self-similarity in an unforced flow of inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell

We investigate quasi-two-dimensional relaxation, by surface tension, of a long straight stripe of inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. Combining analytical and numerical solutions, we describe the emergence of a self-similar dumbbell shape and find non-trivial dynamic exponents that characterize scaling behavior of the dumbbell dimensions.

physics.flu-dyn↗

arXiv2005

How a Long Bubble Shrinks: a Numerical Method for an Unforced Hele-Shaw Flow

We develop a numerical method for solving a free boundary problem which describes shape relaxation, by surface tension, of a long and thin bubble of an inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. The method of solution of the exterior Dirichlet problem employs a classical boundary integral formulation. Our version of the numerical method is especially advantageous for following the dynamics of a very long and thin bubble, for which an asymptotic scaling theory has been recently developed. Because of the very large aspect ratio of the bubble, a direct implementation of the boundary integral algorithm would be impractical. We modify the algorithm by introducing a new approximation of the integrals which appear in the Fredholm integral equation and in the integral expression for the normal derivative of the pressure at the bubble interface. The new approximation allows one to considerably reduce the number of nodes at the almost flat part of the bubble interface, while keeping a good accuracy. An additional benefit from the new approximation is in that it eliminates numerical divergence of the integral for the tangential derivative of the harmonic conjugate. The interface's position is advanced in time by using explicit node tracking, whereas the larger node spacing enables one to use larger time steps. The algorithm is tested on two model problems, for which approximate analytical solutions are available.

physics.comp-ph↗

arXiv2001

Area-preserving dynamics of a long slender finger by curvature: a test case for the globally conserved phase ordering

A long and slender finger can serve as a simple ``test bed'' for different phase ordering models. In this work, the globally-conserved, interface-controlled dynamics of a long finger is investigated, analytically and numerically, in two dimensions. An important limit is considered when the finger dynamics are reducible to the area-preserving motion by curvature. A free boundary problem for the finger shape is formulated. An asymptotic perturbation theory is developed that uses the finger aspect ratio as a small parameter. The leading-order approximation is a modification of ``the Mullins finger" (a well-known analytic solution) which width is allowed to slowly vary with time. This time dependence is described, in the leading order, by an exponential law with the characteristic time proportional to the (constant) finger area. The subleading terms of the asymptotic theory are also calculated. Finally, the finger dynamics is investigated numerically, employing the Ginzburg-Landau equation with a global conservation law. The theory is in a very good agreement with the numerical solution.

cond-mat.dis-nn↗