arXiv subjects

Pavel V. Sasorov

Publications and source records attributed to Pavel V. Sasorov.

At least 19 recordsLinked to original sources

arXiv2026

Speed fluctuations of a stochastic Huxley-Zel'dovich front

The empirical speed of travelling reaction-diffusion fronts fluctuates due to the intrinsic shot noise of the reactions and diffusion. Here we study the long-time front speed fluctuations of a stochastic Huxley-Zel'dovich front. It involves a population of particles $A$ which perform a fast continuous-time random walk on a one-dimensional lattice and undergo reversible on-site reactions $2A \rightleftarrows 3A$. This front describes an invasion of $A$-particles into an initially empty region of space which, in a deterministic description, is marginally stable but nonlinearly unstable with a zero instability threshold. Typical fluctuations of this front can be described as front diffusion in a reference frame moving with the average front speed. According to the existing perturbation theory, the shot-noise-induced systematic shift of the average front speed, $δc$, and the front diffusion coefficient, $D_f$, are both expected to scale with $N$ as $1/N$, where $N \gg 1$ is the typical number of particles in the transition region. Furthermore, $D_f$ can be determined perturbatively in the small parameter $1/\sqrt{N}$. Our Monte Carlo simulations support these asymptotic results, but also reveal a long-lived anomalous behavior of the first few particles before they reach the expected diffusion asymptotic. We also study large deviations of the empirical speed of the front at long times. These are dominated by optimal histories of the system in the form of a propagating front which travels with a speed different from the average speed, or even travel in the wrong direction.

cond-mat.stat-mech↗

arXiv2026

Fractional Brownian Motion with Negative Hurst Exponent

Fractional Brownian motion (fBm) is an important scale-invariant Gaussian non-Markovian process with stationary increments, which serves as a prototypical example of a system with long-range temporal correlations and anomalous diffusion. The fBm is traditionally defined for the Hurst exponent $H$ in the range $0<H<1$. Here we extend this definition to the regime $-1/2<H<0$. The extended fBm is not a pointwise process, so we regularize it via a local temporal averaging with a narrow filter. The resulting process is both very rough and persistent, that is long-range positively correlated. In addition, this process is stationary. The stationarity implies that diffusion is completely suppressed in this region of $H$. We also study another closely related Gaussian process: the stationary fractional Ornstein--Uhlenbeck (fOU) process, extended to the range $-1/2<H<0$ and smoothed in the same way as the fBm. Remarkably, the smoothed fOU process is asymptotically insensitive to the strength of the confining potential. Finally, we determine the optimal paths of the fBm and fOU processes for $-1/2<H<0$, conditioned on reaching a specified value when starting from zero. In the marginal case $H=0$, our results match continuously with known results for the traditionally defined fBm and fOU processes.

cond-mat.stat-mech↗

arXiv2024

Negative large deviations of the front velocity of $N$-particle branching Brownian motion

We study negative large deviations of the long-time empirical front velocity of the center of mass of the one-sided $N$-BBM ($N$-particle branching Brownian motion) system in one dimension. Employing the macroscopic fluctuation theory, we study the probability that $c$ is smaller than the limiting front velocity $c_0$, predicted by the deterministic theory, or even becomes negative. To this end we determine the optimal path of the system, conditioned on the specified $c$. We show that for $c_0-c\ll c_0$ the properly defined rate function $s(c)$, coincides, up to a non-universal numerical factor, with the universal rate functions for front models belonging to the Fisher-Kolmogorov-Petrovsky-Piscounov universality class. For sufficiently large negative values of $c$, $s(c)$ approaches a simple bound, obtained under the assumption that the branching is completely suppressed during the whole time. Remarkably, for all $c\leq c_*$, where $c_*<0$ is a critical value that we find numerically, the rate function $s(c)$ is \emph{equal} to the simple bound. At the critical point $c=c_*$ the character of the optimal path changes, and the rate function exhibits a dynamical phase transition of second order.

cond-mat.stat-mech↗

arXiv2024

Fractional Brownian motion in confining potentials: non-equilibrium distribution tails and optimal fluctuations

At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann) steady state. Here we consider scale-invariant power-law potentials $V(x)\sim |x|^m$, where $m>0$, and employ the optimal fluctuation method (OFM) to determine the large-$|x|$ tails of the steady-state probability distribution $\mathcal{P}(x)$ of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large-$|x|$ tails of $\ln \mathcal{P}(x)$ up to a dimensionless factor $α(H,m)$, where $0<H<1$ is the Hurst exponent. We determine $α(H,m)$ analytically (i) in the limits of $H\to 0$ and $H\to 1$, and (ii) for $m=2$ and arbitrary $H$, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact $\mathcal{P}(x)$ and autocovariance. The form of the tails of $\mathcal{P}(x)$ yields exact conditions, in terms of $H$ and $m$, for the particle confinement in the potential. For $H\neq 1/2$, the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for $m=2$. To compute the optimal paths and the factor $α(H,m)$ for arbitrary permissible $H$ and $m$, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which accounts analytically for an intrinsic cusp singularity of the optimal paths for $H<1/2$.

cond-mat.stat-mech↗

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.

cond-mat.stat-mech↗

arXiv2019

Synergic Cherenkov-Compton Radiation

In vacuum with an induced by strong electromagnetic wave refraction index larger than unity, an ultra-relativistic electron emits Cherenkov radiation. During the interaction with this wave the electron also radiates photons via the Compton scattering. Synergic Cherenkov-Compton process can be observed by colliding laser accelerated electrons with a high intensity electromagnetic pulse. Extremely high energy photons cannot be emitted via the Cherenkov radiation because the vacuum refraction index tends to unity at these energies. Experiments on studying these phenomena will reveal the properties of vacuum predicted by nonlinear quantum electrodynamics.

physics.plasm-ph↗

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

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

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

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↗

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↗

arXiv2013

Extreme Current Fluctuations in a Nonstationary Stochastic Heat Flow

We employ the Hamiltonian formalism of macroscopic fluctuation theory to study large deviations of integrated current in the Kipnis-Marchioro-Presutti (KMP) model of stochastic hear flow when starting from a step-like initial condition. The KMP model belongs to the hyperbolic universality class where diffusion remains relevant no matter how large the fluctuating current is. The extreme current statistics for the KMP model turns out to be sub-Gaussian, as distinguished from the super-Gaussian statistics found for the Symmetric Simple Exclusion Process and other models of the elliptic class. The most probable time history of the system, which dominates the extreme current statistics of the KMP model, involves two large-amplitude solitary pulses: of the energy density field and of the conjugate "momentum" field. The coupled pulses propagate with a constant speed, but their amplitudes slowly grow with time, as the energy density pulse collects most of the available energy on its way.

cond-mat.stat-mech↗

arXiv2013

Extreme Current Fluctuations in Lattice Gases: Beyond Nonequilibrium Steady States

We use the macroscopic fluctuation theory (MFT) to study large current fluctuations in non-stationary diffusive lattice gases. We identify two universality classes of these fluctuations which we call elliptic and hyperbolic. They emerge in the limit when the deterministic mass flux is small compared to the mass flux due to the shot noise. The two classes are determined by the sign of compressibility of \emph{effective fluid}, obtained by mapping the MFT into an inviscid hydrodynamics. An example of the elliptic class is the Symmetric Simple Exclusion Process where, for some initial conditions, we can solve the effective hydrodynamics exactly. This leads to a super-Gaussian extreme current statistics conjectured by Derrida and Gerschenfeld (2009) and yields the optimal path of the system. For models of the hyperbolic class the deterministic mass flux cannot be neglected, leading to a different extreme current statistics.

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↗

arXiv2012

Void formation in diffusive lattice gases

What is the probability that a macroscopic void will spontaneously arise, at a specified time T, in an initially homogeneous gas? We address this question for diffusive lattice gases, and also determine the most probable density history leading to the void formation. We employ the macroscopic fluctuation theory by Bertini et al and consider both annealed and quenched averaging procedures (the initial condition is allowed to fluctuate in the annealed setting). We show that in the annealed case the void formation probability is given by the equilibrium Boltzmann-Gibbs formula, so the probability is independent of T (and also of the void shape, as only the volume matters). In the quenched case, which is intrinsically non-equilibrium, we evaluate the void formation probability analytically for non-interacting random walkers and probe it numerically for the simple symmetric exclusion process. For voids that are small compared with the diffusion length, the equilibrium result for the void formation probability is recovered. We also re-derive our main results for non-interacting random walkers from an exact microscopic analysis.

cond-mat.stat-mech↗

arXiv2011

Velocity fluctuations of population fronts propagating into metastable states

The position of propagating population fronts fluctuates because of the discreteness of the individuals and stochastic character of processes of birth, death and migration. Here we consider a Markov model of a population front propagating into a metastable state, and focus on the weak noise limit. For typical, small fluctuations the front motion is diffusive, and we calculate the front diffusion coefficient. We also determine the probability distribution of rare, large fluctuations of the front position and, for a given average front velocity, find the most likely population density profile of the front. Implications of the theory for population extinction risk are briefly considered.

cond-mat.stat-mech↗

arXiv2011

Extinction rates of established spatial populations

This paper deals with extinction of an isolated population caused by intrinsic noise. We model the population dynamics in a "refuge" as a Markov process which involves births and deaths on discrete lattice sites and random migrations between neighboring sites. In extinction scenario I the zero population size is a repelling fixed point of the on-site deterministic dynamics. In extinction scenario II the zero population size is an attracting fixed point, corresponding to what is known in ecology as Allee effect. Assuming a large population size, we develop WKB (Wentzel-Kramers-Brillouin) approximation to the master equation. The resulting Hamilton's equations encode the most probable path of the population toward extinction and the mean time to extinction. In the fast-migration limit these equations coincide, up to a canonical transformation, with those obtained, in a different way, by Elgart and Kamenev (2004). We classify possible regimes of population extinction with and without an Allee effect and for different types of refuge and solve several examples analytically and numerically. For a very strong Allee effect the extinction problem can be mapped into the over-damped limit of theory of homogeneous nucleation due to Langer (1969). In this regime, and for very long systems, we predict an optimal refuge size that maximizes the mean time to extinction.

cond-mat.stat-mech↗

arXiv2010

Large fluctuations in stochastic population dynamics: momentum space calculations

Momentum-space representation renders an interesting perspective to theory of large fluctuations in populations undergoing Markovian stochastic gain-loss processes. This representation is obtained when the master equation for the probability distribution of the population size is transformed into an evolution equation for the probability generating function. Spectral decomposition then brings about an eigenvalue problem for a non-Hermitian linear differential operator. The ground-state eigenmode encodes the stationary distribution of the population size. For long-lived metastable populations which exhibit extinction or escape to another metastable state, the quasi-stationary distribution and the mean time to extinction or escape are encoded by the eigenmode and eigenvalue of the lowest excited state. If the average population size in the stationary or quasi-stationary state is large, the corresponding eigenvalue problem can be solved via WKB approximation amended by other asymptotic methods. We illustrate these ideas in several model examples.

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