arXiv subjects

Baruch Meerson

Publications and source records attributed to Baruch Meerson.

At least 19 recordsLinked to original sources

arXiv2026

Slow heat-driven flow in a gas of hard disks

We study a slow heat-driven flow in a gas of elastically colliding hard disks confined to a long channel. The initial state consists of two regions with large temperature and density contrasts but nearly equal pressures, leading to a low-Mach-number, nearly isobaric evolution. In the dilute limit, the corresponding isobaric hydrodynamic theory reduces to a previously known ideal-gas description. We extend this theory to finite densities by incorporating a non-ideal equation of state of a hard-disk fluid, and solve the resulting one-dimensional equations numerically. Finite-density effects produce appreciable deviations from the ideal-gas prediction. We then test the theory directly against event-driven molecular dynamics simulations of hard disks and find very good agreement in both the dilute and finite-density regimes. The results provide, to our knowledge, the first particle-level test of isobaric gas dynamics of a strongly inhomogeneous cooling flow.

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arXiv2026

Lattice Brownian bees with cooperative reproduction: steady states, collapse, and spreading

We extend the ``Brownian bees'' model of Berestycki et al. (2021, 2022) to cooperative reproduction, $kA\to(k{+}1)A$, of a population of $N$ symmetric random walkers with removal, at each birth event, of the particle farthest from the origin. Working in the limit $N\to\infty$, we formulate a hydrodynamic free-boundary problem for this model. Using this formalism, we determine steady state population densities for all~$k$ and prove their linear stability for $k\le 2$ and instability for $k\ge 4$. In the marginal case $k=3$, there is a whole continuous family of steady states at a single, critical ratio of the reproduction and diffusion rates. Above criticality the population undergoes an asymptotically self-similar finite-time collapse to the origin. Below the criticality the population spreads diffusively, but the reproduction remains quantitatively relevant. For $k\ge 4$, the unstable steady state separates regimes of a finite-time collapse and a diffusive spreading. Here the collapse dynamics is asymptotically self-similar, and the population density exhibits a scale separation requiring a matched-asymptotic description. Our analytical predictions are confirmed by numerical solutions of the hydrodynamic free-boundary problem and by Monte Carlo simulations of the original microscopic model.

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arXiv2026

Emergent population dynamics of random walkers with cooperative reproduction and spatial selection

We extend the $N$ branching Brownian motions model of population invasion to higher-order asexual reproduction. Increasing reproduction order leads to qualitative changes: invasion fronts generically cease to exist beyond binary reproduction; and in the binary case itself, their speed becomes diffusion-independent. Ternary reproduction shows critical behavior, with collapse into a strongly localized `invasion bullet' in the supercritical regime, diffusive spreading in the subcritical regime, and a continuous family of fronts at criticality. These results suggest that the dominance of division and binary reproduction in nature reflects fundamental constraints on invasion dynamics.

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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, $\delta c$, 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.

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arXiv2026

Short-time statistics of extinction and blowup in reaction kinetics

We study the statistics of extinction and blowup times in well-mixed systems of stochastically reacting particles. We focus on the short-time tail, $T \to 0$, of the extinction- or blowup-time distribution $\mathcal{P}_m(T)$, where $m$ is the number of particles at $t=0$. This tail often exhibits an essential singularity at $T=0$, and we show that the singularity is captured by a time-dependent WKB (Wentzel-Kramers-Brillouin) approximation applied directly to the master equation. This approximation, however, leaves undetermined a large pre-exponential factor. We show how to calculate this factor by applying a leading- and a subleading-order WKB approximation to the Laplace-transformed backward master equation. Accurate asymptotic results can be obtained when this WKB solution can be matched to another approximate solution (the ``inner" solution), valid for not too large $m$. We demonstrate and verify this method on three examples of reactions which are also solvable without approximations.

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arXiv2025

Collective behavior of independent scaled Brownian particles with renewal resetting

We study fluctuations of an ensemble of $N$ independent particles undergoing anomalous diffusion with random renewal resetting. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process, characterized by a power-law time dependence of the diffusion coefficient, $D(t)\sim t^{2H-1}$, where $H>0$. The particles independently reset to the origin, and each particle's clock is set to zero upon spatial resetting. Employing the known steady-state position distribution of a \emph{single} particle undergoing the sBm with renewal resetting [Bodrova et al., Phys. Rev. E \textbf{100}, 012120 (2019)], we study the statistics of the system radius $\ell$ and of the center of mass (COM) of $N\gg 1$ particles. Typical fluctuations of $\ell$ fall under the Gumbel universality class for all $H>0$, and we use extreme value statistics to calculate the moments of $\ell$. We show that, for $H>1/2$, large deviations of the COM exhibit an anomalous scaling behavior. We also uncover a singularity in the corresponding rate function at $N\to\infty$, which is caused by a ``big jump" effect.

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arXiv2025

Age-structured hydrodynamics of ensembles of anomalously diffusing particles with renewal resetting

We develop an age-structured hydrodynamic (HD) theory which describes the collective behavior of $N\gg 1$ anomalously diffusing particles under stochastic renewal resetting. The theory treats the age of a particle -- the time since its last reset -- as an explicit dynamical variable and allows for resetting rules which introduce global inter-particle correlations. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process with independent increments, characterized by a power-law time dependence of the diffusion coefficient, $D(t)\sim t^{2H-1}$, where $H>0$. We apply this theory to three different resetting protocols: independent resetting to the origin (model~A), resetting to the origin of the particle farthest from it (model~B), and a scaled-diffusion extension of the ``Brownian bees" model of Berestycki et al, Ann. Probab. \textbf{50}, 2133 (2022). In all these models non-equilibrium steady states are reached at long times, and we determine the steady-state densities. For model A the (normalized to unity) steady-state density coincides with the steady-state probability density of a single particle undergoing sBM with resetting to the origin. For model B, and for the scaled Brownian bees, the HD steady-state densities are markedly different: in particular, they have compact supports for all $H>0$. The age-structured HD formalism can be extended to other anomalous diffusion processes with renewal resetting protocols which introduce global inter-particle correlations.

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arXiv2025

Anomalous scaling and phase transition in large deviations of dynamical observables of stationary Gaussian processes

We study large deviations, over a long time window $T \to \infty$, of the dynamical observables $A_n = \int_{0}^{T} x^n(t) dt$, $n=3,4,\dots$, where $x(t)$ is a centered stationary Gaussian process in continuous time. We show that, for short-correlated processes the probability density of $A_n$ exhibits an anomalous scaling $P(A_n,T) \sim \exp[-T^{\mu} f_n(\Delta A_n T^{-\nu})]$ at $T\to \infty$ while keeping $\Delta A_n T^{-\nu}$ constant. Here $\Delta A_n$ is the deviation of $A_n$ from its ensemble average. The anomalous exponents $\mu$ and $\nu$ depend on $n$ and are smaller than $1$, whereas the rate function $f_n(z)$ exhibits a first-order dynamical phase transition (DPT) which resembles condensation transitions observed in many systems. The same type of anomaly and DPT, with the same $\mu$ and $\nu$, was previously uncovered for the Ornstein-Uhlenbeck process - the only stationary Gaussian process which is also Markovian. We also uncover an anomalous behavior and a similar DPT in the long-correlated Gaussian processes. However, the anomalous exponents $\mu$ and $\nu$ are determined in this case not only by $n$ but also by the power-law long-time decay $\sim |t|^{-\alpha}$ of the covariance. The different anomalous scaling behavior is a consequence of a faster-than-linear scaling with $T$ of the variance of $A_n$. Finally, for sufficiently long-ranged correlations, $\alpha<2/n$, the DPT disappears, giving way to a smooth crossover between the regions of typical, Gaussian fluctuations and large deviations. The basic mechanism behind the DPT is the existence of strongly localized optimal paths of the process conditioned on very large $A_n$ and coexistence between the localized and delocalized paths of the conditioned process. Our theoretical predictions are corroborated by replica-exchange Wang-Landau simulations where we could probe probability densities down to $10^{-200}$.

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arXiv2025

Short-time blowup statistics of a Brownian particle in repulsive potentials

We study the dynamics of an overdamped Brownian particle in a repulsive scale-invariant potential $V(x) \sim -x^{n+1}$. For $n > 1$, a particle starting at position $x$ reaches infinity in a finite, randomly distributed time. We focus on the short-time tail $T \to 0$ of the probability distribution $P(T, x, n)$ of the blowup time $T$ for integer $n > 1$. Krapivsky and Meerson [Phys. Rev. E \textbf{112}, 024128 (2025)] recently evaluated the leading-order asymptotics of this tail, which exhibits an $n$-dependent essential singularity at $T = 0$. Here we provide a more accurate description of the $T \to 0$ tail by calculating, for all $n = 2, 3, \dots$, the previously unknown large pre-exponential factor of the blowup-time probability distribution. To this end, we apply a WKB approximation -- at both leading and subleading orders -- to the Laplace-transformed backward Fokker--Planck equation governing $P(T, x, n)$. For even $n$, the WKB solution alone suffices. For odd $n$, however, the WKB solution breaks down in a narrow boundary layer around $x = 0$. In this case, it must be supplemented by an ``internal'' solution and a matching procedure between the two solutions in their common region of validity.

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arXiv2025

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.

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arXiv2025

Fluctuations of blowup time in a simple model of a super-Malthusian catastrophe

Motivated by the paradigm of a super-Maltusian population catastrophe, we study a simple stochastic population model which exhibits a finite-time blowup of the population size and is strongly affected by intrinsic noise. We focus on the fluctuations of the blowup time $T$ in the asexual binary reproduction model $2A \to 3A$, where two identical individuals give birth to a third one. We determine exactly the average blowup time as well as the probability distribution $\mathcal{P}(T)$ of the blowup time and its moments. In particular, we show that the long-time tail $\mathcal{P}(T\to \infty)$ is purely exponential. The short-time tail $\mathcal{P}(T\to 0)$ exhibits an essential singularity at $T=0$, and it is dominated by a single (the most likely) population trajectory which we determine analytically.

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arXiv2025

Finite-time blowup of a Brownian particle in a repulsive potential

We consider a Brownian particle performing an overdamped motion in a power-law repulsive potential. If the potential grows with the distance faster than quadratically, the particle escapes to infinity in a finite time. We determine the average blowup time and study the probability distribution of the blowup time. In particular, we show that the long-time tail of this probability distribution decays purely exponentially, while the short-time tail exhibits an essential singularity. These qualitative features turn out to be quite universal, as they occur for all rapidly growing power-law potentials in arbitrary spatial dimensions. The quartic potential is especially tractable, and we analyze it in more detail.

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arXiv2024

Dynamical large deviations of the fractional Ornstein-Uhlenbeck process

The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation $\dot{x}(t)+\gamma x=\sqrt{2 D}\xi(t)$, where $\xi(t)$ is the fractional Gaussian noise with the Hurst exponent $0 1/2$) spectral density at zero frequency. For $H>1/2$, the fOU is long-correlated. Here we study dynamical large deviations of the fOU process and focus on the area $A_n=\int_{-T}^{T} x^n(t) dt$, $n=1,2,\ldots$ over a long time window $2T$. Employing the optimal fluctuation method, we determine the optimal path of the conditioned process, which dominates the large-$A_n$ tail of the probability distribution of the area, $\mathcal{P}(A_n,T)\sim \exp[-S(A_n,T)]$. We uncover a nontrivial phase diagram of scaling behaviors of the optimal paths and of the action $S(A_n\equiv 2 a_n T,T)\sim T^{\alpha(H,n)} a^{2/n}_n$ on the $(H,n)$ plane. The phase diagram includes three regions: (i) $H>1-1/n$, where $\alpha(H,n)=2-2H$, and the optimal paths are delocalized, (ii) $n=2$ and $H\leq \frac{1}{2}$, where $\alpha(H,n)=1$, and the optimal paths oscillate with an $H$-dependent frequency, and (iii) $H\leq 1-1/n$ and $n>2$, where $\alpha(H,n)=2/n$, and the optimal paths are strongly localized. We verify our theoretical predictions in large-deviation simulations of the fOU process. By combining the Wang-Landau Monte-Carlo algorithm with the circulant embedding method of generation of stationary Gaussian fields, we were able to measure probability densities as small as $10^{-170}$. We also generalize our findings to other stationary Gaussian processes with either diverging, or vanishing spectral density at zero frequency.

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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.

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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 $\alpha(H,m)$, where $0<H<1$ is the Hurst exponent. We determine $\alpha(H,m)$ 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 $\alpha(H,m)$ 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 $H<1/2$.

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arXiv2024

Full Statistics of Regularized Local Energy Density in a Freely Expanding Kipnis-Marchioro-Presutti Gas

We combine the Macroscopic Fluctuation Theory and the Inverse Scattering Method to determine the full long-time statistics of the energy density $u(x,t)$ averaged over a given spatial interval, $$U =\frac{1}{2L}\int_{-L}^{L}dx\, u(x,t),$$ in a freely expanding Kipnis-Marchioro-Presutti (KMP) lattice gas on the line, following the release at $t=0$ of a finite amount of energy at the origin. In particular, we show that, as time $t$ goes to infinity at fixed $L$, the large deviation function of $U$ approaches a universal, $L$-independent form when expressed in terms of the energy content of the interval $|x|<L$. A key part of the solution is the determination of the most likely configuration of the energy density at time $t$, conditional on $U$.

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arXiv2024

Relaxation and fluctuations of a mass- and dipole-conserving stochastic lattice gas

Han et al. [Phys. Rev. Lett. \textbf{132}, 137102 (2024)] have recently introduced a classical stochastic lattice gas model which, in addition to particle conservation, also conserves the particles' dipole moment. Because of its intrinsic nonlinearity this model exhibits unusual macroscopic scaling behaviors, different from those of lattice gases that conserve only the number of particles. Here we investigate some basic relaxation and fluctuation properties of this model at large scales and at long times. These properties crucially depend on whether the total number of particles is infinite or finite. We find similarity solutions, describing relaxation of the dipole-conserving gas (DCG) in several standard settings. A major part of our effort is an extension to this model of the macroscopic fluctuation theory (MFT), previously developed for lattice gases where only the number of particles is conserved. We apply the MFT to the calculation of the variance of nonequilibrium fluctuations of the excess number of particles on the positive semi-axis when starting from an (either deterministic, or random) constant density at $t=0$. Using the MFT, we also identify the equilibrium Boltzmann-Gibbs distribution for the DCG. Finally, based on these results, we determine the probability distribution of, and the most probable density history leading to, a large deviation in the form of a macroscopic void of a given size in an initially uniform DCG at equilibrium.

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arXiv2024

Thermally activated particle motion in biased correlated Gaussian disorder potentials

Thermally activated particle motion in disorder potentials is controlled by the large-$\Delta V$ tail of the distribution of height $\Delta V$ of the potential barriers created by the disorder. We employ the optimal fluctuation method to evaluate this tail for correlated quenched Gaussian potentials in one dimension in the presence of a small bias of the potential. We focus on the mean escape time (MET) of overdamped particles averaged over the disorder. We show that the bias leads to a strong (exponential) reduction of the MET in the direction along the bias. The reduction depends both on the bias, and on detailed properties of the covariance of the disorder, such as its derivatives and asymptotic behavior at large distances. We verify our theoretical predictions for the large-$\Delta V$ tail of the barrier height distribution, as well as earlier predictions of this tail for zero bias, by performing large-deviation simulations of the potential disorder. The simulations employ correlated random potential sampling based on the circulant embedding method and the Wang-Landau algorithm, which enable us to probe probability densities smaller than $10^{-1200}$.

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