arXiv subjects

Stanislav Burov

Publications and source records attributed to Stanislav Burov.

At least 19 recordsLinked to original sources

arXiv2026

A Gaussian-Remainder Hierarchy for Sums of Random Variables with Big-Jump Statistics

We develop an exact Gaussian-remainder hierarchy for the probability density of the sum of $N$ independent, identically distributed random variables with broad, finite-variance distribution for the summands. The hierarchy separates the Gaussian fixed-point contribution from residual sectors that retain the original single-summand density. For subexponential densities, the first nontrivial truncation yields a simple finite-$N$ approximation that involves one convolution with a Gaussian background and a subtraction that removes Gaussian overcounting. This approximation captures the Gaussian center, the crossover region, and the big-jump tail within a single expression, as demonstrated numerically for stretched-exponential and finite-variance power-law examples. The same first-order approximation reproduces the known asymptotic anomalous rate function for sums of stretched-exponential random variables and also provides an accurate approximation to the corresponding finite-$N$ rate function.

cond-mat.stat-mech↗

arXiv2026

From Continuous-Time Random Walks to Laplace Tails

During Brownian motion, the displacement is normally distributed, a classical fact aligned with the central limit theorem. However, single particle tracking in complex media such as glasses, living cells, and colloidal suspensions often reveals pronounced exponential decay of the displacement distribution, known as Laplace tails. In a short letter, two of us presented the emergence of Laplace tails in the continuous time random walk (CTRW) framework. Here, a detailed complementary study is presented. By exploring the behavior of $Q_t(n)$, the probability that exactly $n$ renewals occur during time $t$, we develop a rate function-like framework for this quantity, valid for finite $t$. We show that $Q_t(n)$ exhibits exponential tails, which in turn give rise to exponential tails of the positional probability density function $P(x,t)$. Favorable comparison to finite-time numerical simulations and asymptotic large deviation rate functions establishes the validity of our results over a wide temporal range.

cond-mat.stat-mech↗

arXiv2026

Conditional Ergodicity and Universal Fluctuations in Weak Ergodicity Breaking

Time averages extracted from single-particle trajectories in complex media often vary strongly from one trajectory to another, even for long measurement times. Such persistent trajectory-to trajectory scatter is commonly observed in anomalous diffusion and signals weak ergodicity breaking driven by scale-free trapping. Here we identify conditional ergodicity: conditioning on a natural internal clock restores self-averaging of time-averaged observables. Combining conditional ergodicity with the stochastic mapping between the internal clock and physical time implies a universal law: once rescaled by their mean, time-averaged transport coefficients in systems exhibiting weak ergodicity breaking follow the Mittag-Leffler distribution. We demonstrate this universality across multiple models of disordered media displaying anomalous diffusion.

cond-mat.stat-mech↗

arXiv2026

Boundary-Induced Drift and Negative Mobility in Constrained Stochastic Systems

We study overdamped stochastic dynamics confined by hard reflecting boundaries and show that the combination of boundary geometry and an anisotropic diffusion tensor generically generates directed motion. At the level of individual trajectories, the no-flux condition enforces an oblique reflection at the boundary, which produces a systematic drift parallel to the surface. The resulting local velocity takes the general form $v_B(\mathbf{x})=\mathbf{t}(\mathbf{x})^{\!\top}\mathbf{D}\,\mathbf{n}(\mathbf{x})$, determined by the diffusion tensor $\mathbf{D}$ and the local boundary geometry encoded in the normal $\mathbf{n}$ and tangent $\mathbf{t}$. While this boundary-induced drift is local, it can accumulate into a macroscopic response, depending on the statistics of boundary encounters. We illustrate how this local boundary-induced drift gives rise to macroscopic transport using a minimal one-dimensional dimer composed of two particles with unequal diffusion coefficients. The repeated collisions act as reflections in configuration space and lead to sustained center-of-mass motion, including regimes of absolute negative mobility under constant forcing.

cond-mat.stat-mech↗

arXiv2025

Synergy of Doob Transformation and Montroll Defect Theory for Random Walks in External Potentials

We present a systematic method for constructing stochastic processes by modifying simpler, analytically solvable random walks on discrete lattices. Our framework integrates the Doob $h$-transformation with the Montroll defect theory, overcoming the strict constraints associated with each method alone. By combining these two approaches, we map random walks in simple potentials onto processes involving more general external potentials and metastable states. Explicit analytical expressions relate the transformed process to the original one, facilitating direct investigation of exponential decay rates and additional dynamical modes. As an illustrative example, we demonstrate our method by analyzing a random walker in a linear potential modified to include a metastable state, revealing distinct exponential decay regimes relevant to escape dynamics.

cond-mat.stat-mech↗

arXiv2024

The Emergence of Laplace Universality in Correlated Processes

In transport processes across materials like glasses, living cells, and porous media, the probability density function of displacements exhibits exponential decay rather than Gaussian behavior. We show that this universal behavior of rare events, termed Laplace tails, emerges even when correlations and memory affect the dynamics. Using a renormalization-based approach, we demonstrate that correlations and memory do not suppress these tails but rather enhance their visibility, even at short timescales. The developed analytical framework refines the concept of correlations for rare events and enables the computation of effective parameters that govern Laplace tails in correlated processes. These findings suggest that correlations can serve as a tunable parameter to control the behavior of rare events in transport.

cond-mat.stat-mech↗

arXiv2024

Statistics of a Large Number of Renewals in Equilibrium and Non-Equilibrium Renewal Processes

The renewal process is a key statistical model for describing a wide range of stochastic systems in Physics. This work investigates the behavior of the probability distribution of the number of renewals in renewal processes in the short-time limit, with a focus on cases where the number of renewals is large. We find that the specific details of the sojourn time distribution $\phi(\tau)$ in this limit can significantly modify the behavior in the large-number-of-renewals regime. We explore both non-equilibrium and equilibrium renewal processes, deriving results for various forms of $\phi(\tau)$. Using saddle point approximations, we analyze cases where $\phi(\tau)$ follows a power-series expansion, includes a cutoff, or exhibits non-analytic behavior near $\tau = 0$. Additionally, we show how the short-time properties of $\phi(\tau)$ shape the decay of the number of renewals in equilibrium compared to non-equilibrium renewal processes. The probability of the number of renewals plays a crucial role in determining rare event behaviors, such as Laplace tails. The results obtained here are expected to help advance the development of a theoretical framework for rare events in transport processes in complex systems.

cond-mat.stat-mech↗

arXiv2024

Driven Lorentz model in discrete time

We consider a tracer particle performing a random walk on a two-dimensional lattice in the presence of immobile hard obstacles. Starting from equilibrium, a constant force pulling on the particle is switched on, driving the system to a new stationary state. Our study calculates displacement moments in discrete time (number of steps $N$) for an arbitrarily strong constant driving force, exact to first order in obstacle density. We find that for fixed driving force $F$, the approach to the terminal discrete velocity scales as $\sim N^{-1} \exp(- N F^2 / 16)$ for small $F$, differing significantly from the $\sim N^{-1}$ prediction of linear response. Besides a non-analytic dependence on the force and breakdown of Einstein's linear response, our results show that fluctuations in the directions of the force are enhanced in the presence of obstacles. Notably, the variance grows as $\sim N^3$ (superdiffusion) for $F \to \infty$ at intermediate steps, reverting to normal diffusion ($\sim N$) at larger steps, a behavior previously observed in continuous time but demonstrated here in discrete steps for the first time. Unlike the exponential waiting time case, the superdiffusion regime starts immediately at $N=1$. The framework presented allows considering any type of waiting-time distribution between steps and transition to continuous time using subordination methods. Our findings are also validated through computer simulations.

cond-mat.stat-mech↗

arXiv2024

Disorder-Induced Anomalous Mobility Enhancement in Confined Geometries

Strong, scale-free disorder disrupts typical transport properties like the Stokes-Einstein relation and linear response, leading to anomalous, non-diffusive motion observed in amorphous materials, glasses, living cells, and other systems. Our study reveals that the combination of scale-free quenched disorder and geometrical constraints induces unconventional single particle mobility behavior. Specifically, in a $2$-dimensional channel with width $w$, under external drive, tighter geometrical constraints (smaller $w$) enhance mobility. We derive an explicit form of the response to an external force by utilizing the double-subordination approach for the quenched trap model. The observed mobility enhancement occurs in the low-temperature regime where the distribution of localization times is scale-free.

cond-mat.stat-mech↗

arXiv2024

Laplace's first law of errors applied to diffusive motion

In biological, glassy, and active systems, various tracers exhibit Laplace-like, i.e., exponential, spreading of the diffusing packet of particles. The limitations of the central limit theorem in fully capturing the behaviors of such diffusive processes, especially in the tails, have been studied using the continuous time random walk model. For cases when the jump length distribution is super-exponential, e.g., a Gaussian, we use large deviations theory and relate it to the appearance of exponential tails. When the jump length distribution is sub-exponential the packet of spreading particles is described by the big jump principle. We demonstrate the applicability of our approach for finite time, indicating that rare events and the asymptotics of the large deviations rate function can be sampled for large length scales within a reasonably short measurement time.

cond-mat.stat-mech↗

arXiv2023

Continuous Approximation of Stochastic Maps for Modeling Asymmetric Cell Division

Cell size control and homeostasis is a major topic in cell biology yet to be fully understood. Several growth laws like the timer, adder, and sizer were proposed, and mathematical approaches that model cell growth and division were developed. This study focuses on utilizing stochastic map modeling for investigating asymmetric cell division. We establish a mapping between the description of cell growth and division and the Ornstein-Uhlenbeck process with dichotomous noise. We leverage this mapping to achieve analytical solutions and derive a closed-form expression for the stable cell size distribution under asymmetric division. To validate our findings, we conduct numerical simulations encompassing several cell growth scenarios. Our approach allows us to obtain a precise criterion for a bi-phasic behavior of the cell size. While for the case of the sizer scenario, a transition from the uni-modal phase to bi-modal is always possible, given sufficiently large asymmetry at the division, the affine-linear approximation of the adder scenario invariably yields uni-modal distribution.

q-bio.CB↗

arXiv2023

Emergence of Directed Motion in a Crowded Suspension of Overdamped Particles

In this work, we focus on the behavior of a single passive Brownian particle in a suspension of passive particles with short-range repulsive interactions and a larger self-diffusion coefficient. While the forces affecting the single-particle are thermal-like fluctuations and repulsion, due to other particles in the suspension, our numerical simulations show that on intermediate time scales directed motion on a single-particle level emerges. This emergent directional motion leads to a breakdown of the Einstein relation and non-monotonic augmentation of the measured diffusion coefficient. Directional tendency increases with the density of the suspension and leads to growth of the diffusivity with the density of the suspension, a phenomenon recently observed for a system of hard spheres by Ilker, Castellana, and Joanny. Counter-intuitively, the directional flow originates from the tendency of different particles to push each other out of their way. Due to such strictly repulsive interactions, nearby particles form into temporally correlated pairs and move cooperatively, thus creating a preferred direction of motion on intermediate time scales. We show that directional motion emerges when the ratio of the self-diffusion coefficients of the tracked particle and suspension constituents is below a critical value.

cond-mat.stat-mech↗

arXiv2023

Universal to Non-Universal Transition of the statistics of Rare Events During the Spread of Random Walks

Particle hopping is a common feature in heterogeneous media. We explore such motion by using the widely applicable formalism of the continuous time random walk and focus on the statistics of rare events. Numerous experiments have shown that the decay of the positional probability density function P (X, t), describing the statistics of rare events, exhibits universal exponential decay. We show that such universality ceases to exist once the threshold of exponential distribution of particle hops is crossed. While the mean hop is not diverging and can attain a finite value; the transition itself is critical. The exponential universality of rare events arises due to the contribution of all the different states occupied during the process. Once the reported threshold is crossed, a single large event determines the statistics. In this realm, the big jump principle replaces the large deviation principle, and the spatial part of the decay is unaffected by the temporal properties of rare events.

cond-mat.stat-mech↗

arXiv2022

Exponential Tails and Asymmetry Relations for the Spread of Biased Random Walks

Exponential, and not Gaussian, decay of probability density functions was studied by Laplace in the context of his analysis of errors. Such Laplace propagators for the diffusive motion of single particles in disordered media were recently observed in numerous experimental systems. What will happen to this universality when an external driving force is applied? Using the ubiquitous continuous time random walk with bias, and the Crooks relation in conjunction with large deviations theory, we derive two properties of the positional probability density function $P_F(x,t)$ that hold for a wide spectrum of random walk models: (I) Universal asymmetric exponential decay of $P_F(X,t)$ for large $|X|$, and (II) Existence of a time transformation that for large $|X|$ allows to express $P_F(X,t)$ in terms of the propagator of the unbiased process (measured at a shorter time). These findings allow us to establish how the symmetric exponential-like tails, measured in many unbiased processes, will transform into asymmetric Laplace tails when an external force is applied.

cond-mat.stat-mech↗

arXiv2022

Large deviations for continuous time random walks

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e. Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

cond-mat.stat-mech↗

arXiv2021

Self-assembly of two-dimensional, amorphous materials on a liquid substrate

Recent experimental utilization of liquid substrate in the production of two-dimensional crystals, such as graphene, together with a general interest in amorphous materials, raises the following question: is it beneficial to use a liquid substrate to optimize amorphous material production? Inspired by epitaxial growth, we use a two-dimensional coarse-grained model of interacting particles to show that introducing a motion for the substrate atoms improves the self-assembly process of particles that move on top of the substrate. We find that a specific amount of substrate liquidity (for a given sample temperature) is needed to achieve optimal self-assembly. Our results illustrate the opportunities that the combination of different degrees of freedom provides to the self-assembly processes.

cond-mat.stat-mech↗

arXiv2021

The case of the biased quenched trap model in two dimensions with diverging mean dwell times

We investigate the biased quenched trap model on top of a two-dimensional lattice in the case of diverging expected dwell times. By utilizing the double-subordination approach and calculating the return probability in $2$d, we explicitly obtain the disorder averaged probability density function of the particle's position as a function of time (for any given bias) in the limit of large times ($t \rightarrow \infty$). The first and second moments are calculated, and a formula for a general $μ$-th moment is found. The behavior of the first moment, i.e. $\langle x(t)\rangle$, presents non-linear response both in time and in the applied external force $F_0$. While the non-linearity in time occurs for any measurement time $t$, the non-linearity in $F_0$ is expected only when $t\gtrsim \big(F_0 \left| \ln (F_0)\right|\big)^{-2 / α}$ where $α=T/T_g$, for temperatures $T<T_g$. We support our analytic results by comparison to numerical simulations.

cond-mat.stat-mech↗

arXiv2020

Asymptotic $P_N$ Approximation in Radiative Transfer Problems

We study the validity of the time-dependent asymptotic $P_N$ approximation in radiative transfer of photons. The time-dependent asymptotic $P_N$ is an approximation which uses the standard $P_N$ equations with a closure that is based on the asymptotic solution of the exact Boltzmann equation for a homogeneous problem, in space and time. The asymptotic $P_N$ approximation for radiative transfer requires careful treatment regarding the closure equation. Specifically, the mean number of particles that are emitted per collision ($ω_{\mathrm{eff}}$) can be larger than one due to inner or outer radiation sources and the coefficients of the closure must be extended for these cases. Our approximation is tested against a well-known radiative transfer benchmark. It yields excellent results, with almost correct particle velocity that controls the radiative heat-wave fronts.

physics.comp-ph↗