arXiv subjects

Stanislav Burov

Publications and source records attributed to Stanislav Burov.

29 records · Page 2Linked to original sources

arXiv2020

The Time-Dependent Asymptotic $P_N$ Approximation for the Transport Equation

In this study a spatio-temporal approach for the solution of the time-dependent Boltzmann (transport) equation is derived. Finding the exact solution using the Boltzmann equation for the general case is generally an open problem and approximate methods are usually used. One of the most common methods is the spherical harmonics method (the $P_N$ approximation), when the exact transport equation is replaced with a closed set of equations for the moments of the density, with some closure assumption. Unfortunately, the classic $P_N$ closure yields poor results with low-order $N$ in highly anisotropic problems. Specifically, the tails of the particle's positional distribution as attained by the $P_N$ approximation, are inaccurate compared to the true behavior. In this work we present a derivation of a linear closure that even for low-order approximation yields a solution that is superior to the classical $P_N$ approximation. This closure, is based on an asymptotic derivation, both for space and time, of the exact Boltzmann equation in infinite homogeneous media. We test this approximation with respect to the one-dimensional benchmark of the full Green function in infinite media. The convergence of the proposed approximation is also faster when compared to (classic or modified) $P_N$ approximation.

physics.comp-ph↗

arXiv2020

From diffusion in compartmentalized media to non-Gaussian random walks

In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the Brownian yet non-Gaussian diffusion that exhibits linear growth of the mean square displacement joined by the exponential shape of the positional probability density. We explore a microscopic model that gives rise to transient confinement, similar to the one observed for hop-diffusion on top of a cellular membrane. The compartmentalization of the media is achieved by introducing randomly placed, identical barriers. Using this model of a heterogeneous medium we derive a general class of random walks with simple jump rules that are dictated by the geometry of the compartments. Exponential decay of positional probability density is observed and we also quantify the significant decrease of the long time diffusion constant. Our results suggest that the observed exponential decay is a general feature of the transient regime in compartmentalized media.

cond-mat.stat-mech↗

arXiv2020

Limit Forms of the Distribution of the Number of Renewals

In this work the asymptotic properties of $Q_t(N)$ ,the probability of the number of renewals ($N$), that occur during time $t$ are explored. While the forms of the distribution at very long times, i.e. $t\to\infty$, are very well known and are related to the Gaussian Central Limit Theorem or the Lévy stable laws, the alternative limit of large number of renewals, i.e. $N\to\infty$, is much less noted. We address this limit of large $N$ and find that it attains a universal form that solely depends on the analytic properties of the distribution of renewal times. Explicit formulas for $Q_t(N)$ are provided, together with corrections for finite $N$ and the necessary conditions for convergence to the universal asymptotic limit. Our results show that the Large Deviations rate function for $N/t$ exists and attains an universal linear growth (up to logarithmic corrections) in the $N/t\to\infty$ limit. This result holds irrespective of the existence of mean renewal time or presence of power-law statistics.

cond-mat.stat-mech↗

arXiv2020

The Transient Case of The Quenched Trap Model

In this work the diffusion in the quenched trap model with diverging mean waiting times is examined. The approach of randomly stopped time is extensively applied in order to obtain asymptotically exact representation of the disorder averaged positional probability density function. We establish that the dimensionality and the geometric properties of the lattice, on top of which the disorder is imposed, dictate the plausibility of a mean-filed approximation that will only include annealed disorder. Specifically, for any case when the probability to return to the origin ($Q_0$) is less than $1$, i.e. the transient case, the quenched trap model can be mapped on to the continuous time random walk. The explicit form of the mapping is provided. In the case when an external force is applied on a tracer particle in a media described by the quenched trap model, the response to such force is calculated and a non-linear response for sufficiently low dimensionality is observed.

cond-mat.stat-mech↗

arXiv2020

Packets of Diffusing Particles Exhibit Universal Exponential Tails

Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function $P(X,t)$ of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells and bacteria suspensions. We show that such exponential behavior is generally valid in a large class of problems of transport in random media. By extending the Large Deviations approach for a continuous time random walk we uncover a general universal behavior for the decay of the density. It is found that fluctuations in the number of steps of the random walker, performed at finite time, lead to exponential decay (with logarithmic corrections) of $P(X,t)$. This universal behavior holds also for short times, a fact that makes experimental observations readily achievable.

cond-mat.stat-mech↗

arXiv2017

Stochastic Maps, Continuous Approximation and Stable Distribution

A continuous approximation framework for non-linear stochastic as well as deterministic discrete maps is developed. For the stochastic map with uncorelated Gaussian noise, by successively applying the Itô lemma, we obtain a Langevin type of equation. Specifically, we show how non-linear maps give rise to a Langevin description that involves multiplicative noise. The multiplicative nature of the noise induces an additional effective force, not present in the absence of noise. We further exploit the continuum description and provide an explicit formula for the stable distribution of the stochastic map and conditions for its existence. Our results are in good agreement with numerical simulations of several maps.

cond-mat.stat-mech↗

arXiv2017

From Quenched Disorder to Continuous Time Random Walk

This work focuses on quantitative representation of transport in systems with quenched disorder. Explicit mapping of the quenched trap model to continuous time random walk is presented. Linear temporal transformation: $t\to t/Λ^{1/α}$ for transient process on translationally invariant lattice, in the sub-diffusive regime, is sufficient for asymptotic mapping. Exact form of the constant $Λ^{1/α}$ is established. Disorder averaged position probability density function for quenched trap model is obtained and analytic expressions for the diffusion coefficient and drift are provided.

cond-mat.stat-mech↗

arXiv2017

Effective Potential for Cellular Size Control

For various species of biological cells, experimental observations indicate the existence of universal distributions of the cellular size, scaling relations between the cell-size moments and simple rules for the cell-size control. We address a class of models for the control of cell division, and present the steady state distributions. By introducing concepts such as effective force and potential, we are able to address the appearance of scaling collapse of different distributions and the connection between various moments of the cell-size. Our approach allows us to derive strict bounds which a potential cell-size control scenario must meet in order to yield a steady state distribution. The so-called "adder" model for cell-size control exhibits the weakest control that still enables the existence of stable size distribution, a fact that might explain the relative "popularity" of this scenario for different cells.

cond-mat.soft↗

arXiv2016

The Noisy Oscillator : Random Mass and Random Damping

The problem of a linear damped noisy oscillator is treated in the presence of two multiplicative sources of noise which imply a random mass and random damping. The additive noise and the noise in the damping are responsible for an influx of energy to the oscillator and its dissipation to the surrounding environment. A random mass implies that the surrounding molecules not only collide with the oscillator but may also adhere to it, thereby changing its mass. We present general formulas for the first two moments and address the question of mean and energetic stabilities. The phenomenon of stochastic resonance, i.e. the expansion due to the noise of a system response to an external periodic signal, is considered for separate and joint action of two sources of noise and their characteristics.

cond-mat.stat-mech↗

arXiv2016

A cycling state that can lead to glassy dynamics in intracellular transport

Power-law dwell times have been observed for molecular motors in living cells, but the origins of these trapped states are not known. We introduce a minimal model of motors moving on a two-dimensional network of filaments, and simulations of its dynamics exhibit statistics comparable to those observed experimentally. Analysis of the model trajectories, as well as experimental particle tracking data, reveals a state in which motors cycle unproductively at junctions of three or more filaments. We formulate a master equation for these junction dynamics and show that the time required to escape from this vortex-like state can account for the power-law dwell times. We identify trends in the dynamics with the motor valency for further experimental validation. We demonstrate that these trends exist in individual trajectories of myosin II on an actin network. We discuss how cells could regulate intracellular transport and, in turn, biological function, by controlling their cytoskeletal network structures locally.

physics.bio-ph↗

arXiv2014

Scaling laws governing stochastic growth and division of single bacterial cells

Uncovering the quantitative laws that govern the growth and division of single cells remains a major challenge. Using a unique combination of technologies that yields unprecedented statistical precision, we find that the sizes of individual Caulobacter crescentus cells increase exponentially in time. We also establish that they divide upon reaching a critical multiple ($\approx$1.8) of their initial sizes, rather than an absolute size. We show that when the temperature is varied, the growth and division timescales scale proportionally with each other over the physiological temperature range. Strikingly, the cell-size and division-time distributions can both be rescaled by their mean values such that the condition-specific distributions collapse to universal curves. We account for these observations with a minimal stochastic model that is based on an autocatalytic cycle. It predicts the scalings, as well as specific functional forms for the universal curves. Our experimental and theoretical analysis reveals a simple physical principle governing these complex biological processes: a single temperature-dependent scale of cellular time governs the stochastic dynamics of growth and division in balanced growth conditions.

physics.bio-ph↗