The kinetics of two (2D) and three (3D) dimensional diffusion-assisted escaping of Brownian particles from a potential well in the presence of an external force is analyzed in detail. The kinetics is studied within the two-state model (TSM) proposed for processes in the absence of external force. The generalized variant of this model, taking into account the force effect, is proposed which is shown to be quite accurate for some shapes of the well both for 2D and 3D processes. Within the generalized TSM simple expressions for the well depopulation kinetics and, in particular, for the escape rate are obtained. The effect of the force ($F$) is shown to manifest itself in the escape rate dependence on the only parameter $φ= Fa/(2k_b T)$, where $a$ is the Onsager radius of the attractive part of the well $U(r)$, defined by the relation $|U(a)| \approx k_b T$. The limiting behavior of this dependence in the cases of weak and strong force is studied in detail both in 2D and 3D processes. Some applications of obtained results to the analysis of experiments are briefly discussed.
Active particles may happen to be confined in channels so narrow that they cannot overtake each other (Single File conditions). This interesting situation reveals nontrivial physical features as a consequence of the strong inter-particle correlations developed in collective rearrangements. We consider a minimal model for active Brownian particles with the aim of studying the modifications introduced by activity with respect to the classical (passive) Single File picture. Depending on whether their motion is dominated by translational or rotational diffusion, we find that active Brownian partic…
We discuss escape problem with the consideration of both the activity of particles and the roughness of potentials. we derive analytic expressions for the escape rate of a Brownian particle (ABP) in two types of rough potentials by employing the effective equilibrium approach and the Zwanzig method. We find that activity enhances the escape rate, but both the oscillating perturbation and the random amplitude hinder escaping.
Motivated by subdiffusive motion of bio-molecules observed in living cells we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time averaged mean squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular we study the degree of weak ergodicity breaking and scatter between different single trajecto…
Recently, there has been much interest in activity-induced phase separations in concentrated suspensions of "active Brownian particles" (ABPs), self-propelled spherical particles whose direction of motion relaxes through thermal rotational diffusion. To date, almost all these studies have been restricted to 2 dimensions. In this work we study activity-induced phase separation in 3D and compare the results with previous and new 2D simulations. To this end, we performed state-of-the-art Brownian dynamics simulations of up to 40 million ABPs -- such very large system sizes are unavoidable to evad…
The effect of an external force on the kinetics of diffusion-assisted escaping of Brownian particles from a potential well is analyzed in detail. The analysis is made within the two-state model of the process which is known to be valid in the deep well limit in the absence of external force. The generalized variant of this model, taking into account the effect of the force, is shown to be quite accurate as well for some shapes of the well. Within the generalized two-state model simple expressions for the well depopulation kinetics and, in particular, the for the escape rate are obtained. These…
The diffusion of a fractional Brownian particle passing over the saddle point is studied in the field of the metastable potential. The barrier escaping probability is found to be greatly related to the fractional exponent $α$. Properties are revealed to move reversely in the opposite direction of diffusion when $α$ is relatively large despite of the zero-approximating effective friction of the system. This is very anomalous to the standard Brownian motion.
We study non-equilibrium analogues of surface phase transitions in a minimal model of active particles in contact with a purely repulsive potential barrier that mimics a thin porous membrane. Under conditions of bulk motility-induced phase separation, the interaction strength $\varepsilon_w$ of the barrier controls the affinity of the dense phase for the barrier region. We uncover clear signatures of a wetting phase transition as $\varepsilon_w$ is varied. In common with its equilibrium counterpart, the character of this transition depends on the system dimensionality: a continuous transition …
We study here the extreme statistics of Brownian particles escaping from a cusp funnel: the fastest Brownian particles among $n$ follow an ensemble of optimal trajectories located near the shortest path from the source to the target. For the time of such first arrivers, we derive an asymptotic formula that differs from the classical narrow escape and dire strait obtained for the mean first passage time. Consequently, when particles are initially distributed at a given distance from a cusp, the fastest do see some properties characterizing the cusp geometry. Therefore, when many particles diffu…
We consider two random walkers starting at the same time $t=0$ from different points in space separated by a given distance $R$. We compute the average volume of the space visited by both walkers up to time $t$ as a function of $R$ and $t$ and dimensionality of space $d$. For $d<4$, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable $R/\sqrt{t}$. We provide general integral formulas for scaling functions for arbitrary dimensionality $d<4$. In contrast, we show that no scaling function exists for higher dimensionalities $d \geq 4$
Exact dynamics and dimensional effects in torque-driven trapped active Brownian particles
Indian Institute of Science Education and Research Mohali（印度科学教育研究学院莫哈利分校）；Tata Institute of Fundamental Research（塔塔基础研究所）；University of Vermont（佛蒙特大学）；Institute of Physics（物理研究所）；Homi Bhabha National Institute（霍米·巴巴国民研究所）
In this paper, we analyze the scaling properties of a model that has as limiting cases the diffusion-limited aggregation (DLA) and the ballistic aggregation (BA) models. This model allows us to control the radial and angular scaling of the patterns, as well as, their gap distributions. The particles added to the cluster can follow either ballistic trajectories, with probability $P_{ba}$, or random ones, with probability $P_{rw}=1-P_{ba}$. The patterns were characterized through several quantities, including those related to the radial and angular scaling. The fractal dimension as a function of…
We study here the escape time for the fastest diffusing particle from the boundary of an interval with point-sink killing sources. Killing represents a degradation that leads to the probabilistic removal of the moving Brownian particles. We compute asymptotically the mean time it takes for the fastest particle escaping alive and obtain the extreme statistic distribution. These computations relies on an explicit expression for the time dependent flux of the Fokker-Planck equation using the time dependent Green's function and Duhamel's formula. We obtain a general formula for several point-sink …
Chirality, confinement and dimensionality govern re-entrant transitions in active matter
Indian Institute of Science Education and Research Mohali（印度科学教育研究所莫哈利分校）；Tata Institute of Fundamental Research（塔塔基础研究所）；University of Vermont（佛蒙特大学）；Institute of Physics（物理研究所）；Homi Bhabha National Institute（霍米·巴哈国家研究所）
Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting particles, and their large scale behaviour depends on the overall growth geometry. We establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries, which preserves the local scale invariance and is independent of other properties such as the dimensionality. This relation generalizes standard conformal transfo…
Collective behavior of self-propelled particles is observed on a microscale for swimmers such as sperm and bacteria as well as for protein filaments in motility assays. The properties of such systems depend both on their dimensionality and the interactions between their particles. We introduce a model for self-propelled rods in two dimensions that interact via a separation-shifted Lennard-Jones potential. Due to the finite potential barrier, the rods are able to cross. This model allows us to efficiently simulate systems of self-propelled rods that effectively move in two dimensions but can oc…
