Optimal barrier subdivision for Kramers’ escape rate

We examine the effect of subdividing the potential barrier along the reaction coordinate on Kramers’ escape rate for a model potential. Using the known supersymmetric potential approach, we show the existence of an optimal number of subdivisions that maximizes the rate.     

Entropic stochastic resonance of a self-propelled Janus particle

Entropic stochastic resonance is investigated when a self-propelled Janus particle moves in a double-cavity container. Numerical simulation results indicate the entropic stochastic resonance can survive even if there is no symmetry breaking in any direction. This is the essential distinction between the property of a self-propelled Janus particle and that of a passive Brownian particle, for the symmetry breaking is necessary for the entropic stochastic resonance of a passive Brownian particle. With the rotational noise intensity growing at small fixed noise intensity of translational motion, the signal power amplification increases monotonically towards saturation which also can be regarded as a kind of stochastic resonance effect. Besides, the increase in the natural frequency of the periodic driving depresses the degree of the stochastic resonance, whereas the rise in its amplitude enhances and then suppresses the behavior.     

Dynamics of a deformable self-propelled particle under external forcing

We investigate dynamics of a deformable self-propelled particle under external fields in two dimensions based on the time-evolution equations for the center of mass and a tensor variable characterizing deformations. We consider two kinds of external force. One is a gravitational force which enters additively in the time-evolution equation for the center of mass. The other is an electric force supposing that a dipole moment is induced in the particle. This force is added to the equation for the deformation tensor. It is shown that a rich variety of dynamics appears by changing the strength of the forces and the migration velocity of a self-propelled particle. A theoretical analysis is carried out to clarify the dynamics and the bifurcations.     

Cavitation inception on microparticles: a self-propelled particle accelerator

Corrugated, hydrophilic particles with diameters between 30 and 150 microm are found to cause cavitation inception at their surfaces when they are exposed to a short, intensive tensile stress wave. The growing cavity accelerates the particle into translatory motion until the tensile stress decreases, and subsequently the particle separates from the cavity. The cavity growth and particle detachment are modeled by considering the momentum of the particle and the displaced liquid. The analysis suggests that all particles which cause cavitation are accelerated into translatory motion, and separate from the cavities they themselves nucleate. Thus, in the research of cavitation nuclei the link is established between developed cavitation bubbles and their origin.     

Short time scales in the Kramers problem: a stepwise growth of the escape flux

We prove rigorously and demonstrate in simulations that, for a potential system staying initially at the bottom of a well, the escape flux over the barrier grows on times of the order of a period of eigenoscillation in a stepwise manner, provided that friction is small or moderate. If the initial state is not at the bottom of the well, then, typically, some of the steps transform into oscillations. The stepwise/oscillatory evolution at short times appears to be a generic feature of a noise-induced flux.     

Semianalytic solution of the Kramers exit problem for a small ferromagnetic particle

The distribution Q(t) of magnetization reversal times in a small uniaxial particle is computed here directly from Brown's Fokker-Planck equation. Constant applied field and axial symmetry are assumed. The Laplace transform of Q(t) has the form Q(z)=F(1)(z)/F(2)(z) where the regular functions F(i)(z) are defined by a solution of a Volterra integral equation. A separate integral equation is derived for the function dF(2)(z)/dz, and the poles and residues of Q(z) may then be found numerically with arbitary precision.     

Dynamics of a self-propelled particle under different driving modes in a channel flow

In this paper, a model that combines the lattice Boltzmann method with the singularity distribution method is proposed to simulate a self-propelled particle swimming(exhibiting translation and rotation) in a channel flow. The results show that the velocity distribution for a self-propelled particle swimming deviates from a Maxwellian distribution and exhibits highvelocity tails. The influence of an eccentric potential doublet on the translation velocity of the particle is significant. The velocity decay process can be described using a double exponential model form. No large differences in the velocity distribution were observed for different translation Reynolds numbers, rotation Reynolds numbers, or regular intervals.     

Noise-induced escape on time scales preceding quasistationarity: New developments in the Kramers problem

Noise-induced escape from the metastable part of a potential is considered on time scales preceding the formation of quasiequilibrium within that part of the potential. It is shown that, counterintuitively, the escape flux may then depend exponentially strongly, and in a complicated manner, on time and friction. (c) 2001 American Institute of Physics.     

Symmetry properties of the large-deviation function of the velocity of a self-propelled polar particle

A geometrically polar granular rod confined in 2D geometry, subjected to a sinusoidal vertical oscillation, undergoes noisy self-propulsion in a direction determined by its polarity. When surrounded by a medium of crystalline spherical beads, it displays substantial negative fluctuations in its velocity. We find that the large-deviation function (LDF) for the normalized velocity is strongly non-Gaussian with a kink at zero velocity, and that the antisymmetric part of the LDF is linear, resembling the fluctuation relation known for entropy production, even when the velocity distribution is clearly non-Gaussian. We extract an analogue of the phase-space contraction rate and find that it compares well with an independent estimate based on the persistence of forward and reverse velocities.     

State transitions and the continuum limit for a 2D interacting, self-propelled particle system

We study a class of swarming problems wherein particles evolve dynamically via pairwise interaction potentials and a velocity selection mechanism. We find that the swarming system undergoes various changes of state as a function of the self-propulsion and interaction potential parameters. In this paper, we utilize a procedure which connects a class of individual-based models to their continuum formulations and determine criteria for the validity of the latter. H-stability of the interaction potential plays a fundamental role in determining both the validity of the continuum approximation and the nature of the aggregation state transitions. We perform a linear stability analysis of the continuum model and compare the results to the simulations of the individual-based one.     

Kramers problem for a multiwell potential

Fluctuational escape from a multiwell potential is shown to display new features, as compared to the conventional single-well case. The flux J may depend on friction gamma exponentially strongly, over an exponentially long period; for small enough temperatures, J(gamma) undergoes marked oscillations in the range of small gamma, and the time evolution of J changes drastically as gamma exceeds a critical value.     

Kramers Equation and Supersymmetry

Hamilton’s equations with noise and friction possess a hidden supersymmetry, valid for time-independent as well as periodically time-dependent systems. It is used to derive topological properties of critical points and periodic trajectories in an elementary way. From a more practical point of view, the formalism provides new tools to study the reaction paths in systems with separated time scales. A ‘reduced current’ which contains the relevant part of the phase space probability current is introduced, together with strategies for its computation.     

Testing the energy diffusion approximation for the escape of a Brownian particle from a potential pocket

For the first time, the energy diffusion approximation is confronted at the percent level with the exact numerical modeling of thermal decay of a metastable state. This model is useful in many branches of natural sciences: e.g. in biology, nuclear physics, chemistry, etc. The exact (within the statistical errors about 2%) quasistationary decay rates result from the Langevin equations for the coordinate and conjugated momentum. For the energy (or action) diffusion approach, a Langevin-type equation for the action is constructed, validated, and solved numerically. The comparison of these two approaches is performed for four potentials (two of which are anharmonic) in a wide range of two dimensionless scaling parameters: i) the governing parameter G reflecting how high is the barrier with respect to the temperature and ii) the damping parameter φ expressing the friction strength. It turns out that the energy diffusion approach produces the rate which comes into 50%-agreement with the exact one only at φ < 0.02. Thus, we quantify, for the first time by our knowledge, the condition φ ≪ 1 known in the literature.     

粒子在二维开放型四分之一圆形微腔中的逃逸研究

利用半经典理论对粒子在开放型四分之一圆形微腔中的逃逸过程进行了研究,推导出了逃逸几率密度的计算公式.我们研究了一簇从四分之一圆形微腔的左下方的入口出射、并从该微腔右边界逃逸的粒子轨迹.对于粒子的每一条逃逸轨迹,记录下它的传播时间和逃逸的位置.结果发现逃逸时间图随着逃逸点的位置的变化曲线呈现出振荡结构.随着碰撞次数的增加,逃逸点的位置越靠近该腔的右顶端.对一系列的探测点,找到从源点出发到达探测点的轨迹,然后应用半经典理论来构造波函数,进而给出逃逸几率密度的计算公式.研究结果标明,逃逸几率密度与探测平面上逃逸点的位置、粒子的动量、初始出射角及与微腔的碰撞次数有关.为了更清楚的看出量子力学和经典力学之间的联系,我们对体系的半经典波函数进行傅里叶变换,给出了粒子的路径长度谱.路径长度谱的每个峰值对应于一条粒子逃逸轨迹的长度.本文的研究对理解量子力学和经典力学之间的联系以及研究粒子在微腔中的的逃逸和输运过程有一定的参考价值.     

粒子在二维开放型四分之一圆形微腔中的逃逸研究

利用半经典理论对粒子在开放型四分之一圆形微腔中的逃逸过程进行了研究,推导出了逃逸几率密度的计算公式。我们研究了一簇从四分之一圆形微腔的左下方的入口出射、并从该微腔右边界逃逸的粒子轨迹。对于粒子的每一条逃逸轨迹,记录下它的传播时间和逃逸的位置。结果发现逃逸时间图随着逃逸点的位置的变化曲线呈现出振荡结构。随着碰撞次数的增加,逃逸点的位置越靠近该腔的右顶端。对一系列的探测点,找到从源点出发到达探测点的轨迹,然后应用半经典理论来构造波函数,进而给出逃逸几率密度的计算公式。研究结果标明,逃逸几率密度与探测平面上逃逸点的位置、粒子的动量、初始出射角及与微腔的碰撞次数有关。为了更清楚的看出量子力学和经典力学之间的联系,我们对体系的半经典波函数进行傅里叶变换,给出了粒子的路径长度谱。路径长度谱的每个峰值对应于一条粒子逃逸轨迹的长度。本文的研究对理解量子力学和经典力学之间的联系以及研究粒子在微腔中的的逃逸和输运过程可以提供一定的参考价值。     

Crystallization in a dense suspension of self-propelled particles

Using Brownian dynamics computer simulations, we show that a two-dimensional suspension of self-propelled ("active") colloidal particles crystallizes at sufficiently high densities. Compared to the equilibrium freezing of passive particles, the freezing density is both significantly shifted and depends on the structural or dynamical criterion employed. In nonequilibrium the transition is accompanied by pronounced structural heterogeneities. This leads to a transition region between liquid and solid in which the suspension is globally ordered but unordered liquidlike "bubbles" still persist.     

A Remark on the Kramers Problem

We present new point of view on the old problem, the Kramers problem. The passages from the Fokker–Planck equation to the Smoluchowski equation, including corrections to the Smoluchowski current, is treated through an asymptotic expansion of the solution of the stochastic dynamical equations. The case of an extremely weak force of friction is also discussed.     

Structural asymmetry of Kramers clusters as a consequence of time-reversal symmetry

Asymmetry of Kramers magnetic clusters is shown to be dictated by the structure of magnetic four-color point groups. Factorization of the time-reversal operator is performed for Kramers systems.     

Unique definition of the Zeeman-splitting g tensor of a Kramers doublet

It has been widely accepted in the past that the g tensor describing the Zeeman splitting of a Kramers doublet is not uniquely defined. Here we show that with two basic requirements, (i) the invariance of the Zeeman Hamiltonian under symmetry transformations and (ii) its continuous change with the variations of the parameters of the system (geometry and crystal field), a unique determination of the elements of the g tensor is achieved.     

Kramers rate formula with coordinate-dependent mass

We study in this paper the classical stationary fission rate including a coordinate-dependent mass and a realistic potential. The results show that our numerical rate formula is a good approximation to the realistic Langevin simulation even if the nuclear temperature is not smaller than the fission barrier.