This work proposes a method for the two-dimensional simulation of Brownian particles in a fluid with restrictions. The method is based on simple numerical rules between two matrices. One of the matrix represent the identification of all particles over which are adapted statistics rules for particles movement, the results are mapped over other matrix which represent the particles positions. The rules for the movement are established by a statistic mechanism allowing assign random or non-random movement direction. The same probably of movement for each direction for each time step is assumed, in order to be agreed with the physics conditions of Brownian movement in a two dimensional network. The root mean square displacement of all particles was calculated in a large number of simulations, together with the translational velocity of particles in order to compare with theoretical values of diffusion coefficient and the validation of model. Also, the time duration for some simulations vs. the number of particles and concentration was calculated.
We use computer simulations to test a simple idea for mapping between long-time self diffusivities obtained from molecular and Brownian dynamics. The strategy we explore is motivated by the behavior of fluids comprising particles that interact via inverse-power-law pair potentials, which serve as good reference models for dense atomic or colloidal materials. Based on our simulation data, we present an empirical expression that semi-quantitatively describes the "atomic" to "colloidal" diffusivity mapping for inverse-power-law fluids, but also for model complex fluids with considerably softer (s…
It is well known that path probabilities of Brownian motion correspond to the equilibrium configurational probabilities of flexible Gaussian polymers, while those of active Brownian motion correspond to in-extensible semiflexible polymers. Here we investigate the properties of the equilibrium polymer that corresponds to the trajectories of particles acted on simultaneously by both Brownian as well as active noise. Through this mapping we can see interesting crossovers in mechanical properties of the polymer with changing contour length. The polymer end-to-end distribution exhibits Gaussian beh…
The solution of a Caputo time fractional diffusion equation of order $0<α<1$ is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\left(N^2\right)$ to $O\left(N^α\right)$, given a precomputation of $O\left(N^{1+α}\ln N\right)$. The mapping is applied successfully …
We look at the equilibrium of a Brownian particle in an inhomogeneous space following the alternative approach proposed in ref.[1]. We consider a coordinate dependent damping that makes the stochastic dynamics the one with multiplicative noise. Here we show that the mapping to an additive noise gives the equilibrium distribution of the generalized Langevin dynamics of a particle with mass. The procedure does not need inclusion of any ad hoc current cancelling term in the Langevin dynamics. The result shows a modified Maxwell-Boltzmann distribution with a damping dependent amplitude.
We report a theoretical study of an overdamped Brownian particle dynamics in the presence of both a spatially modulated one-dimensional periodic potential and a periodic alternating force (AF). As the periodic potential has a low symmetry (a ratchet potential) the Brownian particle displays a broken symmetry motion with a nonzero time average velocity. By making use of the Green function method and a mapping to the theory of Brillouin bands the probability distribution of the particle coordinate is derived and the nonlinear dependence of the macroscopic velocity on the frequency and the amplit…
We explore the relation between active Brownian particles, a minimal particle-based model for active matter, and scalar field theories. Both show a liquid-gas-like phase transition towards stable coexistence of a dense liquid with a dilute active gas that terminates in a critical point. However, a comprehensive mapping between the particle-based model parameters and the effective coefficients governing the field theories has not been established yet. We discuss conflicting recent numerical results for the critical exponents of active Brownian particles in two dimensions. Starting from the inte…
Simulation studies of the phase diagram of repulsive active Brownian particles in three dimensions reveal that the region of motility-induced phase separation between a high and low density phase is enclosed by a region of gas-crystal phase separation. Near-critical loci and structural crossovers can additionally be identified in analogy with simple fluids. Motivated by the striking similarity to the behaviour of equilibrium fluids with short-ranged pair-wise attractions, we show that a direct mapping to pair potentials in the dilute limit implies interactions that are insufficiently attractiv…
We revisit the problem of Brownian diffusion with drift in order to study finite-size effects in the geometric Galton-Watson branching process. This is possible because of an exact mapping between one-dimensional random walks and geometric branching processes, known as the Harris walk. In this way, first-passage times of Brownian particles are equivalent to sizes of trees in the branching process (up to a factor of proportionality). Brownian particles that reach a distant boundary correspond to percolating trees, and those that do not correspond to non-percolating trees. In fact, both systems …
Recently, we have derived an effective Cahn-Hilliard equation for the phase separation dynamics of active Brownian particles by performing a weakly non-linear analysis of the effective hydrodynamic equations for density and polarization [Phys. Rev. Lett. 112, 218304 (2014)]. Here we develop and explore this strategy in more detail and show explicitly how to get to such a large-scale, mean-field description starting from the microscopic dynamics. The effective free energy emerging from this approach has the form of a conventional Ginzburg-Landau function. On the coarsest scale, our results thus…
We numerically study active Brownian particles that can respond to environmental cues through a small set of actions (switching their motility and turning left or right with respect to some direction) which are motivated by recent experiments with colloidal self-propelled Janus particles. We employ reinforcement learning to find optimal mappings between the state of particles and these actions. Specifically, we first consider a predator-prey situation in which prey particles try to avoid a predator. Using as reward the squared distance from the predator, we discuss the merits of three state-ac…
We consider chirality in active systems by exemplarily studying the phase behavior of planar systems of interacting Brownian circle swimmers with a spherical shape. Continuing previous work presented in [G.-J. Liao, S. H. L. Klapp, Soft Matter, 2018, 14, 7873-7882], we derive a predictive field theory that is able to describe the collective dynamics of circle swimmers. The theory yields a mapping between circle swimmers and noncircling active Brownian particles and predicts that the angular propulsion of the particles leads to a suppression of their motility-induced phase separation, being in …
Active Brownian Dynamics in Channels: First-Passage and Spatiotemporal Properties via Siegmund Duality
Max-Planck-Institut für Physik komplexer Systeme（马克斯·普朗克复杂系统物理研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Cluster of Excellence, Physics of Life, TU Dresden（德累斯顿工业大学生命物理卓越集群）
We consider any dynamical system that starts from a given ensemble of configurations and evolves in time until the system reaches a certain fixed stopping criterion, with the mean first-passage time the quantity of interest. We present a general method, Projective Dynamics, which maps the multi-dimensional dynamics of the system onto an arbitrary discrete set of states ${ζ_k}$, subject only to the constraint that the dynamics is restricted to transitions not further than the neighboring states $ζ_{k\pm 1}$. We prove that with this imposed condition there exists a master equation with nearest-n…
Brownian simulations can be used to generate statistics relevant for studying molecular interactions or trafficking. However, the concurrent simulation of many Brownian trajectories at can become computationally intractable. Replacing detailed Brownian simulations by a rate model was the basis of Gillespie's algorithm, but requires one to disregard spatial information. However, this information is crucial in molecular and cellular biology. Alternatively one can use a hybrid approach, generating Brownian paths only in a small region where the spatial organization is relevant and avoiding it in …
Modern view of activated rate processes: unidirectional fluxes at equilibrium, correlation functions, and splitting probabilities
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Oden Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；Department of Chemistry, University of Texas at Austin（德克萨斯大学奥斯汀分校化学系）
The adhesive dynamics of a one-dimensional aggregating gas of point particles is rigorously described. The infinite hierarchy of kinetic equations for the distributions of clusters of nearest neighbours is shown to be equivalent to a system of two coupled equations for a large class of initial conditions. The solution to these nonlinear equations is found by a direct construction of the relevant probability distributions in the limit of a continuous initial mass distribution. We show that those limiting distributions are identical to those of the statistics of shocks in the Burgers turbulence.…
A distinguishing feature of active particles is the nature of the non-equilibrium noise driving their dynamics. Control of these noise properties is, therefore, of both fundamental and applied interest. We demonstrate emergent tuning of the active noise of a granular self-propelled particle by confining it to a quasi one-dimensional channel. We find that this particle, moving like an active Brownian particle (ABP) in two-dimensions, displays run-and-tumble (RTP) characteristics in confinement. We show that the dynamics of the relative orientation co-ordinate of the particle maps to that of a B…
Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials
King’s College London（伦敦国王学院）；Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS and Université PSL, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎高等师范学院，巴黎文理研究大学，索邦大学，巴黎西岱大学）
Recent experiments with self-phoretic particles at low concentrations show a pronounced dynamic clustering [I. Theurkauff \emph{et al.}, Phys.\ Rev.\ Lett.\ \textbf{108}, 268303 (2012)]. We model this situation by taking into account the translational and rotational diffusiophoretic motion, which the active particles perform in their self-generated chemical field. Our Brownian dynamics simulations show pronounced dynamic clustering only when these two phoretic contributions give rise to competing attractive and repulsive interactions, respectively. We identify two dynamic clustering states and…
Fourier acceleration has been successfully applied to the simulation of lattice field theories for more than a decade. In this paper, we extend the method to the dynamics of discrete particles moving in continuum. Although our method is based on a mapping of the particles' dynamics to a regular grid so that discrete Fourier transforms may be taken, it should be emphasized that the introduction of the grid is a purely algorithmic device and that no smoothing, coarse-graining or mean-field approximations are made. The method thus can be applied to the equations of motion of molecular dynamics (M…
