We present a numerical method that consistently implements thermal fluctuations and hydrodynamic interactions to the motion of Brownian particles dispersed in incompressible host fluids. In this method, the thermal fluctuations are introduced as random forces acting on the Brownian particles. The hydrodynamic interactions are introduced by directly resolving the fluid motions with the particle motion as a boundary condition to be satisfied. The validity of the method has been examined carefully by comparing the present numerical results with the fluctuation-dissipation theorem whose analytical form is known for dispersions of a single spherical particle. Simulations are then performed for more complicated systems, such as a dispersion composed of many spherical particles and a single polymeric chain in a solvent.
Motions of fluctuating Brownian particles in an incompressible viscous fluid have been studied by coupled simulations of Brownian particles and host fluid. We calculated the velocity autocorrelation functions of Brownian particles and compared them with the theoretical results. Extensive discussions have been made on the time scales for which our numerical model is valid.
Theory and simulation of Brownian colloids suspended in an implicit solvent, with the hydrodynamics of the fluid accounted for by effective interactions between the colloids, are shown to yield a marked and hitherto unobserved discrepancy between the viscosity calculated from the average shear stress under an imposed shear rate in the Stokesian regime and the viscosity extracted by the Green-Kubo formalism from the auto-correlations of thermal stress fluctuations in quiescent equilibrium. We show that agreement between both methods is recovered by accounting for the fluctuating Brownian stress…
In this work, we study the dynamics of a single active Brownian particle, as well as the collective behavior of interacting active Brownian particles, in a fluctuating heterogeneous environment. We employ a variant of the diffusing diffusivity model where the equation of motion of the active particle involves a time-dependent motility and diffusivities. Within our model, those fluctuations are coupled to each other. Using analytical methods, we obtain the probability distribution function of particle displacement and its moments for a single particle. We then investigate the impact of the envi…
The traction on the surface of a spherical active colloid in a thermally fluctuating Stokesian fluid contains passive, active, and Brownian contributions. Here we derive these three parts systematically, by "projecting out" the fluid using the boundary-domain integral representation of slow viscous flow. We find an exact relation between the statistics of the Brownian traction and the thermal forces in the fluid and derive, thereby, fluctuation-dissipation relations for every irreducible tensorial harmonic traction mode. The first two modes give the Brownian force and torque, from which we con…
Fluctuating hydrodynamics has been successfully combined with several computational methods to rapidly compute the correlated random velocities of Brownian particles. In the overdamped limit where both particle and fluid inertia are ignored, one must also account for a Brownian drift term in order to successfully update the particle positions. In this paper, we present an efficient computational method for the dynamic simulation of Brownian suspensions with fluctuating hydrodynamics that handles both computations and provides a similar approximation as Stokesian Dynamics for dilute and semidil…
We develop an inertial coupling method for modeling the dynamics of point-like 'blob' particles immersed in an incompressible fluid, generalizing previous work for compressible fluids. The coupling consistently includes excess (positive or negative) inertia of the particles relative to the displaced fluid, and accounts for thermal fluctuations in the fluid momentum equation. The coupling between the fluid and the blob is based on a no-slip constraint equating the particle velocity with the local average of the fluid velocity, and conserves momentum and energy. We demonstrate that the formulati…
We study the statistical properties of many Brownian particles under the We study the statistical properties of many Brownian particles under the influence of both a spatially homogeneous driving force and a periodic potential with period $\ell$ in a two-dimensional space. In particular, we focus on two asymptotic cases, $\ell_{\rm int} \ll \ell$ and $\ell_{\rm int} \gg \ell$, where $\ell_{\rm int}$ represents the interaction length between two particles. We derive fluctuating hydrodynamic equations describing the evolution of a coarse-grained density field defined on scales much larger than $…
The active Brownian particle (ABP) model describes a swimmer, synthetic or living, whose direction of swimming is a Brownian motion. The swimming is due to a propulsion force, and the fluctuations are typically thermal in origin. We present a 2D model where the fluctuations arise from nonthermal noise in a propelling force acting at a single point, such as that due to a flagellum. We take the overdamped limit and find several modifications to the traditional ABP model. Since the fluctuating force causes a fluctuating torque, the diffusion tensor describing the process has a coupling between tr…
We establish an effective Markov theory for the rotational Brownian motion of hot nanobeads and nanorods. Compact analytical expressions for the effective temperature and friction are derived from the fluctuating hydrodynamic equations of motion. They are verified by comparison with recent measurements and with GPU powered parallel molecular dynamics simulations over a wide temperature range. This provides unique insights into the physics of hot Brownian motion and an excellent starting point for further experimental tests and applications involving laser-heated nanobeads, nanorods and Janus p…
Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking
Tokyo University of Science（东京科学大学）；Pohang University of Science and Technology（浦项科技大学）；Asia Pacific Centre for Theoretical Physics（亚太理论物理中心）；University of Potsdam（波茨坦大学）；Wakayama University（和歌山大学）；Nagoya University（名古屋大学）；Keio University（庆应义塾大学）
The Brownian motion of a hot nanoparticle is described by an effective Markov theory based on fluctuating hydrodynamics. Its predictions are scrutinized over a wide temperature range using large-scale molecular dynamics simulations of a hot nanoparticle in a Lennard-Jones fluid. The particle positions and momenta are found to be Boltzmann distributed according to distinct effective temperatures $T_\mathrm{HBM}$ and $T_\mathrm{k}$ . For $T_\mathrm{HBM}$ we derive a formally exact theoretical prediction and establish a generalised Einstein relation that links it to directly measurable quantities…
We revisit the effective parameter description of hot Brownian motion -- a scenario where a colloidal particle is kept at an elevated temperature than the ambient fluid. Due to the time scale separation between heat diffusion and particle motion, a stationary halo of hot fluid is carried along with the particle, resulting in a spatially varying comoving temperature and viscosity profile. The resultant Brownian motion in the overdamped limit can be well described by a Langevin equation with effective parameters such as effective temperature $T_{\rm HBM}$ and friction coefficient $ζ_{\rm HBM}$ t…
We develop a Fluctuating Immersed Boundary (FIB) method for performing Brownian dynamics simulations of confined particle suspensions. Unlike traditional methods which employ analytical Green's functions for Stokes flow in the confined geometry, the FIB method uses a fluctuating finite-volume Stokes solver to generate the action of the response functions "on the fly". Importantly, we demonstrate that both the deterministic terms necessary to capture the hydrodynamic interactions among the suspended particles, as well as the stochastic terms necessary to generate the hydrodynamically-correlated…
The resolution of Brownian motion in simulations of micro-particle suspensions can be crucial to reproducing the correct dynamics of individual particles, as well as providing an accurate characterisation of suspension properties. Including these effects in simulations, however, can be computationally intensive due to the configuration dependent random displacements that would need to be determined at every time step. In this paper, we introduce the fluctuating force-coupling method (FCM) to overcome this difficulty, providing a fast approach to simulate colloidal suspensions at large-scale. W…
We show that the correlated stochastic fluctuation of the friction coefficient can give rise to long-range directional motion of a particle undergoing Brownian random walk in a constant periodic energy potential landscape. The occurrence of this motion requires the presence of two additional independent bodies interacting with the particle via friction and via the energy potential, respectively, which can move relative to each other. Such three-body system generalizes the classical Brownian ratchet mechanism, which requires only two interacting bodies. In particular, we describe a simple two-l…
We introduce a Rigid-Body Fluctuating Immersed Boundary (RB-FIB) method to perform large-scale Brownian dynamics simulations of suspensions of rigid particles in fully confined domains, without any need to explicitly construct Green's functions or mobility operators. In the RB-FIB approach, discretized fluctuating Stokes equations are solved with prescribed boundary conditions in conjunction with a rigid-body immersed boundary method to discretize arbitrarily-shaped colloidal particles with no-slip or active-slip prescribed on their surface. We design a specialized Split--Euler--Maruyama tempo…
Symmetries constrain dynamics. We test this fundamental physical principle, experimentally and by molecular dynamics simulations, for a hot Janus swimmer operating far from thermal equilibrium. Our results establish scalar and vectorial steady-state fluctuation theorems and a thermodynamic uncertainty relation that link the fluctuating particle current to its entropy production at an effective temperature. A Markovian minimal model elucidates the underlying non-equilbrium physics.
The lateral diffusion coefficient of a Brownian particle on a two-dimensional random surface is studied in the quenched limit for which the surface configuration is time-independent. We start with the stochastic equation of motion for a Brownian particle on a fluctuating surface, which has been derived by Naji and Brown. The mean square displacement of the particle projected on a base plane is calculated exactly under the condition that the surface with a constant shape has no spatial correlation. We prove that the obtained lateral diffusion coefficient is in between the known upper and lower …
Within Kirkwood theory, we study the translational diffusion coefficient of a single polymer chain in dilute solution, and focus on the small difference between the short--time Kirkwood value $D^{(K)}$ and the asymptotic long--time value $D$. We calculate this correction term by highly accurate large--scale Brownian Dynamics simulations, and show that it is in perfect agreement with the rigorous variational result $D < D^{(K)}$, and with Fixman's Green--Kubo formula, which is re--derived. This resolves the puzzle posed by earlier numerical results (Rey {\em et al.}, Macromolecules 24, 4666 (19…
We study the diffusion of a Brownian particle quadratically coupled to a thermally fluctuating field. In the weak coupling limit, a path-integral formulation allows to compute the effective diffusion coefficient in the cases of an active particle, that tends to suppress the field fluctuations, and of a passive particle, that only undergoes the field fluctuations. We show that the behavior is similar to what was previously found for a linear coupling: an active particle is always slowed down, whereas a passive particle is slowed down in a slow field and accelerated in a fast field. Numerical si…
An alternative derivation of Brownian motion is presented. Instead of supplementing the linearized Navier-Stokes equation with a fluctuating force, we directly assume a Gaussian action functional for solvent velocity fluctuations. Solvating a particle amounts to expelling the solvent and prescribing a boundary condition to the solvent on the interface that is shared with the solute. We study the dynamical effects of this boundary condition on the solvent and derive explicit expressions for the solvent mean flow and velocity correlations. Moreover, we show that the probability to observe solven…
We present a comprehensive computational study of the collective behavior emerging from the competition between self-propulsion, excluded volume interactions and velocity-alignment in a two-dimensionnal model of active particles. We consider an extension of the Active Brownian Particles model where the self-propulsion direction of the particles aligns with the one of their neighbors. We analyze the onset of collective motion (flocking) in a low-density regime (10% surface area) and show that it is mainly controlled by the strength of velocity-alignment interactions: the competition between sel…
The Brownian diffusion of micron-scale inclusions in freely suspended smectic A liquid crystal films a few nanometers thick and several millimeters in diameter depends strongly on the air surrounding the film. Near atmospheric pressure, the three-dimensionally coupled film/gas system is well described by Hughes/Pailthorpe/White hydrodynamic theory but at lower pressure (p < 70 torr), the diffusion coefficient increases substantially, tending in high vacuum toward the two-dimensional limit where it is determined by film size. In the absence of air, the films are found to be a nearly ideal physi…
Brownian dynamics of a self-propelled particle in linear shear flow is studied analytically by solving the Langevin equation and in simulation. The particle has a constant propagation speed along a fluctuating orientation and is additionally subjected to a constant torque. In two spatial dimensions, the mean trajectory and the mean square displacement (MSD) are calculated as functions of time t analytically. In general, the mean trajectories are cycloids that are modified by finite temperature effects. With regard to the MSD different regimes are identified where the MSD scales with t^a with a…
Although the dynamics of colloids in the vicinity of a solid interface has been widely characterized in the past, experimental studies of Brownian diffusion close to an air-water interface are rare and limited to particle-interface gap distances larger than the particle size. At the still unexplored lower distances, the dynamics is expected to be extremely sensitive to boundary conditions at the air-water interface. There, ad hoc experiments would provide a quantitative validation of predictions. Using a specially designed dual wave interferometric setup, the 3D dynamics of 9 micrometers diame…
We demonstrate experimentally that the long-range hydrodynamic interactions in an incompressible quasi 2D isotropic fluid result in an anisotropic viscous drag acting on elongated particles. The anisotropy of the drag is increasing with increasing ratio of the particle length to the hydrodynamic scale given by the Saffman-Delbrück length. The micro-rheology data for translational and rotational drags collected over three orders of magnitude of the effective particle length demonstrate the validity of the current theoretical approaches to the hydrodynamics in restricted geometry. The results al…
We study the one-dimensional motion of a Brownian particle inside a confinement described by two reactive boundaries which can partially reflect or absorb the particle. Understanding the effects of such boundaries is important in physics, chemistry and biology. We compute the probability density of the particle displacement exactly, from which we derive expressions for the survival probability and the mean absorption time as a function of the reactive coefficients. Furthermore, using the Feynman-Kac formalism, we investigate the reaction time profile, which is the fluctuating time spent by the…
The short-time motion of Brownian particles in an incompressible Newtonian fluid under shear, in which the fluid inertia becomes important, was investigated by direct numerical simulation of particulate flows. Three-dimensional simulations were performed, wherein external forces were introduced to approximately form Couette flows throughout the entire system with periodic boundary conditions. In order to examine the validity of the method, the mean square displacement of a single spherical particle in a simple shear flow was calculated, and these results were compared with a hydrodynamic analy…
We propose a stochastic process wherein molecular transport is mediated by asymmetric nucleation of domains on a one-dimensional substrate. Track-driven mechanisms of molecular transport arise in biophysical applications such as Holliday junction positioning and collagenase processivity. In contrast to molecular motors that hydrolyze nucleotide triphosphates and undergo a local molecular conformational change, we show that asymmetric nucleation of hydrolysis waves on a track can also result in directed motion of an attached particle. Asymmetrically cooperative kinetics between ``hydrolyzed'' a…
Brownian oscillator, i.e. a micron-sized or smaller particle trapped in a thermally fluctuating environment is studied. The confining harmonic potential can move with a constant velocity. As distinct from the standard Langevin theory, the chaotic force driving the particle is correlated in time. The dynamics of the particle is described by the generalized Langevin equation with the inertial term, a coloured noise force, and a memory integral. We consider two kinds of the memory in the system. The first one corresponds to the exponentially correlated noise in a weakly viscoelastic fluid and in …
We study diffusion of colloids on a fluid-fluid interface using particle simulations and fluctuating hydrodynamics. Diffusion on a two-dimensional interface with three-dimensional hydrodynamics is known to be anomalous, with the collective diffusion coefficient diverging like the inverse of the wavenumber. This unusual collective effect arises because of the compressibility of the fluid flow in the plane of the interface, and leads to a nonlinear nonlocal convolution term in the diffusion equation for the ensemble-averaged concentration. We extend the previous hydrodynamic theory to account fo…
A bottom-up approach to fluctuating hydrodynamics: Coarse-graining of stochastic lattice gases and the Dean-Kawasaki equation
Tata Institute of Fundamental Research（塔塔基础研究所）；Institut de Physique Théorique, CEA Saclay and CNRS (UMR 3681)（CEA萨克雷和CNRS（UMR 3681）理论物理研究所）；Laboratoire Charles Coulomb (L2C), Université de Montpellier and CNRS (UMR 5221)（蒙彼利埃大学和CNRS（UMR 5221）夏尔·库仑实验室）
We present an entirely microscopic formulation of viscoleasticity of a fluid starting from the microscopic Stokes-Oldroyd B Model assuming instantaneous hydrodynamic friction, and show that linearization leads to a form for the frequency dependent viscosity that can be directly applied to the Langevin equation. Interestingly, the calculated expression of viscosity can be directly mapped to the Jeffreys' model which is essentially macroscopic in nature with the bulk viscoelasticity of the fluid being characterized by a complex elastic modulus G(ω). Further, we demonstrate that the concerned Gre…
We investigate a model for a Stirling-like engine consisting of a passive Brownian particle confined by a harmonic potential and interacting with a suspension of active Brownian particles that self-propel in a viscous solvent, which cyclically operates under isothermal conditions by means of temporal variations of the trap stiffness and the self-propulsion speed of the active particles. We derive an effective stochastic equation of motion of the trapped Brownian particle, which includes a friction memory kernel as well as thermal and active fluctuating forces due to its coupling with the activ…
We have developed a new in situ method to calibrate optical tweezers experiments and simultaneously measure the size of the trapped particle or the viscosity of the surrounding fluid. The positional fluctuations of the trapped particle are recorded with a high-bandwidth photodetector. Next, we compute the mean-square displacement, as well as the velocity autocorrelation function of the sphere and compare it to the theory of Brownian motion including hydrodynamic memory effects. A careful measurement and analysis of the time scales characterizing the dynamics of the harmonically bound sphere fl…
We analyze the dynamics of a Brownian gas in contact with a heat bath in which large temperature fluctuations occur. There are two distinct time scales present, one describes the decay of the fluctuations in the temperature and the other one is associated with the establishment of local equilibrium. Although the gas has reached local equilibrium, there exist large fluctuations in an intensive parameter (temperature) which break the thermodynamic equilibrium with the heat bath. Thus the decay of the fluctuations in the intensive parameter is larger than the characteristic time for the establish…
We perform Brownian dynamics simulations of semiflexible colloidal sheets with hydrodynamic interactions and thermal fluctuations in shear flow. As a function of the ratio of bending rigidity to shear energy (a dimensionless quantity we denote $S$) and the ratio of bending rigidity to thermal energy, we observe a dynamical transition from stochastic flipping to crumpling and continuous tumbling. This dynamical transition is broadened by thermal fluctuations, and the value of $S$ at which it occurs is consistent with the onset of chaotic dynamics found for athermal sheets. The effects of differ…
The probability per unit time for a thermally activated Brownian particle to escape over a potential well is in general well-described by Kramers theory. Kramers showed that the escape time decreases exponentially with increasing barrier height. The dynamics slow down when the particle is charged and subjected to a Lorentz force due to an external magnetic field. This is evident via a rescaling of the diffusion coefficient entering as a prefactor in the Kramers escape rate without any impact on the barrier-height-dependent exponent. Here we show that the barrier height can be effectively chang…
Active Brownian particles (ABPs) and Run-and-Tumble particles (RTPs) both self-propel at fixed speed $v$ along a body-axis ${\bf u}$ that reorients either through slow angular diffusion (ABPs) or sudden complete randomisation (RTPs). We compare the physics of these two model systems both at microscopic and macroscopic scales. Using exact results for their steady-state distribution in the presence of external potentials, we show that they both admit the same effective equilibrium regime perturbatively that breaks down for stronger external potentials, in a model-dependent way. In the presence o…
Observation of the Brownian motion of a small probe interacting with its environment is one of the main strategies to characterize soft matter. Essentially two counteracting forces govern the motion of the Brownian particle. First, the particle is driven by the rapid collisions with the surrounding solvent molecules, referred to as thermal noise. Second, the friction between the particle and the viscous solvent damps its motion. Conventionally, the thermal force is assumed to be random and characterized by a white noise spectrum. Friction is assumed to be given by the Stokes drag, implying tha…
Recently, the Brownian dynamics of self-propelled (active) rod-like particles was explored to model the motion of colloidal microswimmers, catalytically-driven nanorods, and bacteria. Here, we generalize this description to biaxial particles with arbitrary shape and derive the corresponding Langevin equation for a self-propelled Brownian spinning top. The biaxial swimmer is exposed to a hydrodynamic Stokes friction force at low Reynolds numbers, to fluctuating random forces and torques as well as to an external and an internal (effective) force and torque. The latter quantities control its sel…
