Brownian motion occurs in a variety of fluids, from rare gases to liquids. The Langevin equation, describing friction and agitation forces in statistical balance, is one of the most successful ways to treat the phenomenon. In rare gases, it is appropriate to model both friction and agitation in terms of independent molecular impacts with the particle. But in relatively dense fluids, such as water and air at standard temperature and pressure, the mean free path between collisions of fluid molecules is much smaller than the size of the Brownian particle, and the friction is normally treated as a mesoscopic viscous effect described by Stokes' Law which treats the fluid as continuous. The appropriateness of using Stokes' Law will be discussed in terms of recent experimental research in the ballistic or "coasting" phase of motion occurring at a very short time scale. Given the mesoscopic nature of the friction force for relatively dense fluids, we should expect the agitation force to also be mesoscopic. But it is often unrealistically modeled as uncorrelated individual impacts. It has been suggested occasionally that mesoscopic pressure fluctuations are appropriate for denser fluids. The purpose of this paper is to model friction as a result of mesoscopic pressure fluctuations. First, the simple random walk will be used to approximate the time and space scales below which ballistic motion begins and diffusive motion ends. Following that, pressure fluctuations and the associated time scale will be introduced to explain Brownian motion. As successful as the pressure fluctuation model is for many fluids, it will be shown to fail for fluids like glycerin that have viscosities a thousand times and more that of water.
We derive an analytic expression for the mechanical pressure of a generic one-dimensional model of confined active Brownian particles (ABPs) that is valid for all values of Peclet number Pe and all confining scenarios. Our model reproduces the known scaling of bulk pressure with Pe^2 while in strong confinement pressure scales with Pe. Our analytic results are very well reproduced by simulations of ABPs in 2D. We use the pressure formula to calculate both the work performed by an active engine and its efficiency. In particular, efficiency is maximized for work cycles with finite period and not…
The extent to which active matter might be described by effective equilibrium concepts like temperature and pressure is currently being discussed intensely. Here we study the simplest model, an ideal gas of non-interacting active Brownian particles. While the mechanical pressure exerted onto confining walls has been linked to correlations between particles' positions and their orientations, we show that these correlations are entirely controlled by boundary effects. We also consider a definition of local pressure, which describes interparticle forces in terms of momentum exchange between diffe…
We discuss Onsager's thermodynamic formalism for transport coefficients and apply it to the calculation of the shear modulus and shear viscosity of a monodisperse system of repulsive particles. We focus on the concept of extensive "distance" and intensive "field" conjugated via a Fenchel-Legendre transform involving a thermodynamic(-like) potential, which allows to switch ensembles. Employing Brownian dynamics, we calculate both the shear modulus and the shear viscosity from strain fluctuations and show that they agree with direct calculations from strained and non-equilibrium simulations, res…
We study quasi two-dimensional, monodisperse systems of active Brownian particles (ABPs) for a range of activities, stiffnesses, and densities. We develop a microscopic, analytical method for predicting the dense phase structure formed after motility-induced phase separation (MIPS) has occurred, including the dense cluster's area fraction, interparticle pressure, and radius. Our predictions are in good agreement with our Brownian dynamics simulations. We, then, derive a continuum model to investigate the relationship between the predicted interparticle pressure, the swim pressure, and the macr…
Recently, it has been discovered that systems of Active Brownian particles (APB) at high density organise their velocities into coherent domains showing large spatial structures in the velocity field. Such a collective behavior occurs spontaneously, i.e. is not caused by any specific interparticle force favoring the alignment of the velocities. This phenomenon was investigated in the absence of thermal noise and in the overdamped regime where inertial forces could be neglected. In this work, we demonstrate through numerical simulations and theoretical analysis that the velocity alignment is a …
In contrast with a gas at thermodynamic equilibrium, the mean force exerted on a wall by a gas of active particles usually depends on the confining potential, thereby preventing a proper definition of mechanical pressure. In this paper, we investigate numerically the properties of a gas of underdamped self-propelled dumbbells subject to Brownian noise of increasing intensity, in order to understand how the notion of pressure is recovered as noise progressively masks the effects of self-propulsion and the system approaches thermodynamic equilibrium. The simulations performed for a mobile asymme…
We numerically study the relaxation dynamics and associated criticality of non-Brownian frictionless spheres below jamming in spatial dimensions $d=2$, $3$, $4$, and $8$, and in the mean-field Mari-Kurchan model. We discover non-trivial finite-size and volume fraction dependences of the relaxation time associated to the relaxation of unjammed packings. In particular, the relaxation time is shown to diverge logarithmically with system size at any density below jamming, and no critical exponent can characterise its behaviour approaching jamming. In mean-field, the relaxation time is instead well…
This article reports the modeling of inertial rotational Brownian motion as an Ornstein-Uhlenbeck process evolving on the cotangent bundle of the rotation group, SO(3). The benefit of this approach and the use of a different parameterization of rotations allows the handling of particles with arbitrary shapes, without requiring any simplifying assumptions on the shape or the structure of the viscosity tensors. The resultant Fokker-Planck equation for the joint orientation and angular momentum probability distribution can be solved approximately using an `ansatz' Gaussian distribution in exponen…
An alternative approach to the phonon theory of liquids: An analytical study of Frenkel frequency and heat capacity as a function of pressure and temperature
The motion of a Brownian particle in the presence of Coulomb friction and an asymmetric spatial potential was evaluated in this study. The system exhibits a ratchet effect, i.e., an average directed motion even in the absence of an external force, induced by the coupling of non-equilibrium conditions with the spatial asymmetry. Both the average motion and the fluctuations of the Brownian particle were analysed. The stationary velocity shows a non-monotonic behaviour as a function of both the temperature and the viscosity of the bath. The diffusion properties of the particle, which show several…
