We explore the transport features of a Brownian particle that walks in a periodic ratchet potential that is coupled with a spatially varying temperature background. Since the viscous friction of the medium decreases as the temperature of the medium increases, any reasonable exploration regarding the thermodynamic features of the Brownian engine should take into account the role of temperature on the viscosity of the fluid. In this work, we study this effect of temperature by considering a viscous friction that decreases exponentially as the background temperature increases. Our result depicts that the Brownian particle exhibits a fast unidirectional motion when the viscous friction is temperature dependent than that of constant viscous friction. Moreover the efficiency of this motor is considerably enhanced when the viscous friction is temperature dependent. On the hand, the motor exhibits a higher performance of the refrigerator when the viscose friction is taken to be constant.
We revisit the problem of the overdamped (large friction) limit of the Brownian dynamics in an inhomogeneous medium characterized by a position-dependent friction coefficient and a multiplicative noise (local temperature) in one space dimension. Starting from the Kramers equation and analyzing it through the expansion in terms of eigenfunctions of a quantum harmonic oscillator, we derive analytically the corresponding Fokker-Planck equation in the overdamped limit. The result is fully consistent with the previous finding by Sancho, San Miguel, and Dürr \cite{Sanc82}. Our method allows us to ge…
Exact power spectra of Brownian motion with solid friction
研究了一个朗之万方程，该方程描述了一个受粘性阻力、外力和固体库仑摩擦力作用的物体的布朗运动。在以前的工作[H。Touchette，E. Van der Straeten，W. J. Phys. A：Math. Theor. 43，445002，2010]，我们已经基于相应的Fokker-Planck方程的谱分解给出了该方程的速度传播子的精确解。在这里，我们提出了一个替代的，精确的解决方案的基础上，拉普拉斯变换的这个方程，它具有的优点是被表示在封闭的形式。从这个解中，我们还获得了速度自相关函数的拉普拉斯变换和功率谱的封闭形式表达式，即，自相关函数的傅里叶变换。作为干摩擦力和外部强迫的函数的功率谱的行为示出了已知在固体摩擦的存在下发生的粘滞和滑移制度之间的明显交叉。
We investigate piecewise-linear stochastic models as with regards to the probability distribution of functionals of the stochastic processes, a question which occurs frequently in large deviation theory. The functionals that we are looking into in detail are related to the time a stochastic process spends at a phase space point or in a phase space region, as well as to the motion with inertia. For a Langevin equation with discontinuous drift, we extend the so-called backward Fokker-Planck technique for nonnegative support functionals to arbitrary support functionals, to derive explicit express…
We present a first-principles thermodynamic approach to provide an alternative to the Langevin equation by identifying the deterministic (no stochastic component) microforce F_{k,BP} acting on a nonequilibrium Brownian particle (BP) in its kth microstate m_{k}. (The prefix micro refers to microstate quantities and carry a suffix k.) The deterministic new equation is easier to solve using basic calculus. Being oblivious to the second law, F_{k,BP} does not always oppose motion but viscous dissipation emerges upon ensemble averaging. The equipartition theorem is always satisfied. We reproduce we…
Considering viscous friction that varies spatially and temporally, the general expressions for entropy production, free energy, and entropy extraction rates are derived to a Brownian particle that walks in an overdamped and underdamped media. Via the well known stochastic approaches to underdamped and overdamped media, the thermodynamic expressions first derived at trajectory level then generalized to an ensemble level. To study the non-equilibrium thermodynamic features of a Brownian particle that hops in a medium where its viscosity varies on time, a Brownian particle that walks on a periodi…
We consider a spring-block model with both dry and viscous frictions, subjected to a periodic driving allowing mechanically stable configurations to be sampled. We show that under strong driving, the scaling of the correlation length with the energy density is incompatible with the prediction of Edwards statistical approach, which assumes a uniform sampling of mechanically stable configurations. A crossover between the Edwards scaling and the non-standard high energy scaling is observed at energy scales that depend on the viscous friction coefficient. Generalizing Edwards thermodynamics, we pr…
We study the Brownian dynamics of a solid particle on a vibrating solid surface. Phenomenologically, the interaction between the two solid surfaces is modeled by solid friction, and the Gaussian white noise models the vibration of the solid surface. The solid friction is proportional to the sign of relative velocity. We derive the Fokker-Planck (FP) equation for the time-dependent probability distribution to find the particle at a given location. We calculate analytically the steady state velocity distribution function, mean-squared velocity and diffusion coefficient in $d$ dimensions. We pres…
Frictional forces affect the rheology of hard-sphere colloids, at high shear rate. Here we demonstrate, via numerical simulations, that they also affect the dynamics of active Brownian particles, and their motility induced phase separation. Frictional forces increase the angular diffusivity of the particles, in the dilute phase, and prevent colliding particles from resolving their collision by sliding one past to the other. This leads to qualitatively changes of motility-induced phase diagram in the volume-fraction motility plane. While frictionless systems become unstable towards phase separa…
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…
The Caughey-Dieness process, also known as the Brownian motion with two valued drift, is used in theoretical physics as an advanced model of the Brownian particle velocity if the resistant force is assumed to be dry friction. This process also appears in many other fields, such as applied physics, mechanics, astrophysics, and pure mathematics. In the present paper we are concerned with a more general process, skew Brownian motion with dry friction. The probability distribution of the process itself and of its occupation time on the positive half line are studied. The approach based on the Puga…
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…
We propose a general framework to study transformations that drive an underdamped Brownian particle in contact with a thermal bath from an equilibrium state to a new one in an arbitrarily short time. To this end, we make use of a time and space-dependent potential, that plays a dual role: confine the particle, and manipulate the system. In the special case of an isothermal compression or decompression of a harmonically trapped particle, we derive explicit protocols that perform this quick transformation, following an inverse engineering method. We focus on the properties of these protocols, wh…
We study noise induced thermally activated barrier crossing of a Brownian particle that hops in a periodic ratchet potential where the ratchet potential is coupled with a spatially uniform temperature. The viscous friction is considered to decrease exponentially when the temperature $T$ of the medium increases as proposed originally by Reynolds. The results obtained in this work show that the mean first passage time of the particle is considerably lower when the viscous friction is temperature dependent than that of the case where the viscous friction is temperature independent. We then explor…
The Klein-Kramers equation, governing the Brownian motion of a classical particle in quantum environment under the action of an arbitrary external potential, is derived. Quantum temperature and friction operators are introduced and at large friction the corresponding Smoluchowski equation is obtained. Introducing the Bohm quantum potential, this Smoluchowski equation is extended to describe the Brownian motion of a quantum particle in quantum environment.
We analytically study the role of nonconservative forces, namely viscous couplings, on the statistical properties of the energy flux between two Brownian particles kept at different temperatures. From the dynamical model describing the system, we identify an energy flow that satisfies a fluctuation theorem both in the stationary and in transient states. In particular, for the specific case of a linear nonconservative interaction, we derive an exact fluctuation theorem that holds for any measurement time in the transient regime, and which involves the energy flux alone. Moreover, in this regime…
The Brownian motion of a light quantum particle in a heavy classical gas is theoretically described and a new expression for the friction coefficient is obtained for arbitrary temperature. At zero temperature it equals to the de Broglie momentum of the mean free path divided by the mean free path. Alternatively, the corresponding mobility of the quantum particle in the classical gas is equal to the square of the mean free path divided by the Planck constant. The Brownian motion of a quantum particle in a quantum environment is also discussed.
Tunable shear thickening in active frictional non-Brownian suspensions
School of Engineering, University of Edinburgh（爱丁堡大学工程学院）；Department of Physics, Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校物理系）
We develop a microscopic approach to the kinetic theory of many-particle systems with dissipative and potential interactions in presence of active fluctuations. The approach is based on a generalization of Bogolyubov--Peletminsky reduced description method applied to the systems of many active particles. It is shown that the microscopic approach developed allows to construct the kinetic theory of two- and three-dimensional systems of active particles in presence of non-linear friction (dissipative interaction) and an external random field with active fluctuations. The kinetic equations for the…
In this review article, we discuss recent studies on drops and bubbles in Hele-Shaw cells, focusing on how scaling laws exhibit crossovers from the three-dimensional counterparts and focusing on topics in which viscosity plays an important role. By virtue of progresses in analytical theory and high-speed imaging, dynamics of drops and bubbles have actively been studied with the aid of scaling arguments. However, compared with three dimensional problems, studies on the corresponding problems in Hele-Shaw cells are still limited. This review demonstrates that the effect of confinement in the Hel…
Phase separation in a low-density gas-like phase and a high-density liquid-like one is a common trait of biological and synthetic self-propelling particles' systems. The competition between motility and stochastic forces is assumed to fix the boundary between the homogeneous and the phase-separated phase. Here we demonstrate that motility does also promote the homogeneous phase allowing particles to resolve their collisions. This new understanding allows quantitatively predicting the spinodal-line of hard self-propelling Brownian particles, the prototypical model exhibiting a motility induced …
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…
