Rectification of thermal fluctuations in ideal gases

Effective Langevin equations leading to large deviation function of time-averaged velocity for a nonequilibrium Rayleigh piston

We study fluctuating dynamics of a freely movable piston that separates an infinite cylinder into two regions filled with ideal gas particles at the same pressure but different temperatures. To investigate statistical properties of the time-averaged velocity of the piston in the long-time limit, we perturbatively calculate the large deviation function of the time-averaged velocity. Then, we derive an infinite number of effective Langevin equations yielding the same large deviation function as in the original model. Finally, we provide two possibilities for uniquely determining the form of the effective model.

Additivity of multiple heat reservoirs in the Langevin equation

The Langevin equation greatly simplifies the mathematical expression of the effects of thermal noise by using only two terms, a dissipation term, and a random-noise term. The Langevin description was originally applied to a system in contact with a single heat reservoir; however, many recent studies have also adopted a Langevin description for systems connected to multiple heat reservoirs. This is accomplished through the introduction of a simple summation for the dissipation and random-noise terms associated with each reservoir. However, the validity of this simple addition has been the focus of only limited discussion and has raised several criticisms. Moreover, this additive description has never been either experimentally or numerically verified, rendering its validity is still an open question. Here we perform molecular dynamics simulations for a Brownian system in simultaneous contact with multiple heat reservoirs to check the validity of this additive approach. Our simulation results confirm that the effect of multiple heat reservoirs is additive in general. A very small deviation in the total amount of dissipation and associated noise is found but seems not significant within statistical errors. We find that the steady-state properties satisfy the additivity perfectly and are not affected by this deviation.

Dynamics of a mechanical system with multiple degrees of freedom out of thermal equilibrium

Out of thermal equilibrium, an environment imposes effective mechanical forces on nanofabricated devices as well as on microscopic chemical or biological systems. Here we address the question of how to calculate these forces together with the response of the system from first principles. We show that an ideal gaslike environment, even near thermal equilibrium, can enforce a specific steady state on the system by creating effective potentials in otherwise homogeneous space. An example of stable and unstable rectifications of thermal fluctuations is presented using a modified Feynman-Smoluchowski ratchet with two degrees of freedom. Moreover, the stability of a steady configuration depends on its chiral symmetry. The transition rate probabilities and the corresponding kinetic equations are derived for a complex mechanical system with arbitrary degrees of freedom. This work, therefore, extends the applicability of mechanical systems as a toy model playground of statistical physics for active and living matter with multiple degrees of freedom.

Modeling multiphase flow using fluctuating hydrodynamics

Fluctuating hydrodynamics provides a model for fluids at mesoscopic scales where thermal fluctuations can have a significant impact on the behavior of the system. Here we investigate a model for fluctuating hydrodynamics of a single-component, multiphase flow in the neighborhood of the critical point. The system is modeled using a compressible flow formulation with a van der Waals equation of state, incorporating a Korteweg stress term to treat interfacial tension. We present a numerical algorithm for modeling this system based on an extension of algorithms developed for fluctuating hydrodynamics for ideal fluids. The scheme is validated by comparison of measured structure factors and capillary wave spectra with equilibrium theory. We also present several nonequilibrium examples to illustrate the capability of the algorithm to model multiphase fluid phenomena in a neighborhood of the critical point. These examples include a study of the impact of fluctuations on the spinodal decomposition following a rapid quench, as well as the piston effect in a cavity with supercooled walls. The conclusion in both cases is that thermal fluctuations affect the size and growth of the domains in off-critical quenches.

Granular Brownian motors: role of gas anisotropy and inelasticity

We investigate the motion of a two-dimensional wedge-shaped object (a granular Brownian motor), which is restricted to move along the x axis and cannot rotate as gas particles collide with it. We show that its steady-state drift, resulting from inelastic gas-motor collisions, is dramatically affected by anisotropy in the velocity distribution of the gas. We identify the dimensionless parameter providing the dependence of this drift on shape, masses, inelasticity, and anisotropy: The anisotropy leads to dramatically enhanced drift of the motor, which should easily be visible in experimental realizations.

Nonequilibrium fluctuations in a frictional granular motor: experiments and kinetic theory

We report the study of an experimental granular Brownian motor, inspired by the one published in Eshuis et al. [Phys. Rev. Lett. 104, 248001 (2010)], but different in some ingredients. As in that previous work, the motor is constituted by a rotating blade, the surfaces of which break the rotation-inversion symmetry through alternated patches of different inelasticity, immersed in a gas of granular particles. The main difference of our experimental setup is in the orientation of the main axis, which is parallel to the (vertical) direction of shaking of the granular fluid, guaranteeing an isotropic distribution for the velocities of colliding grains, characterized by a variance v(0)(2). We also keep the granular system diluted, in order to compare with Boltzmann-equation-based kinetic theory. In agreement with theory, we observe the crucial role of Coulomb friction which induces two main regimes: (i) rare collisions, with an average angular velocity <ω>~v(0)(3), and (ii) frequent collisions (FC), with <ω>~v(0). We also study the fluctuations of the angle spanned in a large-time interval Δθ, which in the FC regime is proportional to the work done upon the motor. We observe that the fluctuation relation is satisfied with a slope which weakly depends on the relative collision frequency.

Ratchet effect driven by Coulomb friction: the asymmetric Rayleigh piston

The effect of Coulomb friction is studied in the framework of collisional ratchets. It turns out that the average drift of these devices can be expressed as the combination of a term related to the lack of equipartition between the probe and the surrounding bath, and a term featuring the average frictional force. We illustrate this general result in the asymmetric Rayleigh piston, showing how Coulomb friction can induce a ratchet effect in a Brownian particle in contact with an equilibrium bath. An explicit analytical expression for the average velocity of the piston is obtained in the rare collision limit. Numerical simulations support the analytical findings.

Stochastic dynamics beyond the weak coupling limit: thermalization

We discuss the structure and asymptotic long-time properties of coupled equations for the moments of a Brownian particle's momentum p(n)((t)) derived microscopically beyond the lowest approximation in the weak coupling parameter λ. Generalized fluctuation-dissipation relations are derived and shown to ensure convergence to thermal equilibrium to any order in λ.

Computer simulation of Feynman's ratchet and pawl system

In this work, we introduce two models of Feynman's ratchet and pawl system. Molecular dynamics is carried out to simulate the two designs for Feynman's ratchet and pawl systems followed by a Langevin dynamics simulation of the reduced system. We find that the ratchet will rotate as designed when the temperature of the pawl chamber is lower than that of the ratchet chamber, which is consistent with the second law of thermodynamics. Different parameters and configurations are tested, and the results show that the efficiency of the ratchet depends on the applied torque. We find further that efficiencies of the Feynman's ratchet and pawl systems depend greatly on the details of the systems.

Rectifying the thermal Brownian motion of three-dimensional asymmetric objects

We extend the analysis of a thermal Brownian motor reported by Van den Broeck [Phys. Rev. Lett. 93, 090601 (2004)] to a three-dimensional configuration. We calculate the friction coefficient, diffusion coefficient, and drift velocity as functions of shape and present estimates based on physically realistic parameter values.

Generalized Fokker-Planck equation, Brownian motion, and ergodicity

Microscopic theory of Brownian motion of a particle of mass M in a bath of molecules of mass m<<M is considered beyond lowest order in the mass ratio mM . The corresponding Langevin equation contains nonlinear corrections to the dissipative force, and the generalized Fokker-Planck equation involves derivatives of order higher than 2. These equations are derived from first principles with coefficients expressed in terms of correlation functions of microscopic force on the particle. The coefficients are evaluated explicitly for a generalized Rayleigh model with a finite time of molecule-particle collisions. In the limit of a low-density bath, we recover the results obtained previously for a model with instantaneous binary collisions. In the general case, the equations contain additional corrections, quadratic in bath density, originating from a finite collision time. These corrections survive to order (m/M)2 and are found to make the stationary distribution non-Maxwellian. Some relevant numerical simulations are also presented.

Inertial effects in Büttiker-Landauer motor and refrigerator at the overdamped limit

We investigate the energetics of a Brownian motor driven by position-dependent temperature, commonly known as the Büttiker-Landauer motor. Overdamped models (M=0) predict that the motor can attain Carnot efficiency. However, the overdamped limit (M-->0) contradicts the previous prediction due to the kinetic energy contribution to the heat transfer. Using molecular dynamics simulation and numerical solution of the inertial Langevin equation, we confirm that the motor can never achieve Carnot efficiency and verify that the heat flow via kinetic energy diverges as M{-1/2} in the overdamped limit. The reciprocal process of the motor, namely, the Büttiker-Landauer refrigerator, is also examined. In this case, the overdamped approach succeeds in predicting the heat transfer only when there is no temperature gradient. Its found that the Onsager symmetry between the motor and refrigerator does not suffer from the singular behavior of the kinetic energy contribution.

Nonlinear dissipation effect in Brownian relaxation

In an ensemble of noninteracting Brownian particles, a finite systematic average velocity may temporarily develop, even if it is zero initially. The effect originates from a small nonlinear correction to the dissipative force, causing the equation for the first moment of velocity to couple to moments of higher order. The effect may be relevant when a complex system dissociates in a viscous medium under strongly nonequilibrium conditions.

Numerical methods for the stochastic Landau-Lifshitz Navier-Stokes equations

The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several computational fluid dynamics approaches are considered (including MacCormack's two-step Lax-Wendroff scheme and the piecewise parabolic method) and are found to give good results for the variance of momentum fluctuations. However, neither of these schemes accurately reproduces the fluctuations in energy or density. We introduce a conservative centered scheme with a third-order Runge-Kutta temporal integrator that does accurately produce fluctuations in density, energy, and momentum. A variety of numerical tests, including the random walk of a standing shock wave, are considered and results from the stochastic LLNS solver are compared with theory, when available, and with molecular simulations using a direct simulation Monte Carlo algorithm.

Maxwell's demon and Smoluchowski's trap door

A simulation has been performed to reveal the detailed dynamics and statistical behavior of a Maxwell demon of the simplest kind, a trap door held over by a spring inside a box filled with gas molecules. The role of such a demon can be controlled by tuning Smoluchowski's fluctuations. When the demon is in thermal equilibrium with the rest of the system, it fails to function as designed, and when it is separately subjected to a thermal bath with a different temperature, it creates a temperature or density gradient between the two chambers of the box it divides. As a Maxwell demon, the trap-door device creates more readily a density gradient than that of temperature.

Piston dynamics from a microcanonical ensemble

The dynamical behavior of a system consisting of a (heavy) piston and two rectangular boxes, each containing two hard disks and in contact with the piston, is studied based on a projection operator method for a microcanonical ensemble. We derive a coupled set of nonlinear equations for slow variables of the system and solve it to confirm that our theory with no adjustable parameters reproduces experimental results fairly well. Some limitations of the theory are discussed from the viewpoint of the separation of slow and fast time scales and ergodicity.

Molecular dynamics simulation of ratchet motion in an asymmetric nanochannel

The persistence of ratchet effects, i.e., nonzero mass flux under a zero-mean time-dependent drive, when many-body interactions are present, is studied via molecular dynamics (MD) simulations of a simple liquid flowing in an asymmetric nanopore. The results show that (i) ratchet effects persist under many-body density correlations induced by the forcing; (ii) two distinct linear responses (flux proportional to the drive amplitude) appear under strong loads. One regime has the same conductivity of linear response theory up to a forcing of about 10 kT, while the second displays a smaller conductivity, the difference in responses is due to geometric effects alone. (iii) Langevin simulations based on a naive mapping of the many-body equilibrium bulk diffusivity, D, onto the damping rate, gamma are also found to yield two distinct linear responses. However, in both regimes, the flux is significantly smaller than the one of MD simulations.

Brownian refrigerator

Onsager symmetry implies that a Brownian motor, driven by a temperature gradient, will also perform a refrigerator function upon loading. We analytically calculate the corresponding heat flow for an exactly solvable microscopic model and compare it with molecular dynamics simulations.

Fluctuation and dissipation of work in a Joule experiment

We elucidate the connection between various fluctuation theorems by a microcanonical version of the Crooks relation. We derive the microscopically exact expression for the work distribution in an idealized Joule experiment, namely, for a convex object moving at constant speed through an ideal gas. Analytic results are compared with molecular dynamics simulations of a hard disk gas.

Thermal Brownian motor

Recently, a thermal Brownian motor was introduced (Van den Broeck et al 2004 Phys. Rev. Lett. 93 090601), for which an exact microscopic analysis is possible. The purpose of this paper is to review some further properties of this construction, and to discuss in particular specific issues including the relation with macroscopic response and the efficiency at maximum power.