Abstract. We investigate the distribution of work performed on a Brownian particle in a time-dependent asymmetric potential well. The potential has a harmonic component with time-dependent force constant and a time-independent logarithmic barrier at the origin. For arbitrary driving protocol, the problem of solving the FokkerPlanck equation for the joint probability density of work and particle position is reduced to the solution of the Riccati differential equation. For a particular choice of the driving protocol, an exact solution of the Riccati equation is presented. Asymptotic analysis of the resulting expression yields the tail behavior of the work distribution for small and large work values. In the limit of vanishing logarithmic barrier, the work distribution for the breathing parabola model is obtained.

Abstract:The stochastic model of the Feynman-Smoluchowski ratchet is proposed and solved using generalization of the Fick-Jacobs theory. The theory fully captures nonlinear response of the ratchet to the difference of heat bath temperatures. The ratchet performance is discussed using the mean velocity, the average heat flow between the two heat reservoirs and the figure of merit, which quantifies energetic cost for attaining a certain mean velocity. Limits of the theory are tested comparing its predictions to numerics. We also demonstrate connection between the ratchet effect emerging in the model and rotations of the probability current and explain direction of the mean velocity using simple discrete analogue of the model.

This paper presents an exact probabilistic description of the work done by an external agent on a two-level system. We first develop a general scheme which is suitable for the treatment of functionals of the time-inhomogeneous Markov processes. Subsequently, we apply the procedure to the analysis of the isothermal-work probability density and we obtain its exact analytical forms in two specific settings. In both models, the state energies change with a constant velocity. On the other hand, the two models differ in their interstate transition rates. The explicit forms of the probability density allow a detailed discussion of the mean work. Moreover, we discuss the weight of the trajectories which display a smaller value of work than the corresponding equilibrium work. The results are controlled by a single dimensionless parameter which expresses the ratio of two underlying timescales: the velocity of the energy changes and the relaxation time in the case of frozen energies. If this parameter is large, the process is a strongly irreversible one and the work probability density differs substantially from a Gaussian curve.

Abstract. We discuss two-dimensional diffusion of a Brownian particle confined to a periodic asymmetric channel with soft walls modeled by a parabolic potential. In the channel, the particle experiences different thermal noise intensities, or temperatures, in the transversal and longitudinal directions. The model is inspired by the famous Feynman's ratchet and pawl. Although the standard Fick-Jacobs approximation predicts correctly the effective diffusion coefficient, it absolutely fails to capture the ratchet effect. Deriving a correction, which breaks the local detailed balance with the transversal noise source, we obtain a correct mean velocity of the particle and a stationary probability density in the potential unit cell. The derived results are exact for small channel width. Yet, we check by exact numerical calculation that they qualitatively describe the ratchet effect observed for an arbitrary width of the channel.

We investigate a microscopic motor based on an externally controlled two-level system. One cycle of the motor operation consists of two strokes. Within each stroke, the two-level system is in contact with a given thermal bath and its energy levels are driven with a constant rate. The time evolution of the occupation probabilities of the two states are controlled by one rate equation and represent the system's response with respect to the external driving. We give the exact solution of the rate equation for the limit cycle and discuss the emerging thermodynamics: the work done on the environment, the heat exchanged with the baths, the entropy production, the motor's efficiency, and the power output. Furthermore we introduce an augmented stochastic process which reflects, at a given time, both the occupation probabilities for the two states and the time spent in the individual states during the previous evolution. The exact calculation of the evolution operator for the augmented process allows us to discuss in detail the probability density for the performed work during the limit cycle. In the strongly irreversible regime, the density exhibits important qualitative differences with respect to the more common Gaussian shape in the regime of weak irreversibility.

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We study 1-D diffusion of N hard-core interacting Brownian particles driven by the space-and time-dependent external force. We give the exact solution of the N -particle Smoluchowski diffusion equation. In particular, we investigate the nonequilibrium energetics of two interacting particles under the time-periodic driving. The hard-core interaction induces entropic repulsion which differentiates the energetics of the two particles. We present exact time-asymptotic results which describe the mean energy, the accepted work and heat, and the entropy production of interacting particles and we contrast these quantities against the corresponding ones for the non-interacting particles.PACS numbers: 87.10.Mn, 87.15.hg, 02.50.Ga, 05.40.Jc, 05.10.Gg, 82.75.Jn * rjabov.a@gmail.com 2 Introduction: Stochastic dynamics of interacting particles in a one-dimensional environment is both of great practical and theoretical interest. Due to the one-dimensionality of the problem, the inter-particle interactions play a crucial role as they alter qualitative features of the particle dynamics. The type of interaction we deal with in this Letter is a so called hard-core interaction.The diffusion of particles in narrow channels, where the particles cannot pass each other and their relative ordering is conserved, is known as the single-file diffusion (SFD). The concept of SFD was first introduced by Hodgkin and Keynes in relation to the transport of water and ions through the molecular-sized channels in membranes [1]. Since then, numerous examples of SFD in biological, chemical, and physical processes were studied (e.g. transport of adsorbate molecules through zeolites with a one-dimensional channel system [2,3], geometrically constrained nanosized particles in nano-sized pores [4], migration of adsorbed molecules on surfaces [5], diffusion in nanotubes [6,7], a carrier migration in polymers and superionic conductors [8], diffusion of colloids in one-dimensional channels [9][10][11], confined dynamics of millimetric steel balls [12]).While the global properties of a SFD-system are identical to those for the system of independent particles, the dynamics of an individual particle (also called tagged particle or tracer ) is considerably different [13][14][15]. Theoretical description of SFD was introduced by Harris in 1965. In his pioneering study [16] he showed that the mean square displacement of the tagged particle increases with time as t 1/2 (in contrast to its linear increase for the single free particle). This result was subsequently reestablished by many other authors using different mathematical tools (for a comprehensive review cf. the Introduction in [17]). The first exact solutions of the diffusion equation for SFD systems appeared only recently. The solution for an arbitrary number N of identical particles diffusing along the infinite line has been obtained by Rödenbeck et al. in [13] via the reflection principle. Using a different theoretical procedure, the result has been re-derived in [18]. The exact solution for the diffus...

We present a new method for simulating Markovian jump processes with time-dependent transitions rates, which avoids the transformation of random numbers by inverting time integrals over the rates. It relies on constructing a sequence of random time points from a homogeneous Poisson process, where the system under investigation attempts to change its state with certain probabilities. With respect to the underlying master equation the method corresponds to an exact formal solution in terms of a Dyson series. Different algorithms can be derived from the method and their power is demonstrated for a set of interacting two-level systems that are periodically driven by an external field.