Petr Chvosta

Attempt time Monte Carlo: An alternative for simulation of stochastic jump processes with time-dependent transition rates

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.

Work distribution in a time-dependent logarithmic?harmonic potential: exact results and asymptotic analysis

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 a time-dependent force constant and a time-independent logarithmic barrier at the origin. For an arbitrary driving protocol, the problem of solving the Fokker–Planck 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. An asymptotic analysis of the resulting expression yields the tail behavior of the work distribution for small and large work values. In the limit of a vanishing logarithmic barrier, the work distribution for the breathing parabola model is obtained.

Single-file diffusion of externally driven particles.

We study one-dimensional 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 noninteracting particles.

Energetics and performance of a microscopic heat engine based on exact calculations of work and heat distributions

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 at a constant rate. The time evolutions of the occupation probabilities of the two states are controlled by one rate equation and represent the systems 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 motors 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 work performed 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.

Tracer dynamics in a single-file system with absorbing boundary

The paper addresses the single-file diffusion in the presence of an absorbing boundary. The emphasis is on an interplay between the hard-core interparticle interaction and the absorption process. The resulting dynamics exhibits several qualitatively new features. First, starting with the exact probability density function for a given particle (a tracer), we study the long-time asymptotics of its moments. Both the mean position and the mean-square displacement are controlled by dynamical exponents which depend on the initial order of the particle in the file. Second, conditioning on nonabsorption, we study the distribution of long-living particles. In the conditioned framework, the dynamical exponents are the same for all particles, however, a given particle possesses an effective diffusion coefficient which depends on its initial order. After performing the thermodynamic limit, the conditioned dynamics of the tracer is subdiffusive, the generalized diffusion coefficient D(1/2) being different from that reported for the system without absorbing boundary.

Survival of interacting Brownian particles in crowded one-dimensional environment

We investigate a diffusive motion of a system of interacting Brownian particles in quasi-one-dimensional micropores. In particular, we consider a semi-infinite 1D geometry with a partially absorbing boundary and the hard-core inter-particle interaction. Due to the absorbing boundary the number of particles in the pore gradually decreases. We present the exact analytical solution of the problem. Our procedure merely requires the knowledge of the corresponding single-particle problem. First, we calculate the simultaneous probability density of having still a definite number (N - k) of surviving particles at definite coordinates. Focusing on an arbitrary tagged particle, we derive the exact probability density of its coordinate. Second, we present a complete probabilistic description of the emerging escape process. The survival probabilities for the individual particles are calculated, the first and the second moments of the exit times are discussed. Generally speaking, although the original inter-particle interaction possesses a point-like character, it induces entropic repulsive forces which, e.g., push the leftmost (rightmost) particle towards (opposite) the absorbing boundary thereby accelerating (decelerating) its escape. More importantly, as compared to the reference problem for the non-interacting particles, the interaction changes the dynamical exponents which characterize the long-time asymptotic dynamics. Interesting new insights emerge after we interpret our model in terms of (a) diffusion of a single particle in a N-dimensional space, and (b) order statistics defined on a system of N-independent, identically distributed random variables.

Exact analysis of work fluctuations in two-level systems

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.

Resonant activation phenomenon for non-Markovian potential-fluctuation processes

We consider a generalization of the model by Doering and Gadoua to non-Markovian potential-switching generated by arbitrary renewal processes. For the Markovian switching process, we extend the original results by Doering and Gadoua by giving a complete description of the absorption process. For all non-Markovian processes having the first moment of the waiting time distributions, we get qualitatively the same results as in the Markovian case. However, for distributions without the first moment, the mean first passage time curves do not exhibit the resonant activation minimum. We thus come to the conjecture that the generic mechanism of the resonant activation fails for fluctuating processes widely deviating from Markovian.

Transport coefficients for a confined Brownian ratchet operating between two heat reservoirs

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 noise intensities, or temperatures, in the transversal and longitudinal directions. The model is inspired by the famous Feynmans ratchet and pawl. Using ideas of the Fick-Jacobs approximation we derive the mean velocity of the particle, the effective diffusion coefficient, and the 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.

Dynamics under the influence of semi-Markov noise

We analyze the averaged dynamics of a system which jumps within a predefined set of dynamical modes at random time instants. In between the jumps, the dynamics is not necessarily exponential. The residence times are specific for the dynamical modes and they are distributed according to general probability densities. The transitions between the modes are controlled by a matrix of transition probabilities and they generally depend on the residence times in the individual dynamical modes. Our approach is based on the explicit construction of all possible realizations of the system dynamics together with their corresponding probabilities. We give a closed formula for the averaged dynamics and we discuss the physical meaning of the probabilistic elements incorporated. The general scheme is employed in two physically relevant situations, namely, within a model of thermally activated escape including random fluctuations of the barrier height, and in the framework of the optical Bloch equations taking into account fluctuations of the resonance frequency.