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Register a new userUnusual equilibration of a particle in a potential with a thermal wall
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We consider a particle in a one-dimensional box of length $L$ with a Maxwell bath at one end and a reflecting wall at the other end. Using a renewal approach, as well as directly solving the master equation, we show that the system exhibits a slow power law relaxation with a logarithmic correction towards the final equilibrium state. We extend the renewal approach to a class of confining potentials of the form $U(x) propto x^alpha$, $x>0$, where we find that the relaxation is $sim t^{-(alpha+2)/(alpha-2)}$ for $alpha >2$, with a logarithmic correction when $(alpha+2)/(alpha-2)$ is an integer. For $alpha <2$ the relaxation is exponential. Interestingly for $alpha=2$ (harmonic potential) the localised bath can not equilibrate the particle.
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We investigate the persistence probability of a Brownian particle in a harmonic potential, which decays to zero at long times -- leading to an unbounded motion of the Brownian particle. We consider two functional forms for the decay of the confinement, an exponential and an algebraic decay. Analytical calculations and numerical simulations show, that for the case of the exponential relaxation, the dynamics of Brownian particle at short and long times are independent of the parameters of the relaxation. On the contrary, for the algebraic decay of the confinement, the dynamics at long times is determined by the exponent of the decay. Finally, using the two-time correlation function for the position of the Brownian particle, we construct the persistence probability for the Brownian walker in such a scenario.
We analyze the classical problem of the stochastic dynamics of a particle confined in a periodic potential, through the so called Ilin and Khasminskii model, with a novel semi-analytical approach. Our approach gives access to the transient and the asymptotic dynamics in all damping regimes, which are difficult to investigate in the usual Brownian model. We show that the crossover from the overdamped to the underdamped regime is associated with the loss of a typical time scale and of a typical length scale, as signaled by the divergence of the probability distribution of a certain dynamical event. In the underdamped regime, normal diffusion coexists with a non Gaussian displacement probability distribution for a long transient, as recently observed in a variety of different systems. We rationalize the microscopic physical processes leading to the non-Gaussian behavior, as well as the timescale to recover the Gaussian statistics. The theoretical results are supported by numerical calculations and are compared to those obtained for the Brownian model.
In the context of stochastic thermodynamics, a minimal model for non equilibrium steady states has been recently proposed: the Brownian Gyrator (BG). It describes the stochastic overdamped motion of a particle in a two dimensional harmonic potential, as in the classic Ornstein-Uhlenbeck process, but considering the simultaneous presence of two independent thermal baths. When the two baths have different temperatures, the steady BG exhibits a rotating current, a clear signature of non equilibrium dynamics. Here, we consider a time-dependent potential, and we apply a reverse-engineering approach to derive exactly the required protocol to switch from an initial steady state to a final steady state in a finite time $tau$. The protocol can be built by first choosing an arbitrary quasi-static counterpart - with few constraints - and then adding a finite-time contribution which only depends upon the chosen quasi-static form and which is of order $1/tau$. We also get a condition for transformations which - in finite time - conserve internal energy, useful for applications such as the design of microscopic thermal engines. Our study extends finite-time stochastic thermodynamics to transformations connecting non-equilibrium steady states.
We consider motion of an underdamped Brownian particle in a washboard potential that is subjected to an unbiased time-periodic external field. While in the limiting deterministic system in dependence of the strength and phase of the external field directed net motion can exist, for a finite temperature the net motion averages to zero. Strikingly, with the application of an additional time-delayed feedback term directed particle motion can be accomplished persisting up to fairly high levels of the thermal noise. In detail, there exist values of the feedback strength and delay time for which the feedback term performs oscillations that are phase locked to the time-periodic external field. This yields an effective biasing rocking force promoting periods of forward and backward motion of distinct duration, and thus directed motion. In terms of phase space dynamics we demonstrate that with applied feedback desymmetrization of coexisting attractors takes place leaving the ones supporting either positive or negative velocities as the only surviving ones. Moreover, we found parameter ranges for which in the presence of thermal noise the directed transport is enhanced compared to the noise-less case.
We propose a new protocol that ensures the fast equilibration of an overdamped harmonic oscillator by a joint time-engineering of the confinement strength and of the effective temperature of the thermal bath. We demonstrate experimentally the effectiveness of our protocol with an optically trapped Brownian particle and report an equilibrium recovering time reduced by about two orders of magnitude compared to the natural relaxation time. Our scheme paves the way towards reservoir engineering in nano-systems.
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