We consider the paradigm of an overdamped Brownian particle in a potential well, which is modulated through an external protocol, in the presence of stochastic resetting. Thus, in addition to the short range diffusive motion, the particle also experiences intermittent long jumps which reset the particle back at a preferred location. Due to the modulation of the trap, work is done on the system and we investigate the statistical properties of the work fluctuations. We find that the distribution function of the work typically, in asymptotic times, converges to a universal Gaussian form for any protocol as long as that is also renewed after each resetting event. When observed for a finite time, we show that the system does not generically obey the Jarzynski equality which connects the finite time work fluctuations to the difference in free energy, albeit a restricted set of protocols which we identify herein. In stark contrast, the Jarzynski equality is always fulfilled when the protocols continue to evolve without being reset. We present a set of exactly solvable models, demonstrate the validation of our theory and carry out numerical simulations to illustrate these findings.Introduction.-Stochastic thermodynamics is a cornerstone in non-equilibrium statistical physics [1][2][3][4][5]. Microscopic systems satisfy stochastic laws of motion governed by force fields and thermal fluctuations which arise due to the surrounding. The subject then teaches us that thermodynamic observables such as work, heat, entropy production etc. measured along the stochastic trajectories taken from ensembles of such dynamics will fluctuate too. Understanding the distribution and the statistical properties of these fluctuations is of great interest since they hold a treasure trove of information about microscopic systems and how they respond to external perturbation. Indeed there has been a myriad of studies to understand e.g., non-equilibrium dynamics of biopolymers [6,7], colloidal particles [8][9][10][11][12][13], efficiency of molecular bio-motors [14,15] and microscopic engines [16], heat conduction [17,18], electronic transport in quantum systems [19], trapped-ion systems [20] and many more [21]. Despite there exists a long catalogue of such diverse small systems with no apparent similarity, it is quite remarkable to find universal relations which are obeyed regardless. One of the most celebrated ones is perhaps the Jarzynski equality (JE) that relates the non-equilibrium fluctuations of the work to the equilibrium free energy difference [22][23][24]. Universalities of such kind have always been considered as a holy grail in physical sciences and in this paper we seek out for thermodynamic invariant principles in stochastic resetting systems [25].Dynamics with stochastic reset has drawn a lot of attention recently because of its rich non-equilibrium properties [25][26][27][28][29][30][31][32][33][34][35][36][37] and its broad applicability in first passage processes [38][39][40][41][42][43][44][45][46][47]. Nevertheless, thermodyna...

In the past few years, stochastic resetting has become a subject of immense interest. Most of the theoretical studies so far focused on instantaneous resetting which is, however, a major impediment to practical realisation or experimental verification in the field. This is because in the real world, taking a particle from one place to another requires finite time and thus a generalization of the existing theory to incorporate non-instantaneous resetting is very much in need. In this paper, we propose a method of resetting which involves non-instantaneous returns facilitated by an external confining trap potential U(x) centered at the resetting location. We consider a Brownian particle that starts its random motion from the origin. Upon resetting, the trap is switched on and the particle starts experiencing a force towards the center of the trap which drives it to return to the origin. The return phase ends when the particle makes a first passage to this center. We develop a general framework to study such a set up. Importantly, we observe that the system reaches a non-equilibrium steady state which we analyze in full details for two choices of U(x), namely, (i) linear and (ii) harmonic. Finally, we perform numerical simulations and find an excellent agreement with the theory. The general formalism developed here can be applied to more realistic return protocols opening up a panorama of possibilities for further theoretical and experimental applications.

A recently introduced model describing—on a 1d lattice—the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics is described through the corresponding Master Equation for the time evolution of the probability distribution. In the continuum limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to hydrodynamic equations of granular fluids when restricted to the shear modes. Therefore, the homogeneous cooling state, with its linear instability, and other relevant regimes such as the uniform shear flow and the Couette flow states are described. The evolution in time and space of the single particle probability distribution, in all those regimes, is also discussed, showing that the local equilibrium is not valid in general. The noise for the momentum and energy currents, which are correlated, are white and Gaussian. The same is true for the noise of the energy sink, which is usually negligible

Obtaining adiabatic processes that connect equilibrium states in a given time represents a challenge for mesoscopic systems. In this paper, we explicitly show how to build these finite-time adiabatic processes for an overdamped Brownian particle in an arbitrary potential, a system that is relevant at both the conceptual and the practical level. This is achieved by jointly engineering the time evolutions of the binding potential and the fluid temperature. Moreover, we prove that the second principle imposes a speed limit for such adiabatic transformations: there appears a minimum time to connect the initial and final states. This minimum time can be explicitly calculated for a general compression or decompression situation.