A minimal model of an autonomous thermal motor

Abstract

We consider a model of a Brownian motor composed of two coupled overdamped degrees of freedom moving in periodic potentials and driven by two heat reservoirs. This model exhibits a spontaneous breaking of symmetry and gives rise to directed transport in the case of a nonvanishing interparticle interaction strength. For strong coupling between the particles we derive an expression for the propagation velocity valid for arbitrary periodic potentials. In the limit of strong coupling the model is equivalent to the Büttiker-Landauer model [1-3] for a single particle diffusing in an environment with position dependent temperature. By using numerical calculations of the Fokker-Planck equation and simulations of the Langevin equations we study the model for arbitrary coupling, retrieving many features of the strong coupling limit. In particular, directed transport emerges even for symmetric potentials. For distinct heat reservoirs the heat currents are well-defined quantities allowing a study of the motor efficiency. We show that the optimal working regime occurs for moderate coupling. Finally, we introduce a model with discrete phase space which captures the essential features of the continuous model, can be solved in the limit of weak coupling, and exhibits a larger efficiency than the continuous counterpart. IntroductionThere is currently numerous scientific investigations aimed at characterizing the functioning of micro and nano-motors. There has, for example, been a rapid development of various artificial nanomotors with the aim of mimicking the performance of biological machines [4][5][6].From the point of view of man-made engineered micro and nano-motors, ideally one would like to design autonomous machines which are able to cyclically extract energy from the resources available in the environment and convert it to useful work. Similarly to their macroscopic counterparts, such machines must be driven outof-equilibrium by means of one or more thermodynamic forces.In the present paper we focus in particular on a motor driven by temperature gradients. A Brownian motor has long been the paradigmatic model for a microscopic machine, working either in time-dependent or steady state conditions. One well known example is a Brownian particle moving in a periodic and asymmetric potential, a so-called ratchet potential. In such a spatially periodic system, the breaking of the spatial inversion symmetry and of thermal equilibrium, obtained by modulating the force acting on the particle, results in the emergence of directed transport [7][8][9]. Another typical example is represented by a Brownian particle driven by both a periodic temperature variation and an external parameter, periodically changing the system energy [10,11]. This model, which mimics the operation of a heat engine cyclically in contact with different heat reservoirs, has been implemented in a recent experiment [12]. In all these models there is an external agent that changes periodically some parameters, typically a thermodynamic force, according to the moto...

Quantum duets working as autonomous thermal motors

“…Most importantly, we notice that the second order term of the velocity vanishes for a phase shift between the potentials ϕ = lπ with l an integer number. This is consistent with the results for the classical counterpart of the present model found in [18], where it was numerically shown that the velocity vanishes for ϕ = lπ independently of the order of V 0 . In this case the system does not break the spatial symmetry discussed in section I.…”

“…In this section we introduce the autonomous motor model as the quantum conterpart of the classical model introduced in [18]. The system Hamiltonian reads…”

“…Specifically, we propose a minimal model of quantum motor, which is based on the directed transport emerging in systems with both broken spatial symmetry and thermal equilibrium. This model motor, is the quantum counterpart of the classical system considered in [18], and consists of two particles sitting in two periodic shifted potentials, interacting through a third potential, and kept at different temperatures.…”

Quantum current in dissipative systems

“…Following the terminology of [50], one may say that the classical dynamics is hence only 'weakly' symmetric under global rotations. Now, it can be shown numerically [74] that there is no current in the system whenever f=k π/3 (k is an arbitrary integer), and the current is non-zero whenever k 3 f p ¹ (see figure E1). At the same time, we notice that, whenever f=kπ/3, U j,i =U i, j+k .…”

Current-mediated synchronization of a pair of beating non-identical flagella

“…This is a well-known aspect for stochastic dynamics of coupled components, each evolving at its own temperature (see, e.g. [26][27][28][29][30][31][32][33]). However, in the case at hand this nonzero current has a peculiar form due to the fact that the coupling term in equation (1) is a periodic function of the phase difference.…”