*Article* **Thermodynamics and Steady State of Quantum Motors and Pumps Far from Equilibrium**

### **Raúl A. Bustos-Marún 1,2,\* and Hernán L. Calvo 1,3,\***


Received: 27 June 2019; Accepted: 16 August 2019; Published: 23 August 2019

**Abstract:** In this article, we briefly review the dynamical and thermodynamical aspects of different forms of quantum motors and quantum pumps. We then extend previous results to provide new theoretical tools for a systematic study of those phenomena at far-from-equilibrium conditions. We mainly focus on two key topics: (1) The steady-state regime of quantum motors and pumps, paying particular attention to the role of higher order terms in the nonadiabatic expansion of the current-induced forces. (2) The thermodynamical properties of such systems, emphasizing systematic ways of studying the relationship between different energy fluxes (charge and heat currents and mechanical power) passing through the system when beyond-first-order expansions are required. We derive a general order-by-order scheme based on energy conservation to rationalize how every order of the expansion of one form of energy flux is connected with the others. We use this approach to give a physical interpretation of the leading terms of the expansion. Finally, we illustrate the above-discussed topics in a double quantum dot within the Coulomb-blockade regime and capacitively coupled to a mechanical rotor. We find many exciting features of this system for arbitrary nonequilibrium conditions: a definite parity of the expansion coefficients with respect to the voltage or temperature biases; negative friction coefficients; and the fact that, under fixed parameters, the device can exhibit multiple steady states where it may operate as a quantum motor or as a quantum pump, depending on the initial conditions.

**Keywords:** quantum thermodynamics; steady-state dynamics; nonlinear transport; adiabatic quantum motors; adiabatic quantum pumps; quantum heat engines; quantum refrigerators; transport through quantum dots
