Abstract
Self-oscillation is a phenomenon studied across many scientific disciplines, including the engineering of efficient heat engines and electric generators. We investigate the single electron shuttle, a model nano-scale system that exhibits a spontaneous transition towards self-oscillation, from a thermodynamic perspective. We analyse the model at three different levels of description: The fully stochastic level based on Fokker–Planck and Langevin equations, the mean-field (MF) level, and a perturbative solution to the Fokker–Planck equation that works particularly well for small oscillation amplitudes. We provide consistent derivations of the laws of thermodynamics for this model system at each of these levels. At the MF level, an abrupt transition to self-oscillation arises from a Hopf bifurcation of the deterministic equations of motion. At the stochastic level, this transition is smeared out by noise, but vestiges of the bifurcation remain visible in the stationary probability density. At all levels of description, the transition towards self-oscillation is reflected in thermodynamic quantities such as heat flow, work and entropy production rate. Our analysis provides a comprehensive picture of a nano-scale self-oscillating system, with stochastic and deterministic models linked by a unifying thermodynamic perspective.
Highlights
Self-oscillation has been described as ‘the generation and maintenance of a periodic motion by a source of power that lacks a corresponding periodicity’ [1]
Our analysis provides a comprehensive picture of a nano-scale self-oscillating system, with stochastic and deterministic models linked by a unifying thermodynamic perspective
We investigate the electron shuttle at three different levels of description: the fully stochastic level modelled by a Fokker–Planck equation (FPE) and the equivalent Langevin equation, a mean-field (MF) model described as a deterministic dynamical system, and an intermediate perturbative model based on multiple scale (MS) perturbation theory, containing both deterministic and stochastic elements
Summary
Self-oscillation has been described as ‘the generation and maintenance of a periodic motion by a source of power that lacks a corresponding periodicity’ [1]. By converting direct current into stable oscillations, self-oscillatory systems provide a useful transduction mechanism for the design of autonomous motors and heat engines. One interesting system exhibiting self-oscillation is the electron shuttle, first proposed by Gorelik et al [5], where the mechanical oscillation of a metallic grain is achieved by sequential electron tunnelling between the grain and two connecting leads. This coupled system of mechanical and electronic degrees of freedom has drawn considerable theoretical and experimental attention since its original proposal. Reviews on the electron shuttle can be found in [24,25,26,27,28,29]
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