Suppression of work fluctuations by optimal control: An approach based on Jarzynski's equality

Abstract

Understanding and manipulating work fluctuations in microscale and nanoscale systems are of both fundamental and practical interest. For example, aspects of work fluctuations will be an important factor in designing nanoscale heat engines. In this work, an optimal control approach directly exploiting Jarzynski's equality is proposed to effectively suppress the fluctuations in the work statistics, for systems (initially at thermal equilibrium) subject to a work protocol but isolated from a bath during the protocol. The control strategy is to minimize the deviations of individual values of e −βW from their ensemble average given by e −β∆F , where W is the work, β is the inverse temperature, and ∆F is the free energy difference between two equilibrium states. It is further shown that even when the system Hamiltonian is not fully known, it is still possible to suppress work fluctuations through a feedback loop, by refining the control target function on the fly through Jarzynski's equality itself. Numerical experiments are based on linear and nonlinear parametric oscillators. Optimal control results for linear parametric oscillators are also benchmarked with early results based on accelerated adiabatic processes.

Principle of minimal work fluctuations

“…Third, we have emphasized above that a quantum adiabatic process may be executed via a control field by taking advantage of STA [7,8,[22][23][24]. Consider, for example, one type of STA, which is also called counter-diabatic driving [22] or transitionless quantum driving [23].…”

“…In our previous study [7] treating classical systems, a classical version of STA to realize classical shortcuts to adiabaticity [8] and a classical OCT were considered to suppression classical work fluctuations. It was found that the performance of OCT, as quantified by the suppression of the variance in e −βW , can at most reach that obtained from classical STA (within statistical error).…”

“…In particular, the quantum energy level index n, which is important in our quantum proof, reminds us of the classical action variable I because in semiclassical quantization, n and I are simply related. Second, in our previous study aiming at the suppression of work fluctuations in a classical system, we observed that an optimal control field can dramatically suppress the fluctuations in e −βW but can never outperform that achieved from a classical (mechanical) adiabatic process in all the cases studied (within statistical error) [7]. Indeed, we are able to explicitly show, at least for onedimensional classical systems, that a classical adiabatic process yields the minimal fluctuations in e −βW .…”

“…[7]. As indicated by the Jarzynski equality, i.e., e −βW = e −β∆F (where W is the work done on a system initially prepared at thermal equilibrium with inverse temperature β, ∆F is the free energy difference between the initial and final equilibrium states of the same β), the mean value of e −βW is determined by β and ∆F , but the fluctuations in the work statistics, as manifested in the variance of e −βW , can still be suppressed by an additional control field [7,8]. Such suppression in the work fluctuations may be beneficial in achieving a better convergence of the statistical average of e −βW towards the theoretical value e −β∆F .…”

Merits and qualms of work fluctuations in classical fluctuation theorems

“…Aspects of work fluctuations continue to be of fundamental interest [47][48][49][50]. Deep insights into work fluctuations are also relevant to designs of energy devices.…”