Optimal finite-time processes in weakly driven overdamped Brownian motion

Abstract

The complete physical understanding of the optimization of the thermodynamic work still is an important open problem in stochastic thermodynamics. We address this issue using the Hamiltonian approach of linear response theory in finite time and weak processes. We derive the Euler-Lagrange equation associated and discuss its main features, illustrating them using the paradigmatic example of driven Brownian motion in overdamped regime. We show that the optimal protocols obtained either coincide, in the appropriate limit, with the exact solutions by stochastic thermodynamics or can be even identical to them, presenting the well-known jumps. However, our approach reveals that jumps at the extremities of the process are a good optimization strategy in the regime of fast but weak processes for any driven system. Additionally, we show that fast-but-weak optimal protocols are time-reversal symmetric, a property that has until now remained hidden in the exact solutions far from equilibrium.

Kibble-Zurek mechanism with time-averaged work

“…In region 3, the so-called arbitrarily far-from-equilibrium processes, both ratios are arbitrary. Linear-response theory can calculate the time-averaged excess works of regions 1 and 2 [8].…”

“…For region 2, using the asymptotic approximation of decorrelation of the time-averaged relaxation function per number of spins [8] lim…”

Quantum Ising chain with time-averaged work in linear response theory

“…In region 3, the so-called arbitrarily far-fromequilibrium processes, both ratios are arbitrary. Linear-response theory can calculate the time-averaged excess works of regions 1 and 2 [23].…”

“…Indeed, for region 1, the time-averaged excess work per number of spins is given by For region 2, using the asymptotic approximation of decorrelation of the time-averaged relaxation function per number of spins [23] lim…”

Analytical solution for optimal protocols of weak drivings

“…Optimization problems, with its plethora of particular study cases, are in this kind of classification [1][2][3][4][5][6][7][8][9][10]. In particular, the problem of finding the optimal protocols to minimize the thermodynamic work and its variance for isothermal, finite-time, and weak drivings shares the same quality [11,12]. Indeed, the solution of the Euler-Lagrange integral equation associated, given by a Fredholm integral equation of first kind and constant limits of integration, highly depends on the kernel proposed [13], becoming a case-to-case study.…”