Abstract
We propose an enlarged framework to study transformations that drive an underdamped Brownian particle in contact with a thermal bath from an equilibrium state to a new one in an arbitrarily short time. To this end, we make use of a time and space-dependent potential, that plays a dual role: confine the particle, and manipulate the system. In the special case of an isothermal compression or decompression of a harmonically trapped particle, we derive explicit protocols that perform this quick transformation, following an inverse engineering method. We focus on the properties of these protocols, which crucially depend on two key dimensionless numbers that characterize the relative values of the three timescales of the problem, associated with friction, oscillations in the confinement and duration of the protocol. In particular, we show that our protocols encompass the known overdamped version of this problem and extend it to any friction for decompression and to a large range of frictions for compression.
Highlights
Shortcuts To Adiabaticity (STA) emerged in quantum mechanics as fast protocols for state-to-state transformations that would otherwise require the slow and time-consuming modification of a control parameter of the system to reach the desired final state following a quasi-adiabatic trajectory [1]
We propose a general framework to study transformations that drive an underdamped Brownian particle in contact with a thermal bath from an equilibrium state to a new one in an arbitrarily short time
Many strategies have been proposed to set up non-adiabatic routes to reach the same final state through the use of dynamical invariants [2], counter adiabatic driving [3–5], reverse engineering methods [6–8], fast-forward techniques [9, 10], Lie algebraic approaches [11, 12], and optimal control [13–16] to name but a few
Summary
Shortcuts To Adiabaticity (STA) emerged in quantum mechanics as fast protocols for state-to-state transformations that would otherwise require the slow and time-consuming modification of a control parameter of the system to reach the desired final state following a quasi-adiabatic trajectory [1]. Thermodynamic transformations that connect two different equilibrium states are not in most cases quasi-static and necessarily visit out-ofequilibrium states Operating such transformations in a finite and short amount of time, potentially much shorter than the relaxation time of the system, is crucial for many applications, in particular in micro and nano devices or engines [32–37], triggering a number of works considering how STA could boost engines, among which [38–41]. We proceed to show that it largely depends on whether the transformation is a compression or a decompression, and work out the various properties of these protocols, such as existence, cross-over to the overdamped regime, position-velocity decoupling or transient negativity of the stiffness We supplement this by a discussion on the shape of the temporal evolution of the stiffness, through the comparison between the relevant timescales of the problem.
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