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Stochastic control methods in non-equilibrium thermodynamics: Fundamental bounds on dissipation and power

  • Author(s): Fu, Rui
  • Advisor(s): Georgiou, Tryphon T.
  • et al.
Abstract

Classical thermodynamics is aimed at quantifying the efficiency of thermodynamic engines by bounding the maximal amount of mechanical energy produced compared to the amount of heat required under quasi-static operation. While this was accomplished early on, by Carnot and Clausius, the more practical problem of quantifying limits of output power remained elusive due to the fact that quasi-static processes require infinitely slow cycling, resulting in a vanishing power. Recent insights, drawn from stochastic thermodynamics, bridge the gap between theory and practice by presenting fresh approaches that lead to general laws applicable to the non-equilibrium system. Remarkably, the problem of minimizing dissipation over a finite time window can be expressed as a stochastic control problem leading to physically meaningful expressions for the dissipation cost in thermodynamic engines.

Building on the framework of stochastic thermodynamics, we derive bounds on the maximal power that can be drawn by cycling an overdamped ensemble of particles via a time-varying potential while alternating contact with heat baths of different temperature (Tc cold, and Th hot). The previous work focused on the setting where the potential is quadratic andthe distributions are Gaussian. Our analysis relaxes this assumption and asks about fundamental bounds on the power under arbitrary distribution. Specifically, first by casting the optimization problem of the power into the Benamou-Brenier formulated Optimal mass transport problem, we show that the power output is bounded by the Fisher information of the boundary distribution, which can become unbounded with complicated and irregular control. However, it is unreasonable to expect technological solutions to such demands, and therefore, a constraint on the complexity of the potential seems meaningful. Assuming a suitable bound M on the spatial gradient of the controlling potential, we show that the maximal achievable power is bounded by M/8(Th/Tc-1). Moreover, we show that this bound can be reached to within a factor of (Th/Tc-1)/(Th/Tc+1) by operating the cyclic thermodynamic process with a quadratic potential.

While the majority of works in the literature are concerned with thermodynamic engines operating in the setting of Carnot's cyclic contact with alternating heat baths, the natural processes in living organisms do not follow this setting. Instead, it is the periodic fluctuations in chemical concentrations in conjunction with the variability of electrochemical potentials that provide the universal source of cellular energy. Thus, energy exchange is often mediated by continuous processes and energy differentials, whereas the Carnot cycle reflects the switching mechanics of an idealized engine. We herein propose to consider thermodynamic processes driven by a heat bath with periodic and continuous temperature profile, and study questions of maximal power and efficiency at maximal power. Our results state that themaximal power satisfies a bound proportional to the average fluctuations in the temperature profile. Moreover, we show the surprising result that the efficiency at maximal power does not depend on the specific temperature profile, only the maximum and minimum.

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