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What is a minimum work transition in stochastic thermodynamics?

Paolo Muratore-Ginanneschi, Julia Sanders

Abstract

We reassess the concept of transition at minimum work in classical stochastic finite-time thermodynamics, when the system dynamics is modelled by a diffusion process. We show that a well-posed formulation of the optimal control problem corresponding to the minimization of the mean work done on the system during a finite-time transition necessarily requires taking into account speed limits on control protocols. This fact has major qualitative consequences. First, it permits to discriminate between optimal swift engineered equilibration and transitions at minimum work. Second, it shows that in the limit when speed limits are removed, only transitions specified by generalized Schrödinger bridges admit a consistent physical interpretation. To illustrate these points, we focus on the simplest model problem: a levitating particle in a Gaussian moving trap.

What is a minimum work transition in stochastic thermodynamics?

Abstract

We reassess the concept of transition at minimum work in classical stochastic finite-time thermodynamics, when the system dynamics is modelled by a diffusion process. We show that a well-posed formulation of the optimal control problem corresponding to the minimization of the mean work done on the system during a finite-time transition necessarily requires taking into account speed limits on control protocols. This fact has major qualitative consequences. First, it permits to discriminate between optimal swift engineered equilibration and transitions at minimum work. Second, it shows that in the limit when speed limits are removed, only transitions specified by generalized Schrödinger bridges admit a consistent physical interpretation. To illustrate these points, we focus on the simplest model problem: a levitating particle in a Gaussian moving trap.
Paper Structure (24 sections, 61 equations, 3 figures)

This paper contains 24 sections, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The work functional.
  3. The “naive” formulation of the work optimal control problem.
  4. The Schrödinger Bridge formulation.
  5. A refined formulation: inclusion of speed limits on controls.
  6. Optimal swift engineered equilibration
  7. The minimum work problem
  8. Synthesis
  9. Softening the hard wall potential on the controls
  10. Transition at minimum work, again.
  11. Conditional work minimization.
  12. Work minimization over the system final state.
  13. Speed limits versus constraints on the controls
  14. Conclusions
  15. Acknowledgments
  16. ...and 9 more sections

Figures (3)

  • Figure 1: The mean value $\mathscr{x}_{t}$ (a) and trap centre $\lambda_{t}$ (b), determined by the boundary conditions (\ref{['naive:xbc']}) and (\ref{['realworkbc']}) with $x_{\mathscr{f}}=2$ and $t_{\mathrm{f}}=1.5$. The blue curves are obtained by imposing speed limits on $\lambda$ of the form \ref{['hard']} with increasing values for $V$, solved by a numeric direct optimisation method (see \ref{['direct_numerics']}). We also show (orange dotted), the results of solving the corresponding transcendental equations arising from synthesis using perturbative expansion around $V=100$ (see \ref{['app:synthesis']}). For larger values of $V$, the quantities stay closer to the turnpike solution (grey, dashed lines) specified by (\ref{['bridgesol']}) for longer during the control horizon.
  • Figure 2: The mean value $\mathscr{x}_{t}$ (a) and trap centre $\lambda_{t}$ (b) determined by the boundary conditions (\ref{['naive:xbc']}) and (\ref{['realworkbc']}) with $x_{\mathscr{f}}=2$ and $t_{\mathrm{f}}=1.5$. The curves are obtained by imposing a harmonic penalty with $V=1$ and $V=10$ as labelled. As $V$ increases, the effect of distinct penalties on $\dot{\lambda}_{t}$ decreases, and $\mathscr{x}_{t}$ and $\lambda_{t}$ stay close to the “turnpike” solution (grey, dashed lines), corresponding to (\ref{['bridgesol']}), for longer during the control horizon. Interestingly, as $V$ tends to infinity the limit values of $\lambda_{t_{\mathrm{f}}}$ tends to $x_{\mathscr{f}}$ to approximate equilibrium. This is a feature of the model because the terminal cost is positive definite.
  • Figure 3: The mean value $\mathscr{x}_{t}$ (a) and trap centre $\lambda_{t}$ (b) determined by the boundary conditions (\ref{['naive:xbc']}), (\ref{['naive:Linitial']}), and (\ref{['realworkbc']}) with $x_{\mathscr{f}}=2$, $\ell=4$ and $t_{\mathrm{f}}=1.5$. Solutions for $V$ are computed with a hard penalty of the form \ref{['hard']} using a direct method. As $V$ increases, $\mathscr{x}_{t}$ and $\lambda_{t}$ approach the solution of the first order conditions in the limit of no speed limits, corresponding to (\ref{['eq:condworksol']}).