We characterize finite-time thermodynamic processes of multidimensional quadratic overdamped systems. Analytic expressions are provided for heat, work, and dissipation for any evolution of the system covariance matrix. The Bures-Wasserstein metric between covariance matrices naturally emerges as the local quantifier of dissipation. General principles of how to apply these geometric tools to identify optimal protocols are discussed. Focusing on the relevant slow-driving limit, we show how these results can be used to analyze cases in which the experimental control over the system is partial.

Thermodynamics and optimal protocols of multidimensional quadratic Brownian systems

We investigate the problem of minimizing the entropy production for a physical process that can be described in terms of a Markov jump dynamics. We show that, without any further constraints, a given time-evolution may be realized at arbitrarily small entropy production, yet at the expense of diverging activity. For a fixed activity, we find that the dynamics that minimizes the entropy production is given in terms of conservative forces. The value of the minimum entropy production is expressed in terms of the graph-distance based Wasserstein distance between the initial and final configuration. This yields a new kind of speed limit relating dissipation, the average number of transitions and the Wasserstein distance. It also allows us to formulate the optimal transport problem on a graph in term of a continuous-time interpolating dynamics, in complete analogy to the continuous space setting. We demonstrate our findings for simple state networks, a time-dependent pump and for spin flips in the Ising model. 

/pdf/minimum-entropy-production-detailed-balance-and-wasserstein-2nkkivri.pdf

Minimum entropy production, detailed balance and Wasserstein distance for continuous-time Markov processes

We investigate the problem of minimizing the entropy production for a physical process that can be described in terms of a Markov jump dynamics. We show that, without any further constraints, a given time-evolution may be realized at arbitrarily small entropy production, yet at the expense of diverging activity. For a fixed activity, we find that the dynamics that minimizes the entropy production is given in terms of conservative forces. The value of the minimum entropy production is expressed in terms of the graph-distance based Wasserstein distance between the initial and final configuration. This yields a new kind of speed limit relating dissipation, the average number of transitions and the Wasserstein distance. It also allows us to formulate the optimal transport problem on a graph in term of a continuous-time interpolating dynamics, in complete analogy to the continuous space setting. We demonstrate our findings for simple state networks, a time-dependent pump and for spin flips in the Ising model.

Based on a minimum entropy production principle, entropy production in out-of-equilibrium systems is decomposed into housekeeping and excess parts. In this Letter, we formulate an intuitive geometrical interpretation of this decomposition and express the excess part in terms of the Wasserstein distance. This formulation allows us to derive generalizations of the thermodynamic uncertainty relation to the housekeeping and excess entropy production. We demonstrate that the entropy production in systems with a time-dependent non-potential force can be obtained by measuring the time-dependent probability density without knowing details of the system.

Geometric decomposition of entropy production in out-of-equilibrium systems

Two qualitatively different ways of driving a physical system out of equilibrium, i.e., time-dependent and nonconservative forcing, are reflected by the decomposition of the system's entropy production into excess and housekeeping parts. We show that the difference between these two types of driving gives rise to a geometric formulation in terms of two orthogonal contributions to the currents in the system. This geometric picture in a natural way leads to variational expressions for both the excess and housekeeping entropy, which allow calculating both contributions independently from the trajectory data of the system. We demonstrate this by calculating the excess and housekeeping entropy of a particle in a time-dependent, tilted periodic potential.Received 26 September 2021Accepted 16 February 2022DOI:https://doi.org/10.1103/PhysRevResearch.4.L012034Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBrownian motionEntropy productionFluctuations & noiseNonequilibrium & irreversible thermodynamicsNonequilibrium statistical mechanicsStatistical Physics 

/pdf/geometric-decomposition-of-entropy-production-in-out-of-1djv51sj.pdf

We study a relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation. We show that the lower bound on the entropy production is the action measured by the path length of the ${L}^{2}$-Wasserstein distance. Because the ${L}^{2}$-Wasserstein distance is a geometric measure of optimal transport theory, our result implies a geometric interpretation of the entropy production. Based on this interpretation, we obtain a thermodynamic trade-off relation between transition time and the entropy production. This thermodynamic trade-off relation is regarded as a thermodynamic speed limit which gives a tighter bound of the entropy production. We also discuss stochastic thermodynamics for the subsystem and derive a lower bound on the partial entropy production as a generalization of the second law of information thermodynamics. Our formalism also provides a geometric picture of the optimal protocol to minimize the entropy production. We illustrate these results by the optimal stochastic heat engine and show a geometrical bound of the efficiency.

Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance

We study a relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation. We show that the lower bound on the entropy production is the action measured by the path length of the $L^2$-Wasserstein distance, which is a measure of optimal transport. Using the geometrical interpretation of the entropy production by the $L^2$-Wasserstein distance, we obtain a trade-off relation between the transition time and the entropy production as a variant of thermodynamic speed limits. We also discuss stochastic thermodynamics for the subsystem. We derive a lower bound on the partial entropy production by the $L^2$-Wasserstein distance as a generalization of the second law of information thermodynamics. We focus on a stochastic heat engine and show a geometrical constraint by the $L^2$-Wasserstein distance. Because the $L^2$-Wasserstein distance is a measure of the optimal transport, our formalism leads to the optimal protocol to minimize the entropy production. We illustrate these results by the Brownian particle in a harmonic trap.

Muka Nakazato

Papers

Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance

Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance