Geometric thermodynamics for the Fokker–Planck equation: stochastic thermodynamic links between information geometry and optimal transport

Abstract

We propose a geometric theory of non-equilibrium thermodynamics, namely geometric thermodynamics, using our recent developments of differential-geometric aspects of entropy production rate in non-equilibrium thermodynamics. By revisiting our recent results on geometrical aspects of entropy production rate in stochastic thermodynamics for the Fokker–Planck equation, we introduce a geometric framework of non-equilibrium thermodynamics in terms of information geometry and optimal transport theory. We show that the proposed geometric framework is useful for obtaining several non-equilibrium thermodynamic relations, such as thermodynamic trade-off relations between the thermodynamic cost and the fluctuation of the observable, optimal protocols for the minimum thermodynamic cost and the decomposition of the entropy production rate for the non-equilibrium system. We clarify several stochastic-thermodynamic links between information geometry and optimal transport theory via the excess entropy production rate based on a relation between the gradient flow expression and information geometry in the space of probability densities and a relation between the velocity field in optimal transport and information geometry in the space of path probability densities.

Energy harvesting from anisotropic fluctuations

“…Thus, the decomposition of entropy production in (18) can be seen as a generalization to anisotropic temperature fields of analogous decompositions discussed in earlier works [11], [13], [30], [31]; these works consider non-conservative forcing and heat bath with uniform temperature (a single heat bath, with T scalar). At the time the present work was being completed, Yoshimura etal.…”

“…Proof: It readily follows by comparing the expression for the least entropy production (8b) over paths ρ(t, •), t ∈ [0,t f ], between end-point states, with the definition of the weighted Wasserstein metric (13).…”

“…Thus, under conservative actuation, minimal entropy production can no longer be expressed in terms of a distance between end-point states, since maintaining a stationary state incurs in positive entropy production. Contribution to entropy production due to circulatory currents that are generated when steering a system out of equilibrium by non-conservative forcing in a uniform heat bath [11], [13], [30] has been referred to as "housekeeping entropy production." We will follow a similar convention and label the second term in (18) as housekeeping, and denote it by S hk .…”

“…In this context, it turns out that dissipation can be expressed as a quadratic cost functional that, in geometric terms, is precisely the Wasserstein length traversed by the state of the system [7]- [9]. Important new insights have been gained in recent years regarding optimal control laws that extract work while alternating contact with different heat sources in the Carnot model [10], quantifying natural time-constants and establishing uncertainty relations [11]- [13], and balancing work with dissipation in finite-time thermodynamic transitions [14], [15]. Within this evolving landscape the study of entropy production in the presence of temperature gradients remains largely unexplored, and constitutes the theme of the present work.…”

A Unified Statistical Relativistic Theory of The Generalized Brownian Motion Manifold

“…The following pivotal definitions are taken from (Mageed et al, 2022;Mageed, 2024;Kouvatsos, 2019, 2021;Mageed et al, 2023(a); Mageed et al, 2023(b); Mageed and Zhang, 2022;Mageed, 2023;Minyoung et al, 2022;Parr et al, 2020;Ito and Dechant, 2020;Di Giulio and Tonni, 2020;Barbaresco, 2021;Thiruthummal and Kim, 2022;Ito, 2023;Li, 2022). K G defines the Gaussian Curvature and ℛ is two-dimensional…”