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.

46 pages, 0 figures, Some errors in the published version are corrected

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• Infinite variety of thermodynamic speed limits with general activities
• Analytical solution for optimal protocols of weak drivings
• Geometric thermodynamics of reaction-diffusion systems: Thermodynamic trade-off relations and optimal transport for pattern formation
• Speed limits to fluctuation dynamics
• Force-current structure in Markovian open quantum systems and its applications: geometric housekeeping-excess decomposition and thermodynamic trade-off relations
• Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transport
• Inferring entropy production in many-body systems using nonequilibrium maximum entropy
• Experimentally achieving minimal dissipation via thermodynamically optimal transport
• Role of interactions in non-equilibrium transformations
• Generalized free energy and excess/housekeeping decomposition in nonequilibrium systems: from large deviations to thermodynamic speed limits
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• Decomposition of metric tensor in thermodynamic geometry in terms of relaxation timescales
• From the Fokker-Planck equation to a contact Hamiltonian system
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