Thermodynamics of small systems has become an important field of statistical physics. They are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in Cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a non-equilibrium transition in finite time is solved by Burgers equation of Cosmology and mass transport by the Burgers velocity field. Our contribution hence considerably extends the range of solvable optimization problems in small system thermodynamics.
Many versions of Thermodynamic Uncertainty Relations (TUR) have recently been discovered, which impose lower bounds on relative fluctuations of integrated currents in irreversible dissipative processes, and suggest that there may be fundamental limitations on the precision of small scale machines and heat engines. In this work we rigorously demonstrate that TUR can be evaded by using dynamic protocols that are asymmetric under time-reversal. We illustrate our results using a model heat engine using two-level systems, and also discuss heuristically the fundamental connections between TUR and ti…
Thermodynamic uncertainty relations quantifying a trade-off between current fluctuation and entropy production have been found in various stochastic systems. Herein, we study the thermodynamic uncertainty relations for Langevin systems driven by an external control protocol. Using information-theoretic techniques, we derive the uncertainty relations for arbitrary observables satisfying a scaling condition in both overdamped and underdamped regimes. We prove that the observable fluctuation is constrained by both entropy production and a kinetic term. The derived bounds are applicable to both cu…
Controlling thermodynamic cycles to minimize the dissipated heat is a longstanding goal in thermodynamics, and more recently, a central challenge in stochastic thermodynamics for nanoscale systems. Here, we introduce a theoretical and computational framework for optimizing nonequilibrium control protocols that can transform a system between two distributions in a minimally dissipative fashion. These protocols optimally transport a system along paths through the space of probability distributions that minimize the dissipative cost of a transformation. Furthermore, we show that the thermodynamic…
The thermodynamic uncertainty relation (TUR) for underdamped dynamics has intriguing problems while its counterpart for overdamped dynamics has recently been derived. Even for the case of steady states, a proper way to match underdamped and overdamped TURs has not been found. We derive the TUR for underdamped systems subject to general time-dependent protocols, that covers steady states, by using the Cramér-Rao inequality. We show the resultant TUR to give rise to the inequality of the product of the variance and entropy production. We prove it to approach to the known overdamped result for la…
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.
In this work we explore the use of thermodynamic length to improve the performance of experimental protocols. In particular, we implement Landauer erasure on a driven electron level in a semiconductor quantum dot, and compare the standard protocol in which the energy is increased linearly in time with the one coming from geometric optimisation. The latter is obtained by choosing a suitable metric structure, whose geodesics correspond to optimal finite-time thermodynamic protocols in the slow driving regime. We show experimentally that geodesic drivings minimise dissipation for slow protocols, …
We propose an optimization strategy to control the dynamics of a stochastic system transferred from one thermal equilibrium to another and apply it experimentally to a Brownian particle in an optical trap under compression. Based on a variational principle that treats the transfer duration and the expended work on an equal footing, our strategy leads to a family of protocols that are either optimally cheap for a given duration or optimally fast for a given energetic cost. This approach unveils a universal relation $Δt\,ΔW \ge (Δt\,ΔW)_{\rm opt}$ between the transfer duration and the expended w…
Recently many results namely the Fluctuation theorems (FT), have been discovered for systems arbitrarily away from equilibrium. Many of these relations have been experimentally tested. The system under consideration is usually driven out of equilibrium by an external time-dependent parameter which follows a particular {\it protocol}. One needs to perform several iterations of the same experiment in order to find statistically relevant results. Since the systems are microscopic, fluctuations dominate. Studying the convergence of relevant thermodynamics quantities with number of realizations is …
Thermodynamics serves as a universal means for studying physical systems from an energy perspective. In recent years, with the establishment of the field of stochastic and quantum thermodynamics, the ideas of thermodynamics have been generalized to small fluctuating systems. Independently developed in mathematics and statistics, the optimal transport theory concerns the means by which one can optimally transport a source distribution to a target distribution, deriving a useful metric between probability distributions, called the Wasserstein distance. Despite their seemingly unrelated nature, a…
In many real-world situations, there are constraints on the ways in which a physical system can be manipulated. We investigate the entropy production (EP) and extractable work involved in bringing a system from some initial distribution $p$ to some final distribution $p'$, given that the set of master equations available to the driving protocol obeys some constraints. We first derive general bounds on EP and extractable work, as well as a decomposition of the nonequilibrium free energy into an "accessible free energy" (which can be extracted as work, given a set of constraints) and an "inacces…
Thermodynamic transport phenomena in the system consisting of many hard-disks confined in a circular tube with a temperature difference are discussed. Here, temperatures on parts of the walls of the tube are imposed by stochastic boundary conditions for particles to contact with these thermal walls. In this system, the temperature difference induces, not only energy currents, but also a circulating particle current, inside the tube. Transport properties of these steady currents are discussed in various values of system parameters, such as the temperature difference, the particle density, the w…
Transport coefficients in hard-sphere fluids: thermodynamic versus kinetic descriptions
Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR-CONICET), Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata（马德普拉塔物理研究所（IFIMAR-CONICET），自然科学学院物理系，马德普拉塔国立大学）
The theory of transport phenomena in multicomponent electrolyte solutions is presented here through the integration of continuum mechanics, electromagnetism, and non-equilibrium thermodynamics. The governing equations of irreversible thermodynamics, including balance laws, Maxwell's equations, internal entropy production, and linear laws relating the thermodynamic forces and fluxes, are derived. Green-Kubo relations for the transport coefficients connecting electrochemical potential gradients and diffusive fluxes are obtained in terms of the flux-flux time correlations. The relationship betwee…
Molecular motors work collectively to transport cargo within cells, with anywhere from one to several hundred motors towing a single cargo. For a broad class of collective-transport systems, we use tools from stochastic thermodynamics to derive a new lower bound for the entropy production rate which is tighter than the second law. This implies new bounds on the velocity, efficiency, and precision of general transport systems and a set of analytic Pareto frontiers for identical motors. In a specific model, we identify conditions for saturation of these Pareto frontiers.
How much free energy is irreversibly lost during a thermodynamic process? For deterministic protocols, lower bounds on energy dissipation arise from the thermodynamic friction associated with pushing a system out of equilibrium in finite time. Recent work has also bounded the cost of precisely moving a single degree of freedom. Using stochastic thermodynamics, we compute the total energy cost of an autonomously controlled system by considering both thermodynamic friction and the entropic cost of precisely directing a single control parameter. Our result suggests a challenge to the usual unders…
The two classes of membrane transport, namely, secondary active and passive transport, are understood as different reaction pathways in the same protein structure, described by the 16-state model in this paper. To quantify the thermodynamic difference between secondary active transport and passive transport, we extend the second law of information thermodynamics of the autonomous demon in the four-state model composed of two subsystems to the 16-state model composed of four subsystems representing the membrane transport. We reduce the 16 states to 4 states and derive the coarse-grained second …
Periodically driven coherent conductors provide a universal platform for the development of quantum transport devices. Here, we lay down a comprehensive theory to describe the thermodynamics of these systems. We first focus on moderate thermo-electrical biases and low driving frequencies. For this linear response regime, we establish generalized Onsager-Casimir relations and an extended fluctuation-dissipation theorem. Furthermore, we derive a family of thermodynamic bounds proving that any local matter or heat current puts a non-trivial lower limit on the overall dissipation rate of a coheren…
