Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation–dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production. (Some figures may appear in colour only in the online journal) This article was invited by Erwin Frey.

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Stochastic thermodynamics, fluctuation theorems and molecular machines

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

Stochastic thermodynamics extends the traditional laws of thermodynamics to microscopic systems where thermal and quantum fluctuations cannot be ignored. This review summarizes progress in this field with a look at several experimental and theoretical results and a look toward potential applications in biology and nanotechnology.

https://link.aps.org/pdf/10.1103/PhysRevX.7.021051

Experiments in Stochastic Thermodynamics: Short History and Perspectives

This review presents the effective temperature notion as defined from the deviations from the equilibrium fluctuation–dissipation theorem in out-of-equilibrium systems with slow dynamics. The thermodynamic meaning of this quantity is discussed in detail. Analytic, numeric and experimental measurements are surveyed. Open issues are mentioned.

https://iopscience.iop.org/article/10.1088/1751-8113/44/48/483001/pdf

The effective temperature

Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.

https://iopscience.iop.org/article/10.1088/1361-6633/aab3ed/pdf

Broken detailed balance and non-equilibrium dynamics in living systems: a review

We discuss the relevance of information contained in cross correlations among different degrees of freedom, which is crucial in nonequilibrium systems. In particular we consider a stochastic system where two degrees of freedom X{1} and X{2}-in contact with two different thermostats-are coupled together. The production of entropy and the violation of equilibrium fluctuation-dissipation theorem (FDT) are both related to the cross correlation between X{1} and X{2}. Information about such cross correlation may be lost when single-variable reduced models for X_{1} are considered. Two different procedures are typically applied: (a) one totally ignores the coupling with X{2}; and (b) one models the effect of X{2} as an average memory effect, obtaining a generalized Langevin equation. In case (a) discrepancies between the system and the model appear both in entropy production and linear response; the latter can be exploited to define effective temperatures, but those are meaningful only when time scales are well separated. In case (b) linear response of the model well reproduces that of the system; however the loss of information is reflected in a loss of entropy production. When only linear forces are present, such a reduction is dramatic and makes the average entropy production vanish, posing problems in interpreting FDT violations.

Nonequilibrium and information: the role of cross correlations.

The steady state of a Langevin equation with short-ranged memory and coloured noise is analyzed. When the fluctuation-dissipation theorem of second kind is not satisfied, the dynamics is irreversible, i.e. detailed balance is violated. We show that the entropy production rate for this system should include the power injected by "memory forces". With this additional contribution, the fluctuation relation is fairly verified in simulations. Both dynamics with inertia and overdamped dynamics yield the same expression for this additional power. The role of "memory forces" within the fluctuation-dissipation relation of first kind is also discussed.

https://iopscience.iop.org/article/10.1209/0295-5075/88/30004/pdf

Irreversible effects of memory

We discuss the well known Einstein and the Kubo fluctuation-dissipation relations (FDRs) in the wider framework of a generalized FDR for systems with a stationary probability distribution. A multivariate linear Langevin model, which includes dynamics with memory, is used as a treatable example to show how the usual relations are recovered only in particular cases. This study brings to the fore the ambiguities of a check of the FDR done without knowing the significant degrees of freedom and their coupling. An analogous scenario emerges in the dynamics of diluted shaken granular media. There, the correlation between position and velocity of particles, due to spatial inhomogeneities, induces violation of usual FDRs. The search for the appropriate correlation function which could restore the FDR can be more insightful than a definition of 'non-equilibrium' or 'effective temperatures'.

https://iopscience.iop.org/article/10.1088/1742-5468/2009/07/P07024/pdf

The fluctuation-dissipation relation: how does one compare correlation functions and responses?

A Generalized Langevin Equation with exponential memory is proposed for the dynamics of a massive intruder in a dense granular fluid. The model reproduces numerical correlation and response functions, violating the Equilibrium Fluctuation-Dissipation Relations. The source of memory is identified in the coupling of the tracer velocity V with a spontaneous local velocity field U in the surrounding fluid: fluctuations of this field introduce a new time scale with its associated length scale. Such identification allows us to measure the intruder's fluctuating entropy production as a function of V and U, obtaining a neat verification of the fluctuation relation.

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Irreversible dynamics of a massive intruder in dense granular fluids

We study the stochastic motion of an intruder in a dilute driven granular gas. All particles are coupled to a thermostat, representing the external energy source, which is the sum of random forces and a viscous drag. The dynamics of the intruder, in the large mass limit, is well described by a linear Langevin equation, combining the effects of the external bath and of the 'granular bath'. The drag and diffusion coefficients are calculated under a few assumptions, whose validity is well verified in numerical simulations. We also discuss the non-equilibrium properties of the intruder dynamics, as well as the corrections due to finite packing fraction or finite intruder mass.

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Dario Villamaina

Papers

Nonequilibrium and information: the role of cross correlations.

Irreversible effects of memory

The fluctuation-dissipation relation: how does one compare correlation functions and responses?

Irreversible dynamics of a massive intruder in dense granular fluids

Granular Brownian motion