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.

For stochastic non-equilibrium dynamics like a Langevin equation for a colloidal particle or a master equation for discrete states, entropy production along a single trajectory is studied. It involves both genuine particle entropy and entropy production in the surrounding medium. The integrated sum of both ∆stot is shown to obey a fluctuation theorem exp[−∆stot] = 1 for arbitrary initial conditions and arbitrary time-dependent driving over a finite time interval.PACS numbers: 05.40.-a, 05.70-aIntroduction. -Can the notions appearing in the first and second law of thermodynamics consistently be applied to mesoscopic non-equilibrium processes like dragging a colloidal particle through a viscous fluid [1,2,3,4]? Concerning the first law, Sekimoto interpreted the terms in the standard overdamped Langevin equation as dynamical increments for applied work, internal energy and dissipated heat [5]. For the second law and, in particular, entropy, a proper formulation and interpretation is more subtle. Entropy might be considered as an ensemble property and therefore not to be applicable to a single trajectory. On the other hand, the so-called fluctuation theorem [6,7,8,9,10,11,12,13,14,15,16,17] quite generally relates the probability of entropy generating trajectories to those of entropy annihilating ones which requires obviously a definition of entropy on the level of a single trajectory. While for a colloidal particle immersed in a heat bath it is pretty clear what the entropy change of the bath is, it is less obvious whether or not one should assign an entropy to the particle itself as well.The purpose of this paper is to show that consequent adaption of a previously introduced stochastic entropy [11,18] to the trajectory of a colloidal particle together with the present original discussion of its equation of motion yields a consistent interpretation of entropy production along a single stochastic trajectory. Moreover, it leads to a lucid and concise identification of boundary terms in fluctuation relations. In fact, we show for arbitrary time-dependent driving that the total entropy production obeys an integral fluctuation theorem which is related to but different from Jarzynski's non-equilibrium work relation [19]. The present definition of entropy also implies that the known steady-state fluctuation theorem holds for finite times rather than in the long-time limit only as previously in stochastic dynamics [9,10]. In a final step, this approach is generalized to arbitrary driven stochastic dynamics governed by a master equation with time-dependent rates.Entropy along a trajectory. -As a paradigm, we consider overdamped motion x(τ ) of a particle with mobility µ along a one-dimensional coordinate in the time-interval 0 ≤ τ ≤ t subject to a force

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Vesicle shapes of low energy are studied for two variants of a continuum model for the bending energy of the bilayer: (i) the spontaneous-curvature model and (ii) the bilayer-coupling model, in which an additional constraint for the area difference of the two monolayers is imposed. We systematically investigate four branches of axisymmetric shapes: (i) the prolate-dumbbell shapes; (ii) the pear-shaped vesicles, which are intimately related to budding; (iii) the oblate-discocyte shapes; and (iv) the stomatocytes. These branches end up at limit shapes where either the membrane self-intersects or two (or more) shapes are connected by an infinitesimally narrow neck. The latter limit shape requires a certain condition between the curvatures of the adjacent shape and the spontaneous curvature. For both models, the phase diagram is determined, which is given by the shape of lowest bending energy for a given volume-to-area ratio and a given spontaneous curvature or area difference, respectively. The transitions between different shapes are continuous for the bilayer-coupling model, while most of the transitions are discontinuous in the spontaneous-curvature model. We introduce trajectories into these phase diagrams that correspond to a change in temperature and osmotic conditions. For the bilayer-coupling model, we find extreme sensitivity to an asymmetry in the monolayer expansivity. Both models lead to different predictions for typical trajectories, such as budding trajectories or oblate-stomatocyte transitions. Our study thus should provide the basis for an experimental test of both variants of the curvature model.

Abstract. -We study a class of cyclic Brownian heat engines in the framework of finite-time thermodynamics. For infinitely long cycle times, the engine works at the Carnot efficiency limit producing, however, zero power. For the efficiency at maximum power, we find a universal expression, different from the endoreversible Curzon-Ahlborn efficiency. Our results are illustrated with a simple one-dimensional engine working in and with a time-dependent harmonic potential.

Biomolecular systems like molecular motors or pumps, transcription and translation machinery, and other enzymatic reactions can be described as Markov processes on a suitable network. We show quite generally that in a steady state the dispersion of observables like the number of consumed/produced molecules or the number of steps of a motor is constrained by the thermodynamic cost of generating it. An uncertainty ǫ requires at least a cost of 2kB T /ǫ 2 independent of the time required to generate the output.PACS numbers: 05.70.Ln, Biomolecular processes are generally out of equilibrium and dissipative, with the associated free energy consumption coming most commonly from adenosine triphosphate (ATP) hydrolysis. The role of energy dissipation in a variety of processes related to biological information processing has received much attention recently [1][2][3][4][5][6][7][8][9][10][11][12][13][14], to give just one class of examples for which one tries to uncover fundamental limits involving energy dissipation in biomolecular systems.Chemical reactions catalyzed by enzymes are of central importance for many cellular processes. Prominent examples are molecular motors [15][16][17][18][19], which convert chemical free energy from ATP into mechanical work. In this case an observable of interest is the number of steps the motor made. Another commonly analyzed output in enzymatic kinetics is the number of product molecules generated by an enzymatic reaction, for which the Michaelis-Menten scheme provides a paradigmatic case [2].Quite generally, chemical reactions are well described by stochastic processes. An observable, like the rate of consumed substrate molecules or the number of steps of a motor on a track, is a random variable subjected to thermal fluctuations. Single molecule experiments [20][21][22][23][24] provide detailed quantitative data on such random quantities. Obtaining information about the underlying chemical reaction scheme through the measurement of fluctuations constitutes a field called statistical kinetics [25][26][27][28]. A central result in this field is the fact that the Fano factor quantifying fluctuations provides a lower bound on the number of states involved in an enzymatic cycle [15,28].For a non-zero mean output, the chemical potential difference (or affinity) driving an enzymatic reaction must also be non-zero, leading to a free energy cost. Is there a fundamental relation between the relative uncertainty associated with the observable quantifying the output and the free energy cost of sustaining the biomolecular process generating it?In this Letter, we show that such a general bound does indeed exist. Specifically, for any process running for a time t, we show that the product of the average dissipated heat and the square of the relative uncertainty of a generic observable is independent of t and bounded by 2k B T . This uncertainty relation is valid for general networks and can be proved within linear response theory. Beyond linear response theory, we show it analytically for unicyclic...

Abstract. Stochastic thermodynamics provides a framework for describing small systems like colloids or biomolecules driven out of equilibrium but still in contact with a heat bath. Both, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories. Various exact relations involving the distribution of such quantities like integral and detailed fluctuation theorems for total entropy production and the Jarzynski relation follow from such an approach based on Langevin dynamics. Analogues of these relations can be proven for any system obeying a stochastic master equation like, in particular, (bio)chemically driven enzyms or whole reaction networks. The perspective of investigating such relations for stochastic field equations like the Kardar-Parisi-Zhang equation is sketched as well.

A simple model for the adhesion of vesicles to interfaces and membranes is introduced and theoretically studied. It is shown that adhering {or bound) vesicles can exhibit a large variety of dift'erent shapes. The notion of a contact angle governed by tension is found to be applicable only for a restricted subset of these shapes. Furthermore, the vesicle undergoes a nontrivial adhesion transition from a free to a bound state. This transition is governed by the balance between the overall bending and adhesion energies, and occurs even in the absence of shape fluctuations.Light and electron microscopy have revealed an astonishing complexity of the spatial organization of biological systems. ' There are two basic problems if one tries to un-