Transformation properties of entropy production

Irreversibility of symbolic time series: A cautionary tale

Many empirical time series are genuinely symbolic: Examples range from link activation patterns in network science, to DNA coding or firing patterns in neuroscience, to cryptography or combinatorics on words. In some other contexts, the underlying time series is actually real valued, and symbolization is applied subsequently, as in symbolic dynamics of chaotic systems. Among several time series quantifiers, time series irreversibility-the difference between forward and backward statistics in stationary time series-is of great relevance. However, the irreversible character of symbolized time series is not always equivalent to the one of the underlying real-valued signal, leading to some misconceptions and confusion on interpretability. Such confusion is even bigger for binary time series-a classical way to encode chaotic trajectories via symbolic dynamics. In this paper we aim to clarify some usual misconceptions and provide theoretical grounding for the practical analysis-and interpretation-of time irreversibility in symbolic time series. We outline sources of irreversibility in stationary symbolic sequences coming from frequency asymmetries of nonpalindromic pairs which we enumerate, and prove that binary time series cannot show any irreversibility based on words of length m<4, thus discussing the implications and sources of confusion. We also study irreversibility in the context of symbolic dynamics, and clarify why these can be reversible even when the underlying dynamical system is not, such as the case of the fully chaotic logistic map.

Scaling of entropy production under coarse graining in active disordered media

Entropy production plays a fundamental role in the study of nonequilibrium systems by offering a quantitative handle on the degree of time-reversal symmetry breaking. It depends crucially on the degree of freedom considered as well as on the scale of description. How the entropy production at one resolution of the degrees of freedom is related to the entropy production at another resolution is a fundamental question which has recently attracted interest. This relationship is of particular relevance to coarse-grained and continuum descriptions of a given phenomenon. In this work, we derive the scaling of the entropy production under iterative coarse graining on the basis of the correlations of the underlying microscopic transition rates for noninteracting particles in active disordered media. Our approach unveils a natural criterion to distinguish equilibrium-like and genuinely nonequilibrium macroscopic phenomena based on the sign of the scaling exponent of the entropy production per mesostate.

Non-Markovianity and negative entropy production rates

Entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems. Its positivity can be ensured for a wide class of setups, but the entropy production rate can become negative sometimes. This is often taken as an indicator of non-Markovianity. We make this link precise by showing under which conditions a negative entropy production rate implies non-Markovianity and when it does not. For a system coupled to a single heat bath, this can be established within a unified language for two setups: (i) the dynamics resulting from a coarse-grained description of a Markovian master equation and (ii) the classical Hamiltonian dynamics of a system coupled to a bath. The quantum version of the latter result is shown not to hold despite the fact that the integrated thermodynamic description is formally equivalent to the classical case. The instantaneous fixed point of a non-Markovian dynamics plays an important role in our study. Our key contribution is to provide a consistent theoretical framework to study the finite-time thermodynamics of a large class of dynamics with a precise link to its non-Markovianity.

Local Concentration Deviations in a Glycolytic Model of Nonequilibrium Thermodynamics: Emergence of Bistability from Thermodynamic Theories due to Violation of the Lyapunov Stability Theory

The thermodynamics of nonequilibrium states in the reversible Sel'kov model, a mathematical model of glycolytic oscillations, is reported in terms of the Lyapunov properties of the second differential of its local equilibrium entropy ( S) and that of its entropy production (e.p.) function (σ) per unit volume. This theory interprets the appropriate limit of validity of the generalized Le Chatelier-Braun (LCB) principle for stable nonequilibrium and equilibrium steady states obeying the Lyapunov stability postulate and its implications in terms of the local concentration deviations of the reacting intermediate species S(ATP) and P(ADP) in response to external excitations. The local concentration deviations of the reacting intermediates are reported to be asymmetric for stable steady states in this model system obeying the Lyapunov stability postulate out of equilibrium (both linear and nonlinear domains), whereas symmetrical local concentration deviations prevail at the state of thermodynamic equilibrium, which however never exists in a living cell. In the unstable steady states in this model reaction, as for example, along a limit cycle trajectory not obeying any Lyapunov stability postulate, the two bistable e.p. branches characterized by two forms of deviation product function δ _sδ _p may be identified separated by a large δ _p > 0 value ( p being scaled concentration of the species P) for dσ > 0 and dσ < 0, respectively, which may be generalized as LCB principle for bistability/hysteresis in this model reaction. The method outlined here may be applied safely to other well-known two-variable models of chemistry/biology. For an appropriate three-variable model with one variable in a pseudo-steady-state approximation, this method may be applicable, albeit with some constraints.

Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles

Many single-molecule experiments for molecular motors comprise not only the motor but also large probe particles coupled to it. The theoretical analysis of these assays, however, often takes into account only the degrees of freedom representing the motor. We present a coarse-graining method that maps a model comprising two coupled degrees of freedom which represent motor and probe particle to such an effective one-particle model by eliminating the dynamics of the probe particle in a thermodynamically and dynamically consistent way. The coarse-grained rates obey a local detailed balance condition and reproduce the net currents. Moreover, the average entropy production as well as the thermodynamic efficiency is invariant under this coarse-graining procedure. Our analysis reveals that only by assuming unrealistically fast probe particles, the coarse-grained transition rates coincide with the transition rates of the traditionally used one-particle motor models. Additionally, we find that for multicyclic motors the stall force can depend on the probe size. We apply this coarse-graining method to specific case studies of the F(1)-ATPase and the kinesin motor.

Stochastic thermodynamics, fluctuation theorems and molecular machines

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.

Stochastic thermodynamics under coarse graining

A general formulation of stochastic thermodynamics is presented for open systems exchanging energy and particles with multiple reservoirs. By introducing a partition in terms of "mesostates" (e.g., sets of "microstates"), the consequence on the thermodynamic description of the system is studied in detail. When microstates within mesostates rapidly thermalize, the entire structure of the microscopic theory is recovered at the mesostate level. This is not the case when these microstates remain out of equilibrium, leading to additional contributions to the entropy balance. Some of our results are illustrated for a model of two coupled quantum dots.