Nonadiabatic entropy production for non-Markov dynamics

Dependence of integrated, instantaneous, and fluctuating entropy production on the initial state in quantum and classical processes

We consider the additional entropy production (EP) incurred by a fixed quantum or classical process on some initial state ρ, above the minimum EP incurred by the same process on any initial state. We show that this additional EP, which we term the "mismatch cost of ρ," has a universal information-theoretic form: it is given by the contraction of the relative entropy between ρ and the least-dissipative initial state φ over time. We derive versions of this result for integrated EP incurred over the course of a process, for trajectory-level fluctuating EP, and for instantaneous EP rate. We also show that mismatch cost for fluctuating EP obeys an integral fluctuation theorem. Our results demonstrate a fundamental relationship between thermodynamic irreversibility (generation of EP) and logical irreversibility (inability to know the initial state corresponding to a given final state). We use this relationship to derive quantitative bounds on the thermodynamics of quantum error correction and to propose a thermodynamically operationalized measure of the logical irreversibility of a quantum channel. Our results hold for both finite- and infinite-dimensional systems, and generalize beyond EP to many other thermodynamic costs, including nonadiabatic EP, free-energy loss, and entropy gain.

Entropy production and Onsager reciprocal relations describing the relaxation to equilibrium in stochastic thermodynamics

We study the relation between stochastic thermodynamics and nonequilibrium thermodynamics by evaluating the entropy production and the relation between fluxes and forces in a harmonic system with N particles in contact with N different reservoirs. We suppose that the system is in a nonequilibrium stationary state in a first phase and we study the relaxation to equilibrium in a second phase. During this relaxation, we can identify the linear relation between fluxes and forces satisfying the Onsager reciprocity and we obtain a nonlinear expression for the entropy production. Only when forces and fluxes are small does the entropic production turn into a quadratic form in the forces, as predicted by the Onsager theory.

Heat flow due to time-delayed feedback

Many stochastic systems in biology, physics and technology involve discrete time delays in the underlying equations of motion, stemming, e. g., from finite signal transmission times, or a time lag between signal detection and adaption of an apparatus. From a mathematical perspective, delayed systems represent a special class of non-Markovian processes with delta-peaked memory kernels. It is well established that delays can induce intriguing behaviour, such as spontaneous oscillations, or resonance phenomena resulting from the interplay between delay and noise. However, the thermodynamics of delayed stochastic systems is still widely unexplored. This is especially true for continuous systems governed by nonlinear forces, which are omnipresent in realistic situations. We here present an analytical approach for the net steady-state heat rate in classical overdamped systems subject to time-delayed feedback. We show that the feedback inevitably leads to a finite heat flow even for vanishingly small delay times, and detect the nontrivial interplay of noise and delay as the underlying reason. To illustrate this point, and to provide an understanding of the heat flow at small delay times below the velocity-relaxation timescale, we compare with the case of underdamped motion where the phenomenon of "entropy pumping" has already been established. Application to an exemplary (overdamped) bistable system reveals that the feedback induces heating as well as cooling regimes and leads to a maximum of the medium entropy production at coherence resonance conditions. These observations are, in principle, measurable in experiments involving colloidal suspensions.

Entropy production in a non-Markovian environment

Stochastic thermodynamics and the associated fluctuation relations provide the means to extend the fundamental laws of thermodynamics to small scales and systems out of equilibrium. The fluctuating thermodynamic variables are usually treated in the context of either isolated Hamiltonian evolution, or Markovian dynamics in open systems. However, there is no reason a priori why the Markovian approximation should be valid in driven systems under nonequilibrium conditions. In this work, we introduce an explicitly non-Markovian model of dynamics of an open system, where the correlations between the system and the environment drive a subset of the environment out of equilibrium. Such an environment gives rise to a new type of non-Markovian entropy production term. Such non-Markovian components must be taken into account in order to recover the fluctuation relations for entropy. As a concrete example, we explicitly derive such modified fluctuation relations for the case of an overheated single electron box.

Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities

Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the non-Markovian nature of the dynamics requires a modification of the basic relation linking dissipation to the breaking of time-reversal symmetry at the level of trajectories. The modified relation includes a contribution arising from the acausal character of the reverse process. This, in turn, leads to another second-law-like inequality. We illustrate the general formalism with a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.

Stochastic thermodynamics of a tagged particle within a harmonic chain

We study the stochastic thermodynamics of an overdamped harmonic chain, which can be viewed equivalently as a one-dimensional Rouse chain or as an approximate model of single file diffusion. We discuss mainly two levels of description of this system: the Markovian level for which the trajectories of all the particles of the chain are known and the non-Markovian level in which only the motion of a tagged particle is available. For each case, we analyze the energy dissipation and its dependence on initial conditions. Surprisingly, we find that the average coarse-grained entropy production rate can become transiently negative when an oscillating force is applied to the tagged particle. This occurs due to memory effects as shown in a framework based on path integrals or on a generalized Langevin equation.

Symmetry for the duration of entropy-consuming intervals

We introduce the violation fraction υ as the cumulative fraction of time that a mesoscopic system spends consuming entropy at a single trajectory in phase space. We show that the fluctuations of this quantity are described in terms of a symmetry relation reminiscent of fluctuation theorems, which involve a function Φ, which can be interpreted as an entropy associated with the fluctuations of the violation fraction. The function Φ, when evaluated for arbitrary stochastic realizations of the violation fraction, is odd upon the symmetry transformations that are relevant for the associated stochastic entropy production. This fact leads to a detailed fluctuation theorem for the probability density function of Φ. We study the steady-state limit of this symmetry in the paradigmatic case of a colloidal particle dragged by optical tweezers through an aqueous solution. Finally, we briefly discuss possible applications of our results for the estimation of free-energy differences from single-molecule experiments.

Duration of local violations of the second law of thermodynamics along single trajectories in phase space

We define the violation fraction ν as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. This quantity depends on both the number of degrees of freedom N and the duration of the time interval τ. In the large-τ and large-N limit we show that, for ergodic and microreversible systems, the mean value of ν scales as 〈ν(N,τ)〉 ∼ (τN(1/1+α))(-1). The exponent α is positive and generally depends on the protocol for the external driving forces, being α = 1 for a constant drive. As an example, we study a nontrivial model where the fluctuations of the entropy production are non-Gaussian: an elastic line driven at a constant rate by an anharmonic trap. In this case we show that the scaling of 〈ν〉 with N and τ agrees with our result. Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition.