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The fluctuation relations, which characterize irreversible processes in nature, are among the most important results in non-equilibrium physics. In short, these relations say that it is exponentially unlikely for us to observe a time-reversed process and, thus, establish the thermodynamic arrow of time pointing from low to high entropy. On the other hand, fundamental physical theories are invariant under time-reversal symmetry. Although in Newtonian and quantum physics the emergence of irreversible processes, as well as fluctuation relations, is relatively well understood, many problems arise when relativity enters the game. In this work, by considering a specific class of spacetimes, we explore the question of how the time-dilation effect enters into the fluctuation relations. We conclude that a positive entropy production emerges as a consequence of both the special relativistic and the gravitational (enclosed in the equivalence principle) time-dilation effects. [ABSTRACT FROM AUTHOR]

We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. We analyze exactly solvable oscillator systems and perform simulations of relatively more complex ones. This indicates that the standard statistical mechanical formalism needs to be adjusted in the ever more investigated nano-scale science and technology. In particular, the hypothesis that heat reservoirs be considered infinite and be described by the classical ensembles is found to be critical when exponential quantities are considered since the large size limit may not coincide with the infinite size canonical result. Furthermore, process-dependent emergent irreversibility affects ensemble averages, effectively frustrating, on a statistical level, the time reversal invariance of Hamiltonian dynamics that are used to obtain numerous results. [ABSTRACT FROM AUTHOR]

We consider a general class of Markovian models describing the growth in a randomly fluctuating environment of a clonal biological population having several phenotypes related by stochastic switching. Phenotypes differ e.g. by the level of gene expression for a population of bacteria. The time-averaged growth rate of the population, Λ , is self-averaging in the limit of infinite time; it may be understood as the fitness of the population in a context of Darwinian evolution. The observation time T being however typically finite, the growth rate fluctuates. For T finite but large, we obtain the variance of the time-averaged growth rate as the maximum of a functional based on the stationary probability distribution for the phenotypes. This formula is general. In the case of two states, the stationary probability was computed by Hufton et al. (J Stat Mech 2018:23501, 2018), allowing for an explicit expression which can be checked numerically. Applications of our main formula to the study of survival strategies of biological populations, as developed in the companion article (Dinis et al. in ), are presented here briefly. [ABSTRACT FROM AUTHOR]

We study a class of dynamical semigroups ( n) n ∈ ℕ that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system (ℒ ω n ∘ ⋯ ∘ ℒ ω 1 (ρ ω 0 )) n ∈ ℕ driven by a Markov chain (ω n) n ∈ ℕ . We show that the almost sure large time behavior of the system can be extracted from the large n asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator . As a physical application, we consider the case where the ℒ ω 's are the reduced dynamical maps describing the repeated interactions of a system with thermal probes ℰ ω . We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas. [ABSTRACT FROM AUTHOR]

We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. We analyze exactly solvable oscillator systems and perform simulations of relatively more complex ones. This indicates that the standard statistical mechanical formalism needs to be adjusted in the ever more investigated nano-scale science and technology. In particular, the hypothesis that heat reservoirs be considered infinite and be described by the classical ensembles is found to be critical when exponential quantities are considered since the large size limit may not coincide with the infinite size canonical result. Furthermore, process-dependent emergent irreversibility affects ensemble averages, effectively frustrating, on a statistical level, the time reversal invariance of Hamiltonian dynamics that are used to obtain numerous results.

Thermodynamic probes can be used to deduce microscopic internal dynamics of nanoscale quantum systems. Several direct entropy measurement protocols based on charge transport measurements have been proposed and experimentally applied to single-electron devices. To date, these methods have relied on (quasi-)equilibrium conditions between the nanoscale quantum system and its environment, which constitutes only a small subset of the experimental conditions available. In this paper, we establish a thermodynamic analysis method based on stochastic thermodynamics, that is valid far from equilibrium conditions, is applicable to a broad range of single-electron devices and allows us to find the difference in entropy between the charge states of the nanodevice, as well as a characteristic of any selection rules governing electron transfers. We apply this non-equilibrium entropy measurement protocol to a single-molecule device in which the internal dynamics can be described by a two-site Hubbard model.

We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation–dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory. [ABSTRACT FROM AUTHOR]

We continue the investigation, started in Jakšić et al. in (J. Stat. Phys. 166:926–1015, 2017), of a network of harmonic oscillators driven out of thermal equilibrium by heat reservoirs. We study the statistics of the fluctuations of the heat fluxes flowing between the network and the reservoirs in the nonequilibrium steady state and in the large time limit. We prove a large deviation principle for these fluctuations and derive the fluctuation relation satisfied by the associated rate function. [ABSTRACT FROM AUTHOR]

This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans–Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in Andrieux et al. (in New J Phys 11(4):043014, 2009). We particularly fix some proofs of the latter article where the ultraviolet condition was not mentioned. We detail how these two symmetries lead respectively to fluctuation relations and a statistical refinement of heat conservation for isolated thermodynamic quantum systems. As in Andrieux et al. (in New J Phys 11(4):043014, 2009), we recover the fluctuation–dissipation theorem in the linear response theory, short of Green–Kubo relations. We illustrate the general theory on a number of canonical models. [ABSTRACT FROM AUTHOR]

The so-called fluctuation theorems pushed the study of systems far beyond equilibrium, whose response to thermodynamic forces (affinities) is characterized by the reciprocal and the fluctuation-dissipation relations. All these results rely on the assumption that the observer has complete information about the system: no hidden leakage to the environment, exact evaluation of the thermodynamic cost of processes. Will an observer who has marginal information be able to perform an effective thermodynamic analysis? Given that such observer will only be able to establish local equilibrium, by perturbing the stalling currents will he/she observe equilibrium-like fluctuations? Within the formalism of Markov jump processes on finite networks, we provide a broad theory of the statistical behavior of some out of many currents that flow across a system. In particular (1) There exist effective affinities for which an integral fluctuation relation holds; (2) At stalling, i.e. where the marginal currents vanish, a symmetrized fluctuation-dissipation relation holds; (3) Under reasonable assumptions on the parametrization of the rates, effective affinities can be operationally defined by a procedure of tuning to stalling; (4) There exists a notion of marginally time-reversed process which restores the full-fledged fluctuation relation and reciprocity; (5) There exist fluctuation relations across different levels in the hierarchy of more and more "complete" theories. The above results apply to configuration-space currents, and to their phenomenological linear combinations provided certain symmetries of the effective affinities are respected—a condition whose range of validity we deem the most interesting question left open to future inquiry. [ABSTRACT FROM AUTHOR]

The understanding of out-of-equilibrium fluctuation relations in small open quantum systems has been a focal point of research in recent years. In particular, for systems with adiabatic time-dependent driving, it was shown that the fluctuation relations known from stationary systems do no longer apply due the geometric nature of the pumping current response. However, the precise physical interpretation of the corrected pumping fluctuation relations as well as the role of many-body interactions remained unexplored. Here, we study quantum systems with many-body interactions subject to slow time-dependent driving, and show that fluctuation relations of the charge current can in general not be formulated without taking into account the total energy current put into the system through the pumping process. Moreover, we show that this correction due to the input energy is nonzero only when Coulomb-interactions are present. Thus, fluctuation response relations offer an until now unrevealed opportunity to probe many-body correlations in quantum systems. We demonstrate our general findings at the concrete example of a single-level quantum dot model, and propose a scheme to measure the interaction-induced discrepancies from the stationary case.

We illustrate how, contrary to common belief, transient Fluctuation Relations (FRs) for systems in constant external magnetic field hold without the inversion of the field. Building on previous work providing generalized time-reversal symmetries for systems in parallel external magnetic and electric fields, we observe that the standard proof of these important nonequilibrium properties can be fully reinstated in the presence of net dissipation. This generalizes recent results for the FRs in orthogonal fields—an interesting but less commonly investigated geometry—and enables direct comparison with existing literature. We also present for the first time a numerical demonstration of the validity of the transient FRs with nonzero magnetic field via nonequilibrium molecular dynamics simulations of a realistic model of liquid NaCl.

Levitated Nanoparticles have received much attention for their potential to perform quantum mechanical experiments even at room temperature. However, even in the regime where the particle dynamics are purely classical, there is a lot of interesting physics that can be explored. Here we review the application of levitated nanoparticles as a new experimental platform to explore stochastic thermodynamics in small systems. [ABSTRACT FROM AUTHOR]

In this work we study the distribution function for the total entropy production of a Brownian particle embedded in a non-Markovian thermal bath. The problem is studied in the overdamped approximation of the generalized Langevin equation, which accounts for a friction memory kernel characteristic of a Gaussian colored noise. The problem is studied in two physical situations: (i) when the particle in the harmonic trap is subjected to an arbitrary time-dependent driving force; and (ii) when the minimum of the harmonic trap is arbitrarily dragged out of equilibrium by an external force. By assuming a natural non Markovian canonical distribution for the initial conditions, the distribution function for the total entropy production becomes a non Gaussian one. Its characterization is then given through the first three cumulants.

In this paper we review some general properties of probability distributions which exhibit a singular behavior. After introducing the matter with several examples based on various models of statistical mechanics, we discuss, with the help of such paradigms, the underlying mathematical mechanism producing the singularity and other topics such as the condensation of fluctuations, the relationships with ordinary phase-transitions, the giant response associated to anomalous fluctuations, and the interplay with fluctuation relations.

We present here a set of lecture notes on exact fluctuation relations. We prove the Jarzynski equality and the Crooks fluctuation theorem, two paradigmatic examples of classical fluctuation relations. Finally, we consider their quantum versions, and analyze analogies and differences with the classical case. [ABSTRACT FROM AUTHOR]

We consider a general network of harmonic oscillators driven out of thermal equilibrium by coupling to several heat reservoirs at different temperatures. The action of the reservoirs is implemented by Langevin forces. Assuming the existence and uniqueness of the steady state of the resulting process, we construct a canonical entropy production functional $$S^t$$ which satisfies the Gallavotti-Cohen fluctuation theorem. More precisely, we prove that there exists $$\kappa _c>\frac{1}{2}$$ such that the cumulant generating function of $$S^t$$ has a large-time limit $$e(\alpha )$$ which is finite on a closed interval $$[\frac{1}{2}-\kappa _c,\frac{1}{2}+\kappa _c]$$ , infinite on its complement and satisfies the Gallavotti-Cohen symmetry $$e(1-\alpha )=e(\alpha )$$ for all $$\alpha \in {\mathbb {R}}$$ . Moreover, we show that $$e(\alpha )$$ is essentially smooth, i.e., that $$e'(\alpha )\rightarrow \mp \infty $$ as $$\alpha \rightarrow \tfrac{1}{2}\mp \kappa _c$$ . It follows from the Gärtner-Ellis theorem that $$S^t$$ satisfies a global large deviation principle with a rate function I( s) obeying the Gallavotti-Cohen fluctuation relation $$I(-s)-I(s)=s$$ for all $$s\in {\mathbb {R}}$$ . We also consider perturbations of $$S^t$$ by quadratic boundary terms and prove that they satisfy extended fluctuation relations, i.e., a global large deviation principle with a rate function that typically differs from I( s) outside a finite interval. This applies to various physically relevant functionals and, in particular, to the heat dissipation rate of the network. Our approach relies on the properties of the maximal solution of a one-parameter family of algebraic matrix Riccati equations. It turns out that the limiting cumulant generating functions of $$S^t$$ and its perturbations can be computed in terms of spectral data of a Hamiltonian matrix depending on the harmonic potential of the network and the parameters of the Langevin reservoirs. This approach is well adapted to both analytical and numerical investigations. [ABSTRACT FROM AUTHOR]

We study nonequilibrium statistical mechanics of a Gaussian dynamical system and compute in closed form the large deviation functionals describing the fluctuations of the entropy production observable with respect to the reference state and the nonequilibrium steady state. The entropy production observable of this model is an unbounded function on the phase space, and its large deviation functionals have a surprisingly rich structure. We explore this structure in some detail. [ABSTRACT FROM AUTHOR]

We investigate electron transport in two quantum circuits with mutual Coulomb interaction. The first circuit is a double quantum dot connected to two electron reservoirs, while the second one is a quantum point contact in the weak tunneling limit. The coupling is such that an electron in the first circuit enhances the barrier of the point contact and, thus, reduces its conductivity. While such setups are frequently used as charge monitors, we focus on two different aspects. First, we derive transport coefficients which have recently been employed for testing generalized equilibrium conditions known as exchange fluctuation relations. These formally exact relations allow us to test the consistency of our master equation approach. Second, a biased point contact entails noise on the DQD and induces non-equilibrium phenomena such as a ratchet current. [ABSTRACT FROM AUTHOR]

We obtain the rate function for the level 2.5 of large deviations for pure jump and diffusion processes. This result is proved by two methods: tilting, for which a tilted process with an appropriate typical behavior is considered, and a spectral method, for which the scaled cumulant generating function is used. We also briefly discuss fluctuation relations, pointing out their connection with large deviations at the level 2.5. [ABSTRACT FROM AUTHOR]

In this work we study the distribution function for the total entropy production of a Brownian particle embedded in a non-Markovian thermal bath. The problem is studied in the overdamped approximation of the generalized Langevin equation, which accounts for a friction memory kernel characteristic of a Gaussian colored noise. The problem is studied in two physical situations: (i) when the particle in the harmonic trap is subjected to an arbitrary time-dependent driving force; and (ii) when the minimum of the harmonic trap is arbitrarily dragged out of equilibrium by an external force. By assuming a natural non Markovian canonical distribution for the initial conditions, the distribution function for the total entropy production becomes a non Gaussian one. Its characterization is then given through the first three cumulants. [ABSTRACT FROM AUTHOR]

We exhibit the invariance of cycle affinities in finite state Markov processes under various natural probabilistic constructions, for instance under conditioning and under a new combinatorial construction that we call 'drag and drop'. We show that cycle affinities have a natural probabilistic meaning related to first passage non-equilibrium fluctuation relations that we establish. [ABSTRACT FROM AUTHOR]

Fluctuation theorems establish deep relations between observables away from thermal equilibrium. Until recently, the research on fluctuation theorems was focused on time-reversal-invariant systems. In this review we address some newly discovered fluctuation relations that hold without the time-reversal symmetry, in particular, in the presence of an external magnetic field. One family of relations connects nonlinear transport coefficients in the opposite magnetic fields. Another family relates currents and noises at a fixed direction of the magnetic field in chiral systems, such as the edges of some quantum Hall liquids. We review the recent experimental and theoretical research, including the controversy about the microreversibility without the time-reversal symmetry, consider the applications of fluctuation theorems to the physics of topological states of matter, and discuss open problems. [ABSTRACT FROM AUTHOR]

We discuss work performed on a quantum two-level system coupled to multiple thermal baths. To evaluate the work, a measurement of photon exchange between the system and the baths is envisioned. In a realistic scenario, some photons remain unrecorded as they are exchanged with baths that are not accessible to the measurement, and thus only partial information on work and heat is available. The incompleteness of the measurement leads to substantial deviations from standard fluctuation relations. We propose a recovery of these relations, based on including the mutual information given by the counting efficiency of the partial measurement. We further present the experimental status of a possible implementation of the proposed scheme, i.e. a calorimetric measurement of work, currently with nearly single-photon sensitivity.

It is generally observed that if a dynamical system is sufficiently complex, then as time progresses it will share out energy and other properties amongst its component parts to eliminate any initial imbalances, retaining only fluctuations. This is known as energy dissipation and it is closely associated with the concept of thermodynamic irreversibility, measured by the increase in entropy according to the second law. It is of interest to quantify such behaviour from a dynamical rather than a thermodynamic perspective and to this end stochastic entropy production and the time-integrated dissipation function have been introduced as analogous measures of irreversibility, principally for stochastic and deterministic dynamics, respectively. We seek to compare these measures. First we modify the dissipation function to allow it to measure irreversibility in situations where the initial probability density function (pdf) of the system is asymmetric as well as symmetric in velocity. We propose that it tests for failure of what we call the obversibility of the system, to be contrasted with reversibility, the failure of which is assessed by stochastic entropy production. We note that the essential difference between stochastic entropy production and the time-integrated modified dissipation function lies in the sequence of procedures undertaken in the associated tests of irreversibility. We argue that an assumed symmetry of the initial pdf with respect to velocity inversion (within a framework of deterministic dynamics) can be incompatible with the Past Hypothesis, according to which there should be a statistical distinction between the behaviour of certain properties of an isolated system as it evolves into the far future and the remote past. Imposing symmetry on a velocity distribution is acceptable for many applications of statistical physics, but can introduce difficulties when discussing irreversible behaviour.

Anomalous dynamics characterized by non-Gaussian probability distributions (PDFs) and/or temporal long-range correlations can cause subtle modifications of conventional fluctuation relations (FRs). As prototypes we study three variants of a generic time-fractional Fokker–Planck equation with constant force. Type A generates superdiffusion, type B subdiffusion and type C both super- and subdiffusion depending on parameter variation. Furthermore type C obeys a fluctuation–dissipation relation whereas A and B do not. We calculate analytically the position PDFs for all three cases and explore numerically their strongly non-Gaussian shapes. While for type C we obtain the conventional transient work FR, type A and type B both yield deviations by featuring a coefficient that depends on time and by a nonlinear dependence on the work. We discuss possible applications of these types of dynamics and FRs to experiments.

We consider fluctuation relations between the transport coefficients of a spintronic system where magnetic interactions play a crucial role. We investigate a prototypical spintronic device - a spin-diode - which consists of an interacting resonant level coupled to two ferromagnetic electrodes. We thereby obtain the cumulant generating function for the spin transport in the sequential tunnelling regime. We demonstrate the fulfilment of the nonlinear fluctuation relations when up and down spin currents are correlated in the presence of both spin-flip processes and external magnetic fields. [ABSTRACT FROM AUTHOR]

An open quantum system interacting with its environment can be modeled under suitable assumptions as a Markov process, described by a Lindblad master equation. In this work, we derive a general set of fluctuation relations for systems governed by a Lindblad equation. These identities provide quantum versions of Jarzynski-Hatano-Sasa and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper [M. Colangeli et al., J. Stat. Mech. P04021, (2011)] it was argued that the Fluctuation Relation for the phase space contraction rate could suitably be extended to non-reversible dissipative systems. We strengthen here those arguments, providing analytical and numerical evidence based on the properties of a simple irreversible nonequilibrium baker model. We also consider the problem of response, showing that the transport coefficients are not affected by the irreversibility of the microscopic dynamics. In addition, we prove that a form of detailed balance, hence of equilibrium, holds in the space of relevant variables, despite the irreversibility of the phase space dynamics. This corroborates the idea that the same stochastic description, which arises from a projection onto a subspace of relevant coordinates, is compatible with quite different underlying deterministic dynamics. In other words, the details of the microscopic dynamics are largely irrelevant, for what concerns properties such as those concerning the Fluctuation Relations, the equilibrium behavior and the response to perturbations. [Copyright &y& Elsevier]

We investigate a new symmetry of the large deviation function of certain time-integrated currents in non-equilibrium systems. The symmetry is similar to the well-known Gallavotti-Cohen-Evans-Morriss-symmetry for the entropy production, but it concerns a different functional of the stochastic trajectory. The symmetry can be found in a restricted class of Markov jump processes, where the network of microscopic transitions has a particular structure and the transition rates satisfy certain constraints. We provide three physical examples, where time-integrated observables display such a symmetry. Moreover, we argue that the origin of the symmetry can be traced back to time-reversal if stochastic trajectories are grouped appropriately. [ABSTRACT FROM AUTHOR]

In the context of Markov evolution, we present two original approaches to obtain Generalized Fluctuation-Dissipation Theorems ( GFDT), by using the language of stochastic derivatives and by using a family of exponential martingales functionals. We show that GFDT are perturbative versions of relations verified by these exponential martingales. Along the way, we prove GFDT and Fluctuation Relations ( FR) for general Markov processes, beyond the usual proof for diffusion and pure jump processes. Finally, we relate the FR to a family of backward and forward exponential martingales. [ABSTRACT FROM AUTHOR]

Abstract: Monte Carlo simulations of a sheared Ising model are used to study nonequilibrium fluctuations of mechanical work. The validity of the transient (starting from equilibrium) and the steady state fluctuation relations is verified. A fluctuation relation has been also shown to hold for the mechanical work done on the system, during the transition between two nonequilibrium steady states corresponding to different drivings. [Copyright &y& Elsevier]

Abstract: General aspects of the Fluctuation–Dissipation Relation (FDR), and Response Theory are considered. After analyzing the conceptual and historical relevance of fluctuations in statistical mechanics, we illustrate the relation between the relaxation of spontaneous fluctuations, and the response to an external perturbation. These studies date back to Einstein’s work on Brownian Motion, were continued by Nyquist and Onsager and culminated in Kubo’s linear response theory. The FDR has been originally developed in the framework of statistical mechanics of Hamiltonian systems, nevertheless a generalized FDR holds under rather general hypotheses, regardless of the Hamiltonian, or equilibrium nature of the system. In the last decade, this subject was revived by the works on Fluctuation Relations (FR) concerning far from equilibrium systems. The connection of these works with large deviation theory is analyzed. Some examples, beyond the standard applications of statistical mechanics, where fluctuations play a major role are discussed: fluids, granular media, nanosystems and biological systems. [Copyright &y& Elsevier]

We illustrate how, contrary to common belief, transient Fluctuation Relations (FRs) for systems in constant external magnetic field hold without the inversion of the field. Building on previous work providing generalized time-reversal symmetries for systems in parallel external magnetic and electric fields, we observe that the standard proof of these important nonequilibrium properties can be fully reinstated in the presence of net dissipation. This generalizes recent results for the FRs in orthogonal fields—an interesting but less commonly investigated geometry—and enables direct comparison with existing literature. We also present for the first time a numerical demonstration of the validity of the transient FRs with nonzero magnetic field via nonequilibrium molecular dynamics simulations of a realistic model of liquid NaCl. [ABSTRACT FROM AUTHOR]