For non-equilibrium systems of interacting particles and for interacting diffusions in d dimensions, a novel fluctuation relation is derived. The theorem establishes a quantitative relation between the probabilities of observing two current values in different spatial directions. The result is a consequence of spatial symmetries of the microscopic dynamics, generalizing in this way the Gallavotti-Cohen fluctuation theorem related to the time-reversal symmetry. This new perspective opens up the possibility of direct experimental measurements of fluctuation relations of vectorial observables.
In this short note we consider semi-Markov processes satisfying the condition of direction-time independence (Markov renewal processes). We derive large deviation principles and fluctuation theorems for the empirical current and the empirical currents along cycles. Our derivation is based on the joint LDP for the empirical measure and flow recently proved in \cite{MZ}.
We present a general method to identify an arbitrary number of fluctuating quantities which satisfy a detailed fluctuation theorem for all times within the framework of time-inhomogeneous Markovian jump processes. In doing so we provide a unified perspective on many fluctuation theorems derived in the literature. By complementing the stochastic dynamics with a thermodynamic structure (i.e. using stochastic thermodynamics), we also express these fluctuating quantities in terms of physical observables.
Using the closed time path formalism in thermal field theory, we give a derivation of the fluctuation-dissipation theorem which is based on the unitarity of the $S$-matrix.
We demonstrate that the Gibbs-Shannon entropy is applicable to non-equilibrium systems of any size and boundary conditions. The change in microscopic entropy can be attributed to the stochastic nature of dynamic processes and to the inherent uncertainties of thermodynamic systems. The latter predicts that entropy production is nonnegative on average and varies with different trajectories according to the fluctuation theorem. By contrast, heat is affiliated with stochastic processes underlying particle motions and the ensemble average over all possible trajectories leads to the Clausius inequal…
The cycle current is a crucial quantity in stochastic thermodynamics. The absolute and net cycle currents of a Markovian system can be defined in the loop-erased (LE) or spanning tree (ST) manner. Here we make a comparative study between the large deviations and fluctuation theorems for LE and ST currents, i.e. cycle currents defined in the LE and ST manners. First, we derive the exact joint distribution and large deviation rate function for the LE currents of a system with a cyclic topology and also obtain the exact rate function for the ST currents of a general system. The relationship betwe…
Fluctuation theorems specify the non-zero probability to observe negative entropy production, contrary to a naive expectation from the second law of thermodynamics. For closed particle trajectories in a fluid, Stokes theorem can be used to give a geometric characterization of the entropy production. Building on this picture, we formulate a topological fluctuation theorem that depends only by the winding number around each vortex core and is insensitive to other aspects of the force. The probability is robust to local deformations of the particle trajectory, reminiscent of topologically protect…
Heat fluctuations are studied in a dissipative system with both mechanical and stochastic components for a simple model: a Brownian particle dragged through water by a moving potential. An extended stationary state fluctuation theorem is derived. For infinite time, this reduces to the conventional fluctuation theorem only for small fluctuations; for large fluctuations, it gives a much larger ratio of the probabilities of the particle to absorb rather than supply heat. This persists for finite times and should be observable in experiments similar to a recent one of Wang et al.
It is demonstrated that the "generalized fluctuation-dissipation theorem" [Physica A 106, 443 (1981)] covers the later suggested "fluctuation theorems" and related statistical equalities.
We consider heat fluctuations and fluctuation theorems for systems driven by multiple reservoirs. We establish a fundamental symmetry obeyed by the joint probability distribution for the heat transfers and system coordinates. The symmetry leads to a generalisation of the asymptotic fluctuation theorem for large deviations at large times. As a result the presence of multiple reservoirs influence the tails in the heat distribution. The symmetry, moreover, allows for a simple derivation of a recent exact fluctuation theorem valid at all times. Including a time dependent work protocol we also pres…
In recent letter [Phys.~Rev.~Lett {\bf 123}, 110602 (2019)], Y.~Hasegawa and T.~V.~Vu derived a thermodynamic uncertainty relation. But the bound of their relation is loose. In this comment, through minor changes, an improved bound is obtained. This improved bound is the same as the one obtained in [Phys.~Rev.~Lett {\bf 123}, 090604 (2019)] by A.~M.~Timpanaro {\it et. al.}, but the derivation here is straightforward.
Recently the ``Fluctuation theorem'' has been criticized and incorrect incorrect contents have been atributed to it. Here I reestablish and comment the original statements.
We derive an extended fluctuation theorem for a geometric pumping in a spin-boson system under a periodic control of environmental temperatures by using a Markovian quantum master equation. We perform the Monte-Carlo simulation and obtain the current distribution, the average current and the fluctuation. Using the extended fluctuation theorem we try to explain the results of our simulation. The fluctuation theorem leads to the fluctuation dissipation relations but the absence of the conventional reciprocal relation.
Quantum analogues of the transient fluctuation theorem(TFT) and steady-state fluctuation theorem(SSFT) are investigated for a harmonic oscillator linearly coupled with a harmonic reservoir. The probability distribution for the work done externally is derived and quantum correction for TFT and SSFT are calculated.
The nonequilibrium Time-Reversible Baker Map provides simple illustrations of the Fluctuation Theorem, the Central Limit Theorem, and the Biased Random Walk. This is material in preparation for the Book form of Carol's and my 2016 Kharagpur Lectures.
A generalization of fluctuation theorems in stochastic processes is proposed. The new theorem is written in terms of posterior probabilities, which are introduced via the Bayes theorem. In usual fluctuation theorems, a forward path and its time reversal play an important role, so that a microscopically reversible condition is essential. In contrast, the microscopically reversible condition is not necessary in the new theorem. It is shown that the new theorem adequately recovers various theorems and relations previously known, such as the Gallavotti-Cohen-type fluctuation theorem, the Jarzynski…
There are two related theorems which hold even in far from equilibrium, namely fluctuation theorem and Jarzynski equality. Fluctuation theorem states the existence of symmetry of fluctuation of entropy production, while Jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes. On the other hand, relationship between these theorems was investigated by Crooks for the classical stochastic systems. In this letter, we derive quantum analogues of fluctuation theorem and Jarzynski equality microscopic reversibility condition. In other words, …
The Fluctuation Theorem gives an analytical expression for the probability of observing second law violating dynamical fluctuations, in nonequilibrium systems. At equilibrium statistical mechanical fluctuations are known to be ensemble dependent. In this paper we generalise the Transient and Steady State Fluctuation Theorems to various nonequilibrium dynamical ensembles. The Transient and Steady State Fluctuation Theorem for an isokinetic ensemble of isokinetic trajectories is tested using nonequilibrium molecular dynamics simulations of shear flow.
It is shown that the quantum fluctuation dissipation theorem can be considered as a mathematical formulation in the spectral representation of Onsager hypothesis on the regression of fluctuations in physical systems. It is shown that the quantum fluctuation dissipation theorem can be generalized to an arbitrary stationary state.
We present a simple derivation of the integral fluctuation theorems for excess housekeeping heat for an underdamped Langevin system, without using the concept of dual dynamics. In conformity with the earlier results, we find that the fluctuation theorem for housekeeping heat holds when the steady state distributions are symmetric in velocity, whereas there is no such requirement for the excess heat. We first prove the integral fluctuation theorem for the excess heat, and then show that it naturally leads to the integral fluctuation theorem for housekeeping heat. We also derive the modified det…
In the attempt to derive the regression theorem from the fluctuation dissipation theorem several authors claim the violation of the former theorem in the quantum case. Here we pose the question: does it exists a quantum fluctuation dissipation theorem (QFDT) in its conventional interpretation? It is shown that the relation usually called as the QFDT is the condition of detailed macroscopic energetic balance. Following this interpretation the existing conflict between the two theorems in the quantum case is removed.
We establish a novel generalization of the fluctuation theorem for partially-masked nonequilibrium dynamics. We introduce a partial entropy production with a subset of all possible transitions, and show that the partial entropy production satisfies the integral fluctuation theorem.Our result reveals the fundamental properties of a broad class of autonomous nanomachines as well as non-autonomous ones. In particular, our result gives a unified fluctuation theorem for both autonomous and non-autonomous Maxwell's demons, where mutual information plays a crucial role. Furthermore, we derive a novel…
In this study, we rederive the fluctuation theorems in presence of feedback, by assuming the known Jarzynski equality and detailed fluctuation theorems. We first reproduce the already known work theorems for a classical system, and then extend the treatment to the other classical theorems. For deriving the extended quantum fluctuation theorems, we have considered open systems. No assumption is made on the nature of environment and the strength of system-bath coupling. However, it is assumed that the measurement process involves classical errors.
Fluctuation theorem is derived for a quantum current system around a nonequilibrium steady state. It is demonstrated that the fluctuation theorem can be a part of the generalized Green-Kubo formula or a nonlinear response theory of an external field or a change of the chemical potential difference.
In their recent paper [Phys. Rev. Lett. 98, 094101 (2007)], A. Porporato et al. studied the irreversibility and fluctuation theorem for stationary time series. In this comment, we point out that the fluctuation theorem is in fact the trivial outcome of a symmetry operation, and hence its physical contect is less convincing.
Fluctuation Theorem(FT) has been studied as far from equilibrium theorem, which relates the symmetry of entropy production. To investigate the application of this theorem, especially to biological physics, we consider the FT for tilted rachet system. Under, natural assumption, FT for steady state is derived.
The statistics of heat exchange between two classical or quantum finite systems initially prepared at different temperatures are shown to obey a fluctuation theorem.
Fluctuation Theorems are statements about the entropy of systems far from thermal equilibrium. In this Letter relativistic Fluctuation Theorems for Brownian motion are presented and proven. Though there is a known discretization dilemma leading to a certain class of Brownian processes, the Fluctuation Theorems are shown to cover the entire class, and to determine the physical process within this class. In the general-relativistic case the heat responsible for entropy production is the exchanged 4-momentum projected on the local time axis of the heat bath.
It is demonstrated that today's quantum fluctuation theorems are component part of old quantum fluctuation-dissipation relations [Sov.Phys.-JETP 45, 125 (1977)], and typical misunderstandings in this area are pointed out.
We derive a generalized version of the work fluctuation theorem for nonequilibrium systems with spatio-temporal temperature fluctuations. For chi-square distributed inverse temperature we obtain a generalized fluctuation theorem based on q-exponentials, whereas for other temperature distributions more complicated formulae arise. Since q-exponentials have a power law decay, the decay rate in this generalized fluctuation theorem is much slower than the conventional exponential decay. This implies that work fluctuations can be of relevance for the design of micro and nano structures, since the wo…
In some situations in stochastic thermodynamics not all relevant slow degrees of freedom are accessible. Consequently, one adopts an effective description involving only the visible degrees of freedom. This gives rise to an apparent entropy production that violates standard fluctuation theorems. We present an analytically solvable model illustrating how the fluctuation theorems are modified. Furthermore, we define an alternative to the apparent entropy production: the marginal entropy production which fulfills the fluctuation theorems in the usual form. We show that the non-Markovianity of the…
A unified viewpoint is presented in margin to the ``Work, dissipation and fluctuations in nonequilibrium physics'', Bruxelles 22-25 March, 2006, where the topics were discussed by various authors and it became clear the need that the very different viewpoints be consistently presented by their proponents.
The fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics that is used to derive many important thermodynamic relations, such as the second law of thermodynamics and the Jarzynski equality. Recently, the thermodynamic uncertainty relation was discovered, which states that the fluctuation of observables is lower bounded by the entropy production. In the present Letter, we derive a thermodynamic uncertainty relation from the fluctuation theorem. We refer to the obtained relation as the fluctuation theorem uncertainty relation, and it is valid for arbitrary dynamics, st…
The fluctuation theorem of Gallavotti and Cohen holds for finite systems undergoing Langevin dynamics. In such a context all non-trivial ergodic theory issues are by-passed, and the theorem takes a particularly simple form. As a particular case, we obtain a nonlinear fluctuation-dissipation theorem valid for equilibrium systems perturbed by arbitrarily strong fields.
We examine classical, transient fluctuation theorems within the unifying framework of Langevin dynamics. We explicitly distinguish between the effects of non-conservative forces that violate detailed balance, and non-autonomous dynamics arising from the variation of an external parameter. When both these sources of nonequilibrium behavior are present, there naturally arise two distinct fluctuation theorems.
We analyze the microscopic evolution of a system undergoing a far-from-equilibrium thermodynamic process. Explicitly accounting for the degrees of freedom of participating heat reservoirs, we derive a hybrid result, similar in form to both the fluctuation theorem, and a statement of detailed balance. We relate this result to the steady-state fluctuation theorem, and to a free energy relation valid far from equilibrium.
By taking full advantage of the dynamic property imposed by the detailed balance condition, we derive a new refined unified fluctuation theorem (FT) for general stochastic thermodynamic systems. This FT involves the joint probability distribution functions of the final phase space point and a thermodynamic variable. Jarzynski equality, Crooks fluctuation theorem, and the FTs of heat as well as the trajectory entropy production can be regarded as special cases of this refined unified FT, and all of them are generalized to arbitrary initial distributions. We also find that the refined unified FT…
The Fluctuation Theorem (FT) is a generalisation of the Second Law of Thermodynamics that applies to small systems observed for short times. For thermostatted systems it gives the probability ratio that entropy will be consumed rather than produced. In this paper we derive the Transient and Steady State Fluctuation Theorems using Lyapunov weights rather than the usual Gibbs weights. At long times the Fluctuation Theorems so derived are identical to those derived using the more standard Gibbs weights.
We derive the generalized Green-Kubo relation and an integral form of the fluctuation theorem that apply to uniformly sheared granular systems in which microscopic time-reversal symmetry is broken. The former relation provides an exact representation of nonequilibrium steady-state properties as the integral of the time-correlation function describing transient dynamics from an initial quiescent towards a final sheared steady state. We also investigate implications of the integral fluctuation theorem on the approach towards the steady state and on the possible form of the steady-state distribut…
We re-derive the Nyquist theorem and Callen-Welton fluctuation-dissipation theorem (FDT) as a consequence of detailed balance principle applied to a harmonic oscillator. The usage of electrical notions in the beginning makes the consideration understandable for every physicists. Perhaps it is the simplest derivation of these well-known theorems in statistical physics. The classical limit is understandable as a consequence of Waterston-Herapath equipartition theorem.
The inclusion of the factor $\ln (1/N!)$ in the thermodynamic entropy proposed by Gibbs is shown to be equivalent to the validity of the fluctuation theorem with absolute irreversibility for gas mixing.
Detailed fluctuation theorem, a microscopic version of the steady state fluctuation theorem, has been proposed by Jarzynski and demonstrated in the case of Hamiltonian systems weakly coupled with reservoirs. We show that an identical theorem for phase space compressibility rate can be derived for systems evolving under non-Hamiltonian extended system dynamics, without certain limiting assumptions made in the original work. Our derivation is based on the non-Hamiltonian phase space formulation of statistical mechanics and does not rely on any assumptions of thermodynamic nature. This version of…
We derive a general fluctuation theorem for quantum maps. The theorem applies to a broad class of quantum dynamics, such as unitary evolution, decoherence, thermalization, and other types of evolution for quantum open systems. The theorem reproduces well-known fluctuation theorems in a single and simplified framework and extends the Hatano-Sasa theorem to quantum nonequilibrium processes. Moreover, it helps to elucidate the physical nature of the environment inducing a given dynamics in an open quantum system.
We study the stochastic thermodynamics of resetting systems. Violation of microreversibility means that the well known derivations of fluctuations theorems break down for dynamics with resetting. Despite that we show that stochastic resetting systems satisfy two integral fluctuation theorems. The first is the Hatano-Sasa relation describing the transition between two steady states. The second integral fluctuation theorem involves a functional that includes both dynamical and thermodynamic contributions. We find that the second law-like inequality found by Fuchs et al. for resetting systems [EP…
The fluctuations of a Markovian jump process with one or more unidirectional transitions, where $R_{ij} >0$ but $R_{ji} =0$, are studied. We find that such systems satisfy an integral fluctuation theorem. The fluctuating quantity satisfying the theorem is a sum of the entropy produced in the bidirectional transitions and a dynamical contribution which depends on the residence times in the states connected by the unidirectional transitions. The convergence of the integral fluctuation theorem is studied numerically, and found to show the same qualitative features as in systems exhibiting microre…
Fluctuation theorems are fundamental results in non-equilibrium thermodynamics. Considering the fluctuation theorem with respect to the entropy production and an observable, we derive a new thermodynamic uncertainty relation which also applies to non-cyclic and time-reversal non-symmetric protocols. Furthermore, we investigate the relation between the thermodynamic uncertainty relation and the correlation between the entropy and the observable.
The fluctuation theorem characterizes the distribution of the dissipation in nonequilibrium systems and proves that the average dissipation will be positive. For a large system with no external source of fluctuation, fluctuations in properties will become unobservable and details of the fluctuation theorem are unable to be explored. In this letter, we consider such a situation and show how a fluctuation theorem can be obtained for a small open subsystem within the large system. We find that a correction term has to be added to the large system fluctuation theorem due to correlation of the subs…
A quantum fluctuation theorem for a driven quantum subsystem interacting with its environment is derived based solely on the assumption that its reduced density matrix obeys a closed evolution equation i.e. a quantum master equation (QME). Quantum trajectories and their associated entropy, heat and work appear naturally by transforming the QME to a time dependent Liouville space basis that diagonalizes the instantaneous reduced density matrix of the subsystem. A quantum integral fluctuation theorem, a steady state fluctuation theorem and the Jarzynski relation are derived in a similar way as f…
We present numerical evidence supporting the validity of the Gallavotti-Cohen Fluctuation Theorem applied to the driven Lorentz gas with Nosé-Hoover thermostating. It is moreover argued that the asymptotic form of the fluctuation formula is independent of the amplitude of the driving force, in the limit where it is small.
Fluctuation theorems have a very special place in the study of non equilibrium dynamics of physical systems. The form in which it is used most extensively is the Gallavoti-Cohen Fluctuation Theorem which is in terms of the distribution of the work $p(W)/p(-W)=\exp(αW)$. We derive the general form of the fluctuation theorems for an arbitrary Gaussian Markov process and find conditions when the parameter $α$ becomes a universal parameter $1/kT$. As an application we consider fluctuation theorems for classical cyclotron motion of an electron in a parabolic potential. The motion of the electron is…
The McLennan-Zubarev steady state distribution is studied in the connection with fluctuation theorems. We derive the McLennan-Zubarev steady state distribution from the nonequilibrium detailed balance relation. Then, considering the cumulant function or cumulant functional, two fluctuation theorems for entropy and for currents are proved. Using the fluctuation theorem for currents, the current is expanded in terms of thermodynamic forces. In the lowest order of the thermodynamic force, we find that the transport coefficient satisfies the Onsager's reciprocal relation. In the next order, we der…
For driven open systems in contact with multiple heat reservoirs, we find the marginal distributions of work or heat do not satisfy any fluctuation theorem, but only the joint distribution of work and heat satisfies a family of fluctuation theorems. A hierarchical structure of these fluctuation theorems is discovered from microreversibility of the dynamics by adopting a step-by-step coarse-graining procedure in both classical and quantum regimes. Thus, we put all fluctuation theorems concerning work and heat into a unified framework. We also propose a general method to calculate the joint stat…
We present a generalized integral fluctuation theorem (GIFT) for general diffusion processes using the Feynman-Kac and Cameron-Martin-Girsanov formulas. Existing IFTs can be thought of to be its specific cases. We interpret the origin of this theorem in terms of time-reversal of stochastic systems.
We consider the quantum mechanical generalization of Crooks Fluctuation Theorem and Jarzynski Equality for an open quantum system. The explicit expression for microscopic work for an arbitrary prescribed protocol is obtained, and the relation between quantum Crooks Fluctuation Theorem, quantum Jarzynski Equality and their classical counterparts are clarified. Numerical simulations based on a two-level toy model are used to demonstrate the validity of the quantum version of the two theorems beyond linear response theory regime.
The Fluctuation Theorem describes the probability ratio of observing trajectories that satisfy or violate the second law of thermodynamics. It has been proved in a number of different ways for thermostatted deterministic nonequilibrium systems. In the present paper we show that the Fluctuation Theorem is also valid for a class of stochastic nonequilibrium systems. The Theorem is therefore not reliant on the reversibility or the determinism of the underlying dynamics. Numerical tests verify the theoretical result.
We present a fluctuation theorem for Floquet quantum master equations. This is a detailed version of the famous Gallavotti-Cohen theorem. In contrast to the latter theorem, which involves the probability distribution of the total heat current, the former involves the joint probability distribution of positive and negative heat currents and can be used to derive the latter. A quantum two-level system driven by a periodic external field is used to verify this result.
The fluctuation theorem is a pivotal result of statistical physics. It quantifies the probability of observing fluctuations which are in violation of the second law of thermodynamics. More specifically, it quantifies the ratio of the probabilities of observing entropy-producing and entropy-consuming fluctuations measured over a finite volume and time span in terms of the rate of entropy production in the system, the measurement volume and time. We study the fluctuation theorem in computer simulations of planar shear flow. The simulations are performed employing the method of multiparticle coll…
The probability distribution of the entropy production for the effusion of a relativistic ideal gas is calculated explicitly. This result is then extended to include particle and anti-particle pair production and annihilation. In both cases, the fluctuation theorem is verified.
