Acceleration of relaxation toward a fixed stationary distribution via violation of detailed balance was reported in the context of a Markov chain Monte Carlo method recently. Inspired by this result, systematic methods to violate detailed balance in Langevin dynamics were formulated by using exponential and rotational nonconservative forces. In the present paper, we accentuate that such specific nonconservative forces relate to the large deviation of total heat in an equilibrium state. The response to these nonconservative forces can be described by the intrinsic fluctuation of the total heat in the equilibrium state. Consequently, the fluctuation-dissipation relation for nonequilibrium steady states is derived without recourse to a linear response approximation.
Starting from the pioneering work of G. S. Agarwal [Zeitschrift für Physik 252, 25 (1972)], we present a unified derivation of a number of modified fluctuation-dissipation relations (MFDR) that relate response to small perturbations around non-equilibrium steady states to steady-state correlations. Using this formalism we show the equivalence of velocity forms of MFDR derived using continuum Langevin and discrete master equation dynamics. The resulting additive correction to the Einstein relation is exemplified using a flashing ratchet model of molecular motors.
We present a theoretical framework to understand a modified fluctuation-dissipation theorem valid for systems close to non-equilibrium steady-states and obeying markovian dynamics. We discuss the interpretation of this result in terms of trajectory entropy excess. The framework is illustrated on a simple pedagogical example of a molecular motor. We also derive in this context generalized Green-Kubo relations similar to the ones derived recently by Seifert., Phys. Rev. Lett., 104, 138101 (2010) for more general networks of biomolecular states.
A modified fluctuation-dissipation-theorem (MFDT) for a non-equilibrium steady state (NESS) is experimentally checked by studying the position fluctuations of a colloidal particle whose motion is confined in a toroidal optical trap. The NESS is generated by means of a rotating laser beam which exerts on the particle a sinusoidal conservative force plus a constant non-conservative one. The MFDT is shown to be perfectly verified by the experimental data. It can be interpreted as an equilibrium-like fluctuation-dissipation relation in the Lagrangian frame of the mean local velocity of the particl…
We study the effects of the finite number of experimental data on the computation of a generalized fluctuation-dissipation relation around a nonequilibrium steady state of a Brownian particle in a toroidal optical trap. We show that the finite sampling has two different effects, which can give rise to a poor estimate of the linear response function. The first concerns the accessibility of the generalized fluctuation-dissipation relation due to the finite number of actual perturbations imposed to the control parameter. The second concerns the propagation of the error made at the initial samplin…
Near equilibrium, small current fluctuations are described by a Gaussian with a linear-response variance regulated by the dissipation. Here, we demonstrate that dissipation still plays a dominant role in structuring large fluctuations arbitrarily far from equilibrium. In particular, we prove a linear-response-like bound on the large deviation function for currents in Markov jump processes. We find that nonequilibrium current fluctuations are always more likely than what is expected from a linear-response analysis. As a small-fluctuations corollary, we derive a recently-conjectured uncertainty …
A dynamical theory for the organization and dissipation of a current in a non-equilibrium fluid near equilibrium is presented. This is based on the Lyapunov exponents of the phase space of the system.
A non-equilibrium steady state is characterized by a non-zero steady dissipation rate. Chemical reaction systems under suitable conditions may generate such states. We propose here a method that is able to distinguish states with identical values of the steady dissipation rate. This necessitates a study of the variation of the entropy production rate with the experimentally observable reaction rate in regions close to the steady states. As an exactly-solvable test case, we choose the problem of enzyme catalysis. Link of the total entropy production with the enzyme efficiency is also establishe…
We study a symmetry broken harmonic oscillator in contact with a heat bath characterized by a fixed temperature. The overdampped system is solved exactly to show symmetry broken directed transport raising the question whether fluctuation-dissipation relation does guarantee equilibrium.
The connection between dissipation and symmetry breaking is a long-standing enigma in statistical physics. It is intimately connected to the quest of a non-equilibrium functional whose minimization gives the non-equilibrium steady state (NESS). Writing down such a functional, we show that, in the presence of additive noise, any NESS is characterized by the minimum entropy production compatible with the maximum dissipation along cycles in the trajectory space. This result sheds light on the excess entropy production principle and the onset of chiral symmetry breaking out-of-equilibrium, indicat…
Edges of some quantum Hall liquids and a number of other systems exhibit chiral transport: excitations can propagate in one direction only, e.g., clockwise. We derive a family of fluctuation-dissipation relations in non-equilibrium steady states of such chiral systems. The theorems connect nonlinear response with fluctuations far from thermal equilibrium and hold only in case of chiral transport. They can be used to test chiral or non-chiral character of the system.
Equilibrium is characterized by its fundamental properties such as the detailed balance, the fluctuation-dissipation relation, and no heat dissipation. Based on the stochastic thermodynamics, we show that these three properties are equivalent to each other in conventional Langevin thermal systems with microscopic reversibility. Thus, a conventional steady state has either all three properties (equilibrium) or none of them (nonequilibrium). In contrast, with velocity-dependent forces breaking the microscopic reversibility, we prove that the detailed balance and the fluctuation-dissipation relat…
For systems in equilibrium at a temperature $T$, thermal noise and energy damping are related to $T$ through the fluctuation-dissipation theorem (FDT). We study here an extension of the FDT to an out of equilibrium steady state: a microcantilever subject to a constant heat flux. The resulting thermal profile in this spatially extended system interplays with the local energy dissipation field to prescribe the amplitude of mechanical fluctuations. Using three samples with different damping profiles (localized or distributed), we probe this approach and experimentally demonstrate the link between…
We consider the problem of constructing a thermodynamic theory of non-equilibrium steady states as a formal extension of the equilibrium theory. Specifically, studying a particular system, we attempt to construct a phenomenological theory describing the interplay between heat and mechanical work that takes place during operations through which the system undergoes transitions between non-equilibrium steady states. We find that, in contrast to the case of the equilibrium theory, apparently, there exists no systematic way within a phenomenological formulation to describe the work done by the sys…
Transitions between nonequilibrium steady states obey a generalized Clausius inequality, which becomes an equality in the quasistatic limit. For slow but finite transitions, we show that the behavior of the system is described by a response matrix whose elements are given by a far-from-equilibrium Green-Kubo formula, involving the decay of correlations evaluated in the nonequilibrium steady state. This result leads to a fluctuation-dissipation relation between the mean and variance of the nonadiabatic entropy production, $Δs_{\rm na}$. Furthermore, our results extend -- to nonequilibrium stead…
We show that the system entropy change for the transitions between non-equilibrium steady states arbitrarily far from equilibrium for any constituting process is given by the relative entropy of the distributions of these steady states. This expression is then shown to relate to the dissipation relations of both Vaikuntanathan and Jarzynski [EPL 87, 60005 (2009)] and Kawai, Parrondo and Van den Broeck [Phys. Rev. Lett. 98, 080602 (2007)] in the case of energy-conserving driving.
Using the recently derived Dissipation Theorem and a corollary of the Transient Fluctuation Theorem (TFT), namely the Second Law Inequality, we derive the unique time independent, equilibrium phase space distribution function for an ergodic Hamiltonian system in contact with a remote heat bath. We prove under very general conditions that any deviation from this equilibrium distribution breaks the time independence of the distribution. Provided temporal correlations decay, and the system is ergodic, we show that any nonequilibrium distribution that is an even function of the momenta, eventually…
We present a framework to determine nonequilibrium steady states in strongly correlated electron systems in the presence of dissipation. This is demonstrated for a correlated electron (Falicov-Kimball) model attached to a heat bath and irradiated by an intense pump light, for which an exact solution is obtained with the Floquet method combined with the nonequilibrium dynamical mean-field theory. On top of a Drude-like peak indicative of photometallization as observed in recent pump-probe experiments, new nonequilibrium phenomena are predicted to emerge, where the optical conductivity exhibits …
We consider a three-terminal system with a chiral edge channel connecting the source and drain terminals. Charge can tunnel between the chiral edge and a third terminal. The third terminal is maintained at a different temperature and voltage than the source and drain. We prove a general relation for the current noises detected in the drain and third terminal. It has the same structure as an equilibrium fluctuation-dissipation relation with the nonlinear response in place of the linear conductance. The result applies to a general chiral system and can be useful for detecting "upstream" modes on…
The accurate determination of transport coefficients in numerical simulations is becoming increasingly important in a wide range of applications. Here we consider the linear response in systems driven away from thermal equilibrium into a non-equilibrium steady state with non-vanishing entropy production rate. A simple derivation of a general response formula is presented. For Markov processes we explicitly construct the conjugate quantities and discuss their relation with known response formulas. Emphasis is put on the formal analogy with thermodynamic potentials and some consequences are disc…
An equivalence between non equilibrium steady states (NESS) driven by a time-independent force and stochastic pumps (SP) stirred by a time-varying conservative force is studied for general many-body diffusive systems. When the particle density and current of NESS are imitated by SP time-averaged counterparts, we prove that the entropy production rate in NESS is always greater than that of SP, provided that the conductivity of the particle current is concave as a function of the particle density. Searching for a SP protocol that saturates the entropy production bound reveals an unexpected conne…
When a Brownian object interacts with non-interacting gas particles under non-equilibrium conditions, the energy dissipation associated to the Brownian motion causes an additional force on the object as a `momentum transfer deficit'. This principle is demonstrated first by a new NESS model and then applied to several known models such as adiabatic piston for which simple explanation has been lacking.
Using the Lindblad master equation approach, we investigate the structure of steady-state solutions of open integrable quantum lattice models, driven far from equilibrium by incoherent particle reservoirs attached at the boundaries. We identify a class of boundary dissipation processes which permits to derive exact steady-state density matrices in the form of graded matrix-product operators. All the solutions factorize in terms of vacuum analogues of Baxter's Q-operators which are realized in terms of non-unitary representations of certain finite dimensional subalgebras of graded Yangians. We …
Entropy production is one of the most important characteristics of non-equilibrium steady states. We study here the steady-state entropy production, both at short times as well as in the long-time limit, of two important classes of non-equilibrium systems: transport systems and reaction-diffusion systems. The usefulness of the mean entropy production rate and of the large deviation function of the entropy production for characterizing non-equilibrium steady states of interacting many-body systems is discussed. We show that the large deviation function displays a kink-like feature at zero entro…
We derive generalized Fluctuation-Dissipation Relations (FDR) holding for a general stochastic dynamics that includes as subcases both equilibrium models for passive colloids and non-equilibrium models used to describe active particles. The relations reported here differ from previous formulations of the FDR because of their simplicity: they require only the microscopic knowledge of the dynamics instead of the whole expression of the steady-state probability distribution function that, except for linear interactions, is unknown for systems displaying non-vanishing currents. From the response f…
Long-ranged correlations generically exist in non-equilibrium fluid systems. In the case of a non-equilibrium steady state caused by a temperature gradient the correlations are especially long-ranged and strong. The anomalous light scattering predicted to exist in these systems is well-confirmed by numerous experiments. Recently the Casimir force or pressure due to these fluctuations or correlations have been discussed in great detail. In this paper the notion of a Casimir work is introduced and a novel way to measure the non-equilibrium Casimir force is suggested. In particular, the non-equil…
Motivated by fundamental issues in non-equilibrium statistical mechanics (NESM), we study the venerable susceptible-infected (SIS) model of disease spreading in an idealized, simple setting. Using Monte Carlo and analytic techniques, we consider a fully connected, uni-directional network of odd number of nodes, each having an equal number of in- and out-degrees. With the standard SIS dynamics at high infection rates, this system settles into an active non-equilibrium steady state. We find the exact probability distribution and explore its implications for NESM, such as the presence of persiste…
Stochastic processes that are randomly reset to an initial condition serve as a showcase to investigate non-equilibrium steady states. However, all existing results have been restricted to the special case of memoryless resetting protocols. Here, we obtain the general solution for the distribution of processes in which waiting times between reset events are drawn from an arbitrary distribution. This allows for the investigation of a broader class of much more realistic processes. As an example, our results are applied to the analysis of the efficiency of constrained random search processes.
We study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We establish that the global spectral features, the spectral gap, and the steady-state properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular quantity. Within each regime, we determine the scaling exponents with the dissipation strength and system size. We find that, for two or more dissipation chann…
Near equilibrium, where all currents of a system vanish on average, the fluctuation-dissipation relation (FDR) connects a current's spontaneous fluctuations with its response to perturbations of the conjugate thermodynamic force. Out of equilibrium, fluctuation-response relations generally involve additional nondissipative contributions. Here, in the framework of stochastic thermodynamics, we show that an equilibrium-like FDR holds for internally equilibrated currents, if the perturbing conjugate force only affects the microscopic transitions that contribute to the current. We discuss the phys…
We explore the idea that non-equilibrium steady states breaking detailed balance are obtained by deforming trajectories (lines in space-time) that have been sampled in a reference system with stochastic dynamics obeying detailed balance, and we ask for the work required to perform this task. These geometric deformations are not arbitrary but arise through interactions with the environment, either the manipulation of conserved quantities by an external agent, or by their exchange with a work reservoir. This view allows to consistently model the breaking of detailed balance and the accompanying …
In this work an extremal principle driving the far from equilibrium evolution of a system of structureless particles is derived by using the stochastic quantum hydrodynamic analogy. For a classical phase (i.e., the quantum correlations decay on a distance smaller than the mean inter-molecular distance) the far from equilibrium kinetic equation can be cast in the form of a Fokker-Plank equation whose phase space velocity vector maximizes the dissipation of the energy-type function, named here, stochastic free energy. Near equilibrium the maximum stochastic free energy dissipation is shown to be…
In equilibrium, the fluctuation-dissipation theorem (FDT) expresses the response of an observable to a small perturbation by a correlation function of this variable with another one that is conjugate to the perturbation with respect to \emph{energy}. For a nonequilibrium steady state (NESS), the corresponding FDT is shown to involve in the correlation function a variable that is conjugate with respect to \emph{entropy}. By splitting up entropy production into one of the system and one of the medium, it is shown that for systems with a genuine equilibrium state the FDT of the NESS differs from …
We develop a unified treatment of pumping and nonequilibrium thermodynamics. We show that the pumping current generated through an adiabatic mechanical operation in equilibrium can be expressed in terms of the stationary distribution of the corresponding driven nonequilibrium system. We also show that the total transfer in pumping can be evaluated from the work imported to the driven counterpart. These findings lead us to a unified viewpoint for pumping and nonequilibrium thermodynamics.
We study the steady state of a finite XX chain coupled at its boundaries to quantum reservoirs made of free spins that interact one after the other with the chain. The two-point correlations are calculated exactly and it is shown that the steady state is completely characterized by the magnetization profile and the associated current. Except at the boundary sites, the magnetization is given by the average of the reservoirs' magnetizations. The steady state current, proportional to the difference in the reservoirs' magnetizations, shows a non-monotonous behavior with respect to the system-reser…
We find novel site-dependent Lax operators in terms of which we demonstrate exact solvability of a dissipatively driven XYZ spin-1/2 chain in the Zeno limit of strong dissipation, with jump operators polarizing the boundary spins in arbitrary directions. We write the corresponding nonequilibrium steady state using an inhomogeneous MPA, where the constituent matrices satisfy a simple set of linear recurrence relations. Although these matrices can be embedded into an infinite-dimensional auxiliary space, we have verified that they cannot be simultaneously put into a tridiagonal form, not even in…
In the context of an exactly soluble out of equilibrium (quenched) model, we study an extension of the fluctuation-dissipation relation. This involves a modified differential form of this relation, with an effective temperature which may have an explicit dependence on time scales.
Continuing our work on the nature and existence of fluctuation-dissipation relations (FDR) in linear and nonlinear open quantum systems [1-3], here we consider such relations when a linear system is in a nonequilibrium steady state (NESS). With the model of two-oscillators (considered as a short harmonic chain with the two ends) each connected to a thermal bath of different temperatures we find that when the chain is fully relaxed due to interaction with the baths, the relation that connects the noise kernel and the imaginary part of the dissipation kernel of the chain in one bath does not ass…
