We consider non-equilibrium quantum steady states in conformal field theory (CFT) on star-graph configurations, with a particular, simple connection condition at the vertex of the graph. These steady states occur after a large time as a result of initially thermalizing the legs of the graph at different temperatures, and carry energy flows. Using purely Virasoro algebraic calculations we evaluate the exact long-time cumulant generating function for these flows. We show that this function satisfies a generalization of the usual non-equilibrium fluctuation relations. This extends the results by two of the authors (J. Phys. A 45: 362001, 2012; arXiv:1302.3125) to the cases of more than two legs. It also provides an alternative derivation centered on Virasoro algebra operators rather than local fields, hence an alternative regularization scheme, thus confirming the validity and universality of the long-time cumulant generating function. Our derivation shows how the usual Virasoro algebra leads, in large volumes, to a continuous-index Virasoro algebra for which we develop diagramatic principles, which may be of interest in other non-equilibrium contexts as well. Finally, our results shed light on the Poisson process interpretation of the long-time energy transfer in CFT.
We consider the out-of-equilibrium transport in $T\bar{T}$-deformed (1+1)-dimension conformal field theories (CFTs). The theories admit two disparate approaches, integrability and holography, which we make full use of in order to compute the transport quantities, such as the the exact non-equilibrium steady state currents. We find perfect agreements between the results obtained from these two methods, which serve as the first checks of the $T\bar{T}$-deformed holographic correspondence from the dynamical standpoint. It turns out that integrability also allows us to compute the momentum diffusi…
We present a construction of non-equilibrium steady states within conformal field theory. These states sustain energy flows between two quantum systems, initially prepared at different temperatures, whose dynamical properties are represented by two, possibly different, conformal field theories connected through an impurity. This construction relies on a real time formulation of conformal defect dynamics based on a field scattering picture parallelizing - but yet different from - the Euclidean formulation. We present the basic characteristics of this formulation and give an algebraic constructi…
We present a model independent, operator algebraic approach to non-equilibrium quantum thermodynamics within the framework of two-dimensional Conformal Field Theory. Two infinite reservoirs in equilibrium at their own temperatures and chemical potentials are put in contact through a defect line, possibly by inserting a probe. As time evolves, the composite system then approaches a non-equilibrium steady state that we describe. In particular, we re-obtain recent formulas of Bernard and Doyon.
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.
We provide a pedagogical review of the main ideas and results in non-equilibrium conformal field theory and connected subjects. These concern the understanding of quantum transport and its statistics at and near critical points. Starting with phenomenological considerations, we explain the general framework, illustrated by the example of the Heisenberg quantum chain. We then introduce the main concepts underlying conformal field theory (CFT), the emergence of critical ballistic transport, and the CFT scattering construction of non-equilibrium steady states. Using this we review the theory for …
We develop a hydrodynamic approach to non-equilibrium conformal field theory. We study non-equilibrium steady states in the context of one-dimensional conformal field theory perturbed by the $T\bar T$ irrelevant operator. By direct quantum computation, we show, to first order in the coupling, that a relativistic hydrodynamic emerges, which is a simple modification of one-dimensional conformal fluids. We show that it describes the steady state and its approach, and we provide the main characteristics of the steady state, which lies between two shock waves. The velocities of these shocks are mod…
Entanglement Requires Fluctuations at Conformal Interfaces
Institute for Advanced Study, Kyushu University（九州大学高等研究院）；Department of Physics, Kyushu University（九州大学物理系）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心(iTHEMS)）
We present a generally covariant formulation of conformal higher-order viscoelastic fluid mechanics with strain allowed to take arbitrarily large values. We give a general prescription to determine the dynamics of a relativistic viscoelastic fluid in a way consistent with the hypothesis of local thermodynamic equilibrium and the second law of thermodynamics. We then elaborately study the transient time scales at which the strain almost relaxes and becomes proportional to the gradients of velocity. We particularly show that a conformal second-order fluid with all possible parameters in the cons…
In this note, we present a simple derivation, from time-reversal symmetry, of fluctuation relations for steady-state large deviation functions in non-equilibrium quantum systems. We further show that a condition of pure transmission implies extended fluctuation relations, connecting large deviation functions to mean currents at shifted temperatures and chemical potentials. We illustrate these concepts in various examples, including the interacting resonant level model and conformal or integrable models.
We discuss fluctuation relations in simple cases of non-equilibrium Langevin dynamics. In particular, we show that close to non-equilibrium steady states with non-vanishing probability currents some of these relations reduce to a modified version of the fluctuation-dissipation theorem. The latter may be interpreted as the equilibrium-like relation in the reference frame moving with the mean local velocity determined by the probability current.
In the studies of quantum field theory in curved spacetime, the ambiguous concept of vacuum state and the particle content is a long-standing debatable aspect. So far it is well known to us that in the background of the curved spacetime, some privileged class of observers detect particle production in the suitably chosen vacuum states of the quantum matter fields. In this work we aim to study the characteristics behaviour of these produced particles in the background of the de-Sitter (dS) Friedmann-Lamaître-Robertson-Walker (FLRW) Universe (both for $(1+1)$ and $(3+1)$ dimensions) and $(1+1)$-…
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.
These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow to calculate the fluctuations and large deviations of the density and of the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example non-Gaussian fluctuations of density or non-convexity of the large deviation function which g…
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.
For diffusive systems that can be described by fluctuating hydrodynamics and by the Macroscopic Fluctuation Theory of Bertini et al., the total current fluctuations display universal features when the system is closed and in equilibrium. When the system is taken out of equilibrium by a boundary-drive, current fluctuations, at least for a particular family of diffusive systems, display the same universal features as in equilibrium. To achieve this result, we exploit a mapping between the fluctuations in a boundary-driven nonequilibrium system and those in its equilibrium counterpart. Finally, w…
Thermal fluctuations in non-equilibrium steady states generically lead to power law decay of correlations for conserved quantities. Embedded bodies which constrain fluctuations in turn experience fluctuation induced forces. We compute these forces for the simple case of parallel slabs in a driven diffusive system. The force falls off with slab separation $d$ as $k_B T/d$ (at temperature $T$, and in all spatial dimensions), but can be attractive or repulsive. Unlike the equilibrium Casimir force, the force amplitude is non-universal and explicitly depends on dynamics. The techniques introduced …
The fluctuation relation of the Gallavotti-Cohen Fluctuation Theorem (GCFT) concerns fluctuations in the phase space compression rate of dissipative, reversible dynamical systems. It has been proven for Anosov systems, but it is expected to apply more generally. This raises the question of which non-Anosov systems satisfy the fluctuation relation. We analyze time dependent fluctuations in the phase space compression rate of a class of N-particle systems that are at equilibrium or in near equilibrium steady states. This class does not include Anosov systems or isoenergetic systems, however, it …
Fluctuation theorems impose constraints on the probability of observing negative entropy production in small systems driven out of equilibrium. The range of validity of fluctuation theorems has been extensively tested for transitions between equilibrium and non equilibrium stationary states, but not between general non equilibrium states. Here we report an experimental verification of the detailed fluctuation theorem for the total amount of entropy produced in the isothermal transition between two non-equilibrium states. The experimental setup is a parallel $RC$ circuit driven by an alternatin…
We analyze thermodynamic bounds on equilibrium fluctuations of an order parameter, which are analogous to relations, which have been derived recently in the context of non-equilibrium fluctuations of currents. We discuss the case of {\it global} fluctuations when the order parameter is measured in the full system of interest, and {\it local} fluctuations, when the order parameter is evaluated only in a sub-part of the system. Using isometric fluctuation theorems, we derive thermodynamic bounds on the fluctuations of the order parameter in both cases. These bounds could be used to infer the val…
There exist some boundary-driven open systems with diffusive dynamics whose particle current fluctuations exhibit universal features that belong to the Edwards-Wilkinson universality class. We achieve this result by establishing a mapping, for the system's fluctuations, to an equivalent open --yet equilibrium-- diffusive system. We discuss the possibility of observing dynamic phase transitions using the particle current as a control parameter.
We study the mechanical fluctuations of a micrometer sized silicon cantilever subjected to a strong heat flow, thus having a highly non-uniform local temperature. In this non-equilibrium steady state, we show that fluctuations are equivalent to the thermal noise of a cantilever at equilibrium around room temperature, while its mean local temperature is several hundred of degrees higher. Changing the mechanical dissipation by adding a coating to the cantilever, we recover the expected rise of fluctuations with the mean temperature. Our work demonstrates that inhomogeneous dissipation mechanisms…
The reformulation of nonequilibirum thermodynamics, to include the treatment of thermodynamic fluctuations, is applied to the hydrodynamic fluctuations of a simple fluid. It is shown that the nonequilibrium thermodynamic scheme leads to the explicit form of the Fokker-Planck equation which describes the time behaviour of the probability distribution function of these hydrodynamic fluctuations as well as the irreversible processes which are connected with this behaviour.
We study the dynamics of the entanglement in one dimensional critical quantum systems after a local quench in which two independently thermalized semi-infinite halves are joined to form a homogeneous infinite system and left to evolve unitarily. We show that under certain conditions a nonequilibrium steady state (NESS) is reached instantaneously as soon as the entanglement interval is within the light-cone emanating from the contact point. In this steady state, the exact expressions for the entanglement entropy and the logarithmic negativity are in agreement with the steady state density matri…
We investigate non-equilibrium nature of fluctuations of black hole horizons by applying the fluctuation theorems and the Jarzynski equality developed in the non-equilibrium statistical physics. These theorems applied to space-times with black hole horizons lead to the generalized second law of thermodynamics. It is also suggested that the second law should be violated microscopically so as to satisfy the Jarzynski equality.
In this paper we formulate a dynamical fluctuation theory for stationary non equilibrium states (SNS) which covers situations in a nonlinear hydrodynamic regime and is verified explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Our results include the modification of the Onsager-Machlup theory in the SNS, a general Hamilton-Jacobi equation for the macroscopic entropy and a non equilibrium, non linear fluctuation dissipation relation valid for a wide class of systems.
We discuss the fluctuation properties of equilibrium chaotic systems with constraints such as iso-kinetic and Nosé-Hoover thermostats. Although the dynamics of these systems does not typically preserve phase-space volumes, the average phase-space contraction rate vanishes, so that the stationary states are smooth. Nevertheless finite-time averages of the phase-space contraction rate have non-trivial fluctuations which we show satisfy a simple version of the Gallavotti-Cohen fluctuation theorem, complementary to the usual fluctuation theorem for non-equilibrium stationary states, and appropriat…
Examination of equilibrium radiation in plasma media shows that the spectral the energy distribution of such radiation is different from the Planck equilibrium radiation. Using the previously obtained general relations for the spectral energy density of equilibrium radiation in a system of charged particles, we consider radiation in an electron in the limiting case of an infinitesimal damping. It is shown that zero vacuum fluctuations which are part of the full spectral energy distribution should be renormalized. In this case, the renormalized zero vacuum fluctuations depend on the electron de…
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…
The characteristic function for heat fluctuations in a non equilibrium system is characterised by a large deviation function whose symmetry gives rise to a fluctuation theorem. In equilibrium the large deviation function vanishes and the heat fluctuations are bounded. Here we consider the characteristic function for heat fluctuations in equilibrium, constituting a sub-leading correction to the large deviation behaviour. Modelling the system by an oscillator coupled to an explicit multi-oscillator heat reservoir we evaluate the characteristic function.
The relaxation towards equilibrium of systems with long range interactions is not yet fully understood. As a step towards such a comprehension, we propose the study of the dynamical equilibrium fluctuations in a model system with long range interaction. We compute analytically, from the microscopic dynamics, the autocorrelation function of the order parameter. From this result, we derive analytically a Fokker Planck equation which describes the stochastic process of the impulsion of a single particle in an equilibrium bath. The diffusion coefficient is explicitly computed.
Previously, we have derived a generalization of the canonical fluctuation relation between heat capacity and energy fluctuations $C=β^{2}<δU^{2}>$, which is able to describe the existence of macrostates with negative heat capacities $C<0$. In this work, we extend our previous results for an equilibrium situation with several control parameters to account for the existence of states with anomalous values in other response functions. Our analysis leads to the derivation of three different equilibrium fluctuation theorems: the \textit{fundamental and the complementary fluctuation theorems}, which…
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…
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…
The postulational basis of classical thermodynamics has been expanded to incorporate equilibrium fluctuations. The main additional elements of the proposed thermodynamic theory are the concept of quasi-equilibrium states, a definition of non-equilibrium entropy, a fundamental equation of state in the entropy representation, and a fluctuation postulate describing the probability distribution of macroscopic parameters of an isolated system. Although these elements introduce a statistical component that does not exist in classical thermodynamics, the logical structure of the theory is different f…
While techniques to compute thermal fluctuation induced, or pseudo-Casimir, forces in equilibrium systems are well established, the same is not true for non-equilibrium cases. We present a general formalism that allows us to unambiguously compute non-equilibrium fluctuation induced forces by specifying the energy of interaction of the fluctuating fields with the boundaries. For a general class of classical fields with dissipative dynamics, we derive a very general relation between the Laplace transform of the time-dependent force and the static partition function for a related problem with a d…
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…
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 …
We measure the energy fluctuations of a Brownian particle confined by an optical trap in an aging gelatin after a very fast quench (less than 1 ms). The strong nonequilibrium fluctuations due to the assemblage of the gel, are interpreted, within the framework of fluctuation theorem, as a heat flux from the particle towards the bath. We derive, from a simple model, an analytical expression of the heat probability distribution, which fits the experimental data and satisfies a fluctuation relation similar to that of a system in contact with two baths at different temperatures.
Starting from symmetry considerations, we derive the generic hydrodynamic equation of non-equilibrium $XY$ spin systems with $N$-atic symmetry under weak fluctuations. Through a systematic treatment we demonstrate that, in two dimensions, these systems exhibit two types of scaling behaviours. For $N=1$, they have long-range order and are described by the flocking phase of dry polar active fluids. For all other values of $N$, the systems exhibit quasi long-range order, as in the equilibrium $XY$ model at low temperature.
Fluctuations of the excess heat in an out of equilibrium steady state are experimentally investigated in two stochastic systems : an electric circuit with an imposed mean current and a harmonic oscillator driven out of equilibrium by a periodic torque. In these two linear systems, we study excess heat that represents the difference between the dissipated heat out of equilibrium and the dissipated heat at equilibrium. Fluctuation theorem holds for the excess heat in the two experimental systems for all observation times and for all fluctuation magnitudes.
In this work a generalization of Onsager-Machlup's theory of time-dependent thermal fluctuations of equilibrium systems is proposed, to the case in which the system relaxes irreversibly along a non-equilibrium trajectory that can be approximated as a sequence of stationary states. This generalization is summarized by a canonical description of the dependence of the two-time correlation function C(t+τ,t), and of the equal-time correlation function σ(t)= C(t,t) (the covariance of the fluctuations), on the non-equilibrium relaxation time t.
Near equilibrium, Green-Kubo relations provide microscopic expressions for macroscopic transport coefficients in terms of equilibrium correlation functions. At their core, they are based on the intimate relationship between response and fluctuations as embodied by the equilibrium fluctuation-dissipation theorem, a connection generically broken far from equilibrium. In this work, we identify a class of perturbations whose response around far-from-equilibrium steady states is linked to steady-state correlation functions via an equilibrium-like fluctuation-response equality. We then utilize this …
We experimentally demonstrate the validity of non-equilibrium fluctuation relations by using a quantum coherent conductor. In equilibrium the fluctuation-dissipation relation leads to the correlation between current and current noise at the conductor, namely, Johnson-Nyqusit relation. When the conductor is voltage-biased so that the non-linear regime is entered, the fluctuation theorem has predicted similar non-equilibrium fluctuation relations, which hold true even when the Onsager-Casmir relations are broken in magnetic fields. Our experiments qualitatively validate the predictions as the fi…
We introduce from an experimental point of view the main concepts of fluctuation theorems for work, heat and entropy production in out of equilibrium systems. We will discuss the important difference between the applications of these concepts to stochastic systems and to a second class of systems (chaotic systems) where the fluctuations are induced either by chaotic flows or by fluctuating driving forces. We will mainly analyze the stochastic systems using the measurements performed in two experiments : a) a harmonic oscillator driven out of equilibrium by an external force b) a colloidal part…
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.
Five previously unknown inequalities relating equilibrium free energy differences and non-equilibrium work fluctuations are derived, and lucid path to derivation of many similar inequalities is presented. These results are based upon combined exploitation of the Jarzynski equality and the generalization of the scheme for producing uncertainty-type inequalities due to H. Weyl. The inequalities may possibly lead to better understanding of behavior of the equilibrium free-energy estimators from non-equilibrium experimental data in many important applications concerning biological, chemical, and p…
Condensation of fluctuations is an interesting phenomenon conceptually distinct from condensation on average. One stricking feature is that, contrary to what happens on average, condensation of fluctuations may occurr even in the absence of interaction. The explanation emerges from the duality between large deviation events in the given system and typical events in a new and appropriately biased system. This surprising phenomenon is investigated in the context of the Gaussian model, chosen as paradigmatical non interacting system, before and after an istantaneous temperature quench. It is show…
The various versions of cond-mat/0312353 discuss results obtained by me&coworkers in the last decade. I have received requests to comment on the paper and the comments are collected here, including some that I tried to point in the course of discussions that can be found in the web page http://ipparco.roma1.infn.it
Recent experimental and theoretical works have shown that giant fluctuations are present during diffusion in liquid systems. We use linearized fluctuating hydrodynamics to calculate the net mass transfer due to these non equilibrium fluctuations. Surprisingly the mass flow turns out to coincide with the usual Fick's one. The renormalization of the hydrodynamic equations allows us to quantify the gravitational modifications of the diffusion coefficient induced by the gravitational stabilization of long wavelength fluctuations.
Biological activity gives rise to non-equilibrium fluctuations in the cytoplasm of cells; however, there are few methods to directly measure these fluctuations. Using a reconstituted actin cytoskeleton, we show that the bending dynamics of embedded microtubules can be used to probe local stress fluctuations. We add myosin motors that drive the network out of equilibrium, resulting in an increased amplitude and modified time-dependence of microtubule bending fluctuations. We show that this behavior results from step-like forces on the order of 10 pN driven by collective motor dynamics.
We investigate the dynamics of a generalized survival probability $S(t,R)$ defined with respect to an arbitrary reference level $R$ (rather than the average) in equilibrium step fluctuations. The exponential decay at large time scales of the generalized survival probability is numerically analyzed. $S(t,R)$ is shown to exhibit simple scaling behavior as a function of system-size $L$, sampling time $δt$, and the reference level $R$. The generalized survival time scale, $τ_s(R)$, associated with $S(t,R)$ is shown to decay exponentially as a function of $R$.
We formulate a simple additivity principle allowing to calculate the whole distribution of current fluctuations through a large one dimensional system in contact with two reservoirs at unequal densities from the knowledge of its first two cumulants. This distribution (which in general is non-Gaussian) satisfies the Gallavotti-Cohen symmetry and generalizes the one predicted recently for the symmetric simple exclusion process. The additivity principle can be used to study more complex diffusive networks including loops.
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.
We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability $S(t)$ in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. $S(t)$ is shown to exhibit simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of…
For macroscopic quantum systems, we study what are measured when equilibrium fluctuations of macrovariables are measured in an ideal way that mimics classical ideal measurements as closely as possible. We find that the symmetrized time correlation (symTC) is always obtained for such measurements. As an important consequence, we show that the fluctuation-dissipation theorem (FDT) is partially violated as a relation between observed quantities in macroscopic quantum systems even if measurements are made in such an ideal way.
We point out that the "additivity principle" and "scaling hypothesis" postulated by Bodineau and Derrida in Phys. Rev. Lett 92, 180601 (2004), follow naturally from the saddle point evaluation of a diffusive field theory.
Because of the spatially long-ranged nature of spontaneous fluctuations in thermal non-equilibrium systems, they are affected by boundary conditions for the fluctuating hydrodynamic variables. In this paper we consider a liquid mixture between two rigid and impervious plates with a stationary concentration gradient resulting from a temperature gradient through the Soret effect. For liquid mixtures with a large Lewis number, we are able to obtain explicit analytical expressions for the intensity of the non-equilibrium concentration fluctuations as a function of the frequency $ω$ and the wave nu…
