A thermodynamic-like formalism is developed for superstatistical systems based on conditional entropies. This theory takes into account large-scale variations of intensive variables of systems in nonequilibrium stationary states. Ordinary thermodynamics is recovered as a special case of the present theory, and corrections to it can be systematically evaluated. A generalization of Einstein's relation for fluctuations is presented using a maximum entropy condition.
The existence of fluctuations of temperature has been a somewhat controversial topic in thermodynamics but nowadays it is recognized that they must be taken into account in small, finite systems. Although for nonequilibrium steady states superstatistics is becoming the \textit{de facto} framework for expressing such temperature fluctuations, some recent results put into question the idea of temperature as a phase space observable. In this work we present and explore the statistics that describes a part of an isolated system, small enough to have well-defined uncertainties in energy and tempera…
Nonequilibrium complex systems are often effectively described by the mixture of different dynamics on different time scales. Superstatistics, which is "statistics of statistics" with two largely separated time scales, offers a consistent theoretical framework for such a description. Here, a theory is developed for log-normal superstatistics based on the fluctuation theorem for entropy changes as well as the maximum entropy method. This gives novel physical insight into log-normal statistics, other than the traditional multiplicative random processes. A comment is made on a possible applicatio…
Nonequilibrium systems with large-scale fluctuations of a suitable system parameter are often effectively described by a superposition of two statistics, a superstatistics. Here we illustrate this concept by analysing experimental data of fluctuations in atmospheric wind velocity differences at Florence airport.
We perform a statistical analysis of measured time series of sea levels at various coastal locations in the UK, measured at time differences of 15 minutes over the past 20 years. When the astronomical tide and other deterministic components are subtracted, a stochastic signal remains which is well-described by a superstatistical model. We do various tests on the measured time series, and compare the data of 5 different UK locations. Overall it appears that $χ^2$-superstatistics is best suitable to describe the data, in particular when one looks at the dynamics of sealevel {\em differences} on …
We report a general technique to study a given experimental time series with superstatistics. Crucial for the applicability of the superstatistics concept is the existence of a parameter $β$ that fluctuates on a large time scale as compared to the other time scales of the complex system under consideration. The proposed method extracts the main superstatistical parameters out of a given data set and examines the validity of the superstatistical model assumptions. We test the method thoroughly with surrogate data sets. Then the applicability of the superstatistical approach is illustrated using…
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…
We consider nonequilibrium systems with complex dynamics in stationary states with large fluctuations of intensive quantities (e.g. the temperature, chemical potential, or energy dissipation) on long time scales. Depending on the statistical properties of the fluctuations, we obtain different effective statistical mechanics descriptions. Tsallis statistics is one, but other classes of generalized statistics are obtained as well. We show that for small variance of the fluctuations all these different statistics behave in a universal way.
Superstatistics [C. Beck and E.G.D. Cohen, Physica A 322, 267 (2003)] is a formalism aimed at describing statistical properties of a generic extensive quantity E in complex out-of-equilibrium systems in terms of a superposition of equilibrium canonical distributions weighted by a function P(beta) of the intensive thermodynamic quantity beta conjugate to E. It is commonly assumed that P(beta) is determined by the spatiotemporal dynamics of the system under consideration. In this work we show by examples that, in some cases fulfilling all the conditions for the superstatistics formalism to be ap…
The so-called $χ^{2}$-superstatistics of Beck and Cohen (BC) is employed to investigate the infinite-range Blume-Capel model, a well-known representative system displaying inequivalence of canonical and microcanonical phase diagrams. While not being restricted to any of those particular thermodynamic limits, our analytical result can smoothly recover both canonical and microcanonical ensemble solutions as its nonextensive parameter $q$ is properly tuned. Additionally, we compare our findings to ones previously obtained from a generalized canonical framework named Extended Gaussian ensemble (EG…
The superstatistics concept is a useful statistical method to describe inhomogeneous complex systems for which a system parameter $β$ fluctuates on a large spatio-temporal scale. In this paper we analyze a measured time series of wind speed fluctuations and extract the superstatistical distribution function $f(β)$ directly from the data. We construct suitable Langevin and Fokker-Planck models with a position dependent $β$-field and show that they reduce to standard type of superstatistics in the overdamped limit.
Analytical solutions of the Schrodinger equation for the generalized trigonometric Poschl Teller potential by using an appropriate approximation to the centrifugal term within the framework of the Functional Analysis Approach have been considered. Using the energy equation obtained, the vibrational partition function was calculated and other relevant thermodynamic properties. More so, we use the concept of the superstatistics to also evaluate the thermodynamics properties of the system. It is noted that the well-known normal statistics results are recovered in the absence of the deformation pa…
We provide an overview of Tsallis statistics, presented as a special case of superstatistics and applied to the multiparticle processes described by the statistical cluster model. This model combines Boltzman statistics applied to hadronization of clusters and superstatistics induced by fluctuations of their Lorentz factors. It results in a Tsallis-like distribution of transverse momenta with some scale, the origin of which is discussed. We show that this distribution becomes a Tsallis distribution in a special case, namely when one combines fluctuations of the Lorentz factor and temperature i…
Superstatistical Approach to Turbulent Circulation Fluctuations
Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）；Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）；Instituto de Física, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学物理研究所）；Department of Mechanical and Aerospace Engineering, New York University（纽约大学机械与航空航天工程系）；Department of Physics and Courant Institute of Mathematical Sciences, New York University（纽约大学物理系与库朗数学科学研究所）；Santa Fe Institute（圣塔菲研究所）；Complexity Science Hub Vienna（维也纳复杂性科学中心）；Dipartimento di Fisica e Astronomia Ettore Majorana, University of Catania（卡塔尼亚大学埃托雷·马约拉纳物理与天文学系）
The dynamics of temperature fluctuations of a gas of Brownian particles in local equilibrium with a nonequilibrium heat bath, are described using an approach consistent with Boltzmann-Gibbs statistics (BG). We use mesoscopic nonequilibrium thermodynamics (MNET) to derive a Fokker-Planck equation for the probability distribution in phase space including the local intensive variables fluctuations. We contract the description to obtain an effective probability distribution (EPD) from which the mass density, van Hove's function and the dynamic structure factor of the system are obtained. The main …
A statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards, one with mushroom-like shape and the other from the familiy of the Limacon billiards, is presented. These billiards have mixed regular-chaotic dynamics but different structures in their classical phase spaces. The spectrum of each billiard is represented as a time series where the level order plays the role of time. Two most important findings follow from the time-series analysis. First, the spectra can be characterized by two distinct relaxation lengths. This is a prerequisite for the validity o…
As a main example for the superstatistics approach, we study a Brownian particle moving in a d-dimensional inhomogeneous environment with macroscopic temperature fluctuations. We discuss the average occupation time of the particle in spatial cells with a given temperature. The Fokker-Planck equation for this problem becomes a stochastic partial differential equation. We illustrate our results using experimentally measured time series from hydrodynamic turbulence.
We successfully apply the recent developed superstatistics theory to a temporal series of turbulent wind measurements recorded by the anemometers of Florence airport. Within this approach we can reproduce very well the fluctuations and the pdfs of wind velocity returns and differences.
We study the fluctuation properties of transition intensities applying a recently proposed generalization of the random matrix theory, which is based on Beck and Cohen's superstatistics. We obtain an analytic expression for the distribution of the reduced transition probabilities that applies to systems undergoing a transition out of chaos. The obtained distribution fits the results of a previous nuclear shell model calculations for some electromagnetic transitions that deviate from the Porter-Thomas distribution. It agrees with the experimental reduced transition probabilities for the 26A nuc…
Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures $β$, so that the probability distribution is $p(ε_i) \propto \int_{0}^{\infty} f(β) e^{-βε_i}dβ$, where the `kernel' $f(β)$ is nonnegative and normalized ($\int f(β)d β=1$). We discuss the relation between this distribution and the generalized entropic form $S=\sum_i s(p_i)$. The first three Shannon-Khinchin axioms are assumed to hold. It then turns out that for a given distribution there are two different ways to construct the entropy. One approach uses escort probabilities and the …
In non-equilibrium states, currents are produced by irreversible processes that take a system toward the equilibrium state, where the current vanishes. We demonstrate, in a general setting, that a superstatistics arises when the system relaxes to a (stationary) quasi-equilibrium state instead, where only the \textit{mean} current vanishes because of fluctuations. In particular, we show that a current with Gaussian white noise takes the system to a unique class of quasi-equilibrium states, where the superstatistics coincides with Tsallis escort $q$-distributions. Considering the fully developed…
A recent experiment [Sadoon AA, Wang Y. 2018 Phys. Rev. E 98, 042411] has revealed that nucleoid associated proteins (i.e., DNA-binding proteins) exhibit highly heterogeneous diffusion processes in bacteria where not only the diffusion constant but also the anomalous diffusion exponent fluctuates for the various proteins. The distribution of displacements of such proteins is observed to take a q-Gaussian form, which decays as a power law. Here, a statistical model is developed for the diffusive motion of the proteins within the bacterium, based on a superstatistics with two variables. This mod…
In analogy to superstatistics, which connects Boltzmann-Gibbs statistical mechanics to its generalizations through temperature fluctuations, complex networks are constructed from the fluctuating Erdos-Renyi random graphs. Here, using the quantum mechanical method, the exact analytic formula is presented for the hidden variable distribution, which describes the fluctuation and generates a generic degree distribution through the Poisson transformation. As an example, a static scale-free network is discussed and the corresponding hidden variable distribution is found to decay as a power law.
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.
Recently, there has been a considerable progress on the issue of the thermodynamic second law, which is known as the law of entropy increase or irreversibility. In particular, a novel symmetry known as the Gallavotti-Cohen symmetry is found in nonequilibrium (NEQ) fluctuations, which leads to so-called fluctuation theorems. The thermodynamic second law is a simple corollary of fluctuation theorems, from which one can predict quantitatively how often NEQ processes violate the law of entropy increase. Violations disappear in the thermodynamic limit, but can be observed reasonably well in small s…
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.
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…
We have constructed a nonextensive thermodynamic formalism consisting of two sets of parallel Legendre transformation structures in previous papers. One is the physical set and the other is the Lagrange set. In this paper we study the thermodynamic stability criterion with a dual interpretation of the thermodynamic relations and quantities. By recourse to the assumption that volume in nonextenstive system is nonadditive, we conclude that it is the physical pressure that is responsible for the mechanical balance between any two parts in a given nonextensive system. It is verified that in the ph…
Computer simulations of the Ising model exhibit white noise if thermal fluctuations are governed by Boltzmann's factor alone; whereas we find that the same model exhibits 1/f noise if Boltzmann's factor is extended to include local alignment entropy to all orders. We show that this nonlinear correction maintains maximum entropy during equilibrium fluctuations. Indeed, as with the usual resolution of Gibbs' paradox that avoids net entropy reduction during reversible processes, the correction yields the statistics of indistinguishable particles. The correction also ensures conservation of energy…
The thermodynamic relations in the Tsallis statistics were studied with physical quantities. An additive entropic variable related to the Tsallis entropy was introduced by assuming the form of the first law of the thermodynamics. The fluctuations in the Tsallis statistics were derived with physical quantities with the help of the introduced entropic variable. It was shown that the mean squares of the fluctuations of the physical quantities in the Tsallis statistics are the same as those in the conventional statistics. The mean square of the fluctuation of the Tsallis entropy and the mean squar…
We reformulate stochastic thermodynamics in terms of noise realizations for Langevin systems in contact with multiple reservoirs and investigated the structure of the second laws of thermodynamics. We derive a hierarchy of fluctuation theorems when one degree of freedom of the system is affected by multiple reservoirs simultaneously, that is, when noise mixing occurs. These theorems and the associated second laws of thermodynamics put stricter bounds on the thermodynamics of Langevin systems. We apply our results to a stochastic machine in noise-mixing environments and demonstrate that our new…
We prove the Jarzynski relation for general stochastic processes including non-Markovian systems with memory. The only requirement for our proof is the existence of a stationary state, therefore excluding non-ergodic systems. We then show how the concepts of stochastic thermodynamics can be used to prove further exact non-equilibrium relations like the Crooks relation and the fluctuation theorem on entropy production for non-Markovian dynamics.
The thermodynamic uncertainty relation (TUR) provides a universal entropic bound for the precision of the fluctuation of the charge transfer for example for a class of continuous time stochastic processes. However, its extension to general nonequilibrium dynamics is still an unsolved problem. In this Letter, we show TUR for an arbitrary finite time in terms of exchange fluctuation theorem applied to ensemble of copies of the original system by assuming a physical regularity condition for the probability distribution. As a nontrivial practical consequence, we obtain universal scaling relations …
Large fluctuations have received considerable attention as they encode information on the fine-scale dynamics. Large deviation relations known as fluctuation theorems also capture crucial nonequilibrium thermodynamical properties. Here we report that, using the technique of uniformization, the thermodynamic large deviation functions of continuous-time Markov processes can be obtained from Markov chains evolving in discrete time. This formulation offers new theoretical and numerical approaches to explore large deviation properties. In particular, the time evolution of autonomous and non-autonom…
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…
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.
We compare two approaches to nonequilibrium thermodynamics, the two-generator bracket formulation of time-evolution equations for averages and the macroscopic fluctuation theory, for an isothermal driven diffusive system under steady state conditions. The fluctuation dissipation relations of both approaches play an important role for a detailed comparison. The nonequilibrium Helmholtz free energies introduced in these two approaches differ as a result of boundary conditions. A Fokker-Planck equation derived by projection operator techniques properly reproduces long range fluctuations in nonequ…
A few decades after Hill's work on nano-thermodynamics, the development of a thermodynamic framework, to account consistently for the fluctuations of small systems due to their interactions with the surrounding environment, is still underway. Here we discuss how, in a small system, the interaction energy with the environment may be described through a conjugate pair of intensive and extensive variables, giving rise to a Gibbs thermodynamics with an additional thermodynamic degree of freedom. The relevant thermodynamic potentials help describe the equilibrium conditions and the material propert…
The relationships between reversible Carnot cycles, the absence of perpetual motion machines and the existence of a non-decreasing, globally unique entropy function forms the starting point of many textbook presentations of the foundations of thermodynamics. However, the thermal fluctuation phenomena associated with statistical mechanics has been argued to restrict the domain of validity of this basis of the second law of thermodynamics. Here we demonstrate that fluctuation phenomena can be incorporated into the traditional presentation, extending, rather than restricting, the domain of validi…
We discuss universality of response functions in systems with excited degrees of freedom. We propose a unification of two existing phenomenologies, two-power law decay and deviation from power law due to non-extensivity. A universal curve is derived by maximizing entropy with a non-linear constraint. The same formalism can explain the universal fluctuation curves which have been discovered recently by Bramwell, Holdsworth, and Pinton.
The detailed fluctuation theorem (DFT) is a statement about the asymmetry in the statistics of the entropy production. Consequences of the DFT are the second law of thermodynamics and the thermodynamics uncertainty relation (TUR), which translate into lower bounds for the mean and variance of currents, respectively. However, far from equilibrium, mean and variance are not enough to characterize the underlying distribution of the entropy production. The fluctuations are not necessarily Gaussian (nor symmetric), which means its skewness could be nonzero. We prove that the DFT imposes a negative …
We establish the fluctuation theorem in the presence of information exchange between a nonequilibrium system and other degrees of freedom such as an observer and a feedback controller, where the amount of information exchange is added to the entropy production. The resulting generalized second law sets the fundamental limit of energy dissipation and energy cost during the information exchange. Our results apply not only to feedback-controlled processes but also to a much broader class of information exchanges, and provides a unified framework of nonequilibrium thermodynamics of measurement and…
We extend stochastic thermodynamics by relaxing the two assumptions that the Markovian dynamics must be linear and that the equilibrium distribution must be a Boltzmann distribution. We show that if we require the second law to hold when those assumptions are relaxed, then it cannot be formulated in terms of Shannon entropy. However, thermodynamic consistency is salvaged if we reformulate the second law in terms of generalized entropy; our first result is an equation relating the precise form of the non-linear master equation to the precise associated generalized entropy which results in therm…
The present work extends the well-known thermodynamic relation $C=β^{2}< δ{E^{2}}>$ for the canonical ensemble. We start from the general situation of the thermodynamic equilibrium between a large but finite system of interest and a generalized thermostat, which we define in the course of the paper. The resulting identity $< δβδ{E}> =1+< δ{E^{2}}% > \partial ^{2}S(E) /\partial {E^{2}}$ can account for thermodynamic states with a negative heat capacity $C<0$; at the same time, it represents a thermodynamic fluctuation relation that imposes some restrictions on the determination of the microcano…
Using an additivity property, we study particle-number fluctuations in a system of interacting self-propelled particles, called active Brownian particles (ABPs), which consists of repulsive disks with random self-propulsion velocities. From a fluctuation-response relation - a direct consequence of additivity, we formulate a thermodynamic theory which captures the previously observed features of nonequilibrium phase transition in the ABPs from a homogeneous fluid phase to an inhomogeneous phase of coexisting gas and liquid. We substantiate the predictions of additivity by analytically calculati…
The work approaches the study of the fluctuations for the thermodynamic systems in the presence of the fields. The approach is of phenomenological nature and developed in a Gaussian approximation. The study is exemplified on the cases of a magnetizable continuum in a magnetoquasistatic field, as well as for the so called discrete systems. In the last case one finds that the fluctuations estimators depends both on the intrinsic properties of the system and on the characteristics of the environment. Following some earlier ideas of one of the authors we present a new class of thermodynamic inequa…
Thermodynamic effects of local structure fluctuations in glassformers are analyzed in terms of energy basins and inter-basin hopping. Depending on the time-scale of measurement, one observes short-time thermodynamic properties related to a narrow set of basins, or equilibrium properties that include structural relaxation. The inter-basin hopping is manifested by the fluctuations of the pressure, internal energy, and other thermodynamic short time characteristics. Formulas relating the inter-basin fluctuations of pressure and internal energy to the differences between the long- and short-time s…
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…
Standard results for the fluctuations of thermodynamic quantities are derived under the assumption of sampling identical systems that are in different, not fully equilibrated states. These results apply to fluctuations with time in a particular macroscopic body and can be traced to the fluctuations of thermal occupancy. When many identically prepared - but not identical - systems are studied, mesoscopic fluctuations due to variations from sample to sample contribute to the fluctuations of thermodynamic quantities. We study the combined effect of mesoscopic fluctuations and fluctuations of ther…
We here present the complete analysis of experiments on driven Brownian motion and electric noise in a $RC$ circuit, showing that thermodynamic entropy production can be related to the breaking of time-reversal symmetry in the statistical description of these nonequilibrium systems. The symmetry breaking can be expressed in terms of dynamical entropies per unit time, one for the forward process and the other for the time-reversed process. These entropies per unit time characterize dynamical randomness, i.e., temporal disorder, in time series of the nonequilibrium fluctuations. Their difference…
We illustrate recent results concerning the validity of the work fluctuation theorem in open quantum systems [M. Campisi, P. Talkner, and P. Hänggi, Phys. Rev. Lett. {\bf 102}, 210401 (2009)], by applying them to a solvable model of an open quantum system. The central role played by the thermodynamic partition function of the open quantum system, -- a two level fluctuator with a strong quantum nondemolition coupling to a harmonic oscillator --, is elucidated. The corresponding quantum Hamiltonian of mean force is evaluated explicitly. We study the thermodynamic entropy and the corresponding sp…
