Thermodynamic length is a path function that generalizes the notion of length to the surface of thermodynamic states. Here, we show how to measure thermodynamic length in far-from-equilibrium single molecule experiments using the work fluctuation relations. For these microscopic systems, it proves necessary to define the thermodynamic length in terms of the Fisher information. Consequently, the thermodynamic length can be directly related to the magnitude of fluctuations about equilibrium. The work fluctuation relations link the work and the free energy change during an external perturbation on a system. We use this result to determine equilibrium averages at intermediate points of the protocol in which the system is out-of-equilibrium. This allows us to extend Bennett's method to determine the potential of mean force, as well as the thermodynamic length, in single molecule experiments.
By considering a thermodynamic force as gauge field, we extend constitutive equations of Onsager's non-equilibrium thermodynamics to non-linear equations. In Onsager's non-equilibrium thermodynamics, the thermodynamic force corresponds to a pure gauge, for which the constitutive equations are obtained by gauge fixing. If we extend the thermodynamic force from pure gauge to physical one, we obtain the non-linear constitutive equations of non-equilibrium thermodynamics.
It is known that equilibrium thermodynamics can be deduced from a constrained Fisher information extemizing process. We show here that, more generally, both non-equilibrium and equilibrium thermodynamics can be obtained from such a Fisher treatment. Equilibrium thermodynamics corresponds to the ground state solution, and non-equilibrium thermodynamics corresponds to excited state solutions, of a Schroedinger wave equation (SWE). That equation appears as an output of the constrained variational process that extremizes Fisher information. Both equilibrium- and non-equilibrium situations can ther…
Macroscopic thermodynamics of equilibrium is constructed for systems obeying power-law canonical distributions. With this, the connection between macroscopic thermodynamics and microscopic statistical thermodynamics is generalized. This is complementary to the Gibbs theorem for the celebrated exponential canonical distributions of systems in contact with a heat bath. Thereby, a thermodynamic basis is provided for power-law phenomena ubiquitous in nature.
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.
We develop the stochastic approach to thermodynamics based on the stochastic dynamics, which can be discrete (master equation) continuous (Fokker-Planck equation), and on two assumptions concerning entropy. The first is the definition of entropy itself and the second, the definition of entropy production rate which is nonnegative and vanishes in thermodynamic equilibrium. Based on these assumptions we study interacting systems with many degrees of freedom in equilibrium or out of thermodynamic equilibrium, and how the macroscopic laws are derived from the stochastic dynamics. These studies inc…
Classical gravitation is treated from the point of view of non-equilibrium thermodynamics. Gravitational potential is a thermodynamic state variable in a weakly nonlocal treatment. Entropy production is calculated and the simplest solution of the inequality with corresponding fluxes and forces is given. In case of ideal gravitation without dissipation one obtains the Poisson equation.
Non-equilibrium states of a thermodynamic statistical system are investigated using the thermodynamic parameter of the system lifetime, first-passage time, the time before degeneration of the system under influence of fluctuations. Statistical distributions that describe the behavior of energy and lifetime are used. Entropy and obtained thermodynamic relations are compared with the results of Extended Irreversible thermodynamics, where as an additional parameter selected fluxes. Explicit expressions are obtained for average lifetime and conjugate thermodynamic quantity. It is shown that, when …
A strengthened Lindblad inequality has been proved. We have applied this result for proving a generalized $H$-theorem in non equilibrium thermodynamics. Information processing also can be considered as some thermodynamic process. From this point of view we have proved a strengthened data processing inequality in quantum information theory.
Small thermodynamic systems exhibit peculiar behavior different from that observed in long-scale systems. Non-equilibrium processes taking place in those systems are strongly influenced by the presence of fluctuations which can be large. Contributions to the free energy which vanish at the infinite number of particles limit cannot be neglected and may exert an important influence on the dynamics. We show that in spite of these important differences, the method of non-equilibrium thermodynamics still applies when reducing the size of the system. By using this method, assumption of local equilib…
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.
Validity of local equilibrium has been questioned for non-equilibrium systems which are characterized by delayed response. In particular, for systems with non-zero thermodynamic inertia, the assumption of local equilibrium leads to negative values of the entropy production, which is in contradiction with the second law of thermodynamics. In this paper we address this question by suggesting a variational formulation of irreversible evolution of a system with non-zero thermodynamic inertia. We introduce the Lagrangian, which depends on the properties of the normal and the so-called "mirror-image…
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…
We study the dynamic and thermodynamic origin of non-canonical equilibria, and we discuss their connection with the generalized central limit theorem and the micro-canonical Boltzmann principle. We reach the conclusion that the zeroth law of thermodynamics and the Boltzmann principle are fulfilled thanks to an apparent fault turned into a benefit: the dynamic approach can only produce a truncated form of inverse power law equilibrium.
Using the recently developed covariant Ito-Langevin dynamics, we develop a non-equilibrium thermodynamic theory for small systems coupled to multiplicative noises. The theory is based on Ito-calculus, and is fully covariant under time-independent nonlinear transformation of variables. Assuming instantaneous detailed balance, we derive expressions for various thermodynamic functions, including work, heat, entropy production, and free energy, both at ensemble level and at trajectory level, and prove the second law of thermodynamics for arbitrary non-equilibrium processes. We relate time-reversal…
Ehrenfest urns with interaction that are connected in a ring is considered as a paradigm model for non-equilibrium thermodynamics and is shown to exhibit two distinct non-equilibrium steady states (NESS) of uniform and non-uniform particle distributions. As the inter-particle attraction varies, a first order non-equilibrium phase transition occurs between these two NESSs characterized by a coexistence regime. The phase boundaries, the NESS particle distributions near saddle points and the associated particle fluxes, average urn population fractions, and the relaxational dynamics to the NESSs a…
We derive a universal thermodynamic uncertainty relation (TUR) that applies to an arbitrary observable in a general Markovian system. The generality of our result allows us to make two findings: (1) for an arbitrary out-of-equilibrium system, both the entropy production and the \textit{degree of non-stationarity} are required to tightly bound the strength of a thermodynamic current; (2) by removing the antisymmetric constraint on observables, the TUR in physics and a fundamental inequality in theoretical finance can be unified in a single framework.
Thermodynamic Length in Stochastic Thermodynamics of Far-From-Equilibrium Systems: Unification of Fluctuation Relation and Thermodynamic Uncertainty Relation
In this series of papers we shall carry out a reconsideration of the thermodynamical behavior of the called HMF model, a paradigmatic ferromagnetic toy model exhibiting many features of the real long-range interacting systems. This first work is devoted to perform the microcanonical description of this model system: the calculation of microcanonical entropy and some fundamental thermodynamic observables, the distribution and correlation functions, as well as the analysis of the thermodynamical stability and the relevant thermodynamic limit.
This work presents a unified dissipaton-equation-of-motion (DEOM) theory and its evaluations on the Helmholtz free energy change due to the isotherm mixing of two isolated subsystems. One is a local impurity and another is a nonlocal Gaussian bath. DEOM constitutes a fundamental theory for such open quantum mixtures. To complete the theory, we construct also the imaginary-time DEOM formalism via an analytical continuation of dissipaton algebra, which would be limited to equilibrium thermodynamics. On the other hand, the real-time DEOM deals with both equilibrium structural and nonequilibrium d…
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…
We discuss inertial effects in systems outside equilibrium within the framework of non-equilibrium thermodynamics. By introducing a Gibbs equation in which the entropy depends on the probability density, we are able to describe a system of Brownian particles immersed in a heat bath in both inertial and diffusion regimes. In the former, a relaxation equation for the diffusion current is obtained whereas in the latter we recover Fick's law. Our approach, which uses the elements of the theory of internal degrees of freedom, constitutes the mesoscopic version of a previous analysis which takes int…
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 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 …
This paper presents a nonequilibrium thermodynamic model for the relaxation of a local, isolated system in nonequilibrium using the principle of steepest entropy ascent (SEA), which can be expressed as a variational principle in thermodynamic state space. The model is able to arrive at the Onsager relations for such a system. Since no assumption of local equilibrium is made, the conjugate fluxes and forces, which result, are intrinsic to the subspaces of the system's state space and are defined using the concepts of hypoequilibrium state and nonequilibrium intensive properties, which describe …
We present a thermodynamic formulation for scale-invariant systems based on the principle of extreme information. We create an analogy between these systems and the well-known thermodynamics of gases and fluids, and study as a compelling case the non-interacting system -the scale-free ideal gas- presenting some empirical evidences of electoral results, city population and total cites of Physics journals that confirm its existence. The empirical class of universality known as Zipf's law is derived from first principles: we show that this special class of power law can be understood as the densi…
A non-equilibrium statistical-thermodynamic approach to the study of a Fröhlich-Bose-Einstein condensation of magnons under radio-frequency radiation pumping is presented. Such a system displays a complex behavior consisting in steady-state conditions to the emergence of a synergetic dissipative structure resembling the Bose-Einstein condensation of systems in equilibium. A kind of "two fluid model" arises: the "normal" non-equilibrium structure and Fröhlich condensate or "non-equilibrium" one, which is shown to be an attractor to the system. We analyze some aspects of the irreversible thermod…
Thermodynamic length is a metric distance between equilibrium thermodynamic states. Among other interesting properties, this metric asymptotically bounds the dissipation induced by a finite time transformation of a thermodynamic system. It is also connected to the Jensen-Shannon divergence, Fisher information and Rao's entropy differential metric. Therefore, thermodynamic length is of central interest in understanding matter out-of-equilibrium. In this paper, we will consider how to define thermodynamic length for a small system described by equilibrium statistical mechanics and how to measure…
The typicality of the canonical state shows that majority of the states are indistinguishable from equilibrium, and thus the nonequilibrium states are exceptionally rare in the extremely high-dimensional Hilbert space. On the contrary, we can easily apply an external force acting on the system, and then the actual density matrix quantitatively deviates from the canonical state specified by the system Hamiltonian at each instance. To express how the external forcing amounts to the deviation from equilibrium, we give a universal thermodynamic expression of the lower bound of Hilbert-Schmidt dist…
It is generally assumed that the thermodynamic stability of equilibrium state is reflected by the concavity of entropy. We inquire, in the microcanonical picture, on the validity of this statement for systems described by the bi-parametric entropy $S_{_{κ, r}}$ of Sharma-Taneja-Mittal. We analyze the ``composability'' rule for two statistically independent systems, A and B, described by the entropy $S_{_{κ, r}}$ with the same set of the deformed parameters. It is shown that, in spite of the concavity of the entropy, the ``composability'' rule modifies the thermodynamic stability conditions of …
It is shown that the most important feature of Non-Equilibrium Thermodynamics is not the entropy production, but the organization of the currents in order to flow. This is also needed to obtain the maximum entropy in the equilibrium state, as is required by Equilibrium Thermodynamics.
For genuine non-equilibrium states that even at fixed external control parameter exhibit dissipation, we extend the Hatano-Sasa equality to processes with feedback control. The resulting bound on the maximal extractable work is substantially sharper than what would follow from applying the Sagawa-Ueda equality to transitions involving such states. For repeated measurements at short enough intervals, the power thus extracted can even exceed the average cost of driving as demonstrated explicitly with a simple, analytically solvable example.
Thermodynamical arguments are known to be useful in the construction of physically motivated Lyapunov functionals for nonlinear stability analysis of spatially homogeneous equilibrium steady states in thermodynamically isolated systems. Unfortunately, the limitation to thermodynamically isolated systems is essential, and standard arguments are not applicable even for some very simple thermodynamically open systems. On the other hand, the nonlinear stability of thermodynamically open systems is usually investigated using the so-called energy method. Unfortunately, the designation "energy method…
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…
The extension of equilibrium thermodynamics to non-equilibrium systems is based on the assumption of "local equilibrium," followed by the assumption that an entropy-density function may be defined, and that this entropy-density would have the same functional dependence on the local densities, (u,v,n), as the total entropy S on the variables (U,V,N). In this article we question the validity of all the assumptions made in the theory of non-equilibrium thermodynamics. We find that the assumption of "local equilibrium" is ill-founded, that it is not clear how to define an entropy-density function,…
We study the speed of fluctuation of a quantum system around its thermodynamic equilibrium state, and show that the speed will be extremely small for almost all times in typical thermodynamic cases. The setting considered here is that of a quantum system couples to a bath, both jointly described as a closed system. This setting, is the same as the one considered in [N. Linden et al., Phys. Rev. E 79:061103 (2009)] and the ``thermodynamic equilibrium state'' refers to a situation that includes the usual thermodynamic equilibrium case, as well as far more general situations.
On the basis of Langevin equation the optimal SUSY field scheme is formulated to discribe a non-equilibrium thermodynamic system with quenched disorder and non-ergodicity effects. Thermodynamic and isothermal susceptibilities, memory parameter and irreversible response are determined at different temperatures and quenched disorder intensities.
Starting from the second law of thermodynamics applied to an isolated system consisting of the system surrounded by an extremely large medium, we formulate a general non-equilibrium thermodynamic description of the system when it is out of equilibrium. We then apply it to study the structural relaxation in glasses and establish the phenomenology behind the concept of the fictive temperature and of the empirical Tool-Narayanaswamy equation on firmer theoretical foundation.
The theory of phenomenological Non-equilibrium Thermodynamics is extended by includimg stochastic processes in order to account for recently derived thermodynamical relations such as the Jarzynski equality. Four phenomenological axioms are postulated resulting in a phenomenological interpretation of Jarzynski's equality. Especially, considering the class of Jarzynski processes Jarzynski's equality follows from the axiom that the statistical average of the exponential work is protocol independent.
We present our numerical study on quantum thermodynamics of the resonant level model subjected to non-equilibrium condition as well as external driving. Following our previous work on non-equilibrium quantum thermodynamics (Phys. Rev. B 101, 184304 [2020]), we expand the density operator into a series of power in the driving speed, where we can determine the non-adiabatic thermodynamic quantities. Particularly, we calculate the non-equilibrium entropy production rate as well as higher order non-adiabatic corrections to the energy and/or population. In the limit of weak system-bath coupling, ou…
A framework for the non-equilibrium thermodynamics of glasses is discussed. It also explains the non-equilibrium thermodynamics of a black hole isolated from matter. The first and second laws of black dynamics and black hole thermodynamics are shown to coincide, while the third laws deal with different issues.
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically-reacting species is built on the stochastic trajectories of reaction events obtained from the Chemical Master Equation. However, when the molecular populations are large, the discrete Chemical Master Equation can be approximated with a continuous diffusion process, like the Chemical Langevin Equation or Low Noise Approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the Chemical Master Equation. We find that a stochastic-thermodynamic descr…
We consider a closed quantum system, initially at thermal equilibrium, driven by arbitrary external parameters. We derive a lower bound on the entropy production which we express in terms of the Bures angle between the nonequilibrium and the corresponding equilibrium state of the system. The Bures angle is an angle between mixed quantum states and defines a thermodynamic length valid arbitrarily far from equilibrium. As an illustration, we treat the case of a time-dependent harmonic oscillator for which we obtain analytic expressions for generic driving protocols.
Lagrange and Hamilton equations for thermodynamic evolution near equilibrium as well as Schrodinger-like equation for the non-equilibrium case are obtained extending the CPDQ Principle (Constancy of the product momentum-oordinate uncertainty) to a new conjugate pair where the momentum f is the extension of the mechanical f=p.δq product for the case the coordinate q is not directly observable. Applied to the case of a particle in a one dimension box, the results are in agreement with reported calculated and measured thermal conductance.
Non-equilibrium processes in Schottky systems generate by projection onto the equilibrium subspace reversible accompanying processes for which the non-equilibrium variables are functions of the equilibrium ones. The embedding theorem which guarantees the compatibility of the accompanying processes with the non-equilibrium entropy is proved. The non-equilibrium entropy is defined as a state function on the non-equilibrium state space containing the contact temperature as a non-equilibrium variable. If the entropy production does not depend on the internal energy, the contact temperature changes…
Thermodynamic probes can be used to deduce microscopic internal dynamics of nanoscale quantum systems. Several direct entropy measurement protocols based on charge transport measurements have been proposed and experimentally applied to single-electron devices. To date, these methods have relied on (quasi-)equilibrium conditions between the nanoscale quantum system and its environment, which constitutes only a small subset of the experimental conditions available. In this paper, we establish a thermodynamic analysis method based on stochastic thermodynamics, that is valid far from equilibrium c…
The energy conversion efficiency of far-from-equilibrium systems is generally limited by irreversible thermodynamic fluxes that make contact with different heat baths. For complex systems, the states of the maximum efficiency and the minimum entropy production are usually not equivalent. Here we show that the proper adjustments of the interaction between the energy and matter currents offer some important criteria for the performance characterizations of thermal agents, regardless of the system types and transition protocols. The universal thermodynamic coupling rule plays a critical role in i…
We discuss the relevance of geometric concepts in the theory of stochastic differential equations for applications to the theory of non-equilibrium thermodynamics of small systems. In particular, we show how the Eells-Elworthy-Malliavin covariant construction of the Wiener process on a Riemann manifold provides a physically transparent formulation of optimal control problems of finite-time thermodynamic transitions. Based on this formulation, we turn to an evaluative discussion of recent results on optimal thermodynamic control and their interpretation.
Thermodynamics is a well developed tool to study systems in equilibrium but no such general framework is available for non-equilibrium processes. Only hope for a quantitative description is to fall back upon the equilibrium language as often done in biology. This gap is bridged by the work theorem. By using this theorem we show that the Barkhausen-type non-equilibrium noise in a process, repeated many times, can be combined to construct a special matrix ${\cal S}$ whose principal eigenvector provides the equilibrium distribution. For an interacting system ${\cal S}$, and hence the equilibrium …
