The selection of an equilibrium state by maximising the entropy of a system, subject to certain constraints, is often powerfully motivated as an exercise in logical inference, a procedure where conclusions are reached on the basis of incomplete information. But such a framework can be more compelling if it is underpinned by dynamical arguments, and we show how this can be provided by stochastic thermodynamics, where an explicit link is made between the production of entropy and the stochastic dynamics of a system coupled to an environment. The separation of entropy production into three components allows us to select a stationary state by maximising the change, averaged over all realisations of the motion, in the principal relaxational or nonadiabatic component, equivalent to requiring that this contribution to the entropy production should become time independent for all realisations. We show that this recovers the usual equilibrium probability density function (pdf) for a conservative system in an isothermal environment, as well as the stationary nonequilibrium pdf for a particle confined to a potential under nonisothermal conditions, and a particle subject to a constant nonconservative force under isothermal conditions. The two remaining components of entropy production account for a recently discussed thermodynamic anomaly between over- and underdamped treatments of the dynamics in the nonisothermal stationary state.
We discuss peculiar aspects of the first law of thermodynamics for systems characterized by the presence of meta-equilibrium quasi-stationary states for which the pertinent phase/configuration spaces is generally inhomogeneous. As a consequence, the naive additivity requirement for thermodynamic quantities ceases to be satisfied.
The average thermodynamic power of a time-dependent external potential in the white-noise Langevin model is derived using a Green's function solution. The power appears as a driving term in the differential equation for the average energy and determines whether the solution is stationary or non-stationary. Different dynamics are illustrated with explicit models: a linear potential with a static magnetic field, a linear potential perturbed with an oscillating component and a magnetic field switch modeled using a $\tanh$ protocol.
Nonanalyticities of thermodynamic functions are studied by adopting an approach based on stationary points of the potential energy. For finite systems, each stationary point is found to cause a nonanalyticity in the microcanonical entropy, and the functional form of this nonanalytic term is derived explicitly. With increasing system size, the order of the nonanalytic term grows, leading to an increasing differentiability of the entropy. It is found that only "asymptotically flat" stationary points may cause a nonanalyticity that survives in the thermodynamic limit, and this property is used to…
We discuss the consequences of a variant of the Hatano-Sasa relation in which a non-stationary distribution is used in place of the usual stationary one. We first show that this non-stationary distribution is related to a difference of traffic between the direct and dual dynamics. With this formalism, we extend the definition of the adiabatic and non-adiabatic entropies introduced by M. Esposito and C. Van den Broeck in Phys. Rev. Lett. 104, 090601 (2010) for the stationary case. We also obtain interesting second-law like inequalities for transitions between non-stationary states.
There is a long-standing question as to whether and to what extent it is possible to describe nonequilibrium systems in stationary states in terms of global thermodynamic functions. The positive answers have been obtained only for isothermal systems or systems with small temperature differences. We formulate thermodynamics of the stationary states of the ideal gas subjected to heat flow in the form of the zeroth, first, and second law. Surprisingly, the formal structure of steady state thermodynamics is the same as in equilibrium thermodynamics. We rigorously show that $U$ satisfies the follow…
We focus our attention on dynamical processes characterized by an entropic index Q<1. According to the probabilistic arguments of Tsallis and Bukman [C.Tsallis, D.J. Bukman, Phys. Rev. E 54,R2197 (1996)] these processes are subdiffusional in nature. The non-extensive generalization of the Kolmogorov-Sinai entropy yielding the same entropic index implies the stationary condition. We note, on the other hand, that enforcing the stationary property on subdiffusion has the effect of producing a localization process occurring within a finite time scale. We thus conclude that the stationary dynamic p…
A Unified Nonequilibrium Framework: Thermodynamic Distance, Dissipation, and Stationary Laws via Effective State Count, Variational Stationarity, and Thermodynamic Bounds
A thermodynamics for systems at a stationary states is formulated. It is based upon the assumption of the existence of local equilibrium in phase space which enables one to interpret the probability density ans its conjugated nonequilibrium chemical potential as a mesoscopic thermodynamic variables. The probability current is obtained from the entropy production related to the probability diffusion process and leads to the formulation of the Fokker-Planck equation. For the case of a gas of Brownian particles under steady flow in the dilute and concentrated regimes we derive nonequilibrium equa…
In order to better understand the occurrence of phase transitions, we adopt an approach based on the study of energy landscapes: The relation between stationary points of the potential energy landscape of a classical many-particle system and the analyticity properties of its thermodynamic functions is studied for finite as well as infinite systems. For finite systems, each stationary point is found to cause a nonanalyticity in the microcanonical entropy, and the functional form of this nonanalytic term can be derived explicitly. With increasing system size, the order of the nonanalytic term gr…
We present a general framework for systems which are prepared in a non-stationary non-equilibrium state in the absence of any perturbation, and which are then further driven through the application of a time-dependent perturbation. We distinguish two different situations depending on the way the non-equilibrium state is prepared, either it is created by some driving; or it results from a relaxation following some initial non-stationary conditions. Our approach is based on a recent generalization of the Hatano-Sasa relation for non-stationary probability distributions. We also investigate wheth…
Aspects of the modern dynamical systems approach to thermodynamics of stationary states out of equilibrium with attention to the original conceptions which arose at the beginnings of Statistical Mechanics
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 …
We consider the asymptotic state after a sudden quench of the magnetic field in the transverse field quantum Ising chain starting from excited states of the pre-quench Hamiltonian. We compute the thermodynamic entropies of the generalised Gibbs and the diagonal ensembles and we find that the generalised Gibbs entropy is always twice the diagonal one. We show that particular care should be taken in extracting the thermodynamic limit since different averages of equivalent microstates give different results for the entropies.
Entropy might be a not well defined concept if the system can undergo transformations involving stationary nonequilibria. It might be analogous to the heat content (once called ``caloric'') in transformations that are not isochoric (i.e. which involve mechanical work): it could be just a quantity that can be transferred or created, like heat in equilibrium. The text first reviews the philosophy behind a recently proposed definition of entropy production in nonequilibrium stationary systems. A detailed technical attempt at defining the entropy of a stationary states via their variational proper…
We analyze the diffusion of a Brownian particle in a fluid under stationary flow. By using the scheme of non-equilibrium thermodynamics in phase space, we obtain the Fokker-Planck equation which is compared with others derived from kinetic theory and projector operator techniques. That equation exhibits violation of the fluctuation dissipation-theorem. By implementing the hydrodynamic regime described by the first moments of the non-equilibrium distribution, we find relaxation equations for the diffusion current and pressure tensor, allowing us to arrive at a complete description of the system…
Development of thermodynamic induction up to second order gives a dynamical bifurcation for thermodynamic variables and allows for the prediction and detailed explanation of nonequilibrium phase transitions with associated spontaneous symmetry breaking. By taking into account nonequilibrium fluctuations, long range order is analyzed for possible pattern formation. Consolidation of results up to second order produces thermodynamic potentials that are maximized by stationary states of the system of interest. These new potentials differ from the traditional thermodynamic potentials. In particular…
In recent years, the unified theory of information and thermodynamics has been intensively discussed in the context of stochastic thermodynamics. The unified theory reveals that information theory would be useful to understand non-stationary dynamics of systems far from equilibrium. In this letter, we have found a new link between stochastic thermodynamics and information theory well known as information geometry. By applying this link, an information geometric inequality can be interpreted as a thermodynamic uncertainty relationship between speed and thermodynamic cost. We have numerically ap…
A system of globally coupled rotors is studied in a unified framework of microcanonical and canonical ensembles. We consider the Fokker-Planck equation governing the time evolution of the system, and examine various stationary as well as non-stationary solutions. The canonical distribution, describing equilibrium, provides a stationary solution also in the microcanonical ensemble, which leads to order in a system with ferromagnetic coupling at low temperatures. On the other hand, the microcanonical ensemble admits additional stationary and non-stationary solutions; the latter allows dynamical …
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.
Using statistical thermodynamics, we derive a general expression of the stationary probability distribution for thermodynamic systems driven out of equilibrium by several thermodynamic forces. The local equilibrium is defined by imposing the minimum entropy production and the maximum entropy principle under the scale invariance restrictions. The obtained probability distribution presents a singularity that has immediate physical interpretation in terms of the intermittency models. The derived reference probability distribution function is interpreted as time and ensemble average of the real ph…
Long-range interacting Hamiltonian systems are believed to relax generically towards non-equilibrium states called "quasi-stationary" because they evolve towards thermodynamic equilibrium very slowly, on a time-scale diverging with particle number. We show here that, by applying a suitable perturbation operator for a finite time interval, we obtain, in a family of long-range systems, non-equilibrium states which appear to be strictly stationary. They exist even in the case of a harmonic potential, and are characterised by an ordered microscopic phase space structure. We give some simple heuris…
We investigate the nonequilibrium stationary states of systems consisting of chemical reactions among molecules of several chemical species. To this end we introduce and develop a stochastic formulation of nonequilibrium thermodynamics of chemical reaction systems based on a master equation defined on the space of microscopic chemical states, and on appropriate definitions of entropy and entropy production, The system is in contact with a heat reservoir, and is placed out of equilibrium by the contact with particle reservoirs. In our approach, the fluxes of various types, such as the heat and …
We prepare static granular beds under gravity in different stationary states by tapping the system with pulsed excitations of controlled amplitude and duration. The macroscopic state---defined by the ensemble of static configurations explored by the system tap after tap---for a given tap intensity and duration is studied in terms of volume, V, and force moment tensor, Σ. In a previous paper [Pugnaloni et al., Phys. Rev. E 82, 050301(R) (2010)], we reported evidence supporting that such macroscopic states cannot be fully described by using only V or Σ, apart from the number of particles N. In t…
In any Markov chain with finite state space the distribution of transition records always belongs to the exponential family. This observation is used to prove a fluctuation theorem, and to show that the dynamical entropy of a stationary Markov chain is linear in the number of steps. Three applications are discussed. A known result about entropy production is reproduced. A thermodynamic relation is derived for equilibrium systems with Metropolis dynamics. Finally, a link is made with recent results concerning a one-dimensional polymer model.
The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, whilst the second two comprise the house-keeping heat. We denote these two components the transient and generalised house-keeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity.
Systems with long range interactions display some anomalies when its dynamics and thermodynamics are studied below certain conditions. Among these anomalies are the quasi- stationary states, which are exacerbated because of special initial conditions that are used here. We present in this letter a new Hamiltonian whose potential is inspired in the two-dipole interaction. An analytical solution is obtained for the equilibrium in the canonical ensemble that is coincident with the one obtained from computational simulations. However, results from this model presents a kind of nonequivalence of en…
We study stationary metastable states(SMS's)using a restricted partition function formalism. The formalism ensures that SMS free energy exists all the way to T=0, and remains stable. We introduce the concept of the reality condition, according to which the entropy $S(T)$ of a set of coupled degrees of freedom must be non-negative. The entropy crisis, which does not affect stability, is identified as the violation of the reality condition. We identify and validate rigorously, using general thermodynamic arguments, the following general thermodynamic mechanism behind the entropy crisis in SMS. T…
This paper reports on a macroscopic fluctuation theory developed over the last ten years in collaboration with L. Bertini, A. De Sole, D. Gabrielli and C. Landim. This theory has been inspired by and tested on stochastic models of interacting particles (stochastic lattice gases). It is the basis for a new approach to the study of stationary non equilibrium states applicable to a large class of systems. This overview emphasizes general ideas and for the details I refer to the published papers.
The stochastic thermodynamics provides a framework for the description of systems that are out of thermodynamic equilibrium. It is based on the assumption that the elementary constituents are acted by random forces that generate a stochastic dynamics, which is here represented by a Fokker-Planck-Kramers equation. We emphasize the role of the irreversible probability current, the vanishing of which characterizes the thermodynamic equilibrium and yields a special relation between fluctuation and dissipation. The connection to thermodynamics is obtained by the definition of the energy function an…
Deterministic diffusive systems such as the periodic Lorentz gas, multi-baker map, as well as spatially periodic systems of interacting particles, have non-equilibrium stationary states with fractal properties when put in contact with particle reservoirs at their boundaries. We study the macroscopic limits of these systems and establish a correspondence between the thermodynamics of the macroscopic diffusion process and the fractality of the stationary states that characterize the phase-space statistics. In particular the entropy production rate is recovered from first principles using a forma…
Assuming the existence of stationary metastable states (SMS's), and using general thermodynamic arguments, we prove that a positive Kauzmann temperature exists for SMS's provided the ideal glass energy is higher than the crystal energy at absolute zero. We confirm the general predictions by two exact calculations, one of which is not mean-field.
A basic task of information processing is information transfer (flow). Here we study a pair of Brownian particles each coupled to a thermal bath at temperature $T_1$ and $T_2$, respectively. The information flow in such a system is defined via the time-shifted mutual information. The information flow nullifies at equilibrium, and its efficiency is defined as the ratio of flow over the total entropy production in the system. For a stationary state the information flows from higher to lower temperatures, and its the efficiency is bound from above by $\frac{{\rm max}[T_1,T_2]}{|T_1-T_2|}$. This u…
We introduce the concept of stationary metastable states (SMS's) in the presence of another more stable state. The stationary nature allows us to study SMS's by using a restricted partition function formalism as advocated by Penrose and Lebowitz and requires continuing the free energy. The formalism ensures that SMS free energy satisfies the requirement of thermodynamic stability everywhere including T=0, but need not represent a pysically observable metastable state over the range where the entropy under continuation becomes negative. We consider a 1-dimensional m-component axis-spin model in…
We analyze the entanglement properties of the asymptotic steady state after a quench from free to hard-core bosons in one dimension. The Rényi and von Neumann entanglement entropies are found to be extensive, and the latter coincides with the thermodynamic entropy of the Generalized Gibbs Ensemble (GGE). Computing the spectrum of the two-point function, we provide exact analytical results both for the leading extensive parts and the subleading terms for the entropies as well as for the cumulants of the particle number fluctuations. We also compare the extensive part of the entanglement entropy…
A statistical theory is presented which allows to calculate the stationary state achieved by a driven system after a process of collisionless relaxation. The theory is applied to study an electron beam driven by an external electric field. The Vlasov equation with appropriate boundary conditions is solved analytically and compared with the molecular dynamics simulation. A perfect agreement is found between the theory and the simulations. The full current-voltage phase diagram is constructed.
We connect the rare fluctuations of an Equilibrium (EQ) process and the typical fluctuations of a nonequilibrium (NE) stationary process. In the framework of large deviation theory, this observation allows us to introduce NE thermodynamic potentials. For continuous-time Markov chains, we identify the relevant pairs of conjugated variables and propose two NE ensembles: one with fixed dynamics and fluctuating time-averaged variables, and another with fixed time-averaged variables, but a fluctuating dynamics. Accordingly, we show that NE processes are equivalent to conditioned EQ processes ensuri…
We study the d-HMF model proposed by Atenas and Curilef, a mean field model with long-range interactions inspired by the dipole-dipole interaction. Among the challenges of this thesis is: the resolution of the d-HMF model in the canonical and microcanonical ensembles and the analytical and numerical description of the distribution function of the system. This model has been studied both in equilibrium and out of equilibrium. In the equilibrium, analytical solutions have been found for internal energy per particle and temperature using standard procedures through statistical mechanics such as t…
In recent years it has been found that quantum systems can posses entanglement in equilibrium thermal states provided temperature is low enough. In the present work we explore a possibility of having entanglement in nonequilibrium stationary states. We show analytically that, in a simple one-dimensional spin chain, there is entanglement even at highest attainable energies; that is, starting from an equilibrium state at infinite temperature, a sufficiently strong driving can induce entanglement, even in the thermodynamic limit. We also show that dissipative dephasing, on the other hand, destroy…
In this article the Gordan theorem is applied to the thermodynamics of a chemical reaction network at steady state. From a theoretical viewpoint it is equivalent to the Clausius formulation of the second law for the out of equilibrium steady states of chemical networks, i.e. it states that the exclusion (presence) of closed reactions loops makes possible (impossible) the definition of a thermodynamic potential and vice versa. On the computational side, it reveals that calculating reactions free energy and searching infeasible loops in flux states are dual problems whose solutions are alternati…
We study a paradigmatic system with long-range interactions: the Hamiltonian Mean-Field Model (HMF). It is shown that in the thermodynamic limit this model does not relax to the usual equilibrium Maxwell-Boltzmann distribution. Instead, the final stationary state has a peculiar core-halo structure. In the thermodynamic limit, HMF is neither ergodic nor mixing. Nevertheless, we find that using dynamical properties of Hamiltonian systems, it is possible to quantitatively predict both the spin distribution and the velocity distribution functions in the final stationary state, without any adjustab…
An ensemble of particles in thermal equilibrium at temperature $T$, modeled by Nosè-Hoover dynamics, moves on a triangular lattice of oriented semi-disk elastic scatterers. Despite the scatterer asymmetry a directed transport is clearly ruled out by the second law of thermodynamics. Introduction of a polarized zero mean monochromatic field creates a directed stationary flow with nontrivial dependence on temperature and field parameters. We give a theoretical estimate of directed current induced by a microwave field in an antidot superlattice in semiconductor heterostructures.
We use mesoscopic non-equilibrium thermodynamics theory to describe RNA unfolding under tension. The theory introduces reaction coordinates, characterizing a continuum of states for each bond in the molecule. The unfolding considered is so slow that one can assume local equilibrium in the space of the reaction coordinates. In the quasi-stationary limit of high sequential barriers, our theory yields the master equation of a recently proposed sequential-step model. Non-linear switching kinetics is found between open and closed states. Our theory unifies the thermodynamic and kinetic descriptions…
We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic "equation of state" of the entire system, which exhibits a universal ideal thermodynamic b…
It has been observed empirically that two dimensional vortices tend to cluster forming a giant vortex. To account for this observation Onsager introduced a concept of negative absolute temperature in equilibrium statistical mechanics. In this Letter we will show that in the thermodynamic limit a system of interacting vortices does not relax to the thermodynamic equilibrium, but becomes trapped in a non-equilibrium stationary state. We will show that the vortex distribution in this non-equilibrium stationary state has a characteristic core-halo structure, which can be predicted {\it a priori}. …
We study, using both theory and molecular dynamics simulations, the relaxation dynamics of a microcanonical two dimensional self-gravitating system. After a sufficiently large time, a gravitational cluster of N particles relaxes to the Maxwell-Boltzmann distribution. The time to reach the thermodynamic equilibrium, however, scales with the number of particles. In the thermodynamic limit, $N\to\infty$ at fixed total mass, equilibrium state is never reached and the system becomes trapped in a non-ergodic stationary state. An analytical theory is presented which allows us to quantitatively descri…
