Events in mesoscopic systems often take place at first-passage times, as is for instance the case for a colloidal particle that escapes a metastable state. An interesting question is how much work an external agent has done on a particle when it escapes a metastable state. We develop a thermodynamic theory for processes in mesoscopic systems that terminate at stopping times, which generalize first-passage times. This theory implies a thermodynamic bound, reminiscent of the second law of thermodynamics, for the work exerted by an external protocol on a mesoscopic system at a stopping time. As an illustration, we use this law to bound the work required to stretch a polymer to a certain length or to let a particle escape from a metastable state.
A stopping time $T$ is the first time when a trajectory of a stochastic process satisfies a specific criterion. In this paper, we use martingale theory to derive the integral fluctuation relation $\langle e^{-S_{\rm tot}(T)}\rangle=1$ for the stochastic entropy production $S_{\rm tot}$ in a stationary physical system at stochastic stopping times $T$. This fluctuation relation implies the law $\langle S_{\rm tot}(T)\rangle\geq 0$, which states that it is not possible to reduce entropy on average, even by stopping a stochastic process at a stopping time, and which we call the second law of therm…
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.
We show that the zeroth law of thermodynamics holds within an alternative version of nonextensive statistical mechanics based on {\it incomplete probability distribution}. The generalized zeroth law leads to a generalized definition of thermodynamic functions which are possible to be used for systems with important nonextensivity (nonadditivity) in energy, volume or other external variables.
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…
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.
In classical phenomenological thermodynamics the first and second laws can be regarded as independent statements. Statistical mechanics provides a microscopic substratum that explains thermodynamics in probabilistic terms via a microstate probability distribution ${p_i}$. We study here a hitherto unexplored microscopic connection between the two laws. Given an information measure (or entropic form), each of the two laws implies the other through the process $p_i \to p_i+dp_i$.
The classical thermodynamic laws fail to capture the behavior of systems with energy Hamiltonian which is an explicit function of the temperature. Such Hamiltonian arises, for example, in modeling information processing systems, like communication channels, as thermal systems. Here we generalize the second thermodynamic law to encompass systems with temperature-dependent energy levels, $dQ=TdS+<d\mathcal{E}/dT>dT$, where $<\cdot>$ denotes averaging over the Boltzmann distribution and reveal a new definition to the basic notion of temperature. This generalization enables to express, for instanc…
The second law of ordinary thermodynamics and the second law of steady state thermodynamics, as proposed by Oono and Paniconi, are investigated from the microscopic point of view for the open quantum system. Based on the H-theorem of Lindblad, we show that both second laws are consistent with the quantum dynamics generated by the completely positive map. In addition, microscopic expressions of entropy production and ``housekeeping heat'' are obtained for some classes of dynamical quantum systems.
We show an attempt to establish the zeroth law of thermodynamics within the framework of nonextensive statistical mechanics based on the classic normalization $\texttt{Tr}\hatρ=1$ and the unnormalized expectation $x=\texttt{Tr}\hatρ^q\hat{x}$. The first law of thermodynamics and the definition of heat and work in this formalism are discussed.
In a macroscopic (quantum or classical) Hamiltonian system, we prove the second law of thermodynamics in the forms of the minimum work principle and the law of entropy increase, under the assumption that the initial state is described by a general equilibrium distribution. Therefore the second law is a logical necessity once we accept equilibrium statistical mechanics.
It is shown that the laws of thermodynamics are extremely robust under generalizations of the form of entropy. Using the Bregman-type relative entropy, the Clausius inequality is proved to be always valid. This implies that thermodynamics is highly universal and does not rule out consistent generalization of the maximum entropy method.
Two important problems existing in Tsallis' statistics are investigated, where one is whether energy is extensive or not, and the other is whether it is necessary to introduce the so-called generalized zeroth law of thermodynamics or not. The results obtained show clearly that like entropy, energy is also nonextensive in Tsallis' statistics, and that the zeroth law of thermodynamics has been implicitly used in Tsallis' statistics since 1988. Moreover, it is expounded that the standard energy additivity rule adopted by a great number of researchers is not suitable in Tsallis' statistics, becaus…
The quantum thermodynamic functions of a harmonic oscillator coupled to a heat bath through velocity-dependent coupling are obtained analytically. It is shown that both the free energy and the entropy decay fast with the temperature in relation to that of the usual coupling from. This implies that the velocity-dependent coupling helps to ensure the third law of thermodynamics.
Several models of quantum open systems are known at present to violate, according to principles of the standard quantum theory of open systems, the second law of thermodynamics. Here, a new and rather trivial model of another type is suggested describing mechanism that violates, according to the same principles, the zeroth and the second laws of thermodynamics simultaneously. Up to a technically minor modification, the model resembles some models already known, solved by standard means, and properly understood. Universal validity of two basic principles of thermodynamics in strictly quantum si…
Tsallis' thermostatistics is by now recognized as a new paradigm for statistical mechanical considerations. However, it is still affected by a serious hindrance: the generalization of thermodynamics' zero-th law to a nonextensive scenario is plagued by difficulties. Here we show how to overcome this problem.
We explore driven lattice gases for the existence of an intensive thermodynamic variable which could determine "equilibration" between two nonequilibrium steady-state systems kept in weak contact. In simulations, we find that these systems satisfy surprisingly simple thermodynamic laws, such as the zero-th law and the fluctuation-response relation between the particle-number fluctuation and the corresponding susceptibility remarkably well. However at higher densities, small but observable deviations from these laws occur due to nontrivial contact dynamics and the presence of long-range spatial…
We present a pedagogical review of the fundamental concepts in thermodynamics of information, by focusing on the second law of thermodynamics and the entropy production. Especially, we discuss the relationship among thermodynamic reversibility, logical reversibility, and heat emission in the context of the Landauer principle and clarify that these three concepts are fundamentally distinct to each other. We also discuss thermodynamics of measurement and feedback control by Maxwell's demon. We clarify that the demon and the second law are indeed consistent in the measurement and the feedback pro…
Based only on classical Hamiltonian dynamics, we prove the maximum work principle in a system where macroscopic dynamical degrees of freedom are intrinsically coupled to microscopic degrees of freedom. Unlike recent identities between irreversible work and free energy, such as in the Jarzynski relation, the macroscopic dynamics is not governed by an external action but undergoes the back reaction of the microscopic degrees of freedom. Our theorems cover such physical situations as impact between macroscopic bodies, thermodynamic machines, and molecular motors.
The second law of thermodynamics in nonextensive statistical mechanics is discussed in the quantum regime. Making use of the convexity property of the generalized relative entropy associated with the Tsallis entropy indexed by q, Clausius' inequality is shown to hold in the range of q between zero and two. This restriction on the range of the entropic index, q, is purely quantum mechanical and there exists no upper bound of q for validity of the second law in classical theory.
It was recently shown, starting from first principles, that thermodynamics' first law (TFL) can be microscopically obtained without need of invoking the adiabatic theorem (AT) [Physica A, {\bf 356}, 167 (2005)]. We show here that a TFL can also be found for Fisher's information measure, following a similar procedure. Further, it is proved that enforcing the Fisher-first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing Fisher's information measure subject to appropriate constraints.
We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport. General discussion is illustrated by numerical analysis of a model for micron-size particle manipulated by optical tweezers.
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…
Bento \textit{et al.} [Phys. Rev. E 91, 022105 (2015)] state that the Tsallis entropy violates the third law of thermodynamics for $q \leq 0$ and $0<q<1$. We show that their results are valid only for $q \geq 1$, since there is no distribution maximizing the Tsallis entropy for the intervals $q \leq 0$ and $0<q<1$ compatible with the system energy expression.
On the basis of the entropy of incomplete statistics (IS) and the joint probability factorization condition, two controversial problems existing in IS are investigated, where one is what the correct expression of the internal energy for a composite system is and the other is whether the zeroth law of thermodynamics is true or not. Some new equivalent expressions of the internal energy of a composite system are derived through a precise mathematical calculation. Moreover, a self-consistent calculation is used to expound that the zeroth law of thermodynamics is also suitable for IS, but it can't…
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.
Within the nonextensive framework, it is shown that zeroth law of thermodynamics determines not only the mapping between Lagrange multipliers and intensive variables, but also the mapping between nonextensive and extensive entropy. The form of constraints decides the form of the extensive entropy, standard averages lead to Boltzmann-Shannon-Gibbs entropy while normalised biased averages lead to Renyi entropy. The mapping between Lagrange multipliers and intensive variables is also discussed in the more general context of composable entropy.
The second law of classical thermodynamics, based on the positivity of the entropy production, only holds for deterministic processes. Therefore the Second Law in stochastic quantum thermodynamics may not hold. By making a fundamental connection between thermodynamics and information theory we will introduce a new way of defining the Second Law which holds for both deterministic classical and stochastic quantum thermodynamics. Our work incorporates information well into the Second Law and also provides a thermodynamic operational meaning for negative and positive entropy production.
A general nonequilibrium thermodynamic theory is developed for time-dependent Langevin dynamics, starting from the common definition of nonequilibrium Gibbs entropy. It is shown that the notations appearing in the First and the Second Law of thermodynamics could be consistently applied to transient nonequilibrium processes. We find out a general equality representing the energy balance relation for the system, and further an explicit mathematical interpretation for the extended form of the Second Law is proposed, which only comes into existence in the time-dependent case. More applications to …
In quantum statistical mechanics, equilibrium states have been shown to be the typical states for a system that is entangled with its environment, suggesting a possible identification between thermodynamic and von Neumann entropies. In this paper, we investigate how the relaxation toward equilibrium is made possible through interactions that do not lead to significant exchange of energy, and argue for the validity of the second law of thermodynamics at the microscopic scale.
The quantum thermodynamic property of the fractional damping system is investigated extensively. A fractional power-law decaying entropy function is revealed which presents another evidence for the validity of the third law of thermodynamics in the quantum dissipative region. Several non-trivial characters are excavated such as that the entropy varies from a non-linear diverging function to a semi-linear decaying function of the fractional exponent as the temperature tends to absolute zero.
We introduce and realize demons that follow a customary gambling strategy to stop a nonequilibrium process at stochastic times. We derive second-law-like inequalities for the average work done in the presence of gambling, and universal stopping-time fluctuation relations for classical and quantum non-stationary stochastic processes. We test experimentally our results in a single-electron box, where an electrostatic potential drives the dynamics of individual electrons tunneling into a metallic island. We also discuss the role of coherence in gambling demons measuring quantum jump trajectories.
A new thermodynamic inequality is derived which leads to the maximum work that can be extracted from multi-heat baths with the assistance of discrete quantum feedback control. The maximum work is determined by the free-energy difference and a generalized mutual information content between the thermodynamic system and the feedback controller. This maximum work can exceed that in conventional thermodynamics and, in the case of a heat cycle with two heat baths, the heat efficiency can be greater than that of the Carnot cycle. The consistency of our results with the second law of thermodynamics is…
Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics. This tutorial focuses on the derivation of the first and second law for closed and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies. The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat and temperature. These definitions are shown to satisfy the phenomenological laws of nonequilibrium thermodynamics for a large class of…
We show that the equations which describe irreversible evolution of a system can be derived from a variational principle. We suggest a Lagrangian, which depends on the properties of the normal and the so-called "mirror-image" system. The Lagrangian is symmetric in time and therefore compatible with microscopic reversibility. The evolution equations in the normal and mirror-imaged systems are decoupled and describe therefore independent irreversible evolution of each of the systems. The second law of thermodynamics follows from a symmetry of the Lagrangian. Entropy increase in the normal system…
We reconsider the basic building blocks of classical phenomenological thermodynamics. While doing so we show that the zeroth law is a redundant postulate for the theory by deriving it from the first and the second laws. This is in stark contrast to the prevalent conception that the three laws, the zeroth, first and second, are all necessary and independent axioms.
Motivated by black holes thermodynamics, we consider the zeroth law of thermodynamics for systems whose entropy is a quasi-homogeneous function of the extensive variables. We show that the generalized Gibbs-Duhem identity and the Maxwell construction for phase coexistence based on the standard zeroth law are incompatible in this case. We argue that the generalized Gibbs-Duhem identity suggests a revision of the zeroth law which in turns permits to reconsider Maxwell's construction in analogy with the standard case. The physical feasibility of our proposal is considered in the particular case o…
