In this paper, we extend Gibbs's approach of quasi-equilibrium thermodynamic processes, and calculate the microscopic expression of entropy for general non-equilibrium thermodynamic processes. Also, we analyze the formal structure of thermodynamic relation in non-equilibrium thermodynamic processes.
We show that the generalized Boltzmann distribution is the only distribution for which the Gibbs-Shannon entropy equals the thermodynamic entropy. This result means that the thermodynamic entropy and the Gibbs-Shannon entropy are not generally equal, but rather than the equality holds only in the special case where a system is in equilibrium with a reservoir.
Using the grand canonical ensemble and the virial theorem, we show that the Gibbs thermodynamic potential of the non-relativistic system of charged particles is uniquely defined by single-particle Green functions of electrons and nuclei. This result is valid beyond the perturbation theory with respect to the interparticle interaction.
General relationship between mean Boltzmann entropy and Gibbs entropy is established. It is found that their difference is equal to fluctuation entropy, which is a Gibbs-like entropy of macroscopic quantities. The ratio of the fluctuation entropy and mean Boltzmann, or Gibbs entropy vanishes in the thermodynamic limit for a system of distinguishable and independent particles. It is argued that large fluctuation entropy clearly indicates the limit where standard statistical approach should be modified, or extended using other methods like renormalization group.
We develop a consistent stochastic thermodynamics for environments composed of thermodynamic reservoirs in an external conservative force field, that is environments described by the Generalized or Gibbs canonical ensemble. We demonstrate that small systems weakly coupled to such reservoirs exchange both heat and work by verifying a local detailed balance relation for the induced stochastic dynamics. Based on this analysis, we help to rationalize the observation that nonthermal reservoirs can increase the efficiency of thermodynamic heat engines.
In recent papers, several authors have claimed that a definition of the thermodynamic entropy in terms of the logarithm of a volume in phase space, originally suggested by Gibbs, is the only valid definition. Arguing from the Gibbs entropy, these authors claim that thermodynamics cannot be extended to include negative temperatures. However, the Gibbs entropy fails to satisfy the postulates of thermodynamics, leading to serious errors. In particular, predictions of the Gibbs entropy for systems with non-monotonic energy densities are incorrect. We show that the correct expression for the equili…
For small thermodynamic systems in contact with a heat bath, we determine the free energy by imposing the following two conditions. First, the quasi-static work in any configuration change is equal to the free energy difference. Second, the temperature dependence of the free energy satisfies the Gibbs-Helmholtz relation. We find that these prerequisites uniquely lead to the free energy of a classical system consisting of $N$-interacting identical particles, up to an additive constant proportional to $N$. The free energy thus determined contains the Gibbs factorial $N!$ in addition to the phase…
We generalize the convex duality symmetry in Gibbs' statistical ensemble formulation, between Massieu's free entropy $Φ_{V,N} (β)$ and the Gibbs entropy $φ_{V,N}(u)$ as a function of mean internal energy $u$. The duality tells us that Gibbs thermodynamic entropy is to the law of large numbers (LLN) for arithmetic sample means what Shannon's information entropy is to the LLN for empirical counting frequencies. Following the same logic, we identify $u$ as the conjugate variable to counting frequency, a Hamilton-Jacobi equation for Shannon entropy as an equation of state, and suggest an eigenvalu…
This contribution is dedicated to dilucidating the role of the Gibbs entropy in the discussion of the emergence of irreversibility in the macroscopic world from the microscopic level. By using an extension of the Onsager theory to the phase space we obtain a generalization of the Liouville equation describing the evolution of the distribution vector in the form of a master equation. This formalism leads in a natural way to the breaking of the BBGKY hierarchy. As a particular case we derive the Boltzmann equation.
Integrable quantum many-body systems are considered to equilibrate to generalized Gibbs ensembles (GGEs) characterized by the expectation values of integrals of motion. We study the dynamics of exactly solvable quadratic bosonic systems in the thermodynamic limit, and show a general mechanism for the relaxation to GGEs, in terms of the diagonal singularity. We show analytically and explicitly that a free bosonic system relaxes from a general (not necessarily Gaussian) initial state under certain physical conditions to a Gaussian GGE. We also show the relaxation to a Gaussian GGE in an exactly …
The second Tisza-Callen postulate of equilibrium thermodynamics states that for any system exists a function of the system's extensive parameters, called entropy, defined for all equilibrium states and having the property that the values assumed by the extensive parameters in the absence of a constraint are those that maximize the entropy over the manifold of constrained equilibrium states. By analyzing the evolution of systems of positive and negative absolute temperatures, we show that this postulate is satisfied by the Boltzmann formula for the entropy and is violated by the Gibbs formula. …
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approac…
Very recently, the validity of the concept of negative temperature has been challenged by several authors since they consider Boltzmann's entropy (that allows negative temperatures) inconsistent from a mathematical and statistical point of view, whereas they consider Gibbs' entropy (that does not admit negative temperatures) the correct definition for microcanonical entropy. In the present paper we prove that for systems with equivalence of the statistical ensembles Boltzmann entropy is the correct microcanonical entropy. Analytical results on two systems supporting negative temperatures, conf…
We propose a generalisation of Gibbs' statistical mechanics into the domain of non-negligible phase space correlations. Derived are the probability distribution and entropy as a generalised ensemble average, replacing Gibbs-Boltzmann-Shannon's entropy definition enabling construction of new forms of statistical mechanics. The general entropy may also be of importance in information theory and data analysis. Application to generalised Lorentzian phase space elements yields the Gibbs-Lorentzian power law probability distribution and statistical mechanics. Details can be found in arXiv:1406.6639
In this work, we demonstrate the inappropriateness of the Boltzmann-Gibbs log-formulation of the physical Clausius entropy $S$ in connecting thermodynamics and phase space statistics. To achieve our goal, we study thermodynamically the simple case of ideal gases embedded in a finite heat bath and compare the derived equations with the phase space statistical ones obtained within the canonical ensemble. We then show that the results of the aforementioned approaches contradict to each other even in the dilute gas limit (infinite heat bath) if we request the identification of the Boltmann-Gibbs f…
We brief{}ly review the connection between statistical mechanics and thermodynamics. We show that, in order to satisfy thermodynamics and its Legendre transformation mathematical frame, the celebrated Boltzmann-Gibbs~(BG) statistical mechanics is suff{}icient but not necessary. Indeed, the $N\to\infty$ limit of statistical mechanics is expected to be consistent with thermodynamics. For systems whose elements are generically independent or quasi-independent in the sense of the theory of probabilities, it is well known that the BG theory (based on the additive BG entropy) does satisfy this expec…
Alternative definitions are given of basic concepts of generalized thermostatistics. In particular, generalizations of Shannon's entropy, of the Boltzmann-Gibbs distribution, and of relative entropy are considered. Particular choices made in Tsallis' nonextensive thermostatistics are questioned.
For a dynamical system far from equilibrium, one has to deal with empirical probabilities defined through time-averages, and the main problem is then how to formulate an appropriate statistical thermodynamics. The common answer is that the standard functional expression of Boltzmann-Gibbs for the entropy should be used, the empirical probabilities being substituted for the Gibbs measure. Other functional expressions have been suggested, but apparently with no clear mechanical foundation. Here it is shown how a natural extension of the original procedure employed by Gibbs and Khinchin in defini…
The local thermodynamics of a system with long-range interactions in d dimensions is studied using the mean-field approximation. Long-range interactions are introduced through pair interaction potentials that decay as a power law in the interparticle distance. We compute the local entropy, Helmholtz free energy, and grand potential per particle in the microcanonical, canonical, and grand canonical ensembles, respectively. From the local entropy per particle we obtain the local equation of state of the system by using the condition of local thermodynamic equilibrium. This local equation of stat…
We prove that information-theoretic maximum entropy (MaxEnt) approach to canonical ensemble is mathematically equivalent to the classic approach of Boltzmann, Gibbs and Darwin-Fowler. The two approaches, however, "interpret" a same mathematical theorem differently; most notably observing mean-energy in the former and energy conservation in the latter. However, applying the same MaxEnt method to grand canonical ensemble fails; while carefully following the classic approach based on Boltzmann's microcanonical {\em equal probability a priori} produces the correct statistics: One does not need to …
We study a classical integrable (Neumann) model describing the motion of a particle on the sphere, subject to harmonic forces. We tackle the problem in the infinite dimensional limit by introducing a soft version in which the spherical constraint is imposed only on average over initial conditions. We show that the Generalized Gibbs Ensemble captures the long-time averages of the soft model. We reveal the full dynamic phase diagram with extended, quasi-condensed, coordinate-, and coordinate and momentum-condensed phases. The scaling properties of the fluctuations allow us to establish in which …
Resolving the Debate Between Boltzmann and Gibbs Entropy: Relative Energy Window Eliminates Thermodynamic Inconsistencies and Allows Negative Absolute Temperatures
The generalized Gibbs-Duhem equation is obtained for systems with long-range interactions in $d$ spatial dimensions. We consider that particles in the system interact through a slowly decaying pair potential of the form $1/r^ν$ with $0\leqν\leq d$. The local equation of state is obtained by computing the local entropy per particle and using the condition of local thermodynamic equilibrium. This local equation of state turns out to be that of an ideal gas. Integrating the relation satisfied by local thermodynamic variables over the volume, the equation involving global magnitudes is derived. Th…
We study the temporal approach to equilibrium of the Gibbs' and conditional entropies for both invertible deterministic dynamics as well as non-invertible stochastic systems in the presence of white noise. The conditional entropy will either remain constant or monotonically increase to its maximum of zero. However, the Gibbs' entropy may have a variety of patterns of approach to its final value ranging from a monotone increase or decrease to an oscillatory approach. We have illustrated all of these behaviors using examples in which both entropy dynamics can be determined analytically.
In 1910 Einstein published a crucial aspect of his understanding of Boltzmann entropy. He essentially argued that the likelihood function of any system composed by two probabilistically independent subsystems {\it ought} to be factorizable into the likelihood functions of each of the subsystems. Consistently he was satisfied by the fact that Boltzmann (additive) entropy fulfills this epistemologically fundamental requirement. We show here that entropies (e.g., the $q$-entropy on which nonextensive statistical mechanics is based) which generalize the BG one through violation of its well known a…
The Boltzmann and Gibbs approaches to statistical mechanics have very different definitions of equilibrium and entropy. The problems associated with this are discussed and it is suggested that they can be resolved, to produce a version of statistical mechanics incorporating both approaches, by redefining equilibrium not as a binary property (being/not being in equilibrium) but as a continuous property (degrees of equilibrium) measured by the Boltzmann entropy and by introducing the idea of thermodynamic-like behaviour for the Boltzmann entropy. The Kac ring model is used as an example to test …
Thermodynamic properties of extensive but nonadditive systems are investigated. The precise definitions of additivity and extensivity are presented, and we will see that additivity derives several important properties including the shape-independence of the thermodynamic functions, the concavity of the entropy, and the equivalence of ensembles. In nonadditive systems, some of the above properties can be violated. It is pointed out that the shape-dependence of the entropy density in a nonadditive system results in the violation of the Gibbs-Duhem equation. As an example, the dipolar gas is nume…
We analyze a contrasting dynamical behavior of Gibbs-Shannon and conditional Kullback-Leibler entropies, induced by time-evolution of continuous probability distributions. The question of predominantly purpose-dependent entropy definition for non-equilibrium model systems is addressed. The conditional Kullback-Leibler entropy is often believed to properly capture physical features of an asymptotic approach towards equilibrium. We give arguments in favor of the usefulness of the standard Gibbs-type entropy and indicate that its dynamics gives an insight into physically relevant, but generally i…
We show how to construct non-equilibrium thermodynamics for systems too small to be considered thermodynamically in a traditional sense. Through the use of a non-equilibrium ensemble of many replicas of the system which can be viewed as a large thermodynamic system, we discuss the validity of non-equilibrium thermodynamics relations and analyze the nature of dissipation in small systems through the entropy production rate. We show in particular that the Gibbs equation, when formulated in terms of average values of the extensive quantities, is still valid whereas the Gibbs-Duhem equation differ…
In a recent paper, Dunkel and Hilbert [Nature Physics 10, 67-72 (2014)] use an entropy definition due to Gibbs to provide a 'consistent thermostatistics' which forbids negative absolute temperatures. Here we argue that the Gibbs entropy fails to satisfy a basic requirement of thermodynamics, namely that when two bodies are in thermal equilibrium, they should be at the same temperature. The entropy definition due to Boltzmann does meet this test, and moreover in the thermodynamic limit can be shown to satisfy Dunkel and Hilbert's consistency criterion. Thus, far from being forbidden, negative t…
Introducing the Boltzmann distribution very early in a statistical thermodynamics course (in the spirit of Feynmann) has many didactic advantages, in particular that of easily deriving the Gibbs entropy formula. In this note, a short derivation is proposed from the fundamental postulate of statistical mechanics and basics calculations accessible to undergraduate students.
Stochastic thermodynamics extends classical thermodynamics to small systems in contact with one or more heat baths. It can account for the effects of thermal fluctuations and describe systems far from thermodynamic equilibrium. A basic assumption is that the expression for Shannon entropy is the appropriate description for the entropy of a nonequilibrium system in such a setting. Here, for the first time, we measure experimentally this function. Our system is a micron-scale colloidal particle in water, in a virtual double-well potential created by a feedback trap. We measure the work to erase …
We consider a new model which consists of a DNA together with a RNA. Here we assume that DNA is from a mammal or bird but RNA comes from a virus. To study thermodynamic properties of this model we use methods of statistical mechanics, namely, the theory of Gibbs measures. We use these measures to describe phases (states) of the DNA-RNA system. Using a Markov chain (corresponding to Gibbs measure) we give conditions (on temperature) of DNA-RNA renaturation.
Dunkel and Hilbert, "Consistent thermostatistics forbids negative absolute temperatures," Nature Physics, {\bf 10}, 67 (2014), and Hilbert, Hänggi, and Dunkel, "Thermodynamic laws in isolated systems," Phys. Rev. E {\bf 90}, 062116 (2014) have presented an unusual view of thermodynamics that sets aside several properties that have traditionally have been assumed to be true. Among other features, their results do not satisfy the postulates of thermodynamics originally proposed by Tisza and Callen. In their theory, differences in the temperatures of two objects cannot determine the direction of …
The inclusion of the factor $\ln (1/N!)$ in the thermodynamic entropy proposed by Gibbs is shown to be equivalent to the validity of the fluctuation theorem with absolute irreversibility for gas mixing.
Gibbs' theorem, which is originally intended for canonical ensembles with complete statistics has been generalized to open systems with incomplete statistics. As a result of this generalization, it is shown that the stationary equilibrium distribution of inverse power law form associated with the incomplete statistics has maximum entropy even for open systems with energy or matter influx. The renormalized entropy definition given in this paper can also serve as a measure of self-organization in open systems described by incomplete statistics.
The $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the $q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$ ($x\otimes_1 y=x y$) emerge naturally within nonextensive statistical mechanics. We show here how they lead to two-parameter (namely, $q$ and $q^\prime$) generalizations of the logarithmic and exponential functions (noted respectively $\ln_{q,q^\prime}x$ and $e_{q,q^\prime}^{x}$), as well as of the Boltzmann-Gibbs-Shannon entropy $S_{BGS}\equiv -k \sum_{i=1}^Wp_i \ln p_i$ (noted $S_{q,q^\prime}$). The remarkable properties of the $(q,q^\prime)$-gene…
Shape-dependent thermodynamics and non-local hydrodynamics are argued to occur in dissipative steady states of driven diffusive systems. These predictions are confirmed by numerical simulations. Unlike power-law correlations, these phenomena cannot be explained by a hypothesis of ``criticality''. Instead, they require the effective Hamiltonian of the system to contain very long-range potentials, making the invariant probability measures formally ``non-Gibbsian''.
We reconsider the Boltzmann-Gibbs statistical ensemble in thermodynamics using the multinomial coefficient approach. We show that an ensemble is defined by the determination of four statistical quantities, the element probabilities $p_i$, the configuration probabilities $P_j$, the entropy $S$ and the extremum constraints (EC). This distinction is of central importance for the understanding of the conditions under which a microcanonical, canonical and macrocanonical ensemble is defined. These three ensembles are characterized by the conservation of their sizes. A variation of the ensemble size …
It is shown that the existence of single critical point of the fluid described by van der Waals equation of state is not the hypothesis and it is the consequence of the thermodynamic conditions of liquid-vapor phase equilibrium. It is also shown that the thermodynamic conditions of liquid-vapor phase equilibrium of the fluid give equalities to zero of first and second partial derivatives at constant temperature of pressure with respect to volume at critical point which are usual conditions of existence of critical point.
We develop the argument that the Gibbs-von Neumann entropy is the appropriate statistical mechanical generalisation of the thermodynamic entropy, for macroscopic and microscopic systems, whether in thermal equilibrium or not, as a consequence of Hamiltonian dynamics. The mathematical treatment utilises well known results [Gib02, Tol38, Weh78, Par89], but most importantly, incorporates a variety of arguments on the phenomenological properties of thermal states [Szi25, TQ63, HK65, GB91] and of statistical distributions[HG76, PW78, Len78]. This enables the identification of the canonical distribu…
This paper introduces the basic concepts of information theory. Based on these concepts, we regard the states in the state space and the types of ideal gases as the symbols in a symbol set to calculate the mixing entropy of ideal gas involved in Gibbs Paradox. The discussion above reveals that the non-need for distinguishing can resolve the contradiction of Gibbs Paradox, implying the introduction of indistinguishability is not necessary. Further analysis shows that the information entropy of gas molecular types does not directly correlate to the energy of a gas system, so it should not be use…
Identical classical particles are distinguishable. This distinguishability affects the number of ways W a macrostate can be realized on the micro-level, and from the relation S = k ln W leads to a non-extensive expression for the entropy. This result is usually considered incorrect because of its inconsistency with thermodynamics. It is sometimes concluded from this inconsistency that identical particles are fundamentally indistinguishable after all; and even that quantum mechanics is indispensable for making sense of this. In contrast, we argue that the classical statistics of distinguishable…
For more than 100 years, one of the central concepts in statistical mechanics has been the microcanonical ensemble, which provides a way of calculating the thermodynamic entropy for a specified energy. A controversy has recently emerged between two distinct definitions of the entropy based on the microcanonical ensemble: (1) The Boltzmann entropy, defined by the density of states at a specified energy, and (2) The Gibbs entropy, defined by the sum or integral of the density of states below a specified energy. A critical difference between the consequences of these definitions pertains to the c…
What is the best description that we can construct of a thermodynamic system that is not in equilibrium, given only one, or a few, extra parameters over and above those needed for a description of the same system at equilibrium? Here, we argue the most appropriate additional parameter is the non-equilibrium entropy of the system, and that we should not attempt to estimate the probability distribution of the system, but rather the metaprobability (or hyperensemble) that the system is described by a particular probability distribution. The result is an entropic distribution with two parameters, …
In ordinary Boltzmann-Gibbs thermostatistics, the relative entropy expression plays the role of generalized free energy, providing the difference between the off-equilibrium and equilibrium free energy terms associated with Boltzmann-Gibbs entropy. In this context, we studied whether this physical meaning can be given to Rényi relative entropy definition found in the literature from a generalized thermostatistical point of view. We find that this is possible only in the limit as $q$ approaches to 1. This shows that Rényi relative entropy has a physical (thermostatistical) meaning only when the…
The Gibbs entropy of a microcanonical network ensemble is the logarithm of the number of network configurations compatible with a set of hard constraints. This quantity characterizes the level of order and randomness encoded in features of a given real network. Here we show how to relate this entropy to large deviations of conjugated canonical ensembles. We derive exact expression for this correspondence using the cavity methods for some hard constraints.
The long standing contrast between Boltzmann's and Gibbs' approach to statistical thermodynamics has been recently rekindled by Dunkel and Hilbert [1], who criticize the notion of negative absolute temperature (NAT), as a misleading consequence of Boltzmann's definition of entropy. A different definition, due to Gibbs, has been proposed, which forbids NAT and makes the energy equipartition rigorous in arbitrary sized systems. The two approaches, however, are shown to converge to the same results in the thermodynamical limit. A vigorous debate followed ref.[1], with arguments against [2,3] and …
On a fine grained scale the Gibbs entropy of an isolated system remains constant throughout its dynamical evolution. This is a consequence of Liouville's theorem for Hamiltonian systems and appears to contradict the second law of thermodynamics. In reality, however, there is no problem since the thermodynamic entropy should be associated with the Boltzmann entropy, which for non-equilibrium systems is different from Gibbs entropy. The Boltzmann entropy accounts for the microstates which are not accessible from a given initial condition, but are compatible with a given macrostate. In a sense th…
In this paper we discuss about the validity of the Shannon entropy functional in connection with the correct Gibbs-Hertz probability distribution function. We show that there is no contradiction in using the Shannon-Gibbs functional and restate the validity of information theory applied to equilibrium statistical mechanics. We show that under these assumptions, entropy is always a monotone function of energy, irrespective to the shape of the density of states, leading always to positive temperatures even in the case of inverted population systems. In the second part we assume the validity of t…
Recently a number of approaches has been developed to connect the microscopic dynamics of particle systems to the macroscopic properties of systems in nonequilibrium stationary states, via the theory of dynamical systems. This way a direct connection between dynamics and Irreversible Thermodynamics has been claimed to have been found. However, the main quantity used in these studies is a (coarse-grained) Gibbs entropy, which to us does not seem suitable, in its present form, to characterize nonequilibrium states. Various simplified models have also been devised to give explicit examples of how…
Thermodynamics as limiting behaviors of statistics is generalized to arbitrary system with probability {\it a priori} where thermodynamic infinite-size limit is replaced by multiple-measurement limit. A duality symmetry between Massieu's and Gibbs' entropy arises in the limit of infinitely repeated observations, yielding the Gibbs equation and Hill-Gibbs-Duhem equation (HGDE) as dual pair. If a system has thermodynamic limit satisfying Callen's postulate, entropy being an Eulerian function, the symmetry is lost: the HGDE reduces to the Gibbs-Duhem equation. This theory provides a de-mechanized…
The generalized Gibbs free energy and enthalpy is derived in the framework of nonextensive thermodynamics by using the so-called physical temperature and the physical pressure. Some thermodynamical relations are studied by considering the difference between the physical temperature and the inverse of Lagrange multiplier. The thermodynamical relation between the heat capacities at a constant volume and at a constant pressure is obtained using the generalized thermodynamical potential, which is found to be different from the traditional one in Gibbs thermodynamics. But, the expressions for the h…
