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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 Hilberts consistency criterion. Thus, far from being forbidden, negative temperatures are inevitable, in systems with bounded energy spectra.
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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 approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
The local physical properties of an isolated quantum statistical system in the stationary state reached long after a quench are generically described by the Gibbs ensemble, which involves only its Hamiltonian and the temperature as a parameter. If the system is instead integrable, additional quantities conserved by the dynamics intervene in the description of the stationary state. The resulting generalized Gibbs ensemble involves a number of temperature-like parameters, the determination of which is practically difficult. Here we argue that in a number of simple models these parameters can be effectively determined by using fluctuation-dissipation relationships between response and correlation functions of natural observables, quantities which are accessible in experiments.
As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) standard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistics.
We discuss how to derive a Langevin equation (LE) in non standard systems, i.e. when the kinetic part of the Hamiltonian is not the usual quadratic function. This generalization allows to consider also cases with negative absolute temperature. We first give some phenomenological arguments suggesting the shape of the viscous drift, replacing the usual linear viscous damping, and its relation with the diffusion coefficient modulating the white noise term. As a second step, we implement a procedure to reconstruct the drift and the diffusion term of the LE from the time-series of the momentum of a heavy particle embedded in a large Hamiltonian system. The results of our reconstruction are in good agreement with the phenomenological arguments. Applying the method to systems with negative temperature, we can observe that also in this case there is a suitable Langevin equation, obtained with a precise protocol, able to reproduce in a proper way the statistical features of the slow variables. In other words, even in this context, systems with negative temperature do not show any pathology.
We present a detailed account of a first-order localization transition in the Discrete Nonlinear Schrodinger Equation, where the localized phase is associated to the high energy region in parameter space. We show that, due to ensemble inequivalence, this phase is thermodynamically stable only in the microcanonical ensemble. In particular, we obtain an explicit expression of the microcanonical entropy close to the transition line, located at infinite temperature. This task is accomplished making use of large-deviation techniques, that allow us to compute, in the limit of large system size, also the subleading corrections to the microcanonical entropy. These subleading terms are crucial ingredients to account for the first-order mechanism of the transition, to compute its order parameter and to predict the existence of negative temperatures in the localized phase. All of these features can be viewed as signatures of a thermodynamic phase where the translational symmetry is broken spontaneously due to a condensation mechanism yielding energy fluctuations far away from equipartition: actually they prefer to participate in the formation of nonlinear localized excitations (breathers), typically containing a macroscopic fraction of the total energy.
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