Statistical thermodynamics for metaequilibrium or metastable states

Abstract

We show how statistical thermodynamics can be formulated in situations in which thermodynamics applies, while equilibrium statistical mechanics does not. A typical case is, in the words of Landau and Lifshitz, that of partial (or incomplete) equilibrium. One has a system of interest in equilibrium with the environment, and measures one of its quantities, for example its specific heat, by raising the temperature of the environment. However, within the observation time the global system settles down to a state of apparent equilibrium, so that the measured value of the specific heat is different from the equilibrium one. In such cases formulae for quantities such as the effective specific heat exist, which are provided by Fluctuation Dissipation theory. However, what is lacking is a proof that internal energy exists, i.e., that the fundamental differential form δQ−δW (difference between the heat absorbed by the system and the work performed by it) is closed. Here we show how the coefficients of the fundamental form can be expressed in such a way that the closure property of the form becomes manifest, so that the first principle is proven. We then show that the second principle too follows, and indeed as a consequence of microscopic time-reversibility. The treatment is given in a classical Hamiltonian setting. One has a global time-independent Hamiltonian system constituted by the system of interest and two auxiliary ones controlling temperature and pressure, and the occurring of a process due to a change in the thermodynamic parameters is implemented by a suitable choice of the measure for the initial data.

Relaxation times and ergodic properties in a realistic ionic-crystal model, and the modern form of the FPU problem

“…It seems that this problem could be solved if the expectations were implemented with respect to a suitable invariant phase-space measure (see for example [22]). So it is of interest to understand whether in our case it is possible to find a measure which reproduces the actually computed time-averages.…”

Freezing of the optical--branch energy in a diatomic FPU chain

“…This is not obvious at all, but in this particular case we can imagine, in a completely heuristic way, a mechanism for which this slow decay of correlations might show up in the measurement of an actual physical quantity, the heat capacity C of the chain. Recall, indeed, that the expression of the heat capacity C of a system put in contact with a thermostat, in the linear response theory approximation (see, for instance, [28]), is the following…”

“…We pass from the variables (ξ ± , η ± ) to (p ± , q ± ) by using (6), then apply (28,32) of Appendix A to pass to the variables (p j , r j ), thus getting…”

Tsallis distributions, their relaxations and the relation $Δt \cdot ΔE \simeq h$, in the dynamical fluctuations of a classical model of a crystal

“…In other terms, the region of a constant-energy surface presenting a distribution different from Maxwell-Boltzmann, has negligible measure. This however is not sufficient for the aims of Statistical Thermodynamics, since many quantities of interest are expressed in terms of correlations of suitable observables (see [2]), computed from initial data of generic type (i.e., with a Maxwell-Boltzmann distribution), and not just in term of averages of the normal-mode energies. An example is given in the works [3,4], where the electric susceptibilty of Lithium Fluoride (LiF) was computed through the Fourier transform of the auto-correlation of the optical normal-mode amplitudes, for generic initial data, and the infrared spectra thus obtained were found to agree very well with the experimental ones at several temperatures (more details will be given later).…”

“…This was inspired to the following naive argument. 2 Assume that, starting from a point x 0 in phase space, the evolved point x 1 def = Φ t x 0 at time t may be considered as "independent" of x 0 , in the sense that the normal-mode energies may be considered as random variables independent of their initial values. Then it will turn out that a Maxwell-Boltzmann distribution occurs not only for the normal-mode energies, both in x 0 and in x 1 , but also (as a simple calculation shows -see Section 3) for the moduli of the energy-changes |E k (t) − E k (0)|.…”