Abstract
We show how statistical thermodynamics can be formulated in situations of metaequilibrium or metastability (as in the cases of supercooled liquids or of glasses respectively). By analogy with phenomenological thermodynamics, the primary quantities considered are the heat Q absorbed and the work W performed by the system of interest. These are defined through the energy exchanges which occur when the system is put in contact with a thermostat and with a barostat, the whole system being dealt with as a global Hamiltonian dynamical system. The coefficients of the fundamental form \(\delta Q-\delta W\) turn out to have such expressions that the closure of the form is manifest: this gives the first principle. A further step is performed by making use of time reversibility. This provides new expressions for the coefficients, such that the second principle in the form of Clausius is also manifest. Such coefficients are expressed in terms of time-autocorrelations of suitable dynamical variables, in a way analogous to that of fluctuation dissipation theory for equilibrium states. All these results are independent of the ergodicity properties of the global dynamical system.
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Notes
In equilibrium statistical mechanics, instead, U is a state function just by assumption, being defined as the canonical mean of the Hamiltonian of the system of interest, and \(\delta Q\) is then defined consequently, as \(\delta Q ={\mathrm {d}}U + \delta W\), with a previous natural definition of \(\delta W\).
Obviously one may think of more realistic models in which each of the two systems separately exchanges heat with the wall. But the example given here may suffice to show that exhanges of energy with no work performed, can be modelel in microscopic terms.
As usual it is understood that, for large systems, the interaction Hamiltonian, although playing an essential dynamical role in allowing for energy exchanges between the two subsystems, can nevertheless be neglected in the computations of mean values of the quantities of interest (such as the energy of a subsystem) with respect to a given probability measure. Indeed such quantities are proportional to the volume of the system of interest, whereas the interaction terms are proportional to the area of the basis of the cylinder.
Equivalence of the two procedures is immediately checked. Indeed, writing mean values in terms of a probability density \(\rho\), for the mean value \(\langle F\rangle\) of F at time t one has with the first procedure
$$\langle F\rangle =\int F(z) \rho (t,z) {\mathrm {d}}z,$$where \(\rho\) evolves according to the Liouville theorem, i.e., with \(\rho (t,z)=\rho (0, \varPhi ^{-t}z)\). On the other hand the Jacobian determinant of \(z'=\varPhi ^{-t}z\) has value 1, so that, by changing variables (and calling again z the new variable \(z'\)) one has the dual form
$$\langle F\rangle =\int F(\varPhi ^t z) \rho (0,z) {\mathrm {d}}z\equiv \int F_t(z) \rho (0,z) {\mathrm {d}}z.$$We are assuming that neglecting the interaction term in the measure introduces only very small errors in computing the mean values of the quantities of interest.
A larger number of internal parameters may be required in more general cases, as for example in the case of a nonhomogeneous fluid.
In order to neglect the interaction Hamiltonian in computing averages one requires to assume not only (as usual) that the system is large, but also that motions are considered in which the piston moves very slowly.
This is the reason why relations (33) hold if one considers \(\varDelta F\) and not just F.
The positiveness of the heat capacity was proven above, making use of the assumption that the marginal measure for the environment be of Gibbs type. However, if one looks at the proof one realizes that the result also holds if any assumption is made which guarantees that the partial derivatives of the density \(\rho\) have negative definite sign. For example, a sufficient condition is that, in formula (31) for the density, instead of the exponential \(\exp ( -\beta H)\) there appears a factor \(f(\beta H)\), with any function f having the property \(f'\le 0\).
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Carati, A., Maiocchi, A. & Galgani, L. Statistical thermodynamics for metaequilibrium or metastable states. Meccanica 52, 1295–1307 (2017). https://doi.org/10.1007/s11012-016-0490-3
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DOI: https://doi.org/10.1007/s11012-016-0490-3