IV Temperature Renormalization

[p. 2472l, §2]
The presence of anequilibrium transitions in a statistical-mechanical system S implies strong fluctuations near the transition point. In fact at the transition point the fluctuations become so strong that a canonical or thermodynamical description of the system becomes impossible because the ensemble-averaged energy or entropy diverges in the infinite system. The underlying microscopic dynamics, however, remains well defined in

[p. 2472r, §2]
terms of a classical or quantum-mechanical microscopic Hamiltonian. The total energy of the system remains defined and conserved and the system can be described microcanonically. If the system S undergoes anequilibrium transitions at TminS and TmaxS then heating or cooling across Tmin or Tmax cannot occur quasistatically and the system must fall out of equilibrium when attempting it.

[p. 2472r, §3]
Consider now the usual setup for the canonical ensemble in which S is weakly coupled to a reservoir R. What happens if the reservoir itself undergoes anequilibrium transitions at TminR and TmaxR? Clearly, the combined system R∪S has a Hamiltonian description and can always be treated in the microcanonical ensemble. But what happens to a canonical description? One expects that the canonical description should remain applicable as long as TminR is very small and TmaxR very large, i. e., for TminR/T≪1≪TmaxR/T, while the temperature dependence of the results should become modified otherwise. The temperature dependence of the results of canonical calculations enters through the Lagrange parameter β⁢T for the average energy. The Lagrange parameter β appearing in the canonical and the grand canonical ensembles is a universal (i. e., S independent) function of absolute temperature T and at the same time a property of the reservoir R. In fact if the reservoir R consists of a large number of weakly interacting subsystems (e. g., particles) then β-1 is proportional to the limiting energy per subsystem of R [30]. Because β is related to the energy density of the reservoir unusual temperature dependence must be expected whenever the reservoir substance itself shows anequilibrium transitions. If the reservoir is described quantum mechanically then T0R defined by Eq. (2.20) will be the lowest temperature of the reservoir corresponding to the ground state of the reservoir Hamiltonian. For classical reservoirs TminR will be the lowest temperature. If the reservoir has no anequilibrium transitions, i. e., if T0R=0 and TmaxR=∞ then the temperature dependence must have the usual universal form,

If, however, T0R>0 and TmaxR<∞ the form (4.1) cannot be correct. The energy per subsystem of the reservoir is proportional to β-1 and it diverges as T approaches Tmax. Thus for finite TmaxR<∞ one must have

and this contradicts (4.1) because 1/kB⁢TmaxR>0. Similarly

also violates (4.1) because T0R>TminR>0. To satisfy the relations (4.2) and (4.3) the temperature T and the parameter β must in general become renormalized into

whenever the reservoir undergoes anequilibrium transitions. This suggests that canonical averages O of an

[p. 2473l, §0]
observable O will in general depend on temperature through (4.4) and (4.5). For classical reservoirs T0R in (4.4) has to be replaced by TminR. Equations (4.4) and (4.5) are general predictions for the temperature dependence of canonical averages in systems with anequilibrium transitions resulting from the requirement of consistency for the interpretation of β. These general results are corroborated by the explicit solution (3.24) for the Gaussian model.

[p. 2473l, §1]
The main result expressed in (4.4) and (4.5) is the fact that the temperature dependence of canonical averages depends on the nature of the reservoir with which the system is equilibrated whenever the reservoir undergoes phase transitions of order less than 1. Note that in every theoretical evaluation of a canonical partition sum it is implicitly assumed that the system can be equilibrated with a reservoir R such as an ideal gas with T0R and TmaxR=∞ not undergoing anequilibrium transitions. This implicit assumption need not be fulfilled in experiment. In fact experimentally the reservoir is very often of the same material as the system itself because the system under study is part of a much larger (ideally infinite) sample. In that case T0R=T0S and TmaxR=TminS and the temperature renormalization (4.4) must be expected to become relevant as T approaches T0S.