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IV Temperature Renormalization

[p. 2472l, §2]
The presence of anequilibrium transitions in a statistical-mechanical system \mathfrak{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 \mathfrak{S} undergoes anequilibrium transitions at T_{{\mathrm{min}}}^{\mathfrak{S}} and T_{{\mathrm{max}}}^{\mathfrak{S}} then heating or cooling across T_{{\mathrm{min}}} or T_{{\mathrm{max}}} 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 \mathfrak{S} is weakly coupled to a reservoir \mathfrak{R}. What happens if the reservoir itself undergoes anequilibrium transitions at T_{{\mathrm{min}}}^{\mathfrak{R}} and T_{{\mathrm{max}}}^{\mathfrak{R}}? Clearly, the combined system \mathfrak{R}\cup\mathfrak{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 T_{{\mathrm{min}}}^{\mathfrak{R}} is very small and T_{{\mathrm{max}}}^{\mathfrak{R}} very large, i. e., for T_{{\mathrm{min}}}^{\mathfrak{R}}/T\ll 1\ll T_{{\mathrm{max}}}^{\mathfrak{R}}/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 \beta(T) for the average energy. The Lagrange parameter \beta appearing in the canonical and the grand canonical ensembles is a universal (i. e., \mathfrak{S} independent) function of absolute temperature T and at the same time a property of the reservoir \mathfrak{R}. In fact if the reservoir \mathfrak{R} consists of a large number of weakly interacting subsystems (e. g., particles) then \beta^{{-1}} is proportional to the limiting energy per subsystem of \mathfrak{R} [30]. Because \beta 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 T^{\mathfrak{R}}_{0} 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 T_{{\mathrm{min}}}^{\mathfrak{R}} will be the lowest temperature. If the reservoir has no anequilibrium transitions, i. e., if T^{\mathfrak{R}}_{0}=0 and T_{{\mathrm{max}}}^{\mathfrak{R}}=\infty then the temperature dependence must have the usual universal form,

\beta(T)=\frac{1}{k_{B}T}\,. (4.1)

If, however, T^{\mathfrak{R}}_{0}>0 and T_{{\mathrm{max}}}^{\mathfrak{R}}<\infty the form (4.1) cannot be correct. The energy per subsystem of the reservoir is proportional to \beta^{{-1}} and it diverges as T approaches T_{{\mathrm{max}}}. Thus for finite T_{{\mathrm{max}}}^{\mathfrak{R}}<\infty one must have

\beta(T_{{\mathrm{max}}}^{\mathfrak{R}})=0 (4.2)

and this contradicts (4.1) because 1/(k_{B}T_{{\mathrm{max}}}^{\mathfrak{R}})>0. Similarly

\beta^{{-1}}(T^{\mathfrak{R}}_{0})=0 (4.3)

also violates (4.1) because T^{\mathfrak{R}}_{0}>T_{{\mathrm{min}}}^{\mathfrak{R}}>0. To satisfy the relations (4.2) and (4.3) the temperature T and the parameter \beta must in general become renormalized into

\displaystyle{}^{\prime}T \displaystyle=T-T^{\mathfrak{R}}_{0}, (4.4)
\displaystyle{}^{\prime}\beta \displaystyle=\beta-\beta(T_{{\mathrm{max}}}^{\mathfrak{R}}), (4.5)

whenever the reservoir undergoes anequilibrium transitions. This suggests that canonical averages \langle\mathcal{O}\rangle of an

[p. 2473l, §0]
observable \mathcal{O} will in general depend on temperature through (4.4) and (4.5). For classical reservoirs T^{\mathfrak{R}}_{0} in (4.4) has to be replaced by T_{{\mathrm{min}}}^{\mathfrak{R}}. 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 \beta. 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 \mathfrak{R} such as an ideal gas with T^{\mathfrak{R}}_{0} and T_{{\mathrm{max}}}^{\mathfrak{R}}=\infty 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 T^{\mathfrak{R}}_{0}=T^{\mathfrak{S}}_{0} and T_{{\mathrm{max}}}^{\mathfrak{R}}=T_{{\mathrm{min}}}^{\mathfrak{S}} and the temperature renormalization (4.4) must be expected to become relevant as T approaches T^{\mathfrak{S}}_{0}.