[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 βT 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/kBTmaxR>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

T′ | =T-T0R, | | (4.4) |

β′ | =β-βTmaxR, | | (4.5) |

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.