[p. 2467l, §1]

Let me begin by recalling the definition of the generalized order of a transition [3, 4]
as well as some of the mathematical requirements of thermodynamics [18, 19, 20, 21, 22].
The energy function must be a single-valued,
convex, monotonically increasing, and almost everywhere
differentiable function which is homogeneous of degree
and has the coordinates , entropy, , volume, and particle number.
Classically the state variables satisfy
, , , and ,
while for quantum systems and must also be bounded

[p. 2467r, §1]

from below.
These conditions are both necessary and
sufficient for thermodynamic stability.

[p. 2467r, §2]

The classification of phase transitions is usually discussed
in terms of the free-energy density or the pressure [1, 3, 4]
because other thermodynamic potentials are continuous
for large classes of interactions [1, 2, 23].
The pressure is the conjugate convex function [24]
to the energy density as a function of
entropy density and particle-number density
according to

(2.1) |

where denotes the chemical potential and the temperature. The existence of phase transitions requires the thermodynamic limit to be taken appropriately. Consider a thermodynamic process parametrized by such that corresponds to a critical point. The classification scheme for phase transitions [3, 4] is based on the fractional derivatives [25]

[p. 2467l, §2]

(2.2) |

when denotes the singular part of the pressure and is the regular part. In Ehrenfestâs original classification scheme [26] a phase transition was defined to be of order iff

(2.3) |

for where and denotes the Heaviside step function defined as for and for . Equation (2.3) expresses a finite jump discontinuity in the th-order derivative of the pressure.

[p. 2467l, §3]

In [3] and [4] the classification scheme of Ehrenfest
was generalized by extending the order from integers to real numbers.
A phase transition was defined to be of order iff

(2.4) |

which is sufficiently general to allow confluent logarithmic singularities. Note that the order will in general depend upon the particular choice of thermodynamic process . A phase transition of order implies that behaves asymptotically like a power function (of index ) upon approach to the critical point [3, 4]. This observation relates the order of the transition to the critical exponents as

(2.5a) | |||

(2.5b) |

where denotes the thermal order and , the order along the direction of the field conjugate to the order parameter.

[p. 2467r, §3]

In Eq. (2.5) and are the
thermodynamic fluctuation exponents for the energy density
and the order-parameter density in Fisherâs notation [6, 27],
where denotes the equation-of-state exponent and the specific-heat exponent.

[p. 2467r, §4]

In this paper a mathematically more refined
classification scheme will be introduced based on the observation
that the function in Ehrenfestâs scheme (2.3)
is a slowly varying function [28] of .
A function is called slowly varying at infinity if it is
real valued, positive, and measurable on for some , and if

(2.6) |

for all . A function is called slowly varying at zero if is slowly varying at infinity [28, 29]. The function in (2.2) is slowly varying for as well as for . Therefore in this paper a phase transition is defined to be of order iff

(2.7) |

for all . This means that varies slowly as a function of for . The generalized order in this refined classification scheme is the same as in the scheme (2.4) because every slowly varying function has the property that and for all .

[p. 2467r, §5]

In the refined classification scheme (2.7) each phase
transition is classified by its generalized left
and right

[p. 2468l, §0]

orders and functions
which are slowly varying at the critical point.
The classification scheme also allows to distinguish differences
between transitions having the same order.
The two-dimensional Ising model is of second order
while the mean-field theory will be classed as second order
where denotes the Heaviside step function defined above.

[p. 2468l, §1]

Phase transitions of order occupy a special place
in the thermodynamic classification scheme because they are self-conjugate
under Legendre transformation as will be shown next.
Consider a thermal phase transition of order for in
where and is constant.
Then behaves as

(2.8) |

for where is slowly varying for . Define a slowly varying function through

(2.9) |

It is a standard result in the theory of slowly varying functions that for the conjugate convex function behaves as

(2.10) |

for where is given by

(2.11) |

and is the slowly varying function conjugate to [29]. For every slowly varying at zero there exists a conjugate slowly varying function which is defined such that

(2.12) | |||

(2.13) | |||

(2.14) |

is asymptotically unique in the sense that if there exists another slowly varying function with the properties (2.12)-(2.14) then for . Thus to every phase transition of order in the pressure there corresponds a conjugate transition in the energy density which is of order with given by (2.11) and related to as to in (2.9). Phase transitions of order are self-conjugate in the sense that . Phase transitions of order are conjugate to transitions of order and represent a special limiting situation.

[p. 2468l, §2]

This section turns to the question posed in the Introduction
whether phase transitions of order are thermodynamically permissible.
Consider for a thermodynamic process
in which the density is kept constant
and which crosses a critical point at .
If the phase transition at is of order
then

[p. 2468r, §2]

has the form

(2.15) |

where denotes the regular part and varies slowly near . Consequently any phase transition with and order violates the requirement of convexity for or the condition (for ), and is thus forbidden by the laws of thermodynamics. This appears to restrict thermodynamically admissible transitions to the range .

[p. 2468r, §3]

Although the restrictions on the thermodynamic state
variables require a finite entropy or energy density, i. e.,
or , the laws of thermodynamics
do not require , i. e., finiteness for the critical point.
In fact the simplest solid-fluid phase diagrams in the plane are consistent
with a critical point at terminating the solid-fluid coexistence.
To exhibit the theoretical possibility of such infinite entropy density transitions
it suffices to consider an explicit example which is compatible
with the mathematical requirements specified in the previous section.
Such an example is given by the following single-valued, continuous,
and differentiable energy-density function

(2.16) |

where and . Clearly and and thus is convex and monotonically increasing. fulfills all requirements for the energy density of a thermodynamically stable system. Note that exhibits transitions of order at . Moreover the thermodynamic system described by Eq. (2.16) has the curious property that the set of possible temperatures is restricted to the range

(2.17) |

The pressure obtained from Eqs. (2.1) and (2.16) reads as

(2.18) |

and it again exhibits the restricted temperature range. The pressure (2.18) has transitions of order at and , respectively. More generally transitions of order in are related to transitions of order

(2.19) |

in [5]. Note that now while .

[p. 2468r, §4]

The simple example (2.16) demonstrates that thermodynamics
allows two fundamentally different types of phase transitions:
On the one hand traditional phase transitions of order
and on the other hand unusual phase transitions of order
for which the set of possible equilibrium temperatures
appears to be restricted to a subset of the absolute temperature scale.
The interest in this observation derives from the fact
that equilibrium thermodynamics formally admits transitions
whose presence would restrict its own applicability in the
sense that the limiting critical temperatures and
cannot be reached in any quasistatic thermodynamic process.
A quasistatic process is a sequence of state changes

[p. 2469l, §0]

which proceeds
infinitely slowly compared to the time scale for the establishment of equilibrium.
This raises the question of whether the identification
of the absolute temperature scale with the ideal-gas temperature
scale remains valid when transitions are present.
In such systems plays the role of absolute zero
and that of .
In [5] it was suggested to circumvent the self-limitation
to a finite temperature range through multivalued thermodynamic potentials
and phase transitions of order were called nonequilibrium phase transitions
because transitions between different sheets cannot occur quasistatically.
The present paper, however, restricts all thermodynamic functions to
remain single valued.
In order to avoid confusion with standard literature
usage of the terminus “nonequilibrium phase transitions”
I will use instead the word anequilibrium phase transition from now on.

[p. 2469l, §1]

The entropy density
derived from (2.18) diverges to as .
Therefore the third law implies the existence
of another special temperature defined by the condition

(2.20) |

of vanishing entropy density. Because the third law is of quantum-mechanical origin the temperature is expected to be the minimal temperature for quantum systems while is the minimal temperature for classical systems. Clearly is always fulfilled.