## A Refined thermodynamic classification scheme

[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 US,V,N must be a single-valued,
convex, monotonically increasing, and almost everywhere
differentiable function which is homogeneous of degree 1
and has the coordinates S, entropy, V, volume, and N particle number.
Classically the state variables satisfy
0≤V<∞, 0≤N<∞, -∞<S<∞, and -∞<U<∞,
while for quantum systems S and U 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 p [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 us,ρ=US/V,1,N/V/V as a function of
entropy density s=S/V and particle-number density
ρ=N/V according to

pT,μ=sups,ρμρ+Ts-us,ρ, | | (2.1) |

where μ denotes the chemical potential and T the temperature.
The existence of phase transitions requires the
thermodynamic limit V→∞ to be taken appropriately.
Consider a thermodynamic process C:R→R2,σ↦Tσ,μσ
parametrized by σ such that T(σ=0)=Tc,μ(σ=0)=μc
corresponds to a critical point.
The classification scheme for phase transitions [3, 4]
is based on the fractional derivatives [25]

[p. 2467l, §2]

IC,q;σ=dqpsngTσ,μσdσq=limN→∞Γ-q-1σN-q∑N-1j=0Γj-qΓj+1psngTσ-jσ4,μσ-jσN, | | (2.2) |

when psng denotes the singular part of the pressure
p=preg+psng and preg is the regular part.
In Ehrenfestâs original classification scheme [26]
a phase transition was defined to be of order n∈N iff

for σ≈0 where A,B∈R and θσ denotes the Heaviside
step function defined as θσ=1 for σ>0 and θσ=0 for σ<0.
Equation (2.3) expresses a finite jump discontinuity
in the nth-order derivative of the pressure.

[p. 2467l, §3]

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

λ±C=supq∈R:limσ→0±IC,q;σ<∞ | | (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 C.
A phase transition of order λ implies that f
behaves asymptotically like a power function (of index λ)
upon approach to the critical point [3, 4].
This observation relates the order of the transition to the critical exponents as

λE | =2-αE=2-α, | | (2.5a) |

λΨ | =2-αΨ=1+1/δ, | | (2.5b) |

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

[p. 2467r, §3]

In Eq. (2.5) αE=α and αΨ=1-1/δ are the
thermodynamic fluctuation exponents for the energy density E
and the order-parameter density Ψ in Fisherâs notation [6, 27],
where δ denotes the equation-of-state exponent and α 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 IC,n;σ in Ehrenfestâs scheme (2.3)
is a slowly varying function [28] of σ.
A function Λx is called slowly varying at infinity if it is
real valued, positive, and measurable on A,∞ for some A>0, and if

for all b>0.
A function Λx is called slowly varying at zero
if Λ1/x is slowly varying at infinity [28, 29].
The function IC,n;σ in (2.2) is slowly varying
for σ→0+ as well as for σ→0-.
Therefore in this paper a phase transition is defined to be of order λ± iff

limσ→±∞IC,λ±;b/σIC,λ±;1/σ=1 | | (2.7) |

for all b>0.
This means that IC,λ±;σ varies slowly
as a function of σ for σ→0±.
The generalized order in this refined classification scheme
is the same as in the scheme (2.4)
because every slowly varying function Λx has the
property that limx→0x-εΛx=∞
and limx→0xεΛx=0 for all ε>0.

[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 λ± and functions Λ±
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 λ,Λ=2,log
while the mean-field theory will be classed as second order
λ,Λ=2,θ where θ denotes the Heaviside step function defined above.

[p. 2468l, §1]

Phase transitions of order λ=2 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 λ,Λ for τ→0+ in pτ,μ
where τ=T-Tc/Tc and μ=μ0 is constant.
Then pτ behaves as

for T→Tc+ where Λτ is slowly varying for τ→0+.
Define a slowly varying function Lx through

It is a standard result in the theory of slowly varying functions
that for λ>1 the conjugate convex function
uσ=supττσ-pτ behaves as

uσ∼1λ*σλ*L*λ*-1/λ*σλ* | | (2.10) |

for σ→0+ where λ*>1 is given by

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

| limx→0LxL*xLx=1, | | (2.12) |

| limx→0L*xLxL*x=1, | | (2.13) |

| L**x∼Lx for x→0. | | (2.14) |

L*x is asymptotically unique in the sense that if there
exists another slowly varying function L′x with the
properties (2.12)-(2.14) then L′x∼L*x for x→0.
Thus to every phase transition of order λ,Λ
in the pressure there corresponds a conjugate transition
in the energy density which is of order λ*,Λ*
with λ* given by (2.11)
and Λ* related to L* as Λ to L in (2.9).
Phase transitions of order λ=2 are self-conjugate in the sense that λ=λ*.
Phase transitions of order λ=1 are conjugate to transitions
of order λ*=∞ and represent a special limiting situation.

## B Anequilibrium phase transitions

[p. 2468l, §2]

This section turns to the question posed in the Introduction
whether phase transitions of order λ<1 are thermodynamically permissible.
Consider us,ρ for a thermodynamic process
in which the density ρ=N/V is kept constant
and which crosses a critical point at sc.
If the phase transition at sc is of order λ=λ+=λ-
then us

[p. 2468r, §2]

has the form

us=uregs+u±ss-scλ, | | (2.15) |

where ureg denotes the regular part and u±s varies slowly near sc.
Consequently any phase transition with sc<∞
and order λ<1 violates the requirement of convexity for u
or the condition u<∞ (for λ<0),
and is thus forbidden by the laws of thermodynamics.
This appears to restrict thermodynamically admissible transitions to the range λ≥1.

[p. 2468r, §3]

Although the restrictions on the thermodynamic state
variables require a finite entropy or energy density, i. e.,
s<∞ or u<∞, the laws of thermodynamics
do not require sc<∞, i. e., finiteness for the critical point.
In fact the simplest solid-fluid phase diagrams in the s,v plane are consistent
with a critical point at sc=∞ 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

us=as+bs2+c21/2, | | (2.16) |

where a,b,c>0 and a>b.
Clearly Ts=∂u/∂s>0 and ∂2u/∂s2
and thus us is convex and monotonically increasing.
us fulfills all requirements for the energy density
of a thermodynamically stable system.
Note that us exhibits transitions of order λu±=1 at sc=±∞.
Moreover the thermodynamic system described by Eq. (2.16)
has the curious property that the set of possible temperatures
is restricted to the range

a-b=Tmin<T<Tmax=a+b. | | (2.17) |

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

and it again exhibits the restricted temperature range.
The pressure (2.18) has transitions of order
λp±=12 at Tmin and Tmax, respectively.
More generally transitions of order λu>0 in u
are related to transitions of order

in p [5].
Note that now 0<λp<1 while 0<λu<∞.

[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 λp±≥1
and on the other hand unusual phase transitions of order 0<λp±<1
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 Tmin and Tmax
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 λ<1 transitions are present.
In such systems Tmin plays the role of absolute zero
and Tmax that of T=∞.
In [5] it was suggested to circumvent the self-limitation
to a finite temperature range through multivalued thermodynamic potentials
and phase transitions of order λ<1 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 sT=∂p/∂Tμ
derived from (2.18) diverges to -∞ as T→Tmin+.
Therefore the third law implies the existence
of another special temperature T0 defined by the condition

of vanishing entropy density.
Because the third law is of quantum-mechanical origin
the temperature T0 is expected to be the minimal temperature
for quantum systems while Tmin is the minimal temperature for classical systems.
Clearly T0>Tmin is always fulfilled.