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II Thermodynamics

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 U(S,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\leq V<\infty, 0\leq N<\infty, -\infty<S<\infty, and -\infty<U<\infty, 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 u(s,\rho)=U(S/V,1,N/V)/V as a function of entropy density s=S/V and particle-number density \rho=N/V according to

p(T,\mu)=\sup _{{s,\rho}}[\mu\rho+Ts-u(s,\rho)], (2.1)

where \mu denotes the chemical potential and T the temperature. The existence of phase transitions requires the thermodynamic limit V\rightarrow\infty to be taken appropriately. Consider a thermodynamic process \mathscr C:\mathbb{R}\rightarrow\mathbb{R}^{2},\,\sigma\mapsto(T(\sigma),\mu(\sigma)) parametrized by \sigma such that T(\sigma=0)=T_{c},\,\mu(\sigma=0)=\mu _{c} corresponds to a critical point. The classification scheme for phase transitions [3, 4] is based on the fractional derivatives [25]

[p. 2467l, §2]

\mathscr I(\mathscr C,q;\sigma)=\frac{d^{q}p_{{\mathrm{sng}}}(T(\sigma),\mu(\sigma))}{d\sigma^{q}}=\lim _{{N\rightarrow\infty}}\Gamma(-q)^{{-1}}\left(\frac{\lvert\sigma\rvert}{N}\right)^{{-q}}\sum^{{N-1}}_{{j=0}}\frac{\Gamma(j-q)}{\Gamma(j+1)}p_{{\mathrm{sng}}}\left(T\left(\sigma-\frac{j\sigma}{4}\right),\mu\left(\sigma-\frac{j\sigma}{N}\right)\right), (2.2)

when p_{{\mathrm{sng}}} denotes the singular part of the pressure p=p_{{\mathrm{reg}}}+p_{{\mathrm{sng}}} and p_{{\mathrm{reg}}} is the regular part. In Ehrenfest’s original classification scheme [26] a phase transition was defined to be of order n\in\mathbb{N} iff

\mathscr I(\mathscr C,n;\sigma)\sim A\theta(\sigma)+B (2.3)

for \sigma\approx 0 where A,B\in\mathbb{R} and \theta(\sigma) denotes the Heaviside step function defined as \theta(\sigma)=1 for \sigma>0 and \theta(\sigma)=0 for \sigma<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 \lambda^{\pm}\in\mathbb{R} iff

\lambda^{\pm}(\mathscr C)=\sup\{ q\in\mathbb{R}:\lim _{{\sigma\rightarrow 0\pm}}\mathscr I(\mathscr C,q;\sigma)<\infty\} (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 \mathscr C. A phase transition of order \lambda implies that f behaves asymptotically like a power function (of index \lambda) upon approach to the critical point [3, 4]. This observation relates the order of the transition to the critical exponents as

\displaystyle\lambda E \displaystyle=2-\alpha E=2-\alpha, (2.5a)
\displaystyle\lambda _{\Psi} \displaystyle=2-\alpha _{\Psi}=1+1/\delta, (2.5b)

where \lambda E denotes the thermal order and \lambda _{\Psi}, the order along the direction of the field conjugate to the order parameter.

[p. 2467r, §3]
In Eq. (2.5) \alpha E=\alpha and \alpha _{\Psi}=1-1/\delta are the thermodynamic fluctuation exponents for the energy density \mathscr E and the order-parameter density \Psi in Fisher’s notation [6, 27], where \delta denotes the equation-of-state exponent and \alpha 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 \mathscr I(\mathscr C,n;\sigma) in Ehrenfest’s scheme (2.3) is a slowly varying function [28] of \sigma. A function \Lambda(x) is called slowly varying at infinity if it is real valued, positive, and measurable on [A,\infty) for some A>0, and if

\lim _{{x\rightarrow\infty}}\frac{\Lambda(bx)}{\Lambda(x)}=1 (2.6)

for all b>0. A function \Lambda(x) is called slowly varying at zero if \Lambda(1/x) is slowly varying at infinity [28, 29]. The function \mathscr I(\mathscr C,n;\sigma) in (2.2) is slowly varying for \sigma\rightarrow 0+ as well as for \sigma\rightarrow 0-. Therefore in this paper a phase transition is defined to be of order \lambda^{\pm} iff

\lim _{{\sigma\rightarrow\pm\infty}}\frac{\mathscr I(\mathscr C,\lambda^{\pm};b/\sigma)}{\mathscr I(\mathscr C,\lambda^{\pm};1/\sigma)}=1 (2.7)

for all b>0. This means that \mathscr I(\mathscr C,\lambda^{\pm};\sigma) varies slowly as a function of \sigma for \sigma\rightarrow 0\pm. The generalized order in this refined classification scheme is the same as in the scheme (2.4) because every slowly varying function \Lambda(x) has the property that \lim _{{x\rightarrow 0}}x^{{-\varepsilon}}\Lambda(x)=\infty and \lim _{{x\rightarrow 0}}x^{\varepsilon}\Lambda(x)=0 for all \varepsilon>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 \lambda^{\pm} and functions \Lambda^{\pm} 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 (\lambda,\Lambda)=(2,\log) while the mean-field theory will be classed as second order (\lambda,\Lambda)=(2,\theta) where \theta denotes the Heaviside step function defined above.

[p. 2468l, §1]
Phase transitions of order \lambda=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 (\lambda,\Lambda) for \tau\rightarrow 0^{+} in p(\tau,\mu) where \tau=(T-T_{c})/T_{c} and \mu=\mu _{0} is constant. Then p(\tau) behaves as

p(\tau)\sim\tau^{\lambda}\Lambda(\tau) (2.8)

for T\rightarrow T^{+}_{c} where \Lambda(\tau) is slowly varying for \tau\rightarrow 0^{+}. Define a slowly varying function L(x) through

\Lambda(x)=\frac{1}{\lambda}L^{{(\lambda-1)/\lambda}}(x^{\lambda}). (2.9)

It is a standard result in the theory of slowly varying functions that for \lambda>1 the conjugate convex function u(\sigma)=\sup _{\tau}[\tau\sigma-p(\tau)] behaves as

u(\sigma)\sim\frac{1}{\lambda^{*}}\sigma^{{\lambda^{*}}}L^{{*(\lambda^{*}-1)/\lambda^{*}}}(\sigma^{{\lambda^{*}}}) (2.10)

for \sigma\rightarrow 0^{+} where \lambda^{*}>1 is given by

\lambda^{*}=\frac{\lambda}{\lambda-1} (2.11)

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

\displaystyle\lim _{{x\rightarrow 0}}L(x)L^{*}(xL(x))=1, (2.12)
\displaystyle\lim _{{x\rightarrow 0}}L^{*}(x)L(xL^{*}(x))=1, (2.13)
\displaystyle L^{{**}}(x)\sim L(x)\text{ for }x\rightarrow 0. (2.14)

L^{*}(x) is asymptotically unique in the sense that if there exists another slowly varying function L^{\prime}(x) with the properties (2.12)-(2.14) then L^{\prime}(x)\sim L^{*}(x) for x\rightarrow 0. Thus to every phase transition of order (\lambda,\Lambda) in the pressure there corresponds a conjugate transition in the energy density which is of order (\lambda^{*},\Lambda^{*}) with \lambda^{*} given by (2.11) and \Lambda^{*} related to L^{*} as \Lambda to L in (2.9). Phase transitions of order \lambda=2 are self-conjugate in the sense that \lambda=\lambda^{*}. Phase transitions of order \lambda=1 are conjugate to transitions of order \lambda^{*}=\infty 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 \lambda<1 are thermodynamically permissible. Consider u(s,\rho) for a thermodynamic process in which the density \rho=N/V is kept constant and which crosses a critical point at s_{c}. If the phase transition at s_{c} is of order \lambda=\lambda^{+}=\lambda^{-} then u(s)

[p. 2468r, §2]
has the form

u(s)=u_{{\mathrm{reg}}}(s)+u^{\pm}(s)\lvert s-s_{c}\rvert^{\lambda}, (2.15)

where u_{{\mathrm{reg}}} denotes the regular part and u^{\pm}(s) varies slowly near s_{c}. Consequently any phase transition with s_{c}<\infty and order \lambda<1 violates the requirement of convexity for u or the condition u<\infty (for \lambda<0), and is thus forbidden by the laws of thermodynamics. This appears to restrict thermodynamically admissible transitions to the range \lambda\geq 1.

[p. 2468r, §3]
Although the restrictions on the thermodynamic state variables require a finite entropy or energy density, i. e., s<\infty or u<\infty, the laws of thermodynamics do not require s_{c}<\infty, 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 s_{c}=\infty 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

u(s)=as+b(s^{2}+c^{2})^{{1/2}}, (2.16)

where a,b,c>0 and a>b. Clearly T(s)=\partial u/\partial s>0 and \partial^{2}u/\partial s^{2} and thus u(s) is convex and monotonically increasing. u(s) fulfills all requirements for the energy density of a thermodynamically stable system. Note that u(s) exhibits transitions of order \lambda^{\pm}_{u}=1 at s_{c}=\pm\infty. 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=T_{{\mathrm{min}}}<T<T_{{\mathrm{max}}}=a+b. (2.17)

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

p(T)=\{ c^{2}[b^{2}-(T-a)^{2}]\}^{{1/2}} (2.18)

and it again exhibits the restricted temperature range. The pressure (2.18) has transitions of order \lambda^{\pm}_{p}=\tfrac{1}{2} at T_{{\mathrm{min}}} and T_{{\mathrm{max}}}, respectively. More generally transitions of order \lambda _{u}>0 in u are related to transitions of order

\lambda _{p}=\frac{\lambda _{u}}{\lambda _{u}+1} (2.19)

in p [5]. Note that now 0<\lambda _{p}<1 while 0<\lambda _{u}<\infty.

[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 \lambda^{\pm}_{p}\geq 1 and on the other hand unusual phase transitions of order 0<\lambda^{\pm}_{p}<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 T_{{\mathrm{min}}} and T_{{\mathrm{max}}} 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 \lambda<1 transitions are present. In such systems T_{{\mathrm{min}}} plays the role of absolute zero and T_{{\mathrm{max}}} that of T=\infty. In [5] it was suggested to circumvent the self-limitation to a finite temperature range through multivalued thermodynamic potentials and phase transitions of order \lambda<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 s(T)=(\partial p/\partial T)_{\mu} derived from (2.18) diverges to -\infty as T\rightarrow T_{{\mathrm{min}}}+. Therefore the third law implies the existence of another special temperature T_{0} defined by the condition

s(T_{0})=0 (2.20)

of vanishing entropy density. Because the third law is of quantum-mechanical origin the temperature T_{0} is expected to be the minimal temperature for quantum systems while T_{{\mathrm{min}}} is the minimal temperature for classical systems. Clearly T_{0}>T_{{\mathrm{min}}} is always fulfilled.