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Physical Review

Why is CV Less Singular than CP near the Critical Point?

Melville S. Green

Published Sep 5, 1969• 6 citations
Publisher

Abstract

The fact noted by Schofield that the specific heat at constant volume, albeit singular, is less singular than the specific heat at constant pressure implies an asymptotic relation between the two-, three-, and four-body distribution functions. The experimental background of this fact is discussed and a fluctuation formula which expresses ${C}_{V}$ in terms of two-, three-, and four-body correlation functions is derived. A heuristic explanation of the asymptotic properties of the distribution functions is given based on the fact that local fluctuations proportional to the critical eigenvector are overwhelmingly the most probable near the critical point. It is shown that if ${C}_{V}$ is indeed infinite there exists a second critical eigenvector linearly independent of the first. Some consequences of the existence of two critical eigenvectors are discussed, and a form for the short-range behavior of the distribution functions near the critical point is conjectured.

DOI
10.1103/physrev.185.176

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The fact noted by Schofield that the specific heat at constant volume, albeit singular, is less singular than the specific heat at constant pressure implies an asymptotic relation between the two-, three-, and four-body distribution functions. The experimental background of this fact is discussed and a fluctuation formula which expresses CV{C}_{V}CV​ in terms of two-, three-, and four-body correlation functions is derived. A heuristic explanation of the asymptotic properties of the distribution functions is given based on the fact that local fluctuations proportional to the critical eigenvector are overwhelmingly the most probable near the critical point. It is shown that if CV{C}_{V}CV​ is indeed infinite there exists a second critical eigenvector linearly independent of the first. Some consequences of the existence of two critical eigenvectors are discussed, and a form for the short-range behavior of the distribution functions near the critical point is conjectured.
There are two independent critical exponents that describe the behavior of systems near their critical point. However, at the critical point only the exponent η, which describes the decay of the correlation function, is usually discussed. We emphasize that there is a second independent exponent η′ that describes the decay of the fourth-order correlation function. The exponent η′ is related to the exponents determining the behavior of thermodynamic functions near criticality via a fluctuation-response equation for the specific heat. We also discuss a scaling law for η′.
A general relation between the critical specific heat above and below a second-order phase transition is derived. It is expressed in terms of the effective four-point coupling u(l)u(l)u(l) of the Ginzburg-Landau Hamiltonian and is valid in the entire region between asymptotic criticality and noncritical background behavior. Application to the $\ensuremath{\lambda}$ transition of 4He^{4}\mathrm{He}4He yields excellent agreement with the fixed-point value u∗=u(0){u}^{*}=u(0)u∗=u(0) predicted by high-order perturbation theory.
A simple generalization of the Griffiths and Wheeler postulates for the thermodynamic behavior near a critical point is introduced to describe fluctuations in local dynamical variables. It is proposed that in the classical Hilbert space of the dynamical variables there are only two linearly independent vectors, related to the shape of the coexistence surface and the critical surface, which contain critical fluctuations. It is shown that, as a consequence of the affine structure of the field space, the above assumption leads to the following asymptotic property for the correlation function ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})$ of any two local operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{b}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$: ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})={a}_{1}(q, \ensuremath{\omega}){b}_{1}(q, \ensuremath{\omega}){f}_{11}(q, \ensuremath{\omega})+{a}_{2}(q, \ensuremath{\omega}){b}_{2}(q, \ensuremath{\omega}){f}_{22}(q, \ensuremath{\omega})+{a}_{3}(q, \ensuremath{\omega}){b}_{3}(q, \ensuremath{\omega}){f}_{33}(q, \ensuremath{\omega})+\dots{}$Here qqq is the wave number, $\ensuremath{\omega}$ is a point in the field space, the coefficients ai{a}_{i}ai​ and bi{b}_{i}bi​ are "generally" well-behaved thermodynamic quantities and the fii{f}_{\mathrm{ii}}fii​, which are universal functions, have the following properties: ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is nondivergent if $i\ensuremath{\ge}3$, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{22}(q, \ensuremath{\omega})$ is weakly divergent, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is strongly divergent. A few typical examples are worked out to show that if this property is postulated, the Griffiths and Wheeler thermodynamic behavior is reproduced.
Using renormalization group arguments the author expands n-point correlation functions (for non-exceptional wavevectors) in expectation values of translational invariant short-range operators O i . He uses the fact that the Fourier components of our operators become negligible for wavevectors q large in comparison to the momentum cut-off. The correlation functions show the same non-analyticities at the critical point as the expectation values (O i ). The expansion coefficients are regular in the thermodynamic variables for q not=0. They can be expressed in terms of (a) functions which become singular at q=0 and yield the scaling behaviour, and (b) functions which are regular at q=0. The expansion coefficients of the two-point correlation function are sums of both types of functions.
On the basis of Fisher's method, singularity of specific heat near the transition point is studied for the Ising or Heisenberg model with an arbitrary value of spin and range of interaction. The following results are obtained for a reasonable distribution of zeros of canonical partition function. (a) If the specific heat has a singularity C+(T-Tc)-α above the transition point Tc, it should have a singularity C-(Tc-T)-α below Tc, C- being not necessarily equal to C+. (b) If the specific heat has a logarithmic singularity -Aln(T-Tc)+B+ above Tc, it should have a singularity -Aln(Tc-T)+B- below Tc, B- being not necessarily equal to B+. Some theoretical and experimental works so far reported are discussed in the light of these results.
The critical phenomenon of the second order phase transition is considered for temperatures above the critical point. A general phenomenological approach is developed. The generalized free energy F[{η q }: T ] is introduced. It is a functional of the Fourier component η q of the local order parameter η( r ). A general property of the correlation function <η q η - q > is established rigorously. The equilibrium free energy and the wave-number dependent susceptibility χ q are expressed in terms of the linked clusters, large clusters corresponding to highr order fluctuation of the order parameter. The general rule for calculating the contribution of each cluster is given. The rule is shown to be very simple. It is found that the specific heat is expressed as a sum of the temperature derivatives of χ q 's. As an example of the application of the linked cluster method, the susceptibility is calculated in the vicinity of the critical point by summing up the clusters that are most divergent when approaching the criticat point. The linked cluster expansion developed here could be a clue to the whole mystery of the singularity at the critical point.