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arXiv:cond-mat/0601088

Is the entropy at the liquid-gas critical point of pure fluids proportional to a master dimensionless constant ?

Yves Garrabos

Published Jan 5, 2006• 2 citations
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Abstract

From a minimal set made of four scale factors defined at the liquid-gas critical point of a pure fluid, and one adjustable parameter which accounts for particle quantum effects, we demonstrate here a master singular behavior of the correlation length for the one-component fluid subclass, using an asymptotic scale dilatation of the physical fields. Such master behavior observed within the preasymtotic domain is in conformity with the renormalized $Φ\_{d = 3}^{4}$ Field Theory predictions at large correlation length scale of the fluctuating order parameter, for the complete universality class of the symmetrical uniaxial 3D-Ising-like systems. The following consequences are discussed: (i) A comparison between the critical state of pure fluids and the zero-temperature state leads to an intuitive analogy with the (Nerst) third law of thermo- dynamics, which authorizes specific master form for hyperscaling within the subclass of pure fluids; (ii) A master constant value of the non-dimensional critical entropy can exist for all the pure fluids at the short-ranged lengthscale of the molecular interaction. From this latter hypothesis, we show that the needed four scale factors are the four preferred directions ex- pressing complete thermodynamic (linear) continuity crossing the liquid-gas critical point on the (pressure, volume, temperature) phase surface.

DOI
10.48550/arxiv.cond-mat/0601088

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From a minimal set made of four scale factors defined at the liquid-gas critical point of a pure fluid, and one adjustable parameter which accounts for particle quantum effects, we demonstrate here a master singular behavior of the correlation length for the one-component fluid subclass, using an asymptotic scale dilatation of the physical fields. Such master behavior observed within the preasymtotic domain is in conformity with the renormalized Φ_d=34Φ\_{d = 3}^{4}Φ_d=34 Field Theory predictions at large correlation length scale of the fluctuating order parameter, for the complete universality class of the symmetrical uniaxial 3D-Ising-like systems. The following consequences are discussed: (i) A comparison between the critical state of pure fluids and the zero-temperature state leads to an intuitive analogy with the (Nerst) third law of thermo- dynamics, which authorizes specific master form for hyperscaling within the subclass of pure fluids; (ii) A master constant value of the non-dimensional critical entropy can exist for all the pure fluids at the short-ranged lengthscale of the molecular interaction. From this latter hypothesis, we show that the needed four scale factors are the four preferred directions ex- pressing complete thermodynamic (linear) continuity crossing the liquid-gas critical point on the (pressure, volume, temperature) phase surface.
Nonuniversal scale transformations of the physical fields are extended to pure quantum fluids and used to calculate the susceptibility, the specific heat, and the order parameter density along the critical isochore of near its liquid-vapor critical point. Within the so-called preasymptotic domain, where the Wegner expansion restricted to the first term of confluent corrections to scaling is expected to be valid, the results are in agreement with the experimental measurements and recent predictions, either based on the minimal-substraction renormalization and the massive renormalization schemes within phi (4)(d = 3) (n = 1)the model, or based on the crossover parametric equation of state for Ising-like systems.
The critical behavior of simple single-component fluids near the liquid-vapor critical point (and of other systems in the same universality class) is examined via an asymmetric spin Hamiltonian in Landau-Ginzburg-Wilson form. The leading important odd terms to be added to an Ising-type system are a five-spin interaction $\ensuremath{\int}\ensuremath{\phi}^{5}(x)$ and a nonlocal cubic interaction $\ensuremath{-}\ensuremath{\int}\ensuremath{\phi}^{2}\ensuremath{\nabla}^{2}\ensuremath{\phi}$. These two operators can be combined into two different eigenoperators with distinct physical interpretations. One combination can be shown to be exactly equivalent to the mixing of field variables (e.g., chemical potential and temperature) in the temperaturelike variable of an Ising or Ising-type system. This mixing has been used phenomenologically, found in certain models, and justified on general geometric grounds. This is the first proof that all asymmetric models, assuming universality and barring accidents, will have such mixing; this implies the universal occurrence of a ${t}^{1\ensuremath{-}a}$ fluid diameter. The second combination is treated by renormalization-group methods and its contributions to the Helmholtz free energy, magnetic equation of state, and correlation length are calculated to $O({\ensuremath{\epsilon}}^{2}), \ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. There are several novel features of this interaction which persist to all orders in perturbation theory. In addition to the expected renormalized M5{M}^{5}M5 contribution to the free energy, there are also terms linear in hhh (the magnetic field) and MMM which could be described as nonanalytic fluctuation-induced shifts in the order parameter and field. These might appear experimentally as apparent shifts in the background terms for hhh and MMM.
A Landau-Ginzburg-Wilson model is derived for a single-component fluid whose particles interact via a two-body potential. The effective Hamiltonian contains both even and odd powers in the order parameter (local fluid density). We study the effect of quintic interactions absent in the Ising model, and characterized by a new exponent ${\ensuremath{\varphi}}_{5}$, on various singular thermodynamic functions near the critical point. Mean-field, scaling, and renormalization-group (in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions) theories are used to evaluate the non-Ising effects due to ${\ensuremath{\varphi}}_{5}$. One of the main results of the asymmetry in this model is that it produces the leading singularity of the liquid-vapor coexistence diameter near the critical point: ${|t|}^{\ensuremath{\beta}\ensuremath{-}{\ensuremath{\varphi}}_{5}}$, where $\ensuremath{\beta}$ is the usual order-parameter exponent and ttt is the reduced tem$\stackrel{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}{\mathrm{p}}$erature. The singularity ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ ($\ensuremath{\alpha}$ being the specific heat exponent) predicted by earlier phenomenological theories is not present in this theory. The problems associated with observing the non-Ising effects are briefly discussed.
The asymptotic and preasymptotic critical behavior in fluids, mixtures, and uniaxial magnets is believed to be described by an effective straight phi(4) scalar field theory with suitable, nonuniversal, coupling constants. The critical parameters as well as the extent of crossovers and corrections to the leading critical behavior in physical systems, crucially depends on the choice of these couplings. Here we propose a new method for deriving the effective scalar field theory appropriate to a microscopic model in this universality class. Use is made of the hierarchical reference theory, which implements the basic ideas of Wilson momentum space renormalization group to microscopic Hamiltonians. The effective low-energy field theory is then analyzed by the minimal subtraction scheme of Schloms and Dohm. We discuss the application of this method to the three-dimensional Ising model and to the liquid-vapor phase transition. We make comparison with high-temperature expansion results and with experimental data for rare gas.
The crossover from ideal-Bose-gas behavior to nonclassical, XY-like criticality in a dilute, interacting Bose fluid is considered in detail with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}$(T), below the transition. A general discussion of crossover scaling in constrained systems shows that ${\ensuremath{\rho}}_{s}$ and other temperature-dependent thermodynamic functions at controlled overall density can be expressed in terms of universal scaling functions. The finite-temperature Green's function formalism is used to calculate the associated linear scaling fields and the crossover exponent exactly for all dimensionalities d>2, scaling predictions, including logarithmic singularities just at d=3, being checked via second-order computations. A mapping of the Bose problem onto the classical s4{s}^{4}s4 spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio m∗{m}^{\mathrm{*}}m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio a∗{a}^{\mathrm{*}}a∗/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.
The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from Tc{T}_{c}Tc​, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near Tc{T}_{c}Tc​, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.