Hidden Variables in the Critical Region*B Y MARSHALL FiXMAKChemistry Dept., Yale University, New Haven, Connecticut, U.S.A.Receiued 19th December, 1966A gas in the one-phase region near the critical point exhibits unusually large fluctuations ofdensity in volume elements much larger in linear dimension than are molecular diameters. Thesefluctuations in density give rise to forces and energy fluxes which may be treated by essentiallymacroscopic equations of thermodynamics and hydrodynamics, because the dominant wavelengthsare so large in a Fourier decomposition of the density fluctuation. The anomalous thermal con-ductivity and shear viscosity of a gas near the critical point have been calculated, and comparedwith experiments on CO,. The comparison of thermal conductivities is satisfactory, but thecomparison of shear viscosities is inconclusive, because of the smallness of the predicted effect andthe difficulty of its measurement.1.INTRODUCTIONIn a variety of relaxation problems, an unexpected dissipation of energy is explainedthrough the existence of an uncontrolled or " hidden " variable, such as an intra-molecular mode of energy, whose probability distribution is unable to remain inequilibrium with the varying hydrodynamic variables. The anomalous ultrasonicabsorption of gases near the critical point has been explained on this basis,l* withthe hidden variables taken to be the spectrum of long-wave-length density fluctuations.The density fluctuations appear in an anomalous contribution to the entropy, fromwhich a complex, dynamic heat capacity was derived and used to calculate the soundabsorption.In the previous work the anomalous entropy varied in time because ofa uniformly varying temperature, but here we consider a non-uniform variation oftemperature and from the variation in entropy derive a formula for the anomalousheat flux. The anomalous shear viscosity of the gas is also calculated, by techniquesmathematically similar to those used for the shear viscosity of the binary mixturein the critical region (although the physical bases are different). The thermalconductivity calculation may significantly be compared with experiment, but theanomalous gas viscosity is calculated to be so small, and the discrepancies betweendifferent experiments are so large, that no useful comparison can be made.Zwanzig and Mountain have also calculated the anomalous thermal conductivityand shear viscosity of a gas.Apparently, their results for the thermal conductivityare qualitatively similar to ours, but they predict no anomaly in the viscosity. Theirmethod of calculation differs from ours in two respects. First, they calculated thetransport coefficients from correlation function expressions (as did K a ~ a s a k i , ~ inhis calculations on binary mixtures). Secondly, they retained explicitly in theirequations of motion an intermolecular potential. The use of correlation functionsmakes unnecessary any search for a coupling between the motion of the fluid and thedriving force for shear flow or thermal conduction. We have thought it worthwhileto find this coupling, as something interesting in itself and also to provide an insight* supported in part by the Public Health Service, GM 13556, and the Alfred P.Sloan Foundation.7M . FIXMAN 71into non-linear transport processes. Regarding the intermolecular potential,Zwanzig and Mountain used a long-range potential to justify some of the manipula-tions. We use an alternative justification which consist in the restriction of densityvariations to those describable by long-wave-length Fourier components. Thepotential itself is presumed to be short range and to manifest itself only implicitlyin the various thermodynamic derivatives that are present. The quasi-thermodynamicforms of our equation facilitates a numerical comparison with experimental results.Only one parameter occurs which is microscopic in nature,6 the short-range correlationlength 1.In principle, 1 may be determined from light scattering, but in practice Iis here determined for COz from a previous comparison of heat capacity predictionswith experiment.The following remarks are essentially a summary of methods and results, thedetails of which must be published elsewhere for lack of space here.2. EQUATIONS OF MOTIONWe assume that at each point a specific energy and specific entropy are known,and that the specific entropy is a function of the local density and temperature, whilethe specific energy contains one ordinary part, a function of the specific entropy andmass density, and another part proportional to the square of the density gradient.ThusI-Je = eL+eI, (2.3)e, = a I Vp 1 ’.eL is the ordinary specific energy, while e, becomes significant in the critical region,not because a is supposed to become large, but because the second derivitive of eLat constant s becomes small near the critical point, and vanishes as the reciprocalcompressibility.For a given state of density fluctuation, the force acting on any unit element ofmass is presumed to be given bydu/dt = p - l V * a0+F, (2.5)whereq) = 2q*(Vu)”Yrn C2.6)and ro is a hypothetical shear viscosity which the fluid would have if density fluctua-tions were not allowed.The force which arises because of the density fluctuationsis designated F in eqn.(2.5).The force F is calculated on the basis of a mechanical variational principle. Theposition ro of any element of mass is taken to be a function of time and the positionw of the element in an initial reference configuration. The positions w are takento be stochastic variables, and an implicit average over their possible values is requiredwhen macroscopic quantities are calculated :7ro = r&(w,t72 HIDDEN VARIABLES I N THE CRITICAL REGIONThe mechanical variational principle is that6E = - F Grodm,dm = p(r,t)dr, swhere Gr, is the variation in the path of a mass element, the variation being at aconstant value of the entropy of the mass element. A straightforward calculationyieldspF = V * G,,, (2.9)(2.10) Gp = -p1+2a[p 1 vp I "p2v2p]1-2ap(vp)(vp).The last term on the right-hand side of eqn.(2.10) is of utmost significance attwo points in the analysis. First, with the consistent neglect of third-order fluctua-tions, and the consistent interpretation of p as an average over a small volume element(with dimensions much larger than the range of the intermolecular potential, butmuch smaller than the range of correlation in the critical region), an ensembleaverage of eqn. (2.5) gives- qo(e,eZ + eZeJ + 2ap lim d 2G(1)(r)/dxdz. (2.11)r-tOIn this equation, i: is at the rate of shear of the fluid in laminar flow, G(I)(r) is theperturbation in the radial correlation function evaluated to first order in i. Thelimiting process in eqn. (2.11) is not taken towards a mathematical zero, but to avalue of r much smaller than the range of the correlation.The second point wherethe last term in eqn. (2.10) plays a major role is in a thermal conduction experiment,where this term provides the coupling between the uniform temperature gradientand the density fluctuations. We derive below an expression for the anomalousthermal conductivity.3. HEAT CONDUCTIONFor the anomalous specific entropy, Botch and Fixman derived an expressionequivalent to(6s) = (*)s:uPi 4<(6P) 2> (3.1)where the subscripts on the right-hand side designate the second derivative of thespecific entropy with respect to the specific volume. Interpretation of p as anaverage over a small volume element led to a transformation of the right-hand sideto an integral over the long wave-length components of the radial correlation function.Then, a differential equation for the radial correlation function, the one introducedbelow, allowed a dynamic heat capacity to be calculated.However, there is anotherway in which the anomalous specific entropy can vary with time, and that is by heatconduction. In the absence of heat sources, an anomalous heat flux can be relatedto (6s) bypoToa(ss>/at = -v . q', (3.2)and eqn. (3.1) and (3.2) together giveq' = T ~ S , , ~ ; ~ Iim u(r),r-rO(3.3)where n(r) is the mean fluid velocity at a point r due to the combined effects of amolecule fixed in location at the origin, and the presence of a temperature gradientM. FIXMAN 73We have calculated u(r) for a fluid in the critical region, subject to a uniformtemperature gradient independent of time.This velocity must be determined fromthe modified Navier-Stokes equation, eqn. (2.5) together with eqn. (2.9) and (2.10)for the forces. The linearized equation of mass conservation shows that the diver-gence of u must vanish in the steady state; that is, u can be written as the curl of avector potential. Therefore any part of eqn. (2.5) which can be written as the gradientof a scalar potential will not contribute to the desired solution. The rejection of suchterms givesq0V2u = 2p&x(ap/aT),(VT)V2Go(r), (3 -4)G"(r) = a e-Kr/r, (3.5)where Go is the equilibrium radial correlation function and K is related to the thermo-dynamic parameters byK~ = -pv/(2ap:), (3.6)or, at the critical density, by Debye's relation,I C ~ = 6( T - Tc)/12Tc.Eqn.(3.4) still includes some scalar potential contributions to u, but these may berejected and a thermal conductivity determined from eqn. (3.3). The result for theanomalous conductivity A' isA' = -(3)Toaq, 1(ap/aT),(a2p/aTa In U ) K - l. (3-7)4. SHEAR VISCOSITYThe fluctuation formula for the radial correlation function,(MrIFjP(r2)) = P m l ~ r 2 ) (4.1)and eqn. (2.5) linearized in the density fluctuations, together with the ordinary thermalconduction equation gives a result, previously derived by Botch and Fixman,hdG(r)/dt = h(aG/at + [u(r) - u(O)] * VG) = V2[x2G - 2VG], (4.2)where ;lo is, like yo, a hypothetical quantity, the thermal conductivity in the absenceof long wavelength density fluctuations.Like yo, Ro must be determined by extra-polation from outside the critical region. The previous expression for h was too largeby a factor of two, due to the neglect of the fact that two time-dependent densityfluctuations enter the formula (4.1) for G. A first-order solution of eqn. (4.2) forthe perturbation induced by simple shear flow is straightforward, and gives onsubstitution into eqn. (2.1 1) a formula for the anomalous part of the shear viscosity :q' = aTop+/(40Ao~). (4.4)A virtually identical formula, with different interpretation of the coefficients, has beenderived for the shear viscosity of a binary liquid mixture by severalThe present method may also be used to reproduce the previous results.* *5.COMPARISON WITH EXPERIMENTFor COz in the critical region all of the thermodynamic and transport coefficientsrequired in eqn. (3.7) and (4.4) are available. The value of I was previously estimate74 HIDDEN VARIABLES I N THE CRITICAL REGIONto be 4.5& and this value has been used here. The numerical predictions are, atthe critical density :y’ = 38( T - Tc)-* micropoise,2.’ = 2-6 x 10-4(T- Tc)-* cal/deg. cin sec.(5.1)(5.2)The predicted y’ is some five times smaller than indicated by the data of Michelset a2.,l0 measured in a capillary viscometer, and some five times larger than indicatedby the data of Kestin et a2.,l1 measured in a rotating disc viscometer. The latterexperimental anomalies were increased by factors of about three in sample correctionsfor density gradients.Eqn. (5.2) has been compared with Sengers’ data,I2 the total conductivity beingtaken to be ;1 = Lo +A’, with A,, = cal/deg. cm sec. Over a 40°C intervalabove T,, the calculated and observed conductivities agree within 10 %, althoughthe observed results show a slightly sharper dependence on T than given by eqn. (5.2).1 W. Botch and M. Fixman, J. Chem. Physics, 1965,42, 196.2 W. Botch and M. Fixman, J. Chem. Physics, 1965, 42, 199.3 M. Fixman, Adv. Chem. Physics, 1963, 6, 175.4 R. Zwanzig and R. D. Mountain, private communication.5 K. Kawasaki, Physic. Rev,, 1966, 150, 291.6 In principle even this parameter may be determined from macroscopic observations on inhomo-7 The explicit computation of averages proceeds in terms of correlation functions which are the8 M. Fixman, J. Chem. Physics, 1962, 36, 310.9 J. M. Deutsch and R. Zwanzig, preprint.10 A. Michels, A. Botzen and W. Schuurman, Physica, 1957, 23, 95.11 J. Kestin, J. H. Whitelaw and T. F. Zien, Physica, 1964, 30, 161.12 J. V. Sengers, Thesis (University of Amsterdam, 1962).geneous systems.solutions of differential equations