aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Van Der Waals Revisited
Klaus B¨arwinkel a , ∗ a Universit¨at Osnabr¨uck, Fachbereich Physik, D-49069 Osnabr¨uck, Germany
J¨urgen Schnack b b Universit¨at Bielefeld, Fakult¨at f¨ur Physik, Postfach 100131, D-33501 Bielefeld,Germany
Abstract
The van-der-Waals version of the second virial coefficient is not far from being exactif the model parameters are appropriately chosen. It is shown how the van-der-Waalsresemblance originates from the interplay of thermal averaging and superposition ofscattering phase shift contributions. The derivation of the two parameters from thequantum virial coefficient reveals a fermion-boson symmetry in non-ideal quantumgases. Numerical details are worked out for the Helium quantum gases.
Key words: van der Waals model, Kinetic theory, Quasi-particle methods,fermion-boson symmetry
PACS:
Occupation number statistics for N = n Ω non-interacting distinguishablequantum particles (Boltzmann statistics) in a volume Ω yields the entropydensity functional s = k B Ω Z d p ρ ( ~p ) ν ~p (1 − ln ν ~p ) (1)where ν ~p is the average occupation number of a single-particle energy eigen-state. These eigenstates are enumerated by corresponding points ~p in momen-tum space, the density of which is ρ ( ~p ). k B denotes Boltzmann’s constant. Forthe more general case of indistinguishable quasi-particles one may consider ∗ Tel: ++49 541 969-2694; fax: -2670; Email: [email protected] ∗∗ Tel: ++49 521 106-6193; fax -6455; Email: [email protected]
Preprint submitted to Physica A 1 December 2018 q. (22) of [1], where the Fermi-Bose functional is stated. Nevertheless, thissimplifies to our eqs. (1) or (5) if ν ~p ≪ ρ ( ~p ) = (Ω − N b ) / (2 π ~ ) . (2)Here the “single-particle volume” b is our first van-der-Waals parameter. Weconsider elementary cells of volume v eℓ = Ω ρ ( ~p ) = (2 π ~ ) − nb (3)in six-dimensional phase space ( µ -space) and the one-particle distributionfunction f ( ~p ) = ν ~p /v eℓ . (4)The entropy density s = k B Z d p f ( ~p ) (1 − ln( f ( ~p ) v eℓ )) (5)is then to be maximized as a functional of f subject to the constraints of fixedparticle density n = Z d p f ( ~p ) (6)and fixed energy density u = Z d p f ( ~p ) ǫ ~p . (7)According to the second van-der-Waals ansatz, each particle has its classi-cal kinetic energy and is in the potential field of interaction with the otherparticles, i.e. ǫ ~p = p m − an , (8)where a is the second van-der-Waals parameter. Clearly, the treatment ofcorrelations is incomplete in this model.The energy density now becomes u = Z d p p m f ( ~p ) − an . (9)From the principle of maximum entropy s and using the temperature definition T = [( ∂s/∂u ) n ] − (10)one finds f to be the Maxwellian f ( ~p ) = n (2 πmk B T ) / exp ( − p mk B T ) . (11)2his leads to s = nk B − ln nλ − nb !! (12)with the thermal wavelength λ = (2 π ~ ) / q πmk B T (13)and to u = 32 nk B T − an . (14)Then the pressure formula P eq = − n ∂u∂n ! s (15)results in the van-der-Waals equation of state: P eq = nk B T − nb − an . (16)Because of the insufficient treatment of two-particle correlations, this formulawill allow a quantitatively satisfying fit for real systems only if nλ is suf-ficiently small. Consequently, the van-der-Waals version of the second virialcoefficient B ( T ) is a good approximation if the temperature is not too low: B ( T ) ≈ B vdW ( T ) = b − ak B T . (17)The appropriate choice of the parameters a and b is dealt with in the followingsections. In particular, it could very well be that both parameters depend onthe fermionic or bosonic nature of the interacting particles. It will turn outthat this is not the case. The van-der-Waals model can be introduced via corresponding approxima-tions to the radial distribution function. To this end consider first the averagepotential energy of N = n Ω mutually interacting classical particles: W pot = 12 < X i = j V ( | ~r i − ~r j | > . (18)With the radial distribution function defined by g ( | ~r ′ − ~r ′′ | > = n − < X i = j δ ( ~r ′ − ~r i ) δ ( ~r ′′ − ~r j ) > (19)3ne gets W pot = N n Z d r g ( r ) V ( r ) . (20)Now g ( r ) has its density expansion g ( r ) = g ( r ) + ng ( r ) + n g ( r ) + . . . . (21)The energy of a single classical particle is therefore – apart from higher-orderdensity contributions – given by eq. (8) with a = − Z d r V ( r ) g ( r ) . (22)In view of the classical limit g ,cl ( r ) = exp ( − V ( r ) k B T ) (23)with a Lennard-Jones potential (see below, eq. (35)), there will be a cut-offradius r ∗ such that the approximation g ( r ) = , for r < r ∗ , for r > r ∗ (24)with a constant r ∗ is applicable in a considerable range of temperature. Thiseventually fixes the parameter a as a = − Z r ≥ r ∗ d r V ( r ) . (25)On the other hand, both the parameters a and b may be introduced by firstexpressing the second virial coefficient in terms of g [2], B ( T ) = 12 Z d r (1 − g ( r )) (26)and employing the closer approximation g ( r ) = , for r < r ∗ − V ( r ) k B T , for r > r ∗ . (27)Comparison with the van-der-Waals version of B ( T ) (eq. (17)) then yields a as given by eq. (25) and b = 2 π r ∗ . (28)The closer approximation of g – if inserted in eq. (22) – would cause a slightdependence of a on temperature. This must be negligible for the van-der-Waalsmodel to be acceptable. 4n the next section, formulae (17), (25), and (28) will be substantiated bynumerical analysis of the exact quantum mechanical virial coefficient, and thecut-off radius r ∗ will be determined. The exact theory [3] for boson or fermion gases with their two-particle interac-tion having, possibly, bound state energies E i gives the second virial coefficientas a sum of four terms: B ( T ) = ∓ − / λ − / λ X i e − E i /k B T + ≪ G ± ≫ + ≪ F ± ≫ k B T (29)with the quantities ≪ G ± ≫ and ≪ F ± ≫ being explained below (eqs. (37),(38)) and with the upper (lower) sign valid for bosons (fermions). The dou-ble bracket is our notation for the thermal average of momentum dependentfunctions, e.g. ≪ Φ ≫ = Z ∞ d p w ( p ) Φ( p ) , (30)with the thermal weight function w ( p ) = 4 π ( πmk B T ) − / p exp ( − p mk B T ) . (31)Evidently, formula (17) is justified if ≪ F ± ≫ and ≪ G ± ≫ prove to bepractically constant, i.e. ≪ F ± ≫ = a and ≪ G ± ≫ = b in a relevant range oftemperature where the other contributions are negligible. This is indeed thecase as will be shown in the following. Moreover, a and b thus defined willexhibit a new kind of fermion-boson symmetry in that they are independentof the specific quantum statistics.The functions F ± and G ± may be expressed in terms of the properly (anti-)symmetrized momentum representation of the two-particle operator T ( z ) = V − V H − z V , T ′ ( z ) = ddz T ( z ) , (32)with H = H kin + V being the Hamiltonian of relative motion:5 ± ( p ) = −
12 (2 π ~ ) ℜ ( < ~p |T ± ( E p + iǫ > | ~p > ) , E p = p m , (33) G ± ( p ) = π π ~ ) Z d q δ ( E p − E q ) · (34) ·ℑ (cid:16) < ~p |T ± ( E q + iǫ ) | ~q >< ~q |T ′± ( E q + iǫ ) | ~p > (cid:17) . Our graphics Fig. 1 for F ± and G ± rely on the numerical evaluation for bosons( He atoms) interacting via a Lennard-Jones potential lacking bound states[4] and fermions (same mass and same interaction as He): V ( r ) = 4 V "(cid:18) σr (cid:19) − (cid:18) σr (cid:19) ; V /k B = 10 . , σ = 2 . . (35)The (anti-) symmetrized T -matrix is – up to a multiplicative constant – noth-ing else but the scattering amplitude f ± ( p, θ ) = − π m ~ < ~p |T ± ( E p + iǫ ) | ~q > , | ~p | = | ~q | , ~p · ~q = p cos θ . (36)An alternative representation for F ± and G ± can therefore be given in termsof scattering phase shifts δ ℓ which complies with the Beth-Uhlenbeck resultfor B ( T ) [5,6]: F ± = 4 π ~ m f ± ( p,
0) = 4 π ~ m ~ p X ℓ ± (2 ℓ + 1) sin 2 δ ℓ ( p ) (37) G ± ( p ) = − π ~ ~ p X ℓ ± (2 ℓ + 1) sin [ δ ℓ ( p )] ∂δ ℓ ( p ) ∂p . (38)The summation runs over even ℓ for bosons and odd ℓ for fermions.A remark on the units and dependencies of F ± , G ± will be fitting here. Let p = ~ k and choose ˜ r = r/σ as the dimensionless radial coordinate. Then thedimensionless version of the radial wave equation with eigenvalue E p becomes u ′′ ℓ (˜ r ) + n Re (˜ r − − ˜ r − ) + ℓ ( ℓ + 1)˜ r − o u (˜ r ) = ( kσ ) u ℓ (˜ r ) (39)where Re = 4 V mσ ~ (40)is the “Reynolds number”. Given the appropriate behavior of u ℓ for ˜ r → r → ∞ exhibits the phase shift δ ℓ : u ℓ (˜ r ) −→ sin( kσ ˜ r − ℓ π + δ ℓ ) . (41)Thus the only dependency of δ ℓ is on Re and kσ . Consequently, with somefunctions ϕ ± = ϕ ± (Re , kσ ) and γ ± = γ ± (Re , kσ ) F ± = 4 π ~ σm · ϕ ± (Re , kσ ) = 16 πV σ ϕ ± (Re , kσ )Re , (42)6 ig. 1. Functions F ± and G ± for bosons ( He atoms) interacting via theLennard-Jones potential (35) and fermions with the same mass and same inter-action. The thermal weight function is given for two temperatures. The horizontalline on the l.h.s. marks the value of a as given by (59) and on the r.h.s. the valueof b as given by (58). G ± = 4 πσ γ ± (Re , kσ ) . (43)In each of our graphics the boson and the fermion function refer to the samevalue of Re. With the potential data of eq. (35) and with m the mass of Hewe obtain Re ≈ . w = w ( p ) (eq. (31)) assumes its maximum at p max = ~ k max with k max σ = · , ! / · (cid:18) T K (cid:19) / (44)and it is easily seen that above T = 100 K the weight functions samples ratherlarge values of kσ ( kσ ' kσ Fig. 1 shows that both F + and F − oscillate about a common approximately constant value. Thisreflects the nearly hard-core likeness of the repulsive part of the Lennard-Jones potential. Let us therefore consider, for comparison, a pure hard-corerepulsion with radius σ .In this special case tan δ ℓ = j ℓ ( kσ ) /y ℓ ( kσ ) (45)holds, with j ℓ and y ℓ denoting the spherical Bessel functions j ℓ ( z ) = z n − z ddz ! n sin zz , y ℓ ( z ) = z n ddz ! n cos zz . (46)Invoking sin 2 δ ℓ = 2 tan δ ℓ δ ℓ (47)7nd the asymptotic behaviorlim z →∞ (cid:16) j ℓ ( z ) + y ℓ ( z ) (cid:17) = 1 /z , (48)one arrives at( F hc + − F hc − ) asy = 4 π ~ σ km ∞ X ℓ =0 ( − ℓ +1 (2 ℓ + 1) j ℓ ( kσ ) y ℓ ( kσ ) . (49)Now – as a marginal case of formula 10.1.46 in [7] – ∞ X ℓ =0 ( − ℓ +1 (2 ℓ + 1) j ℓ ( z ) y ℓ ( z ) = cos 2 z σ (50)and consequently ( F hc − − F hc + ) asy = 2 π ~ σm cos(2 kσ ) . (51)This is the asymptotic ( kσ → ∞ ) result for the hard-core system. It is com-pared with ( F − − F + ) for the Lennard-Jones system in Fig. 2. The behavioris very similar both in terms of amplitude and frequency. A slight decrease of σ with increasing k in formula (51) would still improve the agreement. Thisreflects the fact that the Lennard-Jones potential appears the softer the higherthe particles’ energy is. Fig. 2. Comparison of ( F − − F + ) for the Lennard-Jones system (solid curve) withthe asymptotic result for the hard core system (dashed curve). As for G ± , an oscillation about a common constant value is once again seen,compare the r.h.s. of Fig. 1. In contrast to the case of F ± , however, the oscil-latory amplitude of the difference is clearly decreasing. Not surprisingly, thisfeature can again be derived analytically for a pure hard-core repulsion withradius σ . 8o this end, G ± (eq. (38)) is first rewritten as G ± = 2 πσ ( kσ ) ∂∂ ( kσ ) X ℓ ± (2 ℓ + 1) (cid:18)
12 sin 2 δ ℓ ( kσ ) − δ ℓ ( kσ ) (cid:19) (52)and then employed for the hard-core system. With the aid of eqs. (45), (47),and (48) the asymptotic ( kσ → ∞ ) tail of ( G hc + − G hc − ) is found to be( G hc + − G hc − ) asy = 2 πσ ∞ X ℓ =0 ( − ℓ (2 ℓ + 1) j ℓ ( kσ ) y ′ ℓ ( kσ ) . (53)After replacing (for kσ → ∞ ) y ′ ℓ by j ℓ , one can apply formula 10.1.51 in [7], ∞ X ℓ =0 ( − ℓ (2 ℓ + 1) j ℓ ( z ) = sin 2 z z , (54)and hence ( G hc + − G hc − ) asy = 2 πσ sin 2 kσ kσ . (55)The difference ( G + − G − ) for the Lennard-Jones system is compared with thisresult in Fig. 3. Again, the behavior is very similar both in terms of amplitudeand frequency, also the phase difference increases only slightly. Fig. 3. Comparison of ( G − − G + ) for the Lennard-Jones system (solid curve) withthe asymptotic result for the hard core system (dashed curve). Instead of eq. (27) a continuous ansatz for g ( r ) may be used: g ( r ) = , for r ≤ ∗ − exp (cid:16) − α r − r ∗ r ∗ (cid:17) + V ( r ) − V ( r ∗ ) exp( − α r − r ∗ r ∗ ) k B T , for r ≥ r ∗ (56)9ith the potential V ( r ) according to eq. (35). Equation (27) is reproduced for α → ∞ . The values of r ∗ /σ and α follow from the van-der-Waals condition b − ak B T = 12 Z d r (1 − g ( r )) (57)where b = 0 . πσ (58)and a = 4 π ~ σm = 16 π Re · V σ (59)is estimated in view of Fig. 1 (horizontal lines). Then r ∗ = x − / σ (60)where x is the solution of x = 3Re + 0 . x + 23 x − . x (61)with 0 . x > /
3. The parameter α follows from0 . x − / α + 2 α + 2 α . (62)Then a has the following representation in terms of the Reynolds number Reand the Lennard-Jones parameters V and σ : a = 4 πV σ (cid:18) x − x (cid:19) , (63)where x = x (Re) is that one among the solutions of eq. (61), which allowsa positive α in eq. (62). For Re= 22 . x = 1 .
88 and α = 25 . g ( r ) is close to its limit for α → ∞ , which isgiven in eq. (27). Summary
The summary of our elaboration is that for a large region of temperaturethe functions ≪ F ± ≫ = a and ≪ G ± ≫ = b may be nearly consideredas constants, i.e. not depending on temperature, see Fig. 1. Moreover, therespective kind of statistics (Bose-Einstein or Fermi-Dirac) does not matter.This constitutes a fermion-boson symmetry in non-ideal quantum gases. Forideal quantum gases such symmetries are known for about ten years [8,9].The slight residual dependence of a and b on temperature (cf. Fig. 1) reflectsthe fact that these are parameters not of an exact but of a model theory.10 cknowledgment A long-standing collaboration with our colleague and friend Heinz-J¨urgenSchmidt is gratefully acknowledged. This article is dedicated to him on theoccasion of his 60th birthday.
References [1] K. B¨arwinkel, J. Schnack, U. Thelker, Quasi-particle picture for monatomicgases, Physica A 262 (1999) 496.[2] K. B¨arwinkel, S. Großmann, Pair distribution function of moderately densequantum fluids, Z. Phys. 230 (1970) 141.[3] B. Baumgartl, Second and third virial coefficient of a quantum gas from 2-particlescattering amplitude, Z. Phys. 198 (1967) 148.[4] J. E. Kilpatrick, M. F. Kilpatrick, Discrete energy levels associated with theLennard-Jones potential, J. Chem. Phys. 19 (1951) 930.[5] G. Uhlenbeck, E. Beth, The quantum theory of the non-ideal gas. I. deviationsfrom the classical theory, Physica 3 (1936) 729.[6] G. Uhlenbeck, E. Beth, The quantum theory of the non-ideal gas. II. behaviourat low temperatures, Physica 4 (1937) 915.[7] M. Abramovitz, I. Stegun (Eds.), Handbook of Mathematical Functions, Dover,New York, 1973.[8] M. H. Lee, Equivalence of ideal gases in two dimensions and Landen’s relations,Phys. Rev. E 55 (1997) 1518.[9] H.-J. Schmidt, J. Schnack, Thermodynamic fermion-boson symmetry inharmonic oscillator potentials, Physica A 265 (1999) 584.[1] K. B¨arwinkel, J. Schnack, U. Thelker, Quasi-particle picture for monatomicgases, Physica A 262 (1999) 496.[2] K. B¨arwinkel, S. Großmann, Pair distribution function of moderately densequantum fluids, Z. Phys. 230 (1970) 141.[3] B. Baumgartl, Second and third virial coefficient of a quantum gas from 2-particlescattering amplitude, Z. Phys. 198 (1967) 148.[4] J. E. Kilpatrick, M. F. Kilpatrick, Discrete energy levels associated with theLennard-Jones potential, J. Chem. Phys. 19 (1951) 930.[5] G. Uhlenbeck, E. Beth, The quantum theory of the non-ideal gas. I. deviationsfrom the classical theory, Physica 3 (1936) 729.[6] G. Uhlenbeck, E. Beth, The quantum theory of the non-ideal gas. II. behaviourat low temperatures, Physica 4 (1937) 915.[7] M. Abramovitz, I. Stegun (Eds.), Handbook of Mathematical Functions, Dover,New York, 1973.[8] M. H. Lee, Equivalence of ideal gases in two dimensions and Landen’s relations,Phys. Rev. E 55 (1997) 1518.[9] H.-J. Schmidt, J. Schnack, Thermodynamic fermion-boson symmetry inharmonic oscillator potentials, Physica A 265 (1999) 584.