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A generalized Navier–Stokes equation near the liquid-glass transition
9 April 2001
Physics Letters A 282 (2001) 59–71 www.elsevier.nl/locate/pla
A generalized Navier–Stokes equation near the liquid-glass transition Toyoyuki Kitamura Nagasaki Institute of Applied Science, Nagasaki 851-0193, Japan Received 19 September 2000; accepted 19 February 2001 Communicated by L.J. Sham
Abstract The correlation functions for density fluctuations yield the basic equations in fluids. The density fluctuations consists of intraband and interband density fluctuations which yield the conventional density and the divergence of the displacement of an atom in a localized potential, respectively. The current consists of the corresponding conventional current and displacement current. The basic equations combined with the continuity equation yield a generalized Navier–Stokes equation, which reduces to the Navier–Stokes equation at low frequencies or high temperatures and to the elastic equation at high frequencies or low temperatures. The transport coefficients, the relaxation times and the velocity of modes obey the Vogel–Fulcher law. 2001 Elsevier Science B.V. All rights reserved. PACS: 64.70.Pf; 46.35.+z Keywords: Quantum field theory; Glass transition; Vogel–Fulcher law; Viscoelasticity; Navier–Stokes equation
1. Introduction We have proposed a quantum field theory of transport and relaxation processes near the liquid-glass transition [1–6] using the two band model in the harmonic potential approximation [1–13]. Our idea is as follows: an atom stays temporarily in a harmonic potential and hops to a surrounding harmonic potential. The harmonic potentials are distributed randomly in space and magnitude. We consider the two levels; the ground state and the first excited states. The energy gap (∼ the eigenfrequency of the harmonic potential) plays a role as elasticity and the hopping matrices (∼ the band widths of atoms) as the kinetic motion. The density fluctuations consist of intraband and interband density fluctuations. The correlation functions of the intraband density fluctuations yield the structure of sound and diffusion, while those of the interband density fluctuations the structure of phonons and viscosity. The phonons are elastic waves and the sound is a collision wave. The hopping processes relate to the entropy due to intraband density fluctuations. The entropy manifests that the configurational average of a hopping matrix obeys the Vogel–Fulcher law. The purpose of the present Letter is to derive a generalized Navier–Stokes equation from the basic equations which come from the correlation functions of intraband and interband density fluctuations near the liquid-glass E-mail address: [email protected] (T. Kitamura). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 1 4 9 - 9
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T. Kitamura / Physics Letters A 282 (2001) 59–71
transition [14,15]. The intraband and interband density fluctuations correspond to the Fourier components of the conventional density and the divergence of the displacement of an atom in a localized potential, respectively. The current consists of the respective conventional current and displacement current. They constitute the continuity equation. The generalized Navier–Stokes equation reduces to the Navier–Stokes equation at low frequencies or high temperatures and to the elastic equation at high frequencies or low temperatures.
2. The basic formulation We start with the following Hamiltonian [1–13]: 1 H = d 3 x ψ † (x)h¯ 0 (−i∇)ψ(x) + d 3 x d 3 y n(x)V (x − y)n(y), 2
(2.1)
where ψ is an annihilation field operator of an atom, h¯ 0 (−i∇) is an energy operator of a free atom, n(x) is a density operator of the atom, n(x) = ψ † (x)ψ(x) and V is an interaction potential between atoms. If we introduce an annihilation operator of the atom in the harmonic potential at the site Rm , bmµ , we can write ψ(x) = (2.2) w˜ mµ (x − Rm )bmµ , µ = 0, 1, 2, 3. mµ
Here w˜ mµ is the wavefunction of the µth state at the site Rm . The Heisenberg equation for bmµ is given by 0 ∂ −h¯ bmµ = d 3 x w˜ mµ (x − Rm ) h ¯ (−i∇) + ΦRn (x) w˜ nν (x − Rn )bnν , ∂τ nν ΦRn (x) = d 3 y V (x − y)n(y),
(2.3) (2.4)
{Rn }
where {Rn } means summing the surrounding sites of the atom at Rn in n(y). The term ΦRn means that a potential is determined by the surrounding atoms of the site Rn . If we assume that the atom at the site Rn feels a harmonic potential, we obtain
h¯ 0 (−i∇) + ΦRn (x) w˜ nν (x − Rn ) = h¯ ω˜ nν w˜ nν (x − Rn ),
1 ΦRn (x) ∼ = Mωn2 (x − Rn )2 , 2
(2.5)
where ω˜ nµ and w˜ nµ are the eigenfrequency and the eigenfunction of the µth state at the site Rn ; µ = 0 for the ground state, ω˜ n0 = (3/2)ωn and µ = 1, 2, 3 for the first excited states, w˜ ni = (5/2)ωn , where ωn is an eigenfrequency at the site Rn . Hereafter, µ, ν mean 0, 1, 2, 3 and i, j mean 1, 2, 3. There is a relation between the ground state and the first excited state wavefunctions for the ith Cartesian coordinate: h¯ , wn1 (xi ) = −2ζn ∇i wn0 (xi ), ζn = (2.6) 2Mωn where ζn is the mean width of the zero point motion, M is the mass of an atom. The magnitude of potentials ΦRn is distributed randomly, so is that of the eigenfrequencies. At this stage, we can take the following model Hamiltonian in terms of localized operators: † † h¯ ω˜ mµ bmµ bmµ + Jmµnν bmµ bnν , H= (2.7) mµ
mµnν
T. Kitamura / Physics Letters A 282 (2001) 59–71
where the hopping matrix is given by Jmµnν = d 3 x w˜ mµ (x − Rm ) h¯ 0 (−i∇) + ΦRn (x) w˜ nν (x − Rn ).
61
(2.8)
Note that the interaction Hamiltonian is nonlinear through the term n(y) in ΦRn and the potential ΦRn permits an atom to hop only to a vacancy. Now we first establish the unperturbed Hamiltonian by taking the configurational average of the model Hamiltonian (2.7): † † h¯ ω˜ µ bmµ bmµ + Jµ (Rm − Rn )bmµ bnµ , H0 = H c = (2.9)
µm
ω˜ µ = ω˜ mµ c ,
µmn
Jµ (Rm − Rn ) = Jmµnµ c ,
(2.10)
where . . . = Tr e−β(H −µN) . . . / Tr e−β(H −µN) and the subscript c means the configurational average. The hopping between different states is neglected for simplicity. Next we establish an alternative model Hamiltonian in terms of extended operators. If we introduce the following extended operators: 1 ip·Rm e aµp , µ = 0, 1, 2, 3, bmµ = √ (2.11) N p we obtain H0 =
† aµp , h¯ µp aµp
(2.12)
µp
1 J˜µ (p) ≡ N
h ¯ µp = h¯ ω˜ µ + J˜µ (p),
d 3 R g(R)eip·R Jµ (R),
(2.13)
where g(R) is a pair distribution function. The field operator is written down as 1 ip·Rm e w˜ mµ (x − Rm )aµp . ψ(x) = √ N µpm
(2.14)
From Eq. (2.14), we obtain 2 † 2 w˜ mµ (x − Rm )bmµ bmµ − ζm Bm · ∇ w˜ m0 (x − Rm ) + · · · n(x) = m
µ
1 −iq·Rm 2 † † 2 = e w˜ mµ (x − Rm )ρdµq − ζm ∇i w˜ m0 (x − Rm )ρiq + ··· , N mq µ
(2.15)
i
where † † Bmi ≡ bmi bm0 + bmi b0i ,
† ρdµq =
p
† aµp aµp−q ,
† ρiq =
† † aip a0p−q + a0p aip−q .
(2.16)
p
† † and ρiq are intraband (diagonal) and interband (off-diagonal) density fluctuation operators, respectively. ρdµq Substitution of Eq. (2.15) into Eq. (2.1) yields 1 od 1 d † † † aµp + Vij (q)ρiq ρj q + Vµν (q)ρdµq ρdνq , h¯ µp aµp H= (2.17) 2N 2N µp µνq ij q
Vijod (q) ≡ NVi0,j 0 ,
d Vµν
≡ NVµµ,νν ,
(2.18)
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T. Kitamura / Physics Letters A 282 (2001) 59–71
1 −iq·(Rm−Rn ) Vµµ ,νν (q) = 2 e d 3 x d 3 y w˜ mµ (x − Rm )w˜ mµ (x − Rm )V (x − y) N m=n × w˜ nν (y − Rn )w˜ nν (y − Rn ),
(2.19)
where no cross term of the intraband and interband density operators appears, because of the symmetry of the Here it should be noted that since the unperturbed Hamiltonian H0 in Eq. (2.9) involves the term wavefunctions. d 3 x d 3 y n(x)V (x − y) n(y) c , the interaction Hamiltonian HI must exclude the term. Thus the prime on in Eq. (2.17) means that HI excludes one loop diagrams. First we are concerned with the model Hamiltonian (2.7) in terms of localized operators. The interaction Hamiltonian HI is given by † † bmµ + bnµ , HI = (2.20) h¯ ω˜ mµ − ω˜ µ bmµ h¯ Jmµnµ − Jµ (Rm − Rn ) bmµ m
mnµ
where we make the approximation that the term Jmµnµ is a random c-number. Using Eq. (2.11), we rewrite Eq. (2.20) as
1 † HI = aµp . h¯ ω˜ mµ − ω˜ µ e−i(p−p )·Rm + h¯ Jmµnµ − Jµ (Rm − Rn ) e−ip·Rm +ip ·Rn aµp N m mn pp (2.21) µν
Eqs. (2.20) and (2.21) are used to take into account the randomness. Elemental scattering processes, Uω due to µν the random eigenfrequencies of atoms and UJ due to the random hopping matrices, are shown in Fig. 1 and the corresponding terms are given by 1 ω˜ mµ ω˜ mν c − ω˜ µ ω˜ ν , 2 N m 1 µν UJ (q) = d 3 R g(R)e−iq·R Jmµnµ Jmνnν c − Jµ (R)Jν (R) R=Rm −Rn . N
Uωµν =
(2.22) (2.23)
Here the crosses on the diagrams means summing the scattering processes over random positions. Now we derive the Bethe–Salpeter equation. First, we introduce atom Green’s functions: 1 −iωn (τ −τ ) µ µ † Gpp (τ − τ ) ≡ − Tτ aµp (τ )aµp e Gpp (iωn ). (τ ) = β h¯ iω
(2.24)
n
µν
µν
Fig. 1. Elemental scattering processes, Uω due to the random eigenfrequencies of the harmonic potentials and UJ matrices.
due to the random hopping
T. Kitamura / Physics Letters A 282 (2001) 59–71
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We should not confuse the temperature frequency iωn with the eigenfrequency ωn . We derive the Bethe–Salpeter equation for configurationally averaged two Green’s functions: µ µν φpp (iωn , q) ≡ Gp,p (iωn )Gνp−q,p −q (iωn − iνn ) c .
(2.25)
We obtain the Bethe–Salpeter equation [1–6]:
µν µν µν Upp1 (q)φp1 p (iωn , q) , φpp (iωn , q) = Gµ (p)Gν (p − q) δpp +
(2.26)
p1
where Gµ (p) is configurational average of the Green’s function (2.24):
µ µ Gpp (iωn ) c = Gp (iωn )δpp = Gµ (p)δpp
(2.27)
and the abbreviations p = (p, iωn ), q = (q, iνn ) have been used. Note that p = (p , iωn ). Upp1 is the irreducible vertex part. If we introduce the self-energy parts of the Green’s function as µν
Gµ−1 (p) ≡ iωn − ˜µp − Σ µ (p),
˜µp = µp − µ,
(2.28)
we obtain
µν −iνn + ˜µp − ˜νp−q + /Σ µν (p, q) φ µν (p, q) = /Gµν (p, q) 1 + Upp1 (q)φ µν (p1 , q) ,
(2.29)
p1
/Σ µν (p, q) = Σ µ (p) − Σ ν (p − q), µν φpp (iωn , q). φ µν (p, q) ≡
/Gµν (p, q) = Gµ (p) − Gν (p − q), (2.30)
p
00 = U 0 , U ii = U 1 , U 0i = U i0 = U 01 , Hereafter, we consider isotropic scatterings. So we abbreviate Uω,J ω,J ω,J ω,J ω,J ω,J ω,J 00 0 ii 1 0i 01 i0 10 and φ = φ , φ = φ , φ = φ , φ = φ . Since the magnitude of the wavefunctions of the ground state µν is larger than that of the excited states, |Ji | > |J0 |, and Uω is independent of the states, we can expect UJ1 > 01 0 |UJ | > UJ and Uω0 = Uω1 = Uω01 .
3. Intraband and interband density fluctuations We give a brief survey of intraband and interband density fluctuations [1–6]. Hereafter we are concerned with the low temperature regime: β h¯ ω 1, β|J˜| 1, and β h¯ /τ0 1. First we consider the correlation functions for intraband density fluctuations. Since we confine ourselves to low temperatures, we are concerned with the correlation functions for the ground state: Fq (τ1 − τ2 ) ≡ −
1 1 −iνn (τ1 −τ2 ) † (τ2 ) c ≡ e F (q). Tτ ρd0q (τ1 )ρd0q β h¯ h¯ N iν
(3.1)
n
The model Hamiltonian is expressed in two alternative forms: Eq. (2.7) with Eq. (2.9) in terms of localized operators and Eq. (2.17) with Eq. (2.12) in terms of extended operators. We cannot apply both the interaction Hamiltonians to a calculation simultaneously. At this stage, we consider the model Hamiltonian (2.17) with (2.9), while the model Hamiltonian (2.7) with (2.9) is used to take into account the randomness for the self-energy parts
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T. Kitamura / Physics Letters A 282 (2001) 59–71
and vertex corrections. In the random phase approximation, we obtain d 1 1 (q) c F (q), P0 (q) + P0 (q) V00 h¯ h¯ 1 0 φ (p, q). P0 (q) ≡ ∓ β h¯ N p
F (q) =
The configurationally averaged diagonal potential function is given by d d V (q) ≡ V00 (q) c = d 3 R g(R)e−iq·R V00 (R), d V00 (R) ≡ d 3 x d 3 y w˜ 02 (x)V (x − y)w˜ 02 (y − R),
(3.2) (3.3)
(3.4) (3.5)
where w˜ 0 (x) is configurationally averaged. Then we obtain F (q) =
(1/h¯ )P0 (q) . 1 − (1/h¯ )V (q)P0 (q)
(3.6)
In calculating the vertex correction φ 0 , we make the following approximations: (1) we put Σ 0 (q0 ± iδ) ∼ = ∓i/2τ0 (q0 ) and we neglect the real part of the self-energy. We use i 0 2 ∼ f ˜0p . /G (p, q) = ∓(β h¯ ) q · ∇ ˜p − τ0 iωn
f (x) = df (x)/d(β h¯ x) = −f (x){1 ± f (x)}, f (x) = 1/(eβ h¯ x ∓ 1) and, since p f (˜0p ) ∼ = N , f (˜0p ) ∼ = −2 ∼ ∼ for bosons, f (˜0p ) = 0 for fermions and f (˜0p ) = −1 for the Boltzmann distribution. (2) We retain only the following terms in the Bathe–Salpeter equation: (3.7) φ 0 (p, q) ≡ φ, q · ∇p φ 0 (p, q) = q · . iωn ,p
iωn ,p
(3) We make the ladder approximation to the vertex part for the Green’s functions: 00 0 ∼ 0 Upp (q) = Uω (0) + UJ (0).
(3.8)
The corresponding self-energy part is given by 0 Σ 0 (p0 + iδ) ∼ G (p, p0 + iδ) = Uω0 (0) + UJ0 (0) p
= Uω0 (0) + UJ0 (0)
p
1 , p0 + iδ − ˜0p − Σ 0 (p0 + iδ)
(3.9)
which is illustrated in Fig. 2. Here we consider only the imaginary part of the self-energy, 1/2τ0 (p0 ) = Σ(p0 + iδ). We can expand ˜0p ∼ = |J0 |(1 − sin α/α), α = ap, a: the mean atomic distance. The dominant contribution of wave vectors to the density of states comes from the short wavelength regime, where ˜0p ∼ = = |J0 |. Thus we put p G0 (p, p0 + iδ) ∼ N/{p0 − |J0 | − Σ 0 (p0 + iδ)} in Eq. (3.9). Then we obtain 2 2 1 1 2 1 1 ∼ ∼ (3.10) − p0 − |J0 | = − p02 , = N Uω0 + UJ0 . = 2τ0 (p0 ) 2τ0 2τ0 2τ0
T. Kitamura / Physics Letters A 282 (2001) 59–71
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Fig. 2. The self-energy part corresponding to the ladder approximation to the vertex function is shown. The irreducible vertex function for the ladder approximation is shown in Fig. 1.
Thus we obtain P0 (q) ≡ −β h¯ f
ωq2 /(q0 + i/τ0 ) q0 − ωq2 /(q0
+ i/τ0 )
,
ωq2 ∼ =
q 2 va2 1 , (q · ∇0p )2 = N p 3
va =
1 ∂0p , N p ∂p
(3.11)
where we abbreviate q0 in 1/τ0 (q0 ), q0 + i/τ (q0) → i/τ0 for q0 τ0 1 and q0 + i/τ (q0 ) → q0 for q0 τ0 1. The denominator of Eq. (3.6) yields q0 −
2 ωsq
q0 + i/τ0
= 0,
ωq2 2 , ωsq = ωq2 1 + βV (q) = S(q)
(3.12)
where ωsq is a dispersion of sound, S(q) the static structure factor and va is the mean atomic velocity. For q0 τ0 1, we obtain the diffusion constant D = τ0 cs2 ,
cs2 =
va2 , 3S(0)
(3.13)
where cs is the sound velocity. Next we investigate the freezing point. Since limq0 →0 P0 (q) ∼ = f β h¯ and V (q) has the negative minimum value ∼ ˜ ˜ at K (K = 3π/2a for g(R) = ρδ(R − a), a: the mean atomic distance), the freezing point β0 is determined by
1 ˜ ˜ lim 1 − V K P0 K, q0 ∼ (3.14) = 1 − f β0 V K˜ = 0. q0 →0 h¯ The thermodynamical function due to the intraband density fluctuations, Ωd , is given by [5,6] Ωd =
2 1 iνn2 − ωsq 1 ln 2 − ln 1 − e−β h¯ ωq . 2β q iνn − ωq2 β q
The Contour integration of Eq. (3.15) gives 1 1 1 Ωd = ln 1 − e−β h¯ ωsq − ln 1 − e−β h¯ ωq − (h¯ ωsq − hω ln 1 − e−β h¯ ωq . ¯ q) + 2 q β q β q
(3.15)
(3.16)
The dominant contribution of the intraband density fluctuations to the thermodynamical function comes from ˜ where ln(1 − e−β h¯ ωsq ) − ln(1 − e−β h¯ ωq ) ∼ the regions q ∼ K, = (1/2) ln{1 − f βV (q)}, where we have used the ˜ condition β|J | 1. If we denote the number of q-states near K˜ by N0 , we obtain
T0 1 N0 1 ∼ ln 1 − + (h¯ ωsq − hω ln 1 − e−β h¯ ωsq , Ωd = (3.17) ¯ q) + 2 q 2β T β q
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T. Kitamura / Physics Letters A 282 (2001) 59–71
Table 1 The Vogel–Fulcher law of transport coefficients, relaxation times and velocity of modes. |J0 | ∝ exp{−E0 /(T − T0 )}, Uω0 is given in Eq. (3.35) Transport coefficient
Relaxation time
Velocity
D
νλ
τ0
τM
cs
cλ
Higher T
|J0 |
|J0 |−1
|J0 |−1
|J0 |−1
|J0 |
–
Lower T
J02
−1/2 Uω0
−1/2 Uω0
−1/2 Uω0
|J0 |
–
where the dominant contribution of the last term due to sound comes from the region q ∼ 0. The entropy due to the intraband density fluctuations is given by
1 h¯ ωsq ∂Ωd N0 kB T0 T0 ∼ − kB Sd ∼ + ln ln 1 − e−β h¯ ωsq + . =− =− ∂T V ,µ 2 T − T0 T − T0 T q eβ h¯ ωsq − 1 q (3.18) This thermodynamical meaning of the first term which we denote as Sv is as follows [16]: if we consider vacancies as a body in an external medium in a nonequilibrium state and the entropy Sv ≡ −Rmin /T as the total entropy of the body together with medium, Rmin is the minimum work which must be done from the external object when the total system becomes in an equilibrium state. On the other hand, the exponential of the entropy Sv is proportional to the hopping probability. Intraband density fluctuations represent the randomness of configuration of atoms. The number of vacancies is proportional to the magnitude of the randomness of configuration of atoms. Since the number of states due to the intraband density fluctuations is eSv / kB , the number of vacancies around an atom is proportional to eSv /NkB . Thus we put |Jµ | ∝ eSv /NkB ∼ = e−E0 /(T −T0 ) ,
E0 =
N0 T0 . 2N
(3.19)
Eq. (3.19) is nothing but the Vogel–Fulcher law. |J0 | obeys the Vogel–Fulcher law, but Uω0 is almost constant near the freezing point. Hereafter, we take |J0 | as the standard of the powers of the Vogel–Fulcher law. In Table 1, we show the Vogel–Fulcher law in transport coefficients, relaxation times and velocity of modes. Next we introduce the correlation functions for interband density fluctuations 1 1 −iνn (τ1 −τ2 ) Tτ ρiq (τ1 )ρj†q (τ2 ) c ≡ e Dij (q). Dij q (τ1 − τ2 ) ≡ − (3.20) h¯ N β h¯ iν n
At this stage, we consider the model Hamiltonian (2.17) in extended operators in a similar way to the case of intraband density fluctuations. Thus we make the random phase approximation: 1 1 Dij (q) = Q(q)δij + Q(q) (3.21) Vilod (q) c Dlj (q), h¯ h¯ l 1 10 Q(q) ≡ ∓ (3.22) φ (p, q) + φ 01 (p, q) . β h¯ N p Configurational average of the potential function is given by od h¯ 2 3 iq·R , Vij (R), ζM = Vij (q) c = ζM d R g(R)e 2Mω Vij (R) = d 3 x d 3 y ∇i w˜ 02 (x)V (x − y)∇j w˜ 02 (y − R), √ where w˜ 0 (x) = w˜ m0 (x) c , ζM = h¯ /2Mωm c and ω = ωm c .
(3.23) (3.24)
T. Kitamura / Physics Letters A 282 (2001) 59–71
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In calculating Q(q), we make the approximation to the same accuracy as the intraband density fluctuations: we consider the different intraband hoppings Jµµ ≡ Jµ , and make the ladder approximation; the intraband scatterings µµ µν µν Uω,J for the self-energy part Σ µ (p) and the interband scatterings Uω,J for the irreducible vertex part Upp (q) as shown in Figs. 1 and 2. Thus we obtain 2ω , (q0 + i/2τM )2 − ω2 1 1 1 1 1 ≡ + − o − o, 2τM 2τ0 2τ1 2τ0 2τ1 ν i i µ /Σ µν (p, q) ∼ Gµ (p ) − Uων + UJν G (p − q) ∼ − , = Uωµ + UJ =− 2τ 2τ 0 1 Q(q) =
p
µν /Gµν (p, q)Upp1 ∼ =
p
(3.25) (3.26) (3.27)
p
01
/Gµν (p, q) Uω01 + UJ
p
i i ∼ =− o − o, 2τ0 2τ1
(3.28)
where µν = 01 or 10. The translational invariance requires Dij (0) = 0, which yields the gap equation [1–13]: od 1 1 d 3 R g(R)Vij (R). δij = Q(0) Vij (0) c = − (3.29) 2) h¯ M(ω2 + 4/τM Taking q = qe3 , Eq. (3.21) yields the secular equation:
Q(q) od 1 i 2 δij q0 q0 + − ωλq detδij − Q(q) Vij (q) c = det = 0, 2ω τM h¯ 1 2 ωλq ≡− d 3 R g(R) 1 − eiq·R Vij (R), M
(3.30) (3.31)
where ωλq are a longitudinal and two transverse phonons. Now we rewrite the second equation in Eq. (3.30) as q0 − νλ (q)q 2 = 0,
νλ (q) =
2 ωλq 1 . q0 + i/τM q 2
(3.32)
Here τM will be identified to the Maxwell relaxation time later. The thermodynamical function due to the interband density fluctuations: Ωod =
2 2 1 1 iνn − ωλq 3N ln 1 − e−β h¯ ω = ln 2 − (h¯ ωλq − h¯ ω) + ln 1 − e−β h¯ ωλq . 2 2β iνn − ω β 2 q λq
(3.33)
λq
The entropy due to the interband density fluctuations is given by Sod = −kB
1 h¯ ωλq . ln 1 − e−β h¯ ωλq + T eβ h¯ ωλq − 1 λq λq
(3.34)
This is just the entropy of the phonons state. Judging from the entropy due to the intraband in the equilibrium β h¯ ω − 1) corresponds to the fluctuation entropy, which relates to density fluctuations, the term −(1/T ) λq hω/(e ¯ the configurational average of the absolute value of the deviation of the random eigenfrequencies. Thus we can put
h¯ ω . Uω0 ∝ exp −3β β hω (3.35) e ¯ −1
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T. Kitamura / Physics Letters A 282 (2001) 59–71
The term Uω0 is a decreasing function of the temperature T , while the term UJ0 is a rapidly increasing function starting from T0 in the vicinity of T0 . Thus the terms cross near the T0 and the term Uω0 is almost constant near the T0 . Since J˜0 < 0 and J˜i > 0, τµ are positive in Eq. (3.27) and τµo negative in Eq. (3.28), where µ takes 0 or 1. Thus we obtain τM > 0 in Eq. (3.26). From Eqs. (3.27) and (3.28), we obtain U 01 + UJ01 1 1 = ωµ , µ o τµ Uω + UJ τµ
o τ ∝ τ µ ∝ τ 0 . µ
The Vogel–Fulcher law in the transport coefficients, the relaxation times and velocity of modes are shown in Table 1, where cλ is the phonon velocity with λ-mode. Since UJ1 > UJ0 , the crossover temperature of viscosity is higher than that of diffusion. Phonons exist in both phases.
4. A generalized Navier–Stokes equation The correlation functions for intraband and interband density fluctuations lead to the basic equations (3.12) and (3.32). In order to relate Eqs. (3.12) and (3.32) to the Navier–Stokes equation, first we investigate the continuity equation in our theory. We make the following approximation in n(x) in Eq. (2.15): (1) We confine ourselves to the intraband density fluctuations of the first band ρd0q as the intraband density fluctuations, because those of the second band are less important. Hereafter, we abbreviate the subscript d in ρd0q . (2) We replace √ the random wavefunctions w˜ m0 (x − Rm ) and ζm by the averaged wavefunction w˜ 0 (x − Rm ) and ζM = h¯ /2Mω, respectively. The randomness of the wavefunctions is considered through the interaction Hamiltonian (2.21). (3) We consider the classical limit w˜ 02 (x − Rm ) → δ(x − Rm ). Thus we obtain 1 n(x) = (2π)3
dq e
iq·x
ρ0q − i qi ζM ρiq + · · · .
(4.1)
i
Since the conventional density current comes from v0q = ∇ ˜0q , the continuity equation is given by ∂ρ0q + iq · Jq = 0, ∂t
∂ρiq , jiq ∼ = ρv0iq − ζM ∂t
v0q = ∇ ˜0q ,
(4.2)
where ρ is the density. The first term in the current corresponds to the conventional current and the second term to a Fourier component of the displacement current u. We can rewrite Eq. (4.2) as ∂ρ + ∇ · j = 0, ∂t
j ≡ ρv,
v = v0 +
∂u , ∂t
(4.3)
where ρ is the density in the real space. This current resembles an electric current and a displacement current in electricity. Next we investigate Eq. (3.12). In order to relate the factor {1 + βV (q)} to the static structure factor S(q), we employ the fluctuation and dissipation theorem: S(q, q0 ) = −
h¯ 1 F (q, q0 + iδ), 1 − e−β h¯ q0 π
(4.4)
T. Kitamura / Physics Letters A 282 (2001) 59–71
69
where S(q, q0 ) is the dynamical structure factor and means taking the imaginary part. In the limiting case, 1/τ0 → 0, Eqs. (3.6) and (3.11) lead to F (q, q0 + iδ) → −
βπωq2 2ωsq
δ(q0 − ωsq ) − δ(q0 + ωsq ) .
(4.5)
From Eqs. (4.4) and (4.5), we obtain S(q) =
β h¯ ωq2 2ωsq
coth
β h¯ ωsq . 2
(4.6)
In the case β h¯ ωsq 1 and in the long wavelength regime, we obtain S(q) ∼ =
1 . 1 + βV (q)
(4.7)
At high temperatures, we can put va2 /3M ∼ = kB T in Eq. (3.13). Using thermodynamical relation S(0) = ρkB T χT , we can identify the sound velocity cs with v2 1 ∂p 1 = , cs2 = a ∼ (4.8) = 3S(0) MρχT M ∂ρ T where χT is isothermal compressibility and p is a pressure. Under the condition q0 τ0 1, Eq. (3.12) leads to ∂ 2ρ ∂p (4.9) − ∇ 2 ρ = 0. ∂t 2 ∂ρ T Eqs. (4.3) and (4.9) yield M∇ · ρ(∂v/∂t) = −∇ · (∂p/∂ρ)T ∇ρ. If we consider the pressure gradient comes through the density gradient, (∂p/∂ρ)T ∇ρ = ∇p, we obtain Mρ
∂v = −∇p. ∂t
(4.10)
This equation is valid at high temperatures, va2 /3M ∼ = kB T , and there is no displacement current. Eq. (4.10) is just the Euler equation. In order to relate Eq. (3.26) to the viscoelastic theory, we first estimate phonons or elastic waves in the long wavelength regime. If we put g(R) = ρδ(R − a), a is the mean atomic distance, and consider only Vσ -coupling, ct2
=
ωt2q q2
ω2 a 2 ∼ , = 10
cl2
=
2 ωlq
q2
=
ω2 a 2 , 5
ω2 =
4πa 2 Vσ , 3M
(4.11)
where cl and ct are a longitudinal and two transverse phonons or corresponding elastic wave velocities, respectively. In the viscoelastic theory, since we take q = qez , the instantaneous displacement is given by
∂2 ∂2 Mρ 2 − G∞λ 2 uλ = 0, G∞λ = Mρcλ2 , (4.12) ∂t ∂x ˜ λ
where for λ = t, λ˜ = x, y, G∞t is a high frequency shear modulus and for λ = l, λ˜ = z, G∞l a high frequency bulk modulus. The viscous equation is given by
∂ 2 ∂uλ ∂ = 0, Mρ − νλ 2 (4.13) ∂t ∂x ˜ ∂t λ
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T. Kitamura / Physics Letters A 282 (2001) 59–71
where νt = η is a kinetic shear viscosity and νl = ζ + (4/3)η is a kinetic bulk viscosity. If we interpolate between Eqs. (4.12) and (4.13), we obtain a viscoelastic equation as
∂ 2 ∂uλ 1 ∂ ∂ 1 − = 0. + Mρ (4.14) νλ G∞λ ∂t ∂t ∂x 2˜ ∂t λ
The eigenvalues of Eq. (4.14) are given by G∞λ (4.15) . q0 + iG∞λ /νλ Here we have used the same notation νλ (q0 ) in Eqs. (3.32) and (4.15). If we define the Maxwell relaxation time τM = νλ /G∞λ , we can identify νλ (q0 ) in Eq. (3.32) with that in Eq. (4.15). Thus we obtain q0 − νλ (q0 )q 2 = 0,
G∞λ = Mρcλ2 ,
νλ (q0 ) =
νλ = τM cλ2 .
(4.16)
If we take q in any direction, we can rewrite Eq. (4.14) as 1 ∂u ∂ 2 ∂u ∂ 2 ∂u + τM 2 = η∇ + η + ζ ∇∇ · , Mρ (4.17) ∂t ∂t ∂t ∂t 3 ∂t which yields phonons or elastic waves for q0 τM 1 and kinetic viscosity for τM q0 1. At high temperatures, the viscosity comes mainly from the exchange of the conventional velocity v0 , while at lower temperatures, the viscosity comes mainly from the exchange of the displacement velocity ∂u/∂t. Near the liquid-glass transition, since the viscosity due to the conventional velocity v0 is of the order of τ0 cs2 , the viscosity due to the exchange of v0 is negligible. If we interpolate between Eqs. (4.10) and (4.17), we obtain a generalized Navier–Stokes equation for q0 τ0 > 1: ∂2 1 ∂ + τM 2 v = −∇p + η∇ 2 v + η + ζ ∇∇ · v. Mρ (4.18) ∂t 3 ∂t From Eqs. (3.10) and (3.26), using the relation UJ1 > |UJ01 | > UJ0 , we can expect 1/τM > 1/τ0 . Thus for 1/τM > q0 > 1/τ0 , Eq. (4.18) reduces to the Navier–Stokes equation: ∂v 1 2 = −∇p + η∇ v + η + ζ ∇∇ · v. Mρ (4.19) ∂t 3 For q0 τM > 1, since νt = η = τM ct2 , νl = ζ + (4/3)η = τM cl2 , Eq. (4.18) reduces to the equation of elastic waves [17]: ∂ 2u = ct2 ∇ 2 u + cl2 − ct2 ∇∇ · u. ∂t 2 It should be noted that near the liquid-glass transition, τM ∝ |J0 |−1 . Mρ
(4.20)
5. Concluding remarks The correlation functions for intraband and interband density fluctuations yield the basic equations (3.12) and (3.32). In the classical limit, the number density consists of the intraband density fluctuations ρ0q and the interband density fluctuations −iqi ζM ρiq in Eq. (4.1). ρ0q is a Fourier component of the conventional density ρ, while −ζM ρiq corresponds to a Fourier component of the displacement of an atom u. Thus the current consists of the conventional current ρv0 and the displacement current ρ(∂u/∂t). The basic equation (3.12) yields Eq. (4.10) at high temperatures, while Eq. (3.32) yields Eq. (4.17). The interpolation between Eqs. (4.10) and (4.17) yield a generalized Navier–Stokes equation for q0 > 1/τ0 , which reduces to the Navier–Stokes equation for 1/τM > q0 > 1/τ0 and to the elastic equation for q0 > 1/τM .
T. Kitamura / Physics Letters A 282 (2001) 59–71
Acknowledgements I would like to thank Professor Kyozi Kawasaki and Professor Shozo Takeno for valuable discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
T. Kitamura, Physica A 245 (1997) 543. T. Kitamura, Phys. Lett. A 235 (1997) 515. T. Kitamura, Phys. Lett. A 243 (1998) 325. T. Kitamura, Physica A 262 (1999) 16. T. Kitamura, Physica A 272 (1999) 330. T. Kitamura, S. Takeno, Phys. Lett. A 266 (2000) 198. T. Kitamura, Physica A 167 (1990) 509, 527. T. Kitamura, Phys. Lett. A 147 (1990) 511. T. Kitamura, Physica A 174 (1991) 578. T. Kitamura, Phys. Lett. A 152 (1991) 218. T. Kitamura, Physica A 183 (1992) 187, 201. T. Kitamura, Physica A 223 (1996) 227. T. Kitamura, M. Silbert, Phys. Lett. A 215 (1996) 69. J.P. Hansen, I.R. MacDonald, Theory of Simple Liquids, Academic Press, London, 1990. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1963. L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon Press, London, 1958. L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Pergamon Press, London, 1970.
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Physics Letters A 282 (2001) 59–71 www.elsevier.nl/locate/pla
A generalized Navier–Stokes equation near the liquid-glass transition Toyoyuki Kitamura Nagasaki Institute of Applied Science, Nagasaki 851-0193, Japan Received 19 September 2000; accepted 19 February 2001 Communicated by L.J. Sham
Abstract The correlation functions for density fluctuations yield the basic equations in fluids. The density fluctuations consists of intraband and interband density fluctuations which yield the conventional density and the divergence of the displacement of an atom in a localized potential, respectively. The current consists of the corresponding conventional current and displacement current. The basic equations combined with the continuity equation yield a generalized Navier–Stokes equation, which reduces to the Navier–Stokes equation at low frequencies or high temperatures and to the elastic equation at high frequencies or low temperatures. The transport coefficients, the relaxation times and the velocity of modes obey the Vogel–Fulcher law. 2001 Elsevier Science B.V. All rights reserved. PACS: 64.70.Pf; 46.35.+z Keywords: Quantum field theory; Glass transition; Vogel–Fulcher law; Viscoelasticity; Navier–Stokes equation
1. Introduction We have proposed a quantum field theory of transport and relaxation processes near the liquid-glass transition [1–6] using the two band model in the harmonic potential approximation [1–13]. Our idea is as follows: an atom stays temporarily in a harmonic potential and hops to a surrounding harmonic potential. The harmonic potentials are distributed randomly in space and magnitude. We consider the two levels; the ground state and the first excited states. The energy gap (∼ the eigenfrequency of the harmonic potential) plays a role as elasticity and the hopping matrices (∼ the band widths of atoms) as the kinetic motion. The density fluctuations consist of intraband and interband density fluctuations. The correlation functions of the intraband density fluctuations yield the structure of sound and diffusion, while those of the interband density fluctuations the structure of phonons and viscosity. The phonons are elastic waves and the sound is a collision wave. The hopping processes relate to the entropy due to intraband density fluctuations. The entropy manifests that the configurational average of a hopping matrix obeys the Vogel–Fulcher law. The purpose of the present Letter is to derive a generalized Navier–Stokes equation from the basic equations which come from the correlation functions of intraband and interband density fluctuations near the liquid-glass E-mail address: [email protected] (T. Kitamura). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 1 4 9 - 9
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T. Kitamura / Physics Letters A 282 (2001) 59–71
transition [14,15]. The intraband and interband density fluctuations correspond to the Fourier components of the conventional density and the divergence of the displacement of an atom in a localized potential, respectively. The current consists of the respective conventional current and displacement current. They constitute the continuity equation. The generalized Navier–Stokes equation reduces to the Navier–Stokes equation at low frequencies or high temperatures and to the elastic equation at high frequencies or low temperatures.
2. The basic formulation We start with the following Hamiltonian [1–13]: 1 H = d 3 x ψ † (x)h¯ 0 (−i∇)ψ(x) + d 3 x d 3 y n(x)V (x − y)n(y), 2
(2.1)
where ψ is an annihilation field operator of an atom, h¯ 0 (−i∇) is an energy operator of a free atom, n(x) is a density operator of the atom, n(x) = ψ † (x)ψ(x) and V is an interaction potential between atoms. If we introduce an annihilation operator of the atom in the harmonic potential at the site Rm , bmµ , we can write ψ(x) = (2.2) w˜ mµ (x − Rm )bmµ , µ = 0, 1, 2, 3. mµ
Here w˜ mµ is the wavefunction of the µth state at the site Rm . The Heisenberg equation for bmµ is given by 0 ∂ −h¯ bmµ = d 3 x w˜ mµ (x − Rm ) h ¯ (−i∇) + ΦRn (x) w˜ nν (x − Rn )bnν , ∂τ nν ΦRn (x) = d 3 y V (x − y)n(y),
(2.3) (2.4)
{Rn }
where {Rn } means summing the surrounding sites of the atom at Rn in n(y). The term ΦRn means that a potential is determined by the surrounding atoms of the site Rn . If we assume that the atom at the site Rn feels a harmonic potential, we obtain
h¯ 0 (−i∇) + ΦRn (x) w˜ nν (x − Rn ) = h¯ ω˜ nν w˜ nν (x − Rn ),
1 ΦRn (x) ∼ = Mωn2 (x − Rn )2 , 2
(2.5)
where ω˜ nµ and w˜ nµ are the eigenfrequency and the eigenfunction of the µth state at the site Rn ; µ = 0 for the ground state, ω˜ n0 = (3/2)ωn and µ = 1, 2, 3 for the first excited states, w˜ ni = (5/2)ωn , where ωn is an eigenfrequency at the site Rn . Hereafter, µ, ν mean 0, 1, 2, 3 and i, j mean 1, 2, 3. There is a relation between the ground state and the first excited state wavefunctions for the ith Cartesian coordinate: h¯ , wn1 (xi ) = −2ζn ∇i wn0 (xi ), ζn = (2.6) 2Mωn where ζn is the mean width of the zero point motion, M is the mass of an atom. The magnitude of potentials ΦRn is distributed randomly, so is that of the eigenfrequencies. At this stage, we can take the following model Hamiltonian in terms of localized operators: † † h¯ ω˜ mµ bmµ bmµ + Jmµnν bmµ bnν , H= (2.7) mµ
mµnν
T. Kitamura / Physics Letters A 282 (2001) 59–71
where the hopping matrix is given by Jmµnν = d 3 x w˜ mµ (x − Rm ) h¯ 0 (−i∇) + ΦRn (x) w˜ nν (x − Rn ).
61
(2.8)
Note that the interaction Hamiltonian is nonlinear through the term n(y) in ΦRn and the potential ΦRn permits an atom to hop only to a vacancy. Now we first establish the unperturbed Hamiltonian by taking the configurational average of the model Hamiltonian (2.7): † † h¯ ω˜ µ bmµ bmµ + Jµ (Rm − Rn )bmµ bnµ , H0 = H c = (2.9)
µm
ω˜ µ = ω˜ mµ c ,
µmn
Jµ (Rm − Rn ) = Jmµnµ c ,
(2.10)
where . . . = Tr e−β(H −µN) . . . / Tr e−β(H −µN) and the subscript c means the configurational average. The hopping between different states is neglected for simplicity. Next we establish an alternative model Hamiltonian in terms of extended operators. If we introduce the following extended operators: 1 ip·Rm e aµp , µ = 0, 1, 2, 3, bmµ = √ (2.11) N p we obtain H0 =
† aµp , h¯ µp aµp
(2.12)
µp
1 J˜µ (p) ≡ N
h ¯ µp = h¯ ω˜ µ + J˜µ (p),
d 3 R g(R)eip·R Jµ (R),
(2.13)
where g(R) is a pair distribution function. The field operator is written down as 1 ip·Rm e w˜ mµ (x − Rm )aµp . ψ(x) = √ N µpm
(2.14)
From Eq. (2.14), we obtain 2 † 2 w˜ mµ (x − Rm )bmµ bmµ − ζm Bm · ∇ w˜ m0 (x − Rm ) + · · · n(x) = m
µ
1 −iq·Rm 2 † † 2 = e w˜ mµ (x − Rm )ρdµq − ζm ∇i w˜ m0 (x − Rm )ρiq + ··· , N mq µ
(2.15)
i
where † † Bmi ≡ bmi bm0 + bmi b0i ,
† ρdµq =
p
† aµp aµp−q ,
† ρiq =
† † aip a0p−q + a0p aip−q .
(2.16)
p
† † and ρiq are intraband (diagonal) and interband (off-diagonal) density fluctuation operators, respectively. ρdµq Substitution of Eq. (2.15) into Eq. (2.1) yields 1 od 1 d † † † aµp + Vij (q)ρiq ρj q + Vµν (q)ρdµq ρdνq , h¯ µp aµp H= (2.17) 2N 2N µp µνq ij q
Vijod (q) ≡ NVi0,j 0 ,
d Vµν
≡ NVµµ,νν ,
(2.18)
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T. Kitamura / Physics Letters A 282 (2001) 59–71
1 −iq·(Rm−Rn ) Vµµ ,νν (q) = 2 e d 3 x d 3 y w˜ mµ (x − Rm )w˜ mµ (x − Rm )V (x − y) N m=n × w˜ nν (y − Rn )w˜ nν (y − Rn ),
(2.19)
where no cross term of the intraband and interband density operators appears, because of the symmetry of the Here it should be noted that since the unperturbed Hamiltonian H0 in Eq. (2.9) involves the term wavefunctions. d 3 x d 3 y n(x)V (x − y) n(y) c , the interaction Hamiltonian HI must exclude the term. Thus the prime on in Eq. (2.17) means that HI excludes one loop diagrams. First we are concerned with the model Hamiltonian (2.7) in terms of localized operators. The interaction Hamiltonian HI is given by † † bmµ + bnµ , HI = (2.20) h¯ ω˜ mµ − ω˜ µ bmµ h¯ Jmµnµ − Jµ (Rm − Rn ) bmµ m
mnµ
where we make the approximation that the term Jmµnµ is a random c-number. Using Eq. (2.11), we rewrite Eq. (2.20) as
1 † HI = aµp . h¯ ω˜ mµ − ω˜ µ e−i(p−p )·Rm + h¯ Jmµnµ − Jµ (Rm − Rn ) e−ip·Rm +ip ·Rn aµp N m mn pp (2.21) µν
Eqs. (2.20) and (2.21) are used to take into account the randomness. Elemental scattering processes, Uω due to µν the random eigenfrequencies of atoms and UJ due to the random hopping matrices, are shown in Fig. 1 and the corresponding terms are given by 1 ω˜ mµ ω˜ mν c − ω˜ µ ω˜ ν , 2 N m 1 µν UJ (q) = d 3 R g(R)e−iq·R Jmµnµ Jmνnν c − Jµ (R)Jν (R) R=Rm −Rn . N
Uωµν =
(2.22) (2.23)
Here the crosses on the diagrams means summing the scattering processes over random positions. Now we derive the Bethe–Salpeter equation. First, we introduce atom Green’s functions: 1 −iωn (τ −τ ) µ µ † Gpp (τ − τ ) ≡ − Tτ aµp (τ )aµp e Gpp (iωn ). (τ ) = β h¯ iω
(2.24)
n
µν
µν
Fig. 1. Elemental scattering processes, Uω due to the random eigenfrequencies of the harmonic potentials and UJ matrices.
due to the random hopping
T. Kitamura / Physics Letters A 282 (2001) 59–71
63
We should not confuse the temperature frequency iωn with the eigenfrequency ωn . We derive the Bethe–Salpeter equation for configurationally averaged two Green’s functions: µ µν φpp (iωn , q) ≡ Gp,p (iωn )Gνp−q,p −q (iωn − iνn ) c .
(2.25)
We obtain the Bethe–Salpeter equation [1–6]:
µν µν µν Upp1 (q)φp1 p (iωn , q) , φpp (iωn , q) = Gµ (p)Gν (p − q) δpp +
(2.26)
p1
where Gµ (p) is configurational average of the Green’s function (2.24):
µ µ Gpp (iωn ) c = Gp (iωn )δpp = Gµ (p)δpp
(2.27)
and the abbreviations p = (p, iωn ), q = (q, iνn ) have been used. Note that p = (p , iωn ). Upp1 is the irreducible vertex part. If we introduce the self-energy parts of the Green’s function as µν
Gµ−1 (p) ≡ iωn − ˜µp − Σ µ (p),
˜µp = µp − µ,
(2.28)
we obtain
µν −iνn + ˜µp − ˜νp−q + /Σ µν (p, q) φ µν (p, q) = /Gµν (p, q) 1 + Upp1 (q)φ µν (p1 , q) ,
(2.29)
p1
/Σ µν (p, q) = Σ µ (p) − Σ ν (p − q), µν φpp (iωn , q). φ µν (p, q) ≡
/Gµν (p, q) = Gµ (p) − Gν (p − q), (2.30)
p
00 = U 0 , U ii = U 1 , U 0i = U i0 = U 01 , Hereafter, we consider isotropic scatterings. So we abbreviate Uω,J ω,J ω,J ω,J ω,J ω,J ω,J 00 0 ii 1 0i 01 i0 10 and φ = φ , φ = φ , φ = φ , φ = φ . Since the magnitude of the wavefunctions of the ground state µν is larger than that of the excited states, |Ji | > |J0 |, and Uω is independent of the states, we can expect UJ1 > 01 0 |UJ | > UJ and Uω0 = Uω1 = Uω01 .
3. Intraband and interband density fluctuations We give a brief survey of intraband and interband density fluctuations [1–6]. Hereafter we are concerned with the low temperature regime: β h¯ ω 1, β|J˜| 1, and β h¯ /τ0 1. First we consider the correlation functions for intraband density fluctuations. Since we confine ourselves to low temperatures, we are concerned with the correlation functions for the ground state: Fq (τ1 − τ2 ) ≡ −
1 1 −iνn (τ1 −τ2 ) † (τ2 ) c ≡ e F (q). Tτ ρd0q (τ1 )ρd0q β h¯ h¯ N iν
(3.1)
n
The model Hamiltonian is expressed in two alternative forms: Eq. (2.7) with Eq. (2.9) in terms of localized operators and Eq. (2.17) with Eq. (2.12) in terms of extended operators. We cannot apply both the interaction Hamiltonians to a calculation simultaneously. At this stage, we consider the model Hamiltonian (2.17) with (2.9), while the model Hamiltonian (2.7) with (2.9) is used to take into account the randomness for the self-energy parts
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T. Kitamura / Physics Letters A 282 (2001) 59–71
and vertex corrections. In the random phase approximation, we obtain d 1 1 (q) c F (q), P0 (q) + P0 (q) V00 h¯ h¯ 1 0 φ (p, q). P0 (q) ≡ ∓ β h¯ N p
F (q) =
The configurationally averaged diagonal potential function is given by d d V (q) ≡ V00 (q) c = d 3 R g(R)e−iq·R V00 (R), d V00 (R) ≡ d 3 x d 3 y w˜ 02 (x)V (x − y)w˜ 02 (y − R),
(3.2) (3.3)
(3.4) (3.5)
where w˜ 0 (x) is configurationally averaged. Then we obtain F (q) =
(1/h¯ )P0 (q) . 1 − (1/h¯ )V (q)P0 (q)
(3.6)
In calculating the vertex correction φ 0 , we make the following approximations: (1) we put Σ 0 (q0 ± iδ) ∼ = ∓i/2τ0 (q0 ) and we neglect the real part of the self-energy. We use i 0 2 ∼ f ˜0p . /G (p, q) = ∓(β h¯ ) q · ∇ ˜p − τ0 iωn
f (x) = df (x)/d(β h¯ x) = −f (x){1 ± f (x)}, f (x) = 1/(eβ h¯ x ∓ 1) and, since p f (˜0p ) ∼ = N , f (˜0p ) ∼ = −2 ∼ ∼ for bosons, f (˜0p ) = 0 for fermions and f (˜0p ) = −1 for the Boltzmann distribution. (2) We retain only the following terms in the Bathe–Salpeter equation: (3.7) φ 0 (p, q) ≡ φ, q · ∇p φ 0 (p, q) = q · . iωn ,p
iωn ,p
(3) We make the ladder approximation to the vertex part for the Green’s functions: 00 0 ∼ 0 Upp (q) = Uω (0) + UJ (0).
(3.8)
The corresponding self-energy part is given by 0 Σ 0 (p0 + iδ) ∼ G (p, p0 + iδ) = Uω0 (0) + UJ0 (0) p
= Uω0 (0) + UJ0 (0)
p
1 , p0 + iδ − ˜0p − Σ 0 (p0 + iδ)
(3.9)
which is illustrated in Fig. 2. Here we consider only the imaginary part of the self-energy, 1/2τ0 (p0 ) = Σ(p0 + iδ). We can expand ˜0p ∼ = |J0 |(1 − sin α/α), α = ap, a: the mean atomic distance. The dominant contribution of wave vectors to the density of states comes from the short wavelength regime, where ˜0p ∼ = = |J0 |. Thus we put p G0 (p, p0 + iδ) ∼ N/{p0 − |J0 | − Σ 0 (p0 + iδ)} in Eq. (3.9). Then we obtain 2 2 1 1 2 1 1 ∼ ∼ (3.10) − p0 − |J0 | = − p02 , = N Uω0 + UJ0 . = 2τ0 (p0 ) 2τ0 2τ0 2τ0
T. Kitamura / Physics Letters A 282 (2001) 59–71
65
Fig. 2. The self-energy part corresponding to the ladder approximation to the vertex function is shown. The irreducible vertex function for the ladder approximation is shown in Fig. 1.
Thus we obtain P0 (q) ≡ −β h¯ f
ωq2 /(q0 + i/τ0 ) q0 − ωq2 /(q0
+ i/τ0 )
,
ωq2 ∼ =
q 2 va2 1 , (q · ∇0p )2 = N p 3
va =
1 ∂0p , N p ∂p
(3.11)
where we abbreviate q0 in 1/τ0 (q0 ), q0 + i/τ (q0) → i/τ0 for q0 τ0 1 and q0 + i/τ (q0 ) → q0 for q0 τ0 1. The denominator of Eq. (3.6) yields q0 −
2 ωsq
q0 + i/τ0
= 0,
ωq2 2 , ωsq = ωq2 1 + βV (q) = S(q)
(3.12)
where ωsq is a dispersion of sound, S(q) the static structure factor and va is the mean atomic velocity. For q0 τ0 1, we obtain the diffusion constant D = τ0 cs2 ,
cs2 =
va2 , 3S(0)
(3.13)
where cs is the sound velocity. Next we investigate the freezing point. Since limq0 →0 P0 (q) ∼ = f β h¯ and V (q) has the negative minimum value ∼ ˜ ˜ at K (K = 3π/2a for g(R) = ρδ(R − a), a: the mean atomic distance), the freezing point β0 is determined by
1 ˜ ˜ lim 1 − V K P0 K, q0 ∼ (3.14) = 1 − f β0 V K˜ = 0. q0 →0 h¯ The thermodynamical function due to the intraband density fluctuations, Ωd , is given by [5,6] Ωd =
2 1 iνn2 − ωsq 1 ln 2 − ln 1 − e−β h¯ ωq . 2β q iνn − ωq2 β q
The Contour integration of Eq. (3.15) gives 1 1 1 Ωd = ln 1 − e−β h¯ ωsq − ln 1 − e−β h¯ ωq − (h¯ ωsq − hω ln 1 − e−β h¯ ωq . ¯ q) + 2 q β q β q
(3.15)
(3.16)
The dominant contribution of the intraband density fluctuations to the thermodynamical function comes from ˜ where ln(1 − e−β h¯ ωsq ) − ln(1 − e−β h¯ ωq ) ∼ the regions q ∼ K, = (1/2) ln{1 − f βV (q)}, where we have used the ˜ condition β|J | 1. If we denote the number of q-states near K˜ by N0 , we obtain
T0 1 N0 1 ∼ ln 1 − + (h¯ ωsq − hω ln 1 − e−β h¯ ωsq , Ωd = (3.17) ¯ q) + 2 q 2β T β q
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Table 1 The Vogel–Fulcher law of transport coefficients, relaxation times and velocity of modes. |J0 | ∝ exp{−E0 /(T − T0 )}, Uω0 is given in Eq. (3.35) Transport coefficient
Relaxation time
Velocity
D
νλ
τ0
τM
cs
cλ
Higher T
|J0 |
|J0 |−1
|J0 |−1
|J0 |−1
|J0 |
–
Lower T
J02
−1/2 Uω0
−1/2 Uω0
−1/2 Uω0
|J0 |
–
where the dominant contribution of the last term due to sound comes from the region q ∼ 0. The entropy due to the intraband density fluctuations is given by
1 h¯ ωsq ∂Ωd N0 kB T0 T0 ∼ − kB Sd ∼ + ln ln 1 − e−β h¯ ωsq + . =− =− ∂T V ,µ 2 T − T0 T − T0 T q eβ h¯ ωsq − 1 q (3.18) This thermodynamical meaning of the first term which we denote as Sv is as follows [16]: if we consider vacancies as a body in an external medium in a nonequilibrium state and the entropy Sv ≡ −Rmin /T as the total entropy of the body together with medium, Rmin is the minimum work which must be done from the external object when the total system becomes in an equilibrium state. On the other hand, the exponential of the entropy Sv is proportional to the hopping probability. Intraband density fluctuations represent the randomness of configuration of atoms. The number of vacancies is proportional to the magnitude of the randomness of configuration of atoms. Since the number of states due to the intraband density fluctuations is eSv / kB , the number of vacancies around an atom is proportional to eSv /NkB . Thus we put |Jµ | ∝ eSv /NkB ∼ = e−E0 /(T −T0 ) ,
E0 =
N0 T0 . 2N
(3.19)
Eq. (3.19) is nothing but the Vogel–Fulcher law. |J0 | obeys the Vogel–Fulcher law, but Uω0 is almost constant near the freezing point. Hereafter, we take |J0 | as the standard of the powers of the Vogel–Fulcher law. In Table 1, we show the Vogel–Fulcher law in transport coefficients, relaxation times and velocity of modes. Next we introduce the correlation functions for interband density fluctuations 1 1 −iνn (τ1 −τ2 ) Tτ ρiq (τ1 )ρj†q (τ2 ) c ≡ e Dij (q). Dij q (τ1 − τ2 ) ≡ − (3.20) h¯ N β h¯ iν n
At this stage, we consider the model Hamiltonian (2.17) in extended operators in a similar way to the case of intraband density fluctuations. Thus we make the random phase approximation: 1 1 Dij (q) = Q(q)δij + Q(q) (3.21) Vilod (q) c Dlj (q), h¯ h¯ l 1 10 Q(q) ≡ ∓ (3.22) φ (p, q) + φ 01 (p, q) . β h¯ N p Configurational average of the potential function is given by od h¯ 2 3 iq·R , Vij (R), ζM = Vij (q) c = ζM d R g(R)e 2Mω Vij (R) = d 3 x d 3 y ∇i w˜ 02 (x)V (x − y)∇j w˜ 02 (y − R), √ where w˜ 0 (x) = w˜ m0 (x) c , ζM = h¯ /2Mωm c and ω = ωm c .
(3.23) (3.24)
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In calculating Q(q), we make the approximation to the same accuracy as the intraband density fluctuations: we consider the different intraband hoppings Jµµ ≡ Jµ , and make the ladder approximation; the intraband scatterings µµ µν µν Uω,J for the self-energy part Σ µ (p) and the interband scatterings Uω,J for the irreducible vertex part Upp (q) as shown in Figs. 1 and 2. Thus we obtain 2ω , (q0 + i/2τM )2 − ω2 1 1 1 1 1 ≡ + − o − o, 2τM 2τ0 2τ1 2τ0 2τ1 ν i i µ /Σ µν (p, q) ∼ Gµ (p ) − Uων + UJν G (p − q) ∼ − , = Uωµ + UJ =− 2τ 2τ 0 1 Q(q) =
p
µν /Gµν (p, q)Upp1 ∼ =
p
(3.25) (3.26) (3.27)
p
01
/Gµν (p, q) Uω01 + UJ
p
i i ∼ =− o − o, 2τ0 2τ1
(3.28)
where µν = 01 or 10. The translational invariance requires Dij (0) = 0, which yields the gap equation [1–13]: od 1 1 d 3 R g(R)Vij (R). δij = Q(0) Vij (0) c = − (3.29) 2) h¯ M(ω2 + 4/τM Taking q = qe3 , Eq. (3.21) yields the secular equation:
Q(q) od 1 i 2 δij q0 q0 + − ωλq detδij − Q(q) Vij (q) c = det = 0, 2ω τM h¯ 1 2 ωλq ≡− d 3 R g(R) 1 − eiq·R Vij (R), M
(3.30) (3.31)
where ωλq are a longitudinal and two transverse phonons. Now we rewrite the second equation in Eq. (3.30) as q0 − νλ (q)q 2 = 0,
νλ (q) =
2 ωλq 1 . q0 + i/τM q 2
(3.32)
Here τM will be identified to the Maxwell relaxation time later. The thermodynamical function due to the interband density fluctuations: Ωod =
2 2 1 1 iνn − ωλq 3N ln 1 − e−β h¯ ω = ln 2 − (h¯ ωλq − h¯ ω) + ln 1 − e−β h¯ ωλq . 2 2β iνn − ω β 2 q λq
(3.33)
λq
The entropy due to the interband density fluctuations is given by Sod = −kB
1 h¯ ωλq . ln 1 − e−β h¯ ωλq + T eβ h¯ ωλq − 1 λq λq
(3.34)
This is just the entropy of the phonons state. Judging from the entropy due to the intraband in the equilibrium β h¯ ω − 1) corresponds to the fluctuation entropy, which relates to density fluctuations, the term −(1/T ) λq hω/(e ¯ the configurational average of the absolute value of the deviation of the random eigenfrequencies. Thus we can put
h¯ ω . Uω0 ∝ exp −3β β hω (3.35) e ¯ −1
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The term Uω0 is a decreasing function of the temperature T , while the term UJ0 is a rapidly increasing function starting from T0 in the vicinity of T0 . Thus the terms cross near the T0 and the term Uω0 is almost constant near the T0 . Since J˜0 < 0 and J˜i > 0, τµ are positive in Eq. (3.27) and τµo negative in Eq. (3.28), where µ takes 0 or 1. Thus we obtain τM > 0 in Eq. (3.26). From Eqs. (3.27) and (3.28), we obtain U 01 + UJ01 1 1 = ωµ , µ o τµ Uω + UJ τµ
o τ ∝ τ µ ∝ τ 0 . µ
The Vogel–Fulcher law in the transport coefficients, the relaxation times and velocity of modes are shown in Table 1, where cλ is the phonon velocity with λ-mode. Since UJ1 > UJ0 , the crossover temperature of viscosity is higher than that of diffusion. Phonons exist in both phases.
4. A generalized Navier–Stokes equation The correlation functions for intraband and interband density fluctuations lead to the basic equations (3.12) and (3.32). In order to relate Eqs. (3.12) and (3.32) to the Navier–Stokes equation, first we investigate the continuity equation in our theory. We make the following approximation in n(x) in Eq. (2.15): (1) We confine ourselves to the intraband density fluctuations of the first band ρd0q as the intraband density fluctuations, because those of the second band are less important. Hereafter, we abbreviate the subscript d in ρd0q . (2) We replace √ the random wavefunctions w˜ m0 (x − Rm ) and ζm by the averaged wavefunction w˜ 0 (x − Rm ) and ζM = h¯ /2Mω, respectively. The randomness of the wavefunctions is considered through the interaction Hamiltonian (2.21). (3) We consider the classical limit w˜ 02 (x − Rm ) → δ(x − Rm ). Thus we obtain 1 n(x) = (2π)3
dq e
iq·x
ρ0q − i qi ζM ρiq + · · · .
(4.1)
i
Since the conventional density current comes from v0q = ∇ ˜0q , the continuity equation is given by ∂ρ0q + iq · Jq = 0, ∂t
∂ρiq , jiq ∼ = ρv0iq − ζM ∂t
v0q = ∇ ˜0q ,
(4.2)
where ρ is the density. The first term in the current corresponds to the conventional current and the second term to a Fourier component of the displacement current u. We can rewrite Eq. (4.2) as ∂ρ + ∇ · j = 0, ∂t
j ≡ ρv,
v = v0 +
∂u , ∂t
(4.3)
where ρ is the density in the real space. This current resembles an electric current and a displacement current in electricity. Next we investigate Eq. (3.12). In order to relate the factor {1 + βV (q)} to the static structure factor S(q), we employ the fluctuation and dissipation theorem: S(q, q0 ) = −
h¯ 1 F (q, q0 + iδ), 1 − e−β h¯ q0 π
(4.4)
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where S(q, q0 ) is the dynamical structure factor and means taking the imaginary part. In the limiting case, 1/τ0 → 0, Eqs. (3.6) and (3.11) lead to F (q, q0 + iδ) → −
βπωq2 2ωsq
δ(q0 − ωsq ) − δ(q0 + ωsq ) .
(4.5)
From Eqs. (4.4) and (4.5), we obtain S(q) =
β h¯ ωq2 2ωsq
coth
β h¯ ωsq . 2
(4.6)
In the case β h¯ ωsq 1 and in the long wavelength regime, we obtain S(q) ∼ =
1 . 1 + βV (q)
(4.7)
At high temperatures, we can put va2 /3M ∼ = kB T in Eq. (3.13). Using thermodynamical relation S(0) = ρkB T χT , we can identify the sound velocity cs with v2 1 ∂p 1 = , cs2 = a ∼ (4.8) = 3S(0) MρχT M ∂ρ T where χT is isothermal compressibility and p is a pressure. Under the condition q0 τ0 1, Eq. (3.12) leads to ∂ 2ρ ∂p (4.9) − ∇ 2 ρ = 0. ∂t 2 ∂ρ T Eqs. (4.3) and (4.9) yield M∇ · ρ(∂v/∂t) = −∇ · (∂p/∂ρ)T ∇ρ. If we consider the pressure gradient comes through the density gradient, (∂p/∂ρ)T ∇ρ = ∇p, we obtain Mρ
∂v = −∇p. ∂t
(4.10)
This equation is valid at high temperatures, va2 /3M ∼ = kB T , and there is no displacement current. Eq. (4.10) is just the Euler equation. In order to relate Eq. (3.26) to the viscoelastic theory, we first estimate phonons or elastic waves in the long wavelength regime. If we put g(R) = ρδ(R − a), a is the mean atomic distance, and consider only Vσ -coupling, ct2
=
ωt2q q2
ω2 a 2 ∼ , = 10
cl2
=
2 ωlq
q2
=
ω2 a 2 , 5
ω2 =
4πa 2 Vσ , 3M
(4.11)
where cl and ct are a longitudinal and two transverse phonons or corresponding elastic wave velocities, respectively. In the viscoelastic theory, since we take q = qez , the instantaneous displacement is given by
∂2 ∂2 Mρ 2 − G∞λ 2 uλ = 0, G∞λ = Mρcλ2 , (4.12) ∂t ∂x ˜ λ
where for λ = t, λ˜ = x, y, G∞t is a high frequency shear modulus and for λ = l, λ˜ = z, G∞l a high frequency bulk modulus. The viscous equation is given by
∂ 2 ∂uλ ∂ = 0, Mρ − νλ 2 (4.13) ∂t ∂x ˜ ∂t λ
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where νt = η is a kinetic shear viscosity and νl = ζ + (4/3)η is a kinetic bulk viscosity. If we interpolate between Eqs. (4.12) and (4.13), we obtain a viscoelastic equation as
∂ 2 ∂uλ 1 ∂ ∂ 1 − = 0. + Mρ (4.14) νλ G∞λ ∂t ∂t ∂x 2˜ ∂t λ
The eigenvalues of Eq. (4.14) are given by G∞λ (4.15) . q0 + iG∞λ /νλ Here we have used the same notation νλ (q0 ) in Eqs. (3.32) and (4.15). If we define the Maxwell relaxation time τM = νλ /G∞λ , we can identify νλ (q0 ) in Eq. (3.32) with that in Eq. (4.15). Thus we obtain q0 − νλ (q0 )q 2 = 0,
G∞λ = Mρcλ2 ,
νλ (q0 ) =
νλ = τM cλ2 .
(4.16)
If we take q in any direction, we can rewrite Eq. (4.14) as 1 ∂u ∂ 2 ∂u ∂ 2 ∂u + τM 2 = η∇ + η + ζ ∇∇ · , Mρ (4.17) ∂t ∂t ∂t ∂t 3 ∂t which yields phonons or elastic waves for q0 τM 1 and kinetic viscosity for τM q0 1. At high temperatures, the viscosity comes mainly from the exchange of the conventional velocity v0 , while at lower temperatures, the viscosity comes mainly from the exchange of the displacement velocity ∂u/∂t. Near the liquid-glass transition, since the viscosity due to the conventional velocity v0 is of the order of τ0 cs2 , the viscosity due to the exchange of v0 is negligible. If we interpolate between Eqs. (4.10) and (4.17), we obtain a generalized Navier–Stokes equation for q0 τ0 > 1: ∂2 1 ∂ + τM 2 v = −∇p + η∇ 2 v + η + ζ ∇∇ · v. Mρ (4.18) ∂t 3 ∂t From Eqs. (3.10) and (3.26), using the relation UJ1 > |UJ01 | > UJ0 , we can expect 1/τM > 1/τ0 . Thus for 1/τM > q0 > 1/τ0 , Eq. (4.18) reduces to the Navier–Stokes equation: ∂v 1 2 = −∇p + η∇ v + η + ζ ∇∇ · v. Mρ (4.19) ∂t 3 For q0 τM > 1, since νt = η = τM ct2 , νl = ζ + (4/3)η = τM cl2 , Eq. (4.18) reduces to the equation of elastic waves [17]: ∂ 2u = ct2 ∇ 2 u + cl2 − ct2 ∇∇ · u. ∂t 2 It should be noted that near the liquid-glass transition, τM ∝ |J0 |−1 . Mρ
(4.20)
5. Concluding remarks The correlation functions for intraband and interband density fluctuations yield the basic equations (3.12) and (3.32). In the classical limit, the number density consists of the intraband density fluctuations ρ0q and the interband density fluctuations −iqi ζM ρiq in Eq. (4.1). ρ0q is a Fourier component of the conventional density ρ, while −ζM ρiq corresponds to a Fourier component of the displacement of an atom u. Thus the current consists of the conventional current ρv0 and the displacement current ρ(∂u/∂t). The basic equation (3.12) yields Eq. (4.10) at high temperatures, while Eq. (3.32) yields Eq. (4.17). The interpolation between Eqs. (4.10) and (4.17) yield a generalized Navier–Stokes equation for q0 > 1/τ0 , which reduces to the Navier–Stokes equation for 1/τM > q0 > 1/τ0 and to the elastic equation for q0 > 1/τM .
T. Kitamura / Physics Letters A 282 (2001) 59–71
Acknowledgements I would like to thank Professor Kyozi Kawasaki and Professor Shozo Takeno for valuable discussions.
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