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Statistical derivation of hydrodynamic equations of the Grad-type
Volume
PHYSICS
27A, number 9
References 1. T. Sugawara and H. Eguchi. J. Phys. Sot. Japan 21 (1966) 725. 2. A. S. Edelstein, Phys. Rev. Letters 20 (1968) 1348. 3. Ph. W. Anderson. Phys. Rev. 164 (1967) 352. 4. A. S. Edelstein. Phys. Rev. 164 (1967) 510. 5. A. S. Edelstein, Phys. Rev. Letters 19 (1967) 1184. 6. M.D. Daybell and W. A. Steyert. Phys. Rev. Letters 18 (1967) 398. 7. G. J. Van den Berg. Progress in low temperature physics. Vol. IV! ed. C. J. Gorter (North-Holland. Amsterdam. 1964) p. 194. 8. L. M. Roberts and J. M. Lock. Phil. Mag. 2 (1957) 811.
The author wishes to thank R. J. Gambino who ***
**
DERIVATION
OF
OF THE
1968
prepared the alloy starting material and G. J. Padilla and W. G. Thorn Jr., for their assistance in performing the experiment.
heat above the Neel temperature. The temperatures of these additional anomalies are below those of the resistance maxima. Data taken on a sample containing 30 at % Ce shows no resistivityminimum bur instead there is a plateau at 7.5oK. Using the revised estimate of Tk for the alloys one finds using the argument of ref. 2 that Tk for fi Ce is 1OoK and Tk for LY Ce is 1000oK. Obviously, the interpretation of Tk for @ Ce cannot be used literally since the Neel temperature TN for j3 Ce is 12.5oK. The fact that TN = Tk still does not preclude that resonant scattering is important in the concentrated alloys and /3 Ce since the effects of spin compensation begin above Tk.
STATISTICAL
23 September
LETTERS
HYDRODYNAMIC
EQUATIONS
GRAD-TYPE
T. N. KHAZANOVICH and V. A. SAVCHENKO Academy
Institute of Chemical Physics, of Sciences of the USSR, Moscow, Received
USSR
8 July 1968
Hydrodynamic equations corresponding to Grad’s ten-moment approximation are derived by Zubarev’s method of non-equilibrium statistical distributions. These equations are valid for two limiting cases: a) a dilute gas, b) a viscous liquid.
The Grad moment method [l] for dilute gases permits passing beyond the scope of usual hydrodynamics and treating faster processes. Zubarev’s non-equilibrium statistical distribution [2,3] provides a possibility for obtaining similar generalized equations valid for any fluid by more detailed setting of the system state. Such a possibility had been mentioned by Zubarev [3]. We try to realize this possibility in this paper. The non-equilibrium distribution has the form W = q exp{-A
+B)}
,
A =
Jd3r’
Cf#,t)P&-‘,t)
;
k t B =
ld3rr
s dt’ eXp{E(t’-t)} _m
Tfk(r,t’)Pk(rt9t’)
+
?fk(r’, t’) at, Pk(“*
t’)
,
where [also 41 the set of fk usually consists of inverse temperature, velocity and chemical potential of given point r, and Pk are the conjugate microscopic quantities. Kogan [5] has shown that Grad’s tenmoment equations for a dilute gas may be obtained by using the stress tensor oc,a( r, t) as an additional parameter for a non-equilibrium state description. Accordingly we add the moment flux density in the moving coordinate system Jo,( r, t) to the set Pk, the conjugate thermodynamic parameters .&, being 615
Volume 27A. number
9
PHYSICS
LETTERS
23 September
1968
defined by condition: (J,p) = (J,&, = -ucyp. where (. . . ) L denotes averaging with a local equilibrium distribution WL = ye exp(-A). Assuming as usual that the system is close to equilibrium we make our calculations keeping only terms linear in the thermodynamic parameters gradients and in tap. As a result we obtain that [ok proportional to the viscous stress-tensor uk1y/3 = oop + p&g. where p is the hydrostatic pressure. The ensemble average of the local conservation law (the balance equation) for J,B( r, t) gives the equation for the stress tensor:
where zl,(r.t) is the flow velocity. equation. respectively.
JO,p,/ar,
and G
are the “flux”
@P
and “source”
is
terms
in the balance
s 6(f?i - r )
(3)
J2Uij Go0 = - ;$
c i*j
32
Ia
(pi,, -
pjp) +
‘7Ria2Riy
(Rip - Rjp) (Pi,-
Pjy)
1
(subscript l’s” denotes symmetrisation of the tensor, I$i is the i-th molecule position vector, pi is its moment. UijC IRi - t?j / ) is the interaction energy). The linear approximation gives G (r-t) = (Gop( t.f))~ - (G,p( r.t) AB(t))L, where AB = B -(B)L. A very complicated integro-differentia “P equation for uop is obtained for the general case. It is reduced to a differential equation in two limiting cases: a) a dilute gas. b) a viscous liquid. In both cases there are two strongly different characteristic times: 71 “c 73. For gas ‘1 is the collision time, 73 is the time between collision. For viscous liquid ~1 and 72 are the moment and position correlation times, respectively. As it follows from eq. (3), the Go0 autocorrelation time is 71 in both limiting cases. Assuming that [as variation during the time interval of ‘1 is negligibly small, we can take to6 from under the integral with respect to time. Calculation of the correlation functions considering pair collisions ( a dilute gas) leads to Grad’s ten moment equation of the first order in gradients. In the second limiting case we obtain the equation:
J%,, =
?t
(Po_a)
J&b++!$!L + (PO-b)
(
P
>
(y
2 Y
%YR-
%a
-hu'
6 yy OR *
where PO is the equilibrium pressure, constants c and h being expressed in terms of moment time-correlation functions. In fact eq. (4) is the rheological equation for a Maxwellian fluid. The authors wish to express
their gratitude
to Dr. D. N. Zubarev for helpful discussions.
References 1. 2. 3. 4. 5.
H.Grad. Comm. Pure Appl. Math. 2 (1949) 331. D.N.Zubarev. Dokl. Akad. Nauk SSSR 140 (1961) 92; Dokl. Akad. Nauk SSSR 164 (1965) 537. D.N. Zubarev. Dissertation. Math. Inst. AC. Sci. USSR. 1965. Adv. Chem. Phys. 5 (1963) 266. J.A.McLennan. A.M.Kogan. Dokl. Akad. Nauk SSSR 158 (1964) 1054; Prikl. Matem. i Mekhan. 29 (1965) 122
*****
616
PHYSICS
27A, number 9
References 1. T. Sugawara and H. Eguchi. J. Phys. Sot. Japan 21 (1966) 725. 2. A. S. Edelstein, Phys. Rev. Letters 20 (1968) 1348. 3. Ph. W. Anderson. Phys. Rev. 164 (1967) 352. 4. A. S. Edelstein. Phys. Rev. 164 (1967) 510. 5. A. S. Edelstein, Phys. Rev. Letters 19 (1967) 1184. 6. M.D. Daybell and W. A. Steyert. Phys. Rev. Letters 18 (1967) 398. 7. G. J. Van den Berg. Progress in low temperature physics. Vol. IV! ed. C. J. Gorter (North-Holland. Amsterdam. 1964) p. 194. 8. L. M. Roberts and J. M. Lock. Phil. Mag. 2 (1957) 811.
The author wishes to thank R. J. Gambino who ***
**
DERIVATION
OF
OF THE
1968
prepared the alloy starting material and G. J. Padilla and W. G. Thorn Jr., for their assistance in performing the experiment.
heat above the Neel temperature. The temperatures of these additional anomalies are below those of the resistance maxima. Data taken on a sample containing 30 at % Ce shows no resistivityminimum bur instead there is a plateau at 7.5oK. Using the revised estimate of Tk for the alloys one finds using the argument of ref. 2 that Tk for fi Ce is 1OoK and Tk for LY Ce is 1000oK. Obviously, the interpretation of Tk for @ Ce cannot be used literally since the Neel temperature TN for j3 Ce is 12.5oK. The fact that TN = Tk still does not preclude that resonant scattering is important in the concentrated alloys and /3 Ce since the effects of spin compensation begin above Tk.
STATISTICAL
23 September
LETTERS
HYDRODYNAMIC
EQUATIONS
GRAD-TYPE
T. N. KHAZANOVICH and V. A. SAVCHENKO Academy
Institute of Chemical Physics, of Sciences of the USSR, Moscow, Received
USSR
8 July 1968
Hydrodynamic equations corresponding to Grad’s ten-moment approximation are derived by Zubarev’s method of non-equilibrium statistical distributions. These equations are valid for two limiting cases: a) a dilute gas, b) a viscous liquid.
The Grad moment method [l] for dilute gases permits passing beyond the scope of usual hydrodynamics and treating faster processes. Zubarev’s non-equilibrium statistical distribution [2,3] provides a possibility for obtaining similar generalized equations valid for any fluid by more detailed setting of the system state. Such a possibility had been mentioned by Zubarev [3]. We try to realize this possibility in this paper. The non-equilibrium distribution has the form W = q exp{-A
+B)}
,
A =
Jd3r’
Cf#,t)P&-‘,t)
;
k t B =
ld3rr
s dt’ eXp{E(t’-t)} _m
Tfk(r,t’)Pk(rt9t’)
+
?fk(r’, t’) at, Pk(“*
t’)
,
where [also 41 the set of fk usually consists of inverse temperature, velocity and chemical potential of given point r, and Pk are the conjugate microscopic quantities. Kogan [5] has shown that Grad’s tenmoment equations for a dilute gas may be obtained by using the stress tensor oc,a( r, t) as an additional parameter for a non-equilibrium state description. Accordingly we add the moment flux density in the moving coordinate system Jo,( r, t) to the set Pk, the conjugate thermodynamic parameters .&, being 615
Volume 27A. number
9
PHYSICS
LETTERS
23 September
1968
defined by condition: (J,p) = (J,&, = -ucyp. where (. . . ) L denotes averaging with a local equilibrium distribution WL = ye exp(-A). Assuming as usual that the system is close to equilibrium we make our calculations keeping only terms linear in the thermodynamic parameters gradients and in tap. As a result we obtain that [ok proportional to the viscous stress-tensor uk1y/3 = oop + p&g. where p is the hydrostatic pressure. The ensemble average of the local conservation law (the balance equation) for J,B( r, t) gives the equation for the stress tensor:
where zl,(r.t) is the flow velocity. equation. respectively.
JO,p,/ar,
and G
are the “flux”
@P
and “source”
is
terms
in the balance
s 6(f?i - r )
(3)
J2Uij Go0 = - ;$
c i*j
32
Ia
(pi,, -
pjp) +
‘7Ria2Riy
(Rip - Rjp) (Pi,-
Pjy)
1
(subscript l’s” denotes symmetrisation of the tensor, I$i is the i-th molecule position vector, pi is its moment. UijC IRi - t?j / ) is the interaction energy). The linear approximation gives G (r-t) = (Gop( t.f))~ - (G,p( r.t) AB(t))L, where AB = B -(B)L. A very complicated integro-differentia “P equation for uop is obtained for the general case. It is reduced to a differential equation in two limiting cases: a) a dilute gas. b) a viscous liquid. In both cases there are two strongly different characteristic times: 71 “c 73. For gas ‘1 is the collision time, 73 is the time between collision. For viscous liquid ~1 and 72 are the moment and position correlation times, respectively. As it follows from eq. (3), the Go0 autocorrelation time is 71 in both limiting cases. Assuming that [as variation during the time interval of ‘1 is negligibly small, we can take to6 from under the integral with respect to time. Calculation of the correlation functions considering pair collisions ( a dilute gas) leads to Grad’s ten moment equation of the first order in gradients. In the second limiting case we obtain the equation:
J%,, =
?t
(Po_a)
J&b++!$!L + (PO-b)
(
P
>
(y
2 Y
%YR-
%a
-hu'
6 yy OR *
where PO is the equilibrium pressure, constants c and h being expressed in terms of moment time-correlation functions. In fact eq. (4) is the rheological equation for a Maxwellian fluid. The authors wish to express
their gratitude
to Dr. D. N. Zubarev for helpful discussions.
References 1. 2. 3. 4. 5.
H.Grad. Comm. Pure Appl. Math. 2 (1949) 331. D.N.Zubarev. Dokl. Akad. Nauk SSSR 140 (1961) 92; Dokl. Akad. Nauk SSSR 164 (1965) 537. D.N. Zubarev. Dissertation. Math. Inst. AC. Sci. USSR. 1965. Adv. Chem. Phys. 5 (1963) 266. J.A.McLennan. A.M.Kogan. Dokl. Akad. Nauk SSSR 158 (1964) 1054; Prikl. Matem. i Mekhan. 29 (1965) 122
*****
616