Connection between the local Maxwell and local grand canonical distributions

Volume 51A, number 4

PHYSICS LETTERS

10 March 1975

CONNECTION BETWEEN THE LOCAL MAXWELL AND LOCAL GRAND CANONICAL DISTRIBUTIONS J.A. DOMARADZKI and J. PIASECKI Institute of TheoreticalPhysics, Warsaw University, Warsaw, Poland Received 10 December 1974 The one-particle distribution corresponding to the local grand canonical ensemble is calculated rigorously. It is shown to coincide with the local Maxwell distribution provided the macroscopic parameters characterizing the ensernble are chosen properly. Their physical meaning is discussed.

In the classical Chapman-Enskog development the role of the zeroth order approximation to the solution of the Boltzmann equation is played by the local Maxwell distribution [1] flocfr, v) = n(r) [2irkT(r)jexp [_ 2kT(r) 21 (1) [ m 13/2 1 m(v—u(r))

j

representing the density of particles with velocity v at point r in a system with local number density n(r), velocity u(r) and temperature T(r); m is the particle mass and k the Boltzmann constant. In complete analogy to the Chapman-Enskog procedure a perturbative approach to the Liouville equation has been developed having in view the derivation of the correlation function formulae for the transport coefficients [2] (for further references see [3]). In this general approach the role of the zeroth order term is attributed to the local grand canonical ensemble defined by ploc(rN) = [Qloc]_l exP(_ fdr[13(r)e(riFN)

—

vfr)ñ(riFN)]}.

(2)

Probability density pl0~~(FN) for the occurrence of states rN = (r tions r1

...

rN and velocities V1

...

1 ... rN, V1 ... VN) withN particles having posiVN is written in eq. (2) in terms of microscopic number density

N

fl(rif’N)

=

~ &(r—,~)

(3)

i=1

and energy density

ê(rIFN)=~[+m(v,_u(r))2 i1

i-f ~ V(r11)]&(r_ri)~

(4)

J~i

where V(r~1)is the pair interaction depending on distance r11 = I I; Q10c is the normalization factor. Ensemble (2) can be used for the description of the local equilibrium properties of the steady state [7] (for the discussion of the quantum case see ref. [8]). An interesting question, of fundamental importance for the transport theory, is that of the connection between distributions (1) and (2). It seems strange that despite its basic character this problem has not been yet completely clarffied (see [4]). Our object is~thusto provide a full explanation of the connection between the two distributions. To this end we calculate the reduced one-particle distribution —

Floc(r, v) 196

~

6(r—t~)&(v—Vi)~ioc ~0fd

rN[~ &(r— r~)&(v—vj)] ploc(rN)

(5)

Volume 51A, number 4

where dl’N

=

PHYSICS LETTERS

[Mh3NI

1 dr

1 ... drN dv1 can be written as a product of function

exp

10 March 1975

...

doN, and h is the Planck constant. According to eqs. (2), (4) ploc(rN)

[_fdr(3(r) 2~4m(v~_u(r))2&(r_ ri))

=

exp {-(3(r~)fm(v~ u(r~))2}

(6)

-

and a term which does not depend on particle velocities. Using the identity

fdvj &(r ,~) &(v— v~)H exp { —(3~r~~ +m~v~ —u~rj))2} —

~7)

N =

L

2~

j

exp { +m(3(r)(v 0(r))2) fduj &(r— ,~,)[I exp { —

—

—

f m13(r

2]

1)(v~ 0(r,)) —

we can rewrite eq. Floc(r, v)

=

(5) in the form

[m(3(r)]312 exp {

—

f m13(r)(v— u~r))2}E

fdrN[

~

&(r r —

1)] ploc(rN)

(8)

and in this way we arrive at an important result

312 exp {

—

÷ m13(r)(v —0(r))2)

(9)

Fb0c(r, v) = 1oc[m13(r)] (see eq. (3)). The structure of function Fl0~turns out to be the same as that of function floc. This leads to the conclusion that the local grand canonical ensemble (2) describes a state with local temperature T(r), hydrodynamic velocity u(r) and number density n(r) if and only if

(3(r)

=

l/kT(r),

0(r) =

and the macroscopic parameter v(r) is chosen in such a way as to satisfy the condition <ñ(riI’N))Ice =n(r).

(10)

Indeed, in this case distributions (1) and (9) coincide. There remains the question of the physical signifIcance of parameter v(r). From eq. (10) by functional differ-

entiation we obtain &n(r) = fdr’ [gloc(r r’) &v(r’)

— g~C

(r, r’) &(3(r’)]

(11)

where g~ and g~ are the local density-density and energy-density correlation functions, respectively. Eq. (11) has been extensively discussed in the case when variation &n(r) is calculated around complete equilibrium corresponding to constant temperature T = 1 ikl3, density n and chemical potential p = vi(3 (see [5—7]). In this special, but physically most important situation, on neglecting the spatial variation of macroscopic parameters over distance~ f the order of correlation lengths, one gets

(~)

6n(r) = &v(r) fdr’ g~(r’) &(3(r) fdr’ g~(r’)= &v(r)~(~) &(kr)~ —

—

(12)

which is equivalent to the well known Gibbs-Duhem equation. Therefore, within this approximation we can write

i(r) = (3(r) p(r) 197

Volume 51A, number 4

PHYSICS LETTERS

10 March 1975

where p(r) is the local chemical potential.

References [1] S. Chapman and T.G. Cowling, The mathematical theory of non-uniform gases (Cambridge University Press, London, 1970). [2] H. Mod, Progr. Theoret. Phys. (Kyoto) 33 (1965) 423. [3] M.H. Ernst, L.K. Haines andJ.R. Dorfman, Rev. Mod. Phys. 41(1969)296. [4] P. R~sibois,Bull. Cl.

Sci. Acad. R. Belg.

56 (1970) 160.

LP. Kadanoff and P.C. Martin, Ann. Phys. 24 (1963) 419. [6] J.A. McLennan, Phys. Fluids 3 (1960) 493. [7] D.N. Zubarev, Neravnovesnaya statisti~eskayatermodynamika (Nauka, Moskva, 1971). [8] H.N.V. Temperley, Proc. Phys. Soc. 70 (1957) 577. [5]

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