Physical meaning of δ2z of Glansdorff and Prigogine

Volume 57A, number 3

PHYSICS LETTERS

14 June 1976

PHYSICAL MEANING OF ö2z OF GLANSDORFF AND PRIGOGINE* Y. OONO* Department ofApplied Science, Faculty of Engineering, Kyushu University, Higashi.ku, Fukuoka 812, Japan Received 20 April 1976 It is shown that & 2z of Glansdorff and Prigogine is nothing but ~ 2s, where s is the entropy per unit mass, by taking into account that the inverse process of the dissipation of the macroscopic motion can occur through fluctuation.

It is a notable success that the non-equilibrium thermodynamics due to Glansdorff and Prigogine [I] can handle both thermodynamic and hydrodynamic

Local Gibbs relation is du

=

Tds

—

pdv + ~ j.&~dn 7,

stability questions. As was stressed by Landsberg [2], this is achieved by passing from the entropy s per unit mass to the quantity z = s u• u/2T (1) —

where u is the average macroscopic velocity of a fluid and T the absolute temperature. The application to hydrodynamics was made possible by the extension of the Gibbs—Duhem stability criterion (2)

(4)

‘V

where u is the internal energy per unit mass, v the specific volume, n7 the number of moles of the ychemical species per unit mass, p the pressure, and the chemical potential of the 7-species. Let e be the total energy per unit mass: =

C

U + I)

1)12.

(5)

Then (4) can be rewritten as de=Tds—pdv+ L..Ij2,,dn7 +v~dv.

to

(6)

‘V =

—

~u ~u/2T~ 0.

(3)

Hence the quantity z is of fundamental importance and must have deep physical meanings. However, the physical meaning of z has not been clearly stated. For instance, Glansdorff and Prigogine [1] introduce z only because ~ cannot be a Liapounoff function under the existence of convection. Taking into account that the inverse process of the dissipation of macroscopic motions can occur through fluctuation, we shall show that the criterion (3) is included in the criterion (2), and that the equal sign in (3) is somewhat misleading. The physical meaning of z is also briefly discussed at the end of the present letter.

Since the inverse process of the dissipation of macroscopic motion can occur through fluctuation, e and U can vary independently. Hence e, v, v and can be chosen as independent variables. Therefore, from (6), we obtain 2s = ö(T1)~e+ ~(~ ~ 7/fl~n~), —

~II~ ‘V

—

+

~(v/I’)

‘V

~v = T~[—öThs+ ~5pBo

tS4LLiSfl

‘V

‘V

(7)

—~D.(~U].

Denote the variation under constant v by the suffix u. Then (7) is readily rewritten as 2s = [~2s] ~, ~ v ~v/T. (8) —

~ This 24(1975)51 is the translation [in Japanese]. of the work reported in Busseikenkyu * Present adress: Research Institute of Industrial Science, Kyushu University, Higashi-ku, Fukuoka 812, Japan.

6

In the criterion (3), 62s is, rigorously speaking, [~2~~ and hence the criterion (3) is included in the criterion 207

Volume 57A, number 3

PHYSICS LETTERS

(2). Therefore the quantity z appearing in the criterion (3) is the entropy per unit mass itself. The equation for ~2(ps) corresponding to (8) can also be obtained as follows, where p is the density. Since e = Ts

—

pv +

~‘V~’V+

u v/2,

(9)

we get from (6) d(pe)

Td(ps)

+

14 June 1976

dorff and Prigogine [2]: 62~z) ~2(ps) p~• 6u/T. —

(12)

As has been shown in (8) and (11), the quantity z appearing in (3) and (12) is nothing but s itself. Hence the meaning of z defined by (1) is obscure, but z may be interpreted as an explicit expression of s divided into two parts: the one responding to the velocity variation and the remainder.

~p~d(pn 7)

+

d(pu o/2).

(10)

‘V

Taking pe, ~ and u as independent variables, we obtain from (10) 2(ps) = [~2(ps)]~ pS ii. 6u/T (11) 6which corresponds to the following equation of Glans—

208

References [1] fluctuations P. Glansdorff(Wiley and I. Interscience, Prigogine, Structure, London, stability 1971). and [2] P.T. Landsberg, Nature 238 (1972) 229.