Time-structure invariance criteria for closure approximations
Brian J. Edwards and Hans Christian O
¨
ttinger
Department of Materials, ETH Zu ¨rich, Institute of Polymers, Swiss Federal Institute of Technology Rheocenter,
CH-8092 Zu ¨rich, Switzerland
Received 21 April 1997
In many areas of physics, time evolution equations for moments of distributions are expressed in terms of
higher-order moments. Closure approximations are then introduced in an ad hoc fashion to reduce the higher-
order moments to functions of the lower-order ones. Herein, the time-structure invariance of the Poisson
bracket as manifested through the Jacobi identity is used to derive constraint relationships on these approxi-
mations. These constraints severely limit the allowable functionality of general closures and help to define the
boundaries within which future investigations should concentrate. S1063-651X9710410-X
PACS numbers: 05.70.Ln, 05.60.+w, 51.10.+y
INTRODUCTION
In many areas of physics, time evolution equations for
probability distributions are typically expressed in terms of
moments averaged over the probability space. These areas
include polymer dynamics 1, quantum field theory 2, sus-
pension and colloid fluid mechanics 3,4, kinetics of phase
transitions and spinodal decomposition 5,6, the mechanics
of turbulence 7–9, liquid-crystalline dynamics 1,6, the
fluid mechanics of immiscible blends 10,11, and statistical
mechanics 12,13. These moment evolution equations are
much simpler to solve computationally than the full prob-
ability distribution evolution equation, but typically involve
higher-order moments appearing in the equations for the
lower-order moments. Closure approximations are then in-
troduced in order to reduce the moments of higher order in
terms of lower-order ones, but, in many cases, very little
physical guidance is used in the construction of these closure
approximations, with the result that many of those used dis-
play aphysical behavior under certain circumstances. Fur-
thermore, with the few genuine restrictions on the allowable
functional forms of these approximations currently in use, an
infinite variety of them are available, each of which must be
tested individually, which is an even more computationally
intensive process than solving the full evolution equation for
the probability distribution itself, e.g., by simulation tech-
niques 14. The magnitude of literature devoted to testing
closure approximations is voluminous and testifies not only
to the importance of the issue, but also to the lack of physical
guidance that is available for their construction.
As a specific example of the use of closures in the me-
chanics of a suspension of particles in an incompressible
Newtonian fluid, the evolution equation for the second mo-
ment of the probability density function P( x, t ) is given by
4
DP
Dt
=
1
2
v
-
v
P
+
1
2
v
-
v
P
+
1
2
v
+
v
P
+
1
2
v
+
v
P
-2 Q
v
+2 D
-3 P
, 1
v being the velocity gradient tensor, the particle shape
factor, D the rotational diffusivity, and D (
*
)/ Dt the material
derivative of
*
. The quantities P and Q are defined as
P
p
p
d
3
p , Q
p
p
p
p
d
3
p ,
2
p being the unit vector pointing in the direction of the major
axis of a given particle and ( p, x, t ) the probability density
function. With these definitions, normalization of the distri-
bution function at any position x and time t requires that
P
=1 and Q
= P
.
Equation 1 would be a straightforward expression to
solve, even under general circumstances, were it not for the
fourth moment of the distribution Q appearing on its right-
hand side. Similarly to the procedure outlined above, an evo-
lution equation for the fourth moment can also be derived,
but it turns out to depend upon the sixth moment and so on,
ad infinitum. Thus, in order to gain any practical advantages
from Eq. 1 it is necessary to devise a closure approximation
for the fourth moment in terms of the second.
Another tangible example of the use of closure approxi-
mations is provided by the Doi-Ohta theory of incompress-
ible, immiscible blends 10. In this theory, the droplets of
the dispersed phase are taken to have a surface area per unit
volume Q given by
Q =
f n, x, t d
3
n , 3
where n is the outwardly directed unit vector normal to the
droplet interface. The conservative dynamics of the second
moment of the interphase density function f follow an evo-
lution equation of the form
DN
Dt
=-N
v
-N
v
+Z
v
, 4
where
PHYSICAL REVIEW E OCTOBER 1997 VOLUME 56, NUMBER 4
56 1063-651X/97/564/40977/$10.00 4097 © 1997 The American Physical Society