
direct study of the dynamical system associated with the par-
ticles. For instance, Du, Li, and Kadanoff’s corpuscular sys-
tem does not verify the equipartition of energy, but we do
not need to find out the cause of this behavior. First we
derive the balance equations of mass, momentum, and ki-
netic energy for the same system as in @1#. Then we apply
our method to a very simple situation such that all the terms
in the balance equations can be explicitly determined. The
constitutive relations are directly tested and we prove that in
this case the Fourier law of heat conduction is not valid.
Consider N identical sizeless particles of mass m . We
study the evolution of the system during the time interval I
5@ 0,T # . The particles are confined within the spatial interval
@0,1# by collisions with two walls of infinite mass. The col-
lisions between particles are inelastic and are characterized
by the numerical parameter e 5(1 2r )/2. The restitution co-
efficient r is defined in terms of the particles velocities after
and before the collision by v
1
8 2v
2
8 52r ( v
1
2v
2
). The par-
ticles interact only when they collide and there are no exter-
nal forces acting on the particles, except that due to the col-
lisions with the walls. We assume that the system kinematics
is known, i.e., the position of each particle i <N is a given
function of time x
i
: I →@ 0,1# . When a collision occurs, the
corresponding velocities j
i
5x
˙
i
undergo jumps and between
collisions the motion is uniform. We suppose that the total
number of collisions during I is finite.
Let w
i
( t ), t PI , be the real function of time describing the
variation of an arbitrary physical quantity w attached to the
i th particle. In the following, w
i
will represent only the mass
( m ), the momentum ( m j
i
), and the kinetic energy
( m j
i
2
/2). Since the only variations of the velocity j
i
are the
jumps from one constant value to another, the temporal de-
rivative of w
i
identically vanishes w
˙
i
[0 almost everywhere.
Consider two real parameters 0 ,t ,T /2 and a .0 and de-
fine the function
^
w & ~ x , t ! 5
1
4 t a
(
i 51
N
E
t 2t
t 1t
G
i
~ x , t 8 ! dt 8 , ~1!
where
G
i
~ x , t ! 5w
i
~ t ! H „a 2u x
i
~ t ! 2x u …, ~2!
and H is the left continuous Heaviside function. A nonvan-
ishing contribution to ^w& is due only to particles lying in the
spatial interval ( x 2a , x 1a ) during the temporal interval ( t
2t , t 1t ). Therefore, ^
w & ( x , t ) characterizes the mean dis-
tribution of w about the point x and the time t . It is a coarse-
grained average over the space and time intervals defined by
a and t, i.e., the density of w. Obviously, ^w& also depends on
the parameters a and t, but we do not explicitly write this
dependence. The average ^w& is nonvanishing only if x P
( 2a ,11a ) and the integral interval in Eq. ~1! is contained in
I only if t P( t , T 2t ).
For a given x , the integrand ~2! is a continuous function,
except at a finite number of points where it has discontinui-
ties of jump type. Hence G
i
is Riemann integrable and the
partial derivative with respect to t of ^w& is
]
t
^
w & 5
1
4 t a
(
i 51
N
@ G
i
~ x , t 1t ! 2G
i
~ x , t 2t !# . ~3!
The function ^w& depends on x through the instants u when
the i th particle enters or leaves the interval ( x 2a , x 1a ).
These instants are given by the zeros of the equations
x
i
~ u ! 2x 6a 50,
and using the implicit function theorem we obtain du / dx
51/j
i
( u ). If u P( t 2t , t 1t ), then u occurs as the integra-
tion limit in Eq. ~1! and the derivative of ^w& with respect to
x is
]
x
^
w & 5
1
4 t a
(
i 51
N
F
(
u PU
i
8
w
i
~ u !
j
i
~ u !
2
(
u PU
i
9
w
i
~ u !
j
i
~ u !
G
, ~4!
where U
i
8 ( U
i
9 ) is the set containing the instants when the
i th particle leaves ~enters! the interval ( x 2a , x 1a ) during
the interval ( t 2t , t 1t ). One can prove that the partial de-
rivatives ~3! and ~4! are almost everywhere continuous @11#.
Relation ~3! shows that ]
t
^
w & is related to the change of
G
i
from t 2t to t 1t . Since w
˙
i
[0 and H
˙
[0 almost every-
where, the changes of G
i
are only jumps. When the i th par-
ticle enters @leaves# the interval ( x 2a , x 1a ), the change of
G
i
is 1w
i
( u ) @ 2w
i
( u ) # . According to Eq. ~4!, the corre-
sponding part of ]
t
^
w & is equal to 2]
x
^
wj & . The change of
G
i
due to w
i
occurs when the particles collide inside the
interval ( x 2a , x 1a ). The collision part of ]
t
^
w & is
d
c
w 5
1
4 t a
(
i 51
N
(
s PV
i
@
w
i
~ s 20 ! 2w
i
~ s 10 !# , ~5!
where V
i
is the set containing the instants s when the i th
particle collides inside ( x 2a , x 1a ) during ( t 2t , t 1t ),
w
i
( s 10) is the limit to the left, and w
i
( s 20) is the limit to
the right. We deduce that the relation
]
t
^
w & 1]
x
^
wj & 5d
c
w ~6!
is always true. In the following we show that this identity is
the general form of the balance equations for the corpuscular
system considered.
First we apply the identity ~6! for mass, i.e., w
i
5m . The
space-time average ~1! becomes the mass density ^ m & . Since
the masses of the particles are identical, we have ^ m &
5mc , where c 5^ 1 & is the particle number density or the
concentration obtained from Eq. ~1!, for w
i
[1. The mean
velocity field v is defined by ^
j & 5c v if c 0 and is zero
otherwise. ~In @1# the velocity field is incorrectly defined, but
this oversight has not affected the hydrodynamic equations
for granular flows used in @1#.! The term d
c
w defined by Eq.
~5! vanishes because the collisions do not imply the variation
of the mass of particles or their number. Then relation ~6!
becomes
]
t
c 1]
x
~ c v ! 50, ~7!
which is the continuity equation.
For momentum, we choose w
i
5m j
i
and then ^
w &
5mc v . The second term on the left-hand side of Eq. ~6! can
6278 55 BRIEF REPORTS