46
L. Rondoni
interaction potential as being 0 when the particles centers are at a distance larger
than their diameter, and infinitely high when this distance equals their diameter, so
that particles are neither deformed, nor they penetrate each other, when they collide.
Denote by
f
(N )
N (() = f
(N )
N (q 1 , p 1 ; . . . ;q N , p N ; t)
(1.178)
an initial density in the phase space M, with notation indicating that the joint probability density of N particles out of N is considered. In absence of external forces,
there are no accelerations between collisions, hence ˙
p i = 0 for every particle, and
the Liouville equation can be written as:
∂ f
(N )
N
∂t
+
∂
∂
· ˙
+ ˙
·
∂
∂
f
(N )
N
=
∂ f
(N )
N
∂t
+
N
i=1
p i ·
∂ f
(N )
N
∂q i
= 0
(1.179)
where we have used the fact that the dynamics is Hamiltonian, hence the first term
in brackets is null, and that ˙
= ( ˙
q 1 , 0; . . . ; ˙
q N , 0). If we introduce the s-particle
distribution function:
f
(s)
N (q 1 , p 1 ; . . . ;q s , p s ; t) =
dq s+1 dp s+1 . . . dq N dp N f
(N )
N (q 1 , p 1 ; . . . ; q N , p N ; t)
(1.180)
and we integrate the Liouville equation over the variables q s+1 , p s+1 . . . q N , p N , we
obtain [22]:
∂ f
(s)
N
∂t
+
s
i=1
p i ·
∂ f
(s)
N
∂q i
= F
(s)
( f
(s+1)
N
, f
(s+1)
N
)
(1.181)
where F is a function of the joint (s + 1) particles distribution before collision
f
(s+1)
N
, and of the corresponding distribution after collision f
(s+1)
N
. The important
fact is that (1.181) is not a colsed equation: computing f
(s)
N requires knowledge of
f
(s+1)
N
. In particular, taking s = 1 and making F
(1) explicit by solving the elastic
collision dynamics, one obtains:
∂ f
(1)
N
∂t
+ v ·
∂ f
(1)
N
∂q
= (N − 1)σ
2
[ f
(2)
N − f
(2)
N ] |v · n| dp ∗ dn
(1.182)
for the 1-particle probability distribution function, the result of projecting out all
particles but one, from the phase space probability distribution function f
(N )
N . Here,
σ
2 is the collision cross section, v= p becuase the mass of particles is 1, n is the unit
vector in the direction joining the centers of the two particles concerning f
(2)
N , the
distribution after collision, and f
(2)
N , the dsitribution before collision. In turn p ∗ is
the momentum of the second particle.
In order to solve this equation, one needs an assumption on f
(2)
N , because is
not possible to solve the whole of equations for the f
(s)
N . Boltzmann proposed the
following closure hypothesis:
L. Rondoni
interaction potential as being 0 when the particles centers are at a distance larger
than their diameter, and infinitely high when this distance equals their diameter, so
that particles are neither deformed, nor they penetrate each other, when they collide.
Denote by
f
(N )
N (() = f
(N )
N (q 1 , p 1 ; . . . ;q N , p N ; t)
(1.178)
an initial density in the phase space M, with notation indicating that the joint probability density of N particles out of N is considered. In absence of external forces,
there are no accelerations between collisions, hence ˙
p i = 0 for every particle, and
the Liouville equation can be written as:
∂ f
(N )
N
∂t
+
∂
∂
· ˙
+ ˙
·
∂
∂
f
(N )
N
=
∂ f
(N )
N
∂t
+
N
i=1
p i ·
∂ f
(N )
N
∂q i
= 0
(1.179)
where we have used the fact that the dynamics is Hamiltonian, hence the first term
in brackets is null, and that ˙
= ( ˙
q 1 , 0; . . . ; ˙
q N , 0). If we introduce the s-particle
distribution function:
f
(s)
N (q 1 , p 1 ; . . . ;q s , p s ; t) =
dq s+1 dp s+1 . . . dq N dp N f
(N )
N (q 1 , p 1 ; . . . ; q N , p N ; t)
(1.180)
and we integrate the Liouville equation over the variables q s+1 , p s+1 . . . q N , p N , we
obtain [22]:
∂ f
(s)
N
∂t
+
s
i=1
p i ·
∂ f
(s)
N
∂q i
= F
(s)
( f
(s+1)
N
, f
(s+1)
N
)
(1.181)
where F is a function of the joint (s + 1) particles distribution before collision
f
(s+1)
N
, and of the corresponding distribution after collision f
(s+1)
N
. The important
fact is that (1.181) is not a colsed equation: computing f
(s)
N requires knowledge of
f
(s+1)
N
. In particular, taking s = 1 and making F
(1) explicit by solving the elastic
collision dynamics, one obtains:
∂ f
(1)
N
∂t
+ v ·
∂ f
(1)
N
∂q
= (N − 1)σ
2
[ f
(2)
N − f
(2)
N ] |v · n| dp ∗ dn
(1.182)
for the 1-particle probability distribution function, the result of projecting out all
particles but one, from the phase space probability distribution function f
(N )
N . Here,
σ
2 is the collision cross section, v= p becuase the mass of particles is 1, n is the unit
vector in the direction joining the centers of the two particles concerning f
(2)
N , the
distribution after collision, and f
(2)
N , the dsitribution before collision. In turn p ∗ is
the momentum of the second particle.
In order to solve this equation, one needs an assumption on f
(2)
N , because is
not possible to solve the whole of equations for the f
(s)
N . Boltzmann proposed the
following closure hypothesis: