Advanced
Computational
Physics
Course 17164
Hartmut Ruhl,
LMU, Munich
Lecturer
Advanced Computational Physics
Course 17164
Classical many
particle systems
A gas of hard
spheres
BBGKY-hierarchy
Hartmut Ruhl, LMU, Munich
April 27, May 02, 2017
Liouville equation
Literature
Advanced
Computational
Physics
Course 17164
Lecturer
Hartmut Ruhl,
LMU, Munich
Lecturer
Classical many particle systems
A gas of hard spheres
Classical many
particle systems
A gas of hard
spheres
BBGKY-hierarchy
Liouville equation
BBGKY-hierarchy
Liouville equation
Literature
Literature
Lecturer
Advanced
Computational
Physics
Course 17164
Hartmut Ruhl,
LMU, Munich
Lecturer
Classical many
particle systems
Hartmut Ruhl, ASC, room A 238, phone 089-21804210,
email [email protected].
Patrick Böhl, ASC, room A205, phone 089-21804640,
email [email protected].
A gas of hard
spheres
BBGKY-hierarchy
Liouville equation
Literature
Advanced
Computational
Physics
Course 17164
Classical many particle systems
To derive the Liouville equation we start by assuming that we have complete information about a
mechanical system of N particles. Hence, we require that the positions and momenta of all 6N particles
are known to us leading to the definition of the following probability density
C (~
x1 , ~
p1 , ..., ~
xN , ~
pN , ~
x01 , ~
p01 , ..., ~
x0N , ~
p0N , t)
=
N
3
Πk =1 δ [~
xk
(1)
Hartmut Ruhl,
LMU, Munich
Lecturer
Classical many
particle systems
3
−~
xk (~
x01 , ~
p01 , t)] δ [~
pk − ~
pk (~
x0N , ~
p0N , t)] ,
where the ~
x01~
p01 , ..., ~
x0N ~
p0N are the initial conditions. In what follows we will leave labels for the initial
conditions away. The trajectories ~
xk (t) and ~
pk (t) are known. Hence, we have
A gas of hard
spheres
BBGKY-hierarchy
∂C (~
x1 , ~
p1 , ..., ~
xN , ~
pN , t)
∂t
=
(2)
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−
N h
i
X
∂δ 3 [~
xj − ~
xj (t)] 3
N
3
3
Πk =1,k 6=j δ [~
xk − ~
xk (t)] δ [~
pk − ~
pk (t)] ~
x˙ j (t) ·
δ [~
pj − ~
pj (t)]
~
∂
x
j
j=1
−
N h
i
X
∂δ 3 [~
pj − ~
pj (t)]
N
3
3
3
.
Πk =1,k 6=j δ [~
xk − ~
xk (t)] δ [~
pk − ~
pk (t)] δ [~
xj − ~
xj (t)] ~
p˙ j (t) ·
∂~
pj
j=1
Making use of
3
3
m~
x˙ (t)j δ [~
pj − ~
pj (t)] = ~
pj δ [~
pj − ~
pj (t)]
we find
Liouville equation
(3)
Advanced
Computational
Physics
Course 17164
Classical many particle systems
Hartmut Ruhl,
LMU, Munich
Lecturer
∂C (~
x1 , ~
p1 , ..., ~
xN , ~
pN , t)
(4)
∂t
=
−
N h
i ~
X
pj
∂δ 3 [~
xj − ~
xj (t)] 3
N
3
3
Πk =1,k 6=j δ [~
xk − ~
xk (t)] δ [~
pk − ~
pk (t)]
·
δ [~
pj − ~
pj (t)]
m
∂~
xj
j=1
−
N h
i
X
∂δ 3 [~
pj − ~
pj (t)]
3
N
3
3
Πk =1,k 6=j δ [~
xk − ~
xk (t)] δ [~
pk − ~
pk (t)] δ [~
xj − ~
xj (t)] ~
p˙ j ·
∂~
pj
j=1
Classical many
particle systems
A gas of hard
spheres
BBGKY-hierarchy
Liouville equation
Literature
and
∂C
∂t
+
N
X
j=1
~
vj ·
∂C
∂~
xj
+
N
X
j=1
∂C
~
p˙ j ·
= 0,
∂~
pj
The function C is still a generalized function. Averaging C over all initial conditions compatible with an
experiment we obtain a smooth probability density function for N particles. We call it ρN .
(5)
Advanced
Computational
Physics
Course 17164
Classical many particle systems
A gas of N classical particles is represented in 6N-dimensional phase space. To describe N classical
particles the probability density
ρN (~
x1 , ~
p1 , ..., ~
xN , ~
pN , t)
Z 0
3
3
3
3
=
d x01 d p01 ...d x0N d p0N C (~
x1 , ~
p1 , ..., ~
xN , ~
pN , ~
x01 , ~
p01 , ..., ~
x0N , ~
p0N , t)
Hartmut Ruhl,
LMU, Munich
(6)
Lecturer
Classical many
particle systems
is introduced. The quantity ρN is the probability density. The primed integral represents the integration
over all initial conditions compatible with the experimental conditions.
The probability of finding particle 1 in the phase space volume element d 3 x1 d 3 p1 at ~
x1 , ~
p1 , particle 2 in
the phase space volume element d 3 x2 d 3 p2 at ~
x2 , ~
p2 and so on is given by
A gas of hard
spheres
BBGKY-hierarchy
Liouville equation
d
3N
xd
3N
p ρN (~
x1 , ~
p1 , ..., ~
xN , ~
pN , t) ,
(7)
where the normalization condition is
Z
N =
d
3N
xd
3N
p ρN (~
x1 , ~
p1 , ..., ~
xN , ~
pN , t) .
(8)
The temporal evolution of the probability density ρN is governed by the Liouville equation
dρN
dt
as can be infered from Eqn. (5)
(~
x1 , ~
p1 , ..., ~
xN , ~
pN , t) = 0
(9)
Literature
Advanced
Computational
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Course 17164
A gas of hard spheres
A classical gas consisting of N identical hard spheres with diameter σ is a sufficiently simple system for
educational purposes concerning the Liouville equation. Since the spheres do not interact as long as they
are sufficiently far apart from each other the Liouville equation for the problem is given by
∂t ρN +
N
X
Hartmut Ruhl,
LMU, Munich
Lecturer
~
vi · ∂~x ρN = 0 ,
i
i=1
|~
xi − ~
xj | > σ ,
i, j ∈ {1, ..., N} ,
i 6= j .
(10)
Classical many
particle systems
A gas of hard
spheres
The interaction of the spheres is a contact interaction. It amounts to the formulation of appropriate
boundary conditions, whenever two spheres touch each other. We find
BBGKY-hierarchy
ρN
0
0
... ~
xi , ~
pi ... ~
xj , ~
pj ... tc + = ρN ... ~
xi , ~
pi ... ~
xj , ~
pj ... tc − ,
0
h
i
~
pi = ~
pi − ~
nij ~
nij · (~
pi − ~
pj ) ,
~
vi =
~
nij =
~
pi
,
~
vi =
m
~
xi − ~
xj
|~
xi − ~
xj |
~
pi
0
h
i
~
pj = ~
pj + ~
nij ~
nij · (~
pi − ~
pj ) ,
(11)
(12)
0
m
(13)
(14)
at ~
xi − ~
xj = σ ~
nij for i, j ∈ {1, ..., N} , i 6= j at collision time tc . The ~
xi , ~
vi , ~
pi , i ∈ {1, ..., N} are the
pre-collision positions, velocities, and momenta of the spheres. The time tc − is the pre-collision time.
0
0
The ~
xi , ~
vi , ~
pi , i ∈ {1, ..., N} are the post-collision positions, velocities, momenta and tc + , the
post-collision time. At the volume boundaries ρN is assumed to disappear. This is the case if we assume
mirror reflection for the spheres there.
Liouville equation
Literature
Advanced
Computational
Physics
Course 17164
BBGKY-hierarchy
For hard spheres it can be shown by elementary menas that a hierarchy of equations for so called
reduced probability density functions is obtained. The hierarchy is called the BBGKY hierarchy
(Bogoliubov Born Green Kirkwood Yvon hierarchy). It is given by
∂t f
(s)
(~
x1 , ~
p1 , ..., ~
xs , ~
ps , t) +
s
X
i=1
=
s Z
X
3
Z
d ps+1
(s)
~
vi · ∂~x f (~
x1 , ~
p1 , ..., ~
xs , ~
ps , t)
i
dSis+1 ~
nis+1 · (~
vi − ~
vs+1 ) f
(s+1)
Hartmut Ruhl,
LMU, Munich
(15)
Lecturer
Classical many
particle systems
(~
x1 , ~
p1 , ..., ~
xs+1 , ~
ps+1 , t) ,
A gas of hard
spheres
i=1
where ~
xs+1 = ~
xi − σ ~
nis+1 . The unit vector ~
nis+1 points into the interior of sphere i and is normal to its
surface.
(s)
The quantity ρN is the s-particle probability density defined by
BBGKY-hierarchy
Liouville equation
Literature
f
(s)
(~
x1 , ~
p1 , ..., ~
xs , ~
ps , t)
=
N!
(N − s)!
Z
3
3
Z
d ps+1 d xs+1 ...
3
3
d xN d pN
(16)
s
N
×Πi=1 Πj=s+1 Θ |~
xi − ~
xj | − σ
×ρN (~
x1 , ~
p1 , ..., ~
xN , ~
pN , t) ,
where
1,
Θ |~
xi − ~
xj | − σ =
0,
|~
xi − ~
xj | > σ
else
blocks spherical volumes in configuration space that cannot be occupied by the hard spheres.
(17)
Advanced
Computational
Physics
Course 17164
Liouville equation
Given N point particles Γ = (t, q(t), p(t)) defines the set of all points of the system. The space Γ is
6N + 1 dimensional. The variables q(t) and p(t) are each 3N dimensional. We define a density function
ρ (q, p, t) : Γ → R, where dqdp ρ (q, p, t) is the number of points at time t in the volume dqdp.
Assuming that the number of points in the volume dV = dqdp is conserved it must hold
Hartmut Ruhl,
LMU, Munich
Lecturer
Z
∂t
v
a
Z
Z
a
dV ρ (q, p, t) = −
dSa v ρ (q, p, t) = −
V
(18)
V
∂V
i
j
= ..., q̇ , ..., ṗ , ... ,
a
dV ∂a v ρ (q, p, t) ,
∂a = ..., ∂qi , ..., ∂pj , ... .
(19)
In the limit V → 0 we obtain the continuity equation
(20)
i
ṗ = −∂ i H .
q
(22)
Making use of Eqs. (22) leads to
∂t ρ + ∂ i
q
∂ iH ρ − ∂ i ∂ iH ρ = 0.
p
p
q
Liouville equation
Literature
(21)
Let H (q, p, t) be the Hamiltonian of the system then we have
i
A gas of hard
spheres
BBGKY-hierarchy
a
∂t ρ (q, p, t) + ∂a v ρ (q, p, t) = 0 ,
i
i
∂t ρ (q, p, t) + ∂ i q̇ ρ (q, p, t) + ∂ i ṗ ρ (q, p, t) = 0 .
q
p
q̇ = ∂ i H ,
p
Classical many
particle systems
(23)
Advanced
Computational
Physics
Course 17164
Liouville equation
Hartmut Ruhl,
LMU, Munich
Further simplification leads to
∂t ρ + ∂ i H
∂ iρ − ∂ iH
∂ iρ = 0.
p
q
q
p
(24)
Classical many
particle systems
Introducing the Poisson bracket we obtain the Liouville equation
∂t ρ = {H, ρ} ,
Lecturer
{H, ρ} = ∂ i H
∂ iρ − ∂ iH
∂ iρ .
q
p
p
q
A gas of hard
spheres
(25)
BBGKY-hierarchy
Liouville equation
The reduced s-particle distribution function is defined as
Literature
f
(s)
(1...s) =
N!
Z
(N − s)!
dq
s+1
dp
s+1
N
N
... dq dp ρ (1...N) .
(26)
We assume that the Hamiltionian H can be written as
H (q, p, t) =
N
X
i=1
Ti +
N
X
i=1,j>i
Vij .
(27)
Advanced
Computational
Physics
Course 17164
Liouville equation
We split the Hamiltonian into the following parts

H (q, p, t) = 
s
X
s
X
Ti +
i=1
Hartmut Ruhl,
LMU, Munich


Vij  + 
i=1,j>i
N
X
i=s+1
N
X
Ti +

Vij  +
i=s+1,j>i
s X
N
X
Vij . (28)
i=1 j>s
Lecturer
Classical many
particle systems
We define
s
r
I
H (q, p, t) = H (q, p, t) + H (q, p, t) + V (q, p, t) ,
s
H =
s
X
i=1
r
H =
N
X
i=s+1
I
Vij ,
(30)
i=1,j>i
N
X
Ti +
s X
N
X
(31)
Vij .
(32)
i=1 j>s
We obtain
∂t ρ =
n
Liouville equation
Literature
Vij ,
i=s+1,j>i
V (q, p, t) =
A gas of hard
spheres
BBGKY-hierarchy
s
X
Ti +
(29)
o
s
r
I
H + H + V ,ρ .
(33)
Advanced
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Course 17164
BBGKY hierarchy
Partial integration yields
Hartmut Ruhl,
LMU, Munich
(N − s)! (s)
(s) (N − s)! (s)
∂t
f
− H ,
f
N!
N!
Z
n
o
s+1 s+1
N
N
r
I
=
dq
dp
... dq dp
H + V ,ρ .
(34)
Lecturer
Classical many
particle systems
It can shown
A gas of hard
spheres
Z
dq
s+1
dp
s+1
N
... dq dp
N
n
o
r
H ,ρ = 0.
(35)
BBGKY-hierarchy
Liouville equation
Please proof this as an exercise. We finally obtain
Literature
(N − s)! (s)
(s) (N − s)! (s)
∂t
f
− H ,
f
N!
N!
Z
n
o
s+1 s+1
N
N
I
=
dq
dp
... dq dp
V ,ρ .
(36)
If all particles are equal and cannot be distinguished from each other we find
∂t f
(s)
−
n
H
(s)
,f
(s)
o
=
s Z
X
i=1
dq
s+1
dp
s+1
n
Vis+1 , f
(s+1)
o
.
(37)
Literature
Advanced
Computational
Physics
Course 17164
Hartmut Ruhl,
LMU, Munich
Lecturer
Classical many
particle systems
A gas of hard
spheres
Carlo Cercignani, Mathematical Methods in Kinetic
Theory, Plenum Press.
BBGKY-hierarchy
Liouville equation
Literature