Time evolution of reduced phase-space densities. BBGKY hierarchy

N.BORGHINI Topics in nonequilibrium physics
Remark: In the absence of interactions, the Hamilton function (III.2) reduces obviously to that of
an ideal gas.
III.2 Time evolution of reduced phase-space densities
We now deduce from the Liouville equation (III.1) the equations of motion for the reduced
phase-space densities, first for a system of particles in the absence of vector potential (Sec. III.2.1),
then for charged particles in a external vector potential (Sec. III.2.2).
Throughout this Section, we shall use the shorthand notations
Fi ≡ − ∇ri V (ri), (III.4a)
Kij ≡ − ∇ri W
ri − rj
, (III.4b)
for the forces upon particle i due to the external potential V and to particle j = i, respectively. In
accordance with Newton’s third law,
Kij = − Kji, (III.4c)
which follows from ∇rj Wij = − ∇ri Wij and the relabeling of particles.
III.2.1 System of neutral particles
In the absence of external vector potential, the Hamilton equations (II.4) with the Hamilton
function (III.2) read
˙ri = ∇pi HN = pi
m , pi
˙ = − ∇ri HN = Fi +
Kij. (III.5)
The first equation simply states that linear momentum and canonical momentum coincide, from
where it follows that the second equation is Newton’s second law.
Accordingly, the Liouville equation (III.1) becomes
∂ρN
∂t +
N
vi · ∇ri ρN +
i=1
N
Fi · ∇pi ρN +
i=1
N
i=1
j=i
N
Kij · ∇pi ρN = 0. (III.6)
III.2.1 a BBGKY hierarchy
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
The reduced k-particle phase density fk(t, r1, p1, . . . , rk, pk) follows from the Γ-space probability
density ρN after integrating out the degrees of freedom of the remaining N − k particles. Similarly,
integrating the Liouville equation (III.6) over the positions and momenta rj, pj of N − k particles
gives the evolution equation for fk.
This integration relies on a simple fact, namely that the probability density ρN vanishes when
one of its phase-space variables goes to infinity, to ensure the normalization condition (II.1). As a
consequence, integrals of the type
˙ri · ∇ri ρN d 3
ri or ˙pi · ∇pi ρN d 3 pi
(III.7)
identically vanish. This is here trivial because ˙ ri is independent of ri and ˙ pi is independent of pi,
so that they can be taken out of the respective integrals.
Additionally, we assume that we are allowed to exchange the integration and partial differentiation
operations when needed.
Integrating out N − 1 particles with the proper normalization factor αN,1 = N/(2π) 3 thus
yields for the evolution of the single-particle phase-space density (II.18a) the equation
∂f1(t, r1, p1)
∂t
j=1
j=i
+ v1 · ∇r1 f1(t, r1, p1) + F1 · ∇p1 f1(t, r1, p1)
+
N
j=2
N
(2π) 3
K1j · ∇p1 ρN
t, {ri}, {pi} d 6(N−1) V = 0.
III. Evolution of the reduced classical phase-space densities 37