1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
47
f
(2)
N (q 1 , p 1 ; q ∗ , p ∗ ; t) = f
(1)
N (q 1 , p 1 ; t) f
(1)
N (q ∗ , p ∗ ; t)
(1.183)
which is known in the kinetic theory of gases as the stosszahlansatz, or hypothesis
of molecolar chaos. Indeed, this assumption amounts to state that when two particles
collide, they are indpendent. As the dynamics of particles is deterministic, this statement may only have a statistical meaning, and requires some kind of randomness,
that can be legitimately called chaos.
16 This makes sense for systems with many particles, two of which collide at time t, since they may have hardly interacted before,
and can be considered independent. But clearly this independence does not hold for
particles which have just collided. Moreover, independence is harder to achieve for
s particles if s is larger, because s particles occupy a volume of order O(sσ
3
) and
particles cannot overlap. Therefore, s = 1 is the best candidate for the independence
of particles. Furthermore, for s = 1, the limit N → ∞, σ → 0 should foster the
validity of the stosszahlansatz, when the total cross section for collision does not
vanish, and a randomizing mechanism is in place. Grad identified the scaling regime
in which this makes sense, now known as Boltzmann-Grad limit [23]: it consists in
keeping N σ
2 positive and finite, while N grows, so that N σ
3
→ 0:
N σ
2
= constant > 0 =⇒ σ
2
∼
1
N
, N σ
3
∼ N
−
1
2
(1.184)
This way, there is a net effect coming from particles collisions, necessary to randomize the motion and, at the same time, the excluded volume that may lead to
correlations vanishes, while N can be as large as desired. Physically, this picture
corresponds to a rarefied gas, in which particles collide, but their interaction energy
is negligible compared to their kinetic energy. The resulting equation:
∂ f
∂t
+ v ·
∂ f
∂q
= (N − 1)σ
2
[ f
(1)
f
(1)
∗
− f (1) f
(1)
∗ ] |v · n| dp ∗ dn
(1.185)
is the celebrated Boltzmann equation, where we have simplified notation writing f
in place of f
(1)
N . Its applicability goes well beyond the bounds of its strict derivation,
which is that of rarefied gases in regular containers. For instance, introducing external
electric potentials, it is applied to transport of electrons in solids [24]; adding nuclear
cross sections, it is applied to transport of neutrons in conventional nuclear reactors,
or to cold as well as hot nuclear fusion technology [25, 26]; in linearized and/or
discretized versions, such as those known as lattice Boltzmann models, it is applied
to a great variety of fluids, including blood cells in blood vessels [27].
An important result concerning Eq. (1.185) is that f
(2)
N remains factorized in time,
if it is such at start, keeping the validity of the model in time. More precisely, assuming
that f
(s)
N exists and is well behavied for any fixed s, when N → ∞ and N → σ
2
16 Note: in the theory of dynamical systems, the term chaos is often used to indicate systems that
have at least on positive Lyapunov exponent. This notion was not available to Boltzmann and,
indeed, he did not need this notion of chaos.
47
f
(2)
N (q 1 , p 1 ; q ∗ , p ∗ ; t) = f
(1)
N (q 1 , p 1 ; t) f
(1)
N (q ∗ , p ∗ ; t)
(1.183)
which is known in the kinetic theory of gases as the stosszahlansatz, or hypothesis
of molecolar chaos. Indeed, this assumption amounts to state that when two particles
collide, they are indpendent. As the dynamics of particles is deterministic, this statement may only have a statistical meaning, and requires some kind of randomness,
that can be legitimately called chaos.
16 This makes sense for systems with many particles, two of which collide at time t, since they may have hardly interacted before,
and can be considered independent. But clearly this independence does not hold for
particles which have just collided. Moreover, independence is harder to achieve for
s particles if s is larger, because s particles occupy a volume of order O(sσ
3
) and
particles cannot overlap. Therefore, s = 1 is the best candidate for the independence
of particles. Furthermore, for s = 1, the limit N → ∞, σ → 0 should foster the
validity of the stosszahlansatz, when the total cross section for collision does not
vanish, and a randomizing mechanism is in place. Grad identified the scaling regime
in which this makes sense, now known as Boltzmann-Grad limit [23]: it consists in
keeping N σ
2 positive and finite, while N grows, so that N σ
3
→ 0:
N σ
2
= constant > 0 =⇒ σ
2
∼
1
N
, N σ
3
∼ N
−
1
2
(1.184)
This way, there is a net effect coming from particles collisions, necessary to randomize the motion and, at the same time, the excluded volume that may lead to
correlations vanishes, while N can be as large as desired. Physically, this picture
corresponds to a rarefied gas, in which particles collide, but their interaction energy
is negligible compared to their kinetic energy. The resulting equation:
∂ f
∂t
+ v ·
∂ f
∂q
= (N − 1)σ
2
[ f
(1)
f
(1)
∗
− f (1) f
(1)
∗ ] |v · n| dp ∗ dn
(1.185)
is the celebrated Boltzmann equation, where we have simplified notation writing f
in place of f
(1)
N . Its applicability goes well beyond the bounds of its strict derivation,
which is that of rarefied gases in regular containers. For instance, introducing external
electric potentials, it is applied to transport of electrons in solids [24]; adding nuclear
cross sections, it is applied to transport of neutrons in conventional nuclear reactors,
or to cold as well as hot nuclear fusion technology [25, 26]; in linearized and/or
discretized versions, such as those known as lattice Boltzmann models, it is applied
to a great variety of fluids, including blood cells in blood vessels [27].
An important result concerning Eq. (1.185) is that f
(2)
N remains factorized in time,
if it is such at start, keeping the validity of the model in time. More precisely, assuming
that f
(s)
N exists and is well behavied for any fixed s, when N → ∞ and N → σ
2
16 Note: in the theory of dynamical systems, the term chaos is often used to indicate systems that
have at least on positive Lyapunov exponent. This notion was not available to Boltzmann and,
indeed, he did not need this notion of chaos.