1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
51
The conceptual difference between Eqs. (1.185) and (1.193) is enormous: in the
first case, one considers the fraction of identical abstract systems that enjoy certain
properties to the total of an ideal hypothetical continuum of identical systems; in the
second case, one considers the mass of a single concrete, experimentally observable
system. Clearly, there is no stringent reason for one quite sophisticated abstract
notion to enjoy the same evolution of a materially touchable object. Nevertheless,
this is precisely what happens if the assumptions in the derivations above are valid.
That validity may well be too hard to prove within a mathematical framework, but
experience has demonstrated that it does hold in very many situations. This is one
case in which probability turns material: in some sense, probability can be identified
with mass.
We noted that the structure of Eqs. (1.181) preserves the factorization in s single
particle densities of the multiparticle densities, which is essential for the validity
of the Boltzmann equation. Why should the distribution of pairs of particles be
factorized to start with?
First of all, note that the construction of the Boltzmann equation proceeds from
expressing the evolution of the density due to the collisions among particles. If this
density is smooth at some point before two molecules collide, the question is whether
collisions tend to preserve smoothness or to produce wrinkles and eventually singularities. In other words, whether collisions tend to maximize or reduce the distances
among particles in (q,p) space. Smoothness is indeed required for the equation to
continue to hold in time, since derivatives have to exist. Therefore, because there
are TRI dynamics that produce singularities as well as TRI dynamics that smooth
out the distributions of particles both in phase space and in real space, we conclude
that the Boltzmann equation is suitable only for the second case [22]. Then, repeated
smoothing of the distribution gradually produces a homogenous state, which means
maximum microscopic disorder. In phase space, this corresponds to the uniform
distribution, known as the microcanonical ensmeble, which amounts to lack of correlations and, utlimately to the factorization of the distributions.
In fact, collisions of hard spheres are defocussing, which seems to imply a wider
volume occupied by the colliding particles, hence higher smoothness, after a collision. However, the situation is not that simple: Hamiltonian dynamics are time
reversal invariant (TRI), which is to say that there exists an operator i : M → M,
that anticommutes with the time evoution:
S
t i = i S
−t
, and ii = i
2
= , ∀ ∈ M
(1.195)
For instance, the operator defined by i(q, p) = (q, −p) is the best known time reversal operation. Now, the problem is that reversibility makes possible a focusing collision for each defocusing collision …(Fig. 1.10).
This is the basis of the so-called reversibility objection or Loschmidt paradox,
which states that the H-theorem cannot be a consequence of reversible microscopic
dynamics, such as that of hard spheres. In fact, Loschmidt correctly argued that if H
decreases in a time interval for a given choice of initial conditions, there is another
51
The conceptual difference between Eqs. (1.185) and (1.193) is enormous: in the
first case, one considers the fraction of identical abstract systems that enjoy certain
properties to the total of an ideal hypothetical continuum of identical systems; in the
second case, one considers the mass of a single concrete, experimentally observable
system. Clearly, there is no stringent reason for one quite sophisticated abstract
notion to enjoy the same evolution of a materially touchable object. Nevertheless,
this is precisely what happens if the assumptions in the derivations above are valid.
That validity may well be too hard to prove within a mathematical framework, but
experience has demonstrated that it does hold in very many situations. This is one
case in which probability turns material: in some sense, probability can be identified
with mass.
We noted that the structure of Eqs. (1.181) preserves the factorization in s single
particle densities of the multiparticle densities, which is essential for the validity
of the Boltzmann equation. Why should the distribution of pairs of particles be
factorized to start with?
First of all, note that the construction of the Boltzmann equation proceeds from
expressing the evolution of the density due to the collisions among particles. If this
density is smooth at some point before two molecules collide, the question is whether
collisions tend to preserve smoothness or to produce wrinkles and eventually singularities. In other words, whether collisions tend to maximize or reduce the distances
among particles in (q,p) space. Smoothness is indeed required for the equation to
continue to hold in time, since derivatives have to exist. Therefore, because there
are TRI dynamics that produce singularities as well as TRI dynamics that smooth
out the distributions of particles both in phase space and in real space, we conclude
that the Boltzmann equation is suitable only for the second case [22]. Then, repeated
smoothing of the distribution gradually produces a homogenous state, which means
maximum microscopic disorder. In phase space, this corresponds to the uniform
distribution, known as the microcanonical ensmeble, which amounts to lack of correlations and, utlimately to the factorization of the distributions.
In fact, collisions of hard spheres are defocussing, which seems to imply a wider
volume occupied by the colliding particles, hence higher smoothness, after a collision. However, the situation is not that simple: Hamiltonian dynamics are time
reversal invariant (TRI), which is to say that there exists an operator i : M → M,
that anticommutes with the time evoution:
S
t i = i S
−t
, and ii = i
2
= , ∀ ∈ M
(1.195)
For instance, the operator defined by i(q, p) = (q, −p) is the best known time reversal operation. Now, the problem is that reversibility makes possible a focusing collision for each defocusing collision …(Fig. 1.10).
This is the basis of the so-called reversibility objection or Loschmidt paradox,
which states that the H-theorem cannot be a consequence of reversible microscopic
dynamics, such as that of hard spheres. In fact, Loschmidt correctly argued that if H
decreases in a time interval for a given choice of initial conditions, there is another