54
L. Rondoni
Fig. 1.12 LTE condition: the macroscopic system of interest is subdivided in cells, the circular
volumes, are large compared to the microscopic space scales, so they may contain many particles,
but are small compared to the macroscopic scales, the scales at which measurement takes place. LTE
is established if the state of these cells becomes homogeneous in a time δt large compared to the
microscopic times, but small compared to the macroscopic characterisitc times. The condition δL
ensures that many particles are contained in a volume δL 3 , which experience many collisions in
a short time. Collisions lead to a homogeneous state within the cell, which is then well represented
by a single point, and also help particles in the bulk of a celle to remain inside that cell
Fig. 1.13 Two snapshots of a billiard table with only two balls. It is not possible to distribute the
mass uniformly on the table; equivalently, it is not possible to identify the direction of time: which
snapshot was taken earlier? They are equally plausible
of correlations both in space and in time, with cells surface effects negligible with
respect to their bulk. Then, each δL
3 volume may be considered as a small isolated
equilibrium system. In the case of kinetic theory of gases, is the mean free path
and τ the mean free time, cf. Fig.1.12.
As is well known, a homogeneous probability distribution is achieved in the phase
space of hard spheres, but that does not mean that a homogeneous mass distribution
is automatically achieved in real space. Indeed, even a billiard with only two balls
has the uniform probability distribution in phase space as its invariant probability
density; but it cannot have a uniform mass distribution in real space. For that, large
N is required as illustrated in Figs.1.13 and 1.14.
This example shows that relaxation of mass distribution to the homogeneous state,
which is an irreversible process required for LTE to be defined, is not possible unless
the cells contain many particles, and the particles interact. In fact, in the case of two
balls, while probability irreversibly relaxes to the uniform phase space distribution,
no direction of time can be perceived in the motion of balls Fig.1.13. At the same
time, that balls are many does not guarantee that they converge to a (stable) uniform
mass distribution; that requires collisions. In case of pointlike particles, an initial
inhomogeneous mass distribution may be preserved in time.
This way, we have further demonstrated that the identification of probability in
phase space and mass in real space is a delicate point of statistical mechanics. The
first is an abstract notion referring to an hyothetical ensemble of identical objects;
L. Rondoni
Fig. 1.12 LTE condition: the macroscopic system of interest is subdivided in cells, the circular
volumes, are large compared to the microscopic space scales, so they may contain many particles,
but are small compared to the macroscopic scales, the scales at which measurement takes place. LTE
is established if the state of these cells becomes homogeneous in a time δt large compared to the
microscopic times, but small compared to the macroscopic characterisitc times. The condition δL
ensures that many particles are contained in a volume δL 3 , which experience many collisions in
a short time. Collisions lead to a homogeneous state within the cell, which is then well represented
by a single point, and also help particles in the bulk of a celle to remain inside that cell
Fig. 1.13 Two snapshots of a billiard table with only two balls. It is not possible to distribute the
mass uniformly on the table; equivalently, it is not possible to identify the direction of time: which
snapshot was taken earlier? They are equally plausible
of correlations both in space and in time, with cells surface effects negligible with
respect to their bulk. Then, each δL
3 volume may be considered as a small isolated
equilibrium system. In the case of kinetic theory of gases, is the mean free path
and τ the mean free time, cf. Fig.1.12.
As is well known, a homogeneous probability distribution is achieved in the phase
space of hard spheres, but that does not mean that a homogeneous mass distribution
is automatically achieved in real space. Indeed, even a billiard with only two balls
has the uniform probability distribution in phase space as its invariant probability
density; but it cannot have a uniform mass distribution in real space. For that, large
N is required as illustrated in Figs.1.13 and 1.14.
This example shows that relaxation of mass distribution to the homogeneous state,
which is an irreversible process required for LTE to be defined, is not possible unless
the cells contain many particles, and the particles interact. In fact, in the case of two
balls, while probability irreversibly relaxes to the uniform phase space distribution,
no direction of time can be perceived in the motion of balls Fig.1.13. At the same
time, that balls are many does not guarantee that they converge to a (stable) uniform
mass distribution; that requires collisions. In case of pointlike particles, an initial
inhomogeneous mass distribution may be preserved in time.
This way, we have further demonstrated that the identification of probability in
phase space and mass in real space is a delicate point of statistical mechanics. The
first is an abstract notion referring to an hyothetical ensemble of identical objects;