1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
49
Fig. 1.9 Subdivision of
available physical and
momentum space in discrete
cells, sufficiently large to
contain very many finite
sized particles, but
sufficiently small to look like
a point to one observer
fraction of identical N -particle systems, the ensemble described by f
(N )
N , that have
one particle within dqdp.
Of course, given a single system of interest, there is no need for any of its particles
to actually lie within the volume dqdp, since there is a complementary fraction of the
ensemble whose systems have no particles in dqdp. Therefore, one could conclude
that even the Boltzmann equation is immaterial, and there is no compelling reason
to take it as a description of a given material system. There is, however, a more
physical derivation of the Boltzmann equation, that does not start from a continuos
probability distribution on a very high dimensional continuos set of geometric points
called phase space M. This derivation considers a large, but finite number, of small,
but finite size, particles with positions and velocities in a given volume dqdp. The
distribution of these particles in such a volume is discrete, rather than continuous,
and requires a proper coarse graining to be well represented by a continuous density
of mass [28]. The construction requires the following steps.
Consider one system of N particles of unit mass, inside a given 3-dimensional
container. It is a single macroscopic object, which has got nothing to do with collections of identical systems, or with probability distributions on abstract spaces. Both
the container and the space of velocities are discretized in cells C i j of size qp
centered around a discrete set of lattice of points (q i , p j ), cf. Fig.1.9. The cells must
be sufficiently large that each of them contains a large number of particles n i j , so
that it makes sense to define the mass density ρ within them as the sum of the masses
per unit volume:
n i j (t) = ρ(q i , p j ; t))qp ;
i, j
n i j = N
(1.191)
Then, one may try to approximate this set of discrete mass densities ρ i j with a
continuous function ρ, so that
n i j (t) ≈
C i j
ρ(q, p; t)dqdp ; N =
i, j C i j
ρ(q, p; t)dqdp
(1.192)
49
Fig. 1.9 Subdivision of
available physical and
momentum space in discrete
cells, sufficiently large to
contain very many finite
sized particles, but
sufficiently small to look like
a point to one observer
fraction of identical N -particle systems, the ensemble described by f
(N )
N , that have
one particle within dqdp.
Of course, given a single system of interest, there is no need for any of its particles
to actually lie within the volume dqdp, since there is a complementary fraction of the
ensemble whose systems have no particles in dqdp. Therefore, one could conclude
that even the Boltzmann equation is immaterial, and there is no compelling reason
to take it as a description of a given material system. There is, however, a more
physical derivation of the Boltzmann equation, that does not start from a continuos
probability distribution on a very high dimensional continuos set of geometric points
called phase space M. This derivation considers a large, but finite number, of small,
but finite size, particles with positions and velocities in a given volume dqdp. The
distribution of these particles in such a volume is discrete, rather than continuous,
and requires a proper coarse graining to be well represented by a continuous density
of mass [28]. The construction requires the following steps.
Consider one system of N particles of unit mass, inside a given 3-dimensional
container. It is a single macroscopic object, which has got nothing to do with collections of identical systems, or with probability distributions on abstract spaces. Both
the container and the space of velocities are discretized in cells C i j of size qp
centered around a discrete set of lattice of points (q i , p j ), cf. Fig.1.9. The cells must
be sufficiently large that each of them contains a large number of particles n i j , so
that it makes sense to define the mass density ρ within them as the sum of the masses
per unit volume:
n i j (t) = ρ(q i , p j ; t))qp ;
i, j
n i j = N
(1.191)
Then, one may try to approximate this set of discrete mass densities ρ i j with a
continuous function ρ, so that
n i j (t) ≈
C i j
ρ(q, p; t)dqdp ; N =
i, j C i j
ρ(q, p; t)dqdp
(1.192)