The Physical Basis of The Direction of Time (The Frontiers ...

48 3 The Thermodynamical Arrow of Time
always an appropriate definition of entropy. It will indeed turn out to be insufficient
when correlations between particles become essential, as is the case,
for example, for real gases or solid bodies. Taking them into account requires
more general concepts, which were first proposed by Gibbs. His approach will
also allow us to formulate the exact ensemble dynamics in Γ -space, although
it cannot yet explain the origin of the thermodynamical arrow of time (that
is, of the low-entropy initial conditions).
3.1.2 Γ -Space Dynamics and Gibbs’ Entropy
In the preceding section, Boltzmann’s smooth phase space density ρ µ was justified
by means of small uncertainties in particle positions and momenta. It
describes an infinite number (a continuum) of possible single-particle states,
for example each particle represented by a small volume element ∆V µ .Anobjective
(‘real’) state would instead be described by a point (or a δ-distribution)
in Γ -space, or by a sum over Nδ-functions in µ-space. This would then lead
to an infinite value of Boltzmann’s H-functional, or negative infinite entropy.
However, the finite value of S µ [ρ µ ], derived from the smooth µ-space distribution,
is not just a measure of this arbitrary smoothing procedure (for
example representing the size of the volume elements ∆V µ ). If N points are
replaced by small but overlapping volume elements, this leads to a smooth
distribution ρ µ whose width reflects that of the discrete (real) distribution
of particles. Therefore, S µ characterizes the real physical state. The formal
‘renormalization of entropy’, which is part of this smoothing procedure, adds
an infinite positive contribution to the infinite negative entropy corresponding
to a point in such a way that the finite result S µ [ρ µ ]isphysically meaningful.
The ‘representative ensemble’ obtained in this way defines a finite measure of
probability (in the sense of the introduction to this chapter) for the N! points
in Γ -space. It depends only slightly on the precise smoothing conditions, provided
the discrete µ-space distribution is already smooth in the mean.
The ensemble concept introduced by Josiah Willard Gibbs (1902) differs
from Boltzmann’s at the very outset. He considered probability densities
ρ Γ (p, q) with ∫ ρ Γ (p, q)dp dq = 1 – from now on writing p := p 1 ,...,p 3N ,
q := q 1 ,...,q 3N and dp dq := d 3N pd 3N q for short, which are meant to describe
incomplete information (‘ignorance’) about microscopic degrees of freedom.
For example, a probability density may characterize a macroscopic (incomplete)
preparation procedure. Boltzmann’s H-functional is then replaced by
Gibbs’ formally analogous extension in phase η :
∫
η[ρ Γ ]:=ln ρ Γ = ρ Γ (p, q)lnρ Γ (p, q)dp dq . (3.17)
It leads generically to a finite ensemble entropy S Γ := −kη[ρ Γ ]. For a probability
density that is constant on a phase space volume element of size ∆V Γ (while
vanishing elsewhere), one has η[ρ Γ ]=− ln ∆V Γ . The entropy S Γ = k ln ∆V Γ