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

44 3 The Thermodynamical Arrow of Time
consequences for many interacting particles. However, these uncertainties
cannot be based on the quantum mechanical uncertainty relations with
their corresponding phase space cells of size h 3N , since equivalent problems
reappear in quantum theory if phase space points are consistently
replaced by wave functions (see Sect. 4.1.1).
The time dependence of an individual point {p i (t),q i (t)} in Γ -space (with
i =1,...,3N), described by Hamilton’s equations, is equivalent to the simultaneous
time dependence of all N points in µ-space. Therefore, the time
dependence of an ensemble in Γ -space (represented by a distribution ρ Γ )determines
that of the corresponding density ρ µ . In contrast to the dynamics in
Γ -space (Sect. 3.1.2), however, this dynamics is not ‘autonomous’: the time
derivative of a non-singular density ρ µ is not determined by ρ µ . The reason is
that ρ Γ cannot be recovered from ρ µ in order to determine the latter’s time
derivative from that of the former. The mapping of Γ -space distributions on
µ-space distributions cannot be uniquely inverted, as it destroys information
about correlations between the particles (see also Fig. 3.1 and the subsequent
discussion). The smooth µ-space distribution may, for example, characterize a
‘macroscopic state’ in the sense mentioned in the introduction to this chapter.
Therefore, the envisioned chain of computation
ρ µ −→ ρ Γ
H
−→
dρ Γ
dt
−→ ∂ρ µ
∂t
, (3.2)
which would be required to derive an autonomous dynamics for ρ µ ,isbrokenat
its first link. Boltzmann’s attempt to bridge this gap by statistical arguments
will turn out to be the source of the time direction asymmetry in his statistical
mechanics, and similarly in other formulations of irreversible processes. His
procedure specifies a direction in time in a phenomenologically justified way,
although it was originally meant to represent a general approximation rather
than a modification of the Hamiltonian dynamics. One must then ask under
what circumstances it may be valid.
Boltzmann postulated a stochastic dynamical law of the form
∂ρ µ
∂t
=
{ ∂ρµ
∂t
}
free+ext
+
{ ∂ρµ
∂t
}
collision
. (3.3)
Its first term is defined to describe particle motion under external forces only.
It can be written as a continuity equation in 6-dimensional µ-space:
{ } ∂ρµ
= −div µ j µ := −∇ q·( ˙qρ µ ) −∇ p·(ṗρ µ )
∂t
free+ext
( p
)
= −∇ q·
m ρ µ −∇ p·(F ext ρ µ ) , (3.4)
where j µ is the current density in µ-space. In the absence of particle interactions
this equation would describe the dynamics of the ‘phase space fluid’