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

66 3 The Thermodynamical Arrow of Time
∫
H gen [ρ(α)] := ρ(α)ln ρ(α) dα = ln p. (3.53)
σ(α)
The final form on the RHS is appropriate, since the mean probability p(α)
for individual microscopic states and for given ρ(α) isthenp(α) =ρ(α)/σ(α).
The entropy defined by −kH gen is also known as the relative entropy of ρ(α)
with respect to the measure σ(α). The latter is often introduced ad hoc as
part of a phenomenological description.
Under the approximation w(α ′ ,α)=f(α)δ ′ (α − α ′ ) one now obtains the
deterministic ‘drift’ limit of the master equation (3.51) – usually representing
the first term of (3.44). It defines the first order of the Kramers–Moyal expansion
for w(α, α ′ ), equivalent to an expansion of ρ(α ′ ,t) in terms of powers of
α ′ −α at α ′ = α. The second order, w(α ′ ,α)=f(α)δ ′ (α−α ′ )+g(α)δ ′′ (α−α ′ ),
leads to the Fokker–Planck equation as the lowest non-trivial approximation
that leads to an irreversible equation (see de Groot and Mazur 1962, Röpke
1987). In this respect, it is analogous to the LAD equation as the lowest nontrivial
order in the Taylor expansion of the Caldirola equation (2.31). A master
equation is generally equivalent to a (stochastic) Langevin equation for individual
macroscopic trajectories α(t) which may form a dynamical ensemble
represented by ρ(α, t).
In contrast to the Liouville equation (3.26), the master equation (3.48) or
(3.36) cannot be unitary with respect to the inner product for probability distributions
defined above (3.27). While total probability must be conserved by
these equations, that of the individual trajectories cannot (see also Sect. 3.4).
Information-reducing master equations describe an indeterministic evolution,
which in general only determines an ever-increasing ensemble of different potential
successors for each macroscopic state (such as a point in α-space). 7
As discussed above, this macroscopic indeterminism is compatible with microscopic
determinism if that information which is transformed from relevant
7 The frequently used picture of a ‘fork’ in configuration space, characterizing a
dynamical indeterminism, may be misleading, since it seems to imply unique
predecessors. This would be wrong, as can be recognized, for example, in an
equilibrium situation. In the case of a stochastic dynamical law that is defined on a
finite set of states, a state must in general also have different possible predecessors,
corresponding to an inverse fork. Inverse forks by themselves would represent
a pure forward determinism (a ‘semigroup’, that may describe attractors). All
these structures are meant to characterize the dynamical law. They are neither
properties of the (f)actual history (which is assumed to evolve along a definite
trajectory regardless of the nature of the dynamical law), nor of an evolving
ensemble that represents a specific state of knowledge.
However, only dynamically unique predecessors may give rise to recordable
histories (consisting of ‘facts’ that are redundantly documented). The historical
nature of our world is thus based on a uniquely determined or even overdetermined
macroscopic past – see also footnote 1 of Chap. 2, Fig. 3.8, and Sect. 3.5.
A macroscopic history that was completely determined from its macroscopic past
would be in conflict with the notion of an (apparent) free will.