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

42 3 The Thermodynamical Arrow of Time
in conflict with unbiased statistical reasoning. The widespread ‘double standard’
of readily accepting improbable initial conditions while rejecting similar
final ones has been duly criticized by Price (1996).
Another argument against the statistical interpretation of irreversibility,
the recurrence objection (or Wiederkehreinwand), was raised much later by
Ernst Friedrich Zermelo, a collaborator of Max Planck at a time when the
latter still opposed atomism, and instead supported the ‘energeticists’, who
attempted to understand energy and entropy as fundamental ‘substances’.
This argument is based on a mathematical theorem due to Henri Poincaré,
which states that every bounded mechanical system will return as close as
one wishes to its initial state within a sufficiently large time. The entropy of
a closed system would therefore have to return to its former value, provided
only the function F (z) is continuous. This is a special case of the quasiergodic
theorem which asserts that every system will come arbitrarily close to
any point on the hypersurface of fixed energy (and possibly with other fixed
analytical constants of the motion) within finite time.
While all these theorems are mathematically correct, the recurrence objection
fails to apply to reality for quantitative reasons. The age of our Universe
is much smaller than the Poincaré recurrence times even for a gas consisting
of no more than a few tens of particles. Their recurrence to the vicinity
of their initial states (or their coming close to any other similarly specific
state) can therefore be excluded in practice. Nonetheless, some ‘foundations’
of irreversible thermodynamics in the literature rely on formal idealizations
that would lead to strictly infinite Poincaré recurrence times (for example
the ‘thermodynamical limit’ of infinite particle number). Such assumptions
are not required in our Universe of finite age, and they would not invalidate
the reversibility objection (or the equilibrium expectation, mentioned above).
However, all foundations of irreversible behavior have to presume some very
improbable initial conditions.
The theory of thermodynamically irreversible processes must therefore address
two main problems:
1. The investigation of realistic mechanisms which describe the dynamical
evolution away from certain (presumed) improbable initial states. This is
usually achieved in the form of ‘master equations’, which mimic a law-like
T-asymmetry – analogous to Ritz’s retarded action-at-a-distance in electrodynamics.
In contrast to electrodynamics, they describe the dynamics
of ensembles, equivalent to an effective stochastic dynamics for the individual
states (applicable in the ‘forward’ direction of time). These mechanisms
should be able to justify the representative ensembles (macroscopic
states) and even describe the emergence of order (Sect. 3.4).
2. The precise nature of the required improbable initial states. This leads
again to the quest for an appropriate cosmic initial condition, similar to
the global condition A µ in
= 0 in the early Universe that would be able to
explain the radiation arrow (see Sects. 2.2 and 5.3).