- #106
vanesch
Staff Emeritus
Science Advisor
Gold Member
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Ah, we're getting somewhere Juan !
I'll cite Zeh too, from chapter 3 (the only one that matters here):
About a way of deriving apparent irreversible phenomena:
A bit further:
For all fairness, I add the following criticism:
Concerning your argument about Poincare recurrence, this is recognized NOT to be a valid criticism:
I'll cite Zeh too, from chapter 3 (the only one that matters here):
About a way of deriving apparent irreversible phenomena:
This improbable initial state is called Kaltgeburt.Their overwhelming occurrence in nature can thus be statistically derived
from the presumption of an improbable initial state.
A bit further:
I understand this that POSTULATING an improbable initial state is of course "in conflict with unbiased statistical reasoning" ! But it is *possible*.In order to describe the thermodynamical arrow of time statistically,
one either has to derive the Kaltgeburt in some form from a new and fundamental assumption, or simply to postulate it. The Second Law is by no
means incompatible with deterministic or T-symmetric dynamical laws; it is
just extremely improbable, and therefore in conflict with unbiased statistical
reasoning.
For all fairness, I add the following criticism:
However, we are talking now about HOW REASONABLE IT IS TO POSTULATE SPECIAL INITIAL CONDITIONS. Note that Zeh DOES NOT SAY that he cannot derive irreversibility from special initial conditions (on the contrary!) ; he's discussing about whether this is a reasonable thing to do.The widespread `double standard' of readily accepting improbable
initial conditions while rejecting similar Ønal ones has been duly criticized by
Price (1996).
Concerning your argument about Poincare recurrence, this is recognized NOT to be a valid criticism:
Concerning this objection, Zeh further notices:Another historically relevant objection (Zermelo's Wiederkehreinwand
or recurrence objection) is mathematically correct, but does not apply to a
sufficiently young universe (as ours seems to be). It can be based on a theorem
by Poincare 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 closed systems would therefore have to return to its former values,
provided only the function F(z) is continuous. This is a special case of the
quasi-ergodic theorem which asserts that every system will come arbitrarily
close to any point on the hypersurface of fixed energy (and possibly of fixed
other analytical constants of the motion) within finite time.
While these theorems are mathematically correct, the recurrence objection
fails to apply, since the age of our universe is much smaller than the
Poincare times of a gas consisting of as few as ten or twenty particles. Their
recurrence to the vicinity of their initial states (or their coming close to any
other similarly speciØc state) can therefore be practically excluded.
See, depending on how you read the text, you can illustrate one or the other point. What I read here (what I had read elsewhere and what I find plausible) is that there is no A PRIORI conflict between phenomenological irreversibility and time-symmetric microdynamics, but that this implies the so-called "Kaltgeburt" (improbable initial conditions). You can then go on about the reasonableness to POSTULATE this.Rather, they illustrate that some
kind of Kaltgeburt is required in order to derive the thermodynamical arrow.
The theory of thermodynamically irreversible processes thus has to 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.
Their ensemble dynamics is equivalent to a stochastic process
for individual states (applicable in the `forward' direction of time). These
asymmetric dynamical equations may then even describe the emergence
of order (Sect. 3.4).
2. The precise characterization of the required improbable initial states. This
leads again to the quest for an appropriate cosmic initial condition, similar
to the radiation arrow (cf. Sects. 2.2 and 5.3).