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We know only that the system is in a shaded region (Figure
39). Similarly, if we know some precise initial conditions corresponding
to a point in the system, we don't know the partition
to which it belongs, nor the age of the system. For such
systems we know therefore two complementary descriptions,
and the situation becomes somewhat reminiscent of the one
we described in Chapter VII, when we discussed quantum mechanics.
It is because of the existence of this new alternative, nonlocal
description, that we can make the transition from dynamics
to probabilities. We call the systems for which this is
possible "intrinsically random systems".
In classical deterministic systems, we may use transition
probabilities to go from one point to another on a quite degenerate
sense. This transition probability will be equal to one if
the two points lie on the same dynamic trajectory, or zero if
they are not.
In contrast, in genuine probability theory, we need transition
probabilities which are positive numbers between zero
and one. How is this possible? Here we see in full light the
conflict between subjectivistic views and objective interpretations
of probability. The subjective interpretation corresponds
to the situation where individual trajectories are not known.
Probability (and, eventually, irreversibility, closely related to
it) would originate from our ignorance. But fortunately, there
is another objective interpretation: probability arises as a result
of an alternative description of dynamics, a non-local description
which arises in strongly unstable dynamical systems.
Here, probability becomes an objective property generated
from the inside of dynamics, so to speak, and which expresses
a basic structure of the dynamical system. We have stressed
the importance of Boltzmann's basic discovery: the connection
between entropy and probability. For intrinsic random
systems, the concept of probability acquires a dynamical
meaning. We have now to make the transition from intrinsic
random systems to irreversible systems. We have seen that out
of unstable dynamical processes, we obtain two Markov
chains.
We may see this duality in a different way. Take a distribution
concentrated on a line (instead of being distributed on a
surface). This line may be vertical or horizontal. Let us look at