content: text: Probability, Combinatorics and Control

content: text: between this and the fact that the development of the underlying wave function for
the whole universe is unique.

content: text: Summarizing:

content: text: Definition 1. A universe U is a chain of states (one state Ut for each moment of
time t), with the property that the transition between adjacent states is always
possible.

content: text: Definition 2. A multiverse M is the set of all possible universes U in the sense of
Definition 1 together with a probability measure on this set.

content: text: It may of course be said that quantum mechanics should allow for transitions
between all kinds of states, although the probability for most such transitions may be
extremely small. In this extremely simplified treatment, I will assume that for a given
state at a given moment of time t, the dynamical laws will only permit transitions to a
very limited number of states at the previous and next moments, which will make the
probabilistic part of the investigation particularly simple. However, modifications are
called for near the endpoints (the Big Bang and the Big Crunch); see Section 5.

content: text: As it stands, the model presented so far is too simple to generate any results. In
fact, there are no observable differences at all between the states, which mean that
there are no measurable variables which could be related to the (so far non-
specified) dynamics.

content: text: There are of course many different variables which we can choose to enrich this
structure, and which ones to choose must depend on what properties we want to
explain. For explaining the second law of thermodynamics, the obvious choice is the
entropy.

content: text: 4. Entropy

content: text: According to Boltzmann, the total entropy of a certain macro-state at a certain
time is given by

content: text: S=k_B\ln\Omega,

content: text: (2)

content: text: or inversely

content: text: \Omega=W^S,\text{with}W=e^{1/k_B},

content: text: (3)

content: text: where � denotes the number of corresponding micro-states and kB is
Boltzmann's constant.

content: text: This formula was from the beginning derived for simple cases, like an ideal gas.
Nevertheless, it does represent a kind of universal truth in statistical mechanics: the
number of possible micro-states corresponding to a given macro-state grows expo-
nentially with the entropy. Although there are many complications when one tries
to consider the entropy of the universe as a whole, I will still take it as the starting
point for the discussion that the entropy (at a given time t) is an exponential
function of the total entropy as in (3). A more difficult question is if and how the
constant W may vary with time, but for the purpose of the present paper, I will
simply let it be constant.

content: text: One may of course argue that this can only be true when the universe is still
quite ordered and the entropy is very far from reaching its maximum. But this is
certainly what the situation is like in our universe today, and according to the
computations in [10, 11], it would take an almost incredibly long time to reach such
a state of maximal entropy. Thus, it will in the following be taken for granted that
this time is much longer than the life-span of our universe.