### Entropy as a measure of disorder

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*We can view Ω as a
measure of the disorder in a system. This is reasonable
because what we think of as "ordered" systems tend to
have very few configurational possibilities, and
"disordered" systems have very many. Consider, for
example, a set of 10
coins, each of which is either
heads up or tails up. The most "ordered" macroscopic
states are 10 heads or 10 tails; in either case, there
is exactly one configuration that can produce the
result. In contrast, the most "disordered" state
consists of 5 heads and 5 tails, and there are 252 (10
choose 5) ways to produce this result.*

*Under the statistical
definition of entropy, the second law of thermodynamics
states that the disorder in an isolated system tends to
increase. This can be understood using our coin example.
Suppose that we start off with 10 heads, and re-flip one
coin at random every minute. If we examine the system
after a long time has passed, it is possible that we
will still see 10 heads, or even 10 tails, but that is
not very likely; it is far more probable that we will
see approximately as many heads as tails.*

*Since its discovery, the
idea that disorder tends to increase has been the focus
of a great deal of thought, some of it confused. A chief
point of confusion is the fact that the result ΔS ≥ 0
applies only to isolated systems; notably, the
Earth is not an isolated system because it is
constantly receiving energy in the form of
sunlight. Nevertheless, it has been pointed out that
the
universe may be considered an isolated system, so
that its total disorder should be constantly increasing.
It has been speculated that the universe is fated to a
heat death in which all the energy ends up as a
homogeneous distribution of thermal energy, so that no
more work can be extracted from any source. Recent work,
however, has cast extensive doubt on the heat death
hypothesis and the applicability of any simple
thermodynamical model to the universe in general.
Although entropy does increase in an expanding universe,
the maximum possible entropy rises much more rapidly and
leads to an "entropy gap," thus pushing the system
further away from equilibrium with each time increment.
Furthermore, complicating factors such as the impact of
gravity, energy density of the vacuum (and thus a
hypothesized "antigravity"), and macroscopic
quantum effects under unusual conditions cannot be
reconciled with current thermodynamical models, making
any predictions of large-scale thermodynamics extremely
difficult.*