** Next:** Statistical mechanics
** Up:** Probability theory
** Previous:** The Gaussian distribution

##

The central limit theorem

Now, you may be thinking that we got a little carried away in our discussion of
the Gaussian distribution function.
After all, this distribution only seems to
be relevant to two-state systems. In fact, as we shall see, the Gaussian
distribution is of crucial importance to statistical physics because, under certain
circumstances, it applies to *all* systems.
Let us briefly review how we
obtained the Gaussian distribution function in the
first place. We started from a
very simple system with only two possible outcomes. Of course, the
probability distribution function (for ) for this system did not look
anything like a Gaussian. However, when we combined very many
of these simple systems together,
to produce a complicated system with a great number of possible
outcomes, we found that the resultant probability distribution function
(for )
reduced to a
Gaussian in the limit as the number of simple systems tended
to infinity.
We started from a two outcome
system because it was easy to calculate the final probability
distribution function when a *finite* number of such systems were
combined together. Clearly, if we had started from a more complicated
system then this calculation would have been
far more difficult.

Let me now tell you something which
is quite astonishing! Suppose that we start from *any* system,
with *any* distribution function (for some measurable quantity ). If
we combine a sufficiently large number of
such systems together, the resultant distribution function
(for ) is *always* Gaussian.
This proposition is known as the *central limit theorem*. As far as
Physics is concerned, it is one of
the most important theorems in the whole of mathematics.

Unfortunately, the central limit theorem is notoriously difficult to prove.
A somewhat restricted proof is presented
in Sections 1.10 and 1.11 of Reif.

The central limit theorem guarantees that the probability distribution of
*any* measurable quantity
is Gaussian, provided that a sufficiently large number
of statistically independent observations are made. We can, therefore,
confidently predict that Gaussian distributions are going to crop up
all over the place in
statistical thermodynamics.

** Next:** Statistical mechanics
** Up:** Probability theory
** Previous:** The Gaussian distribution
Richard Fitzpatrick
2006-02-02