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The Boltzmann distribution
Particles, as they approach equilibrium, tend toward lower energy, but thermal energy can keep
some particles above their lowest energy state. The Boltzmann distribution quantifies the
equilibrium distribution of particles in their possible energy states. At equilibrium, the
probability that a given particle will have energy u
i
is
Z
p
e
kT
u
i
i

=
where
Z
is a normalization constant
chosen such that
1
=
∑
j
j
p
.
We do not need to know
Z
if we are only
interested in relative probabilities:


=
kT
u
u
e
p
p
1
2
1
2
In the diagram above, 8 particles have potential energy u
1
and 3 have u
2
. If the system is at
equilibrium, we can calculate the energy difference, u
2
u
1
, corresponding to this population
distribution. Since u
2
>u
1
, the equation gives p
2
<p
1
, as shown in the diagram (p
2
=3/11 and
p
1
=8/11, assuming these are the only two energy levels).
The difference between p
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 Winter '08
 HUNT

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