## The Boltzmann Distribution

The distribution of the energies of a collection of particles depends on the average energy, which depends on the temperature. The most probable distribution of the energies of a collection of particles depends on the temperature therefore. The Boltzmann distribution expresses this dependence.

The Boltzmann distribution relates the number of particles per state at in terms of the energies of the states: where is the number of particles at energy which has degeneracy etc.

Proof: The goal is to find that maximises This expresses the number of ways that a particular distribution of molecules throughout a set of allowed energies can be achieved. Taking logs of this expression, and using Stirling's approximation for large gives We differentiate with respect to general obtaining (1).

We set this derivative equal to zero. The are subject to the constraint is fixed, and the energy over and above the zero point energy is also fixed.

Now use Lagrange multipliers to find the  is the function (2). We differentiate with respect to each forming or for each Substitute from (1) to obtain which rearranges to We can eliminate the factor by considering at two different energies, obtaining Consideration of (2) since but for one degree of freedom so we can write and further analysis gives hence #### Add comment Refresh