# Finding beta for the boltzman distribution.

Hello! I'm trying to do a satisfactory derivation of the boltzmann distribution. By using lagrange multipliers I've come as far as to prove that

$$P(i) = \frac{1}{Z} e^{-\beta E(i)}$$
where
$$Z = \sum_i e^{-\beta E(i)},$$

but how does one actually establish that
$$\beta = 1/kT?$$

vanhees71
Gold Member
2021 Award
Take a monatomic ideal gas and derive the mean energy,

$$U=-\frac{\partial \ln Z}{\partial \beta}$$

and compare with the definition of the temperature,

$$U=\frac{3}{2} N k T.$$

I think that Leonard Susskind does this derivation in one of his lectures on statistical mechanics that is available on youtube.

Last edited by a moderator:
Take a monatomic ideal gas and derive the mean energy
Ah, yes that is certainly a way to go. But how could that result possibly be general? Doesn't the distribution apply to any combination of systems who shares a total energy E and a number of particles N?