# Brillouin function - Classical Limit

The Brillouin function, defined as

$$B_j(x) = \frac{j+1/2}{j} \coth\left(\frac{j+1/2}{j} x\right) - \frac{1}{2j}\coth\left(\frac{1}{2j} x\right),$$ tends to the Langevin funcion

$$\mathcal{L}(x) = \coth(x) - \frac{1}{x},$$

in the limit when $j \rightarrow\infty$. My lecture notes refer to this limit as the classical limit.

This function comes up in the study of spin interaction in the presence of a magnetic field, and is of particular interest in Statistical Mechanics. Here, $j$ is the spin number of a particle.

My question is: how does $j \rightarrow\infty$ make this system classic?

The total angular momentum $j$ is quantized. If you take the limit $j \rightarrow \infty$ then a huge amount of states will become accesible to the system (more specifically 2$j$ + 1 states). Due to this, $j$ (a vector), can now point in almost every direction just like a normal "classical" vector can take on every value. So you can treat the system in a classical way.