# Can detailed balance pricinple(DBP) be derived or is it an axiom?

1. Sep 15, 2010

### kof9595995

In my statistical mechanics class, my prof derived DBP in the following way:
Consider a system with many quantum states |a>, |b>..., keep scattering with each other and in equilibrium the master equation give us :
$$\frac{{d{P_k}}}{{dt}} = \sum\limits_{l \ne k} {{T_{kl}}} {P_l} - {T_{lk}}{P_k} = 0$$
(I adopt the notation from wiki, Pk is the probability for the system to be in the state |k>, Tkl is the transition rate from |k> to |l>)
And Tkl is governed by fermi golden rule
$${T_{kl}} = \frac{{2\pi }}{\hbar }{\rho _0}| < k|{H_I}|l > {|^2}$$
$${H_I}$$ is the interaction Hamiltonian.
Then he said Tkl=Tlk, and they're time-dependent because $${H_I}$$ is time-dependent. Then $$\sum\limits_{l \ne k} {{T_{kl}}} (t){P_l} - {T_{lk}}(t){P_k} = 0$$ requires Tlk*Pk=Tkl*Pl, thus the detailed balance principle.
But from what I read from several resources, they seem to treat DBP as something like an axiom(though it's not addressed explicitly). So I'm quite skeptical about what I learnt in class, can some one clarify for me?