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

[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 :
[tex]\frac{{d{P_k}}}{{dt}} = \sum\limits_{l \ne k} {{T_{kl}}} {P_l} - {T_{lk}}{P_k} = 0[/tex]
(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
[tex]{T_{kl}} = \frac{{2\pi }}{\hbar }{\rho _0}| < k|{H_I}|l > {|^2}[/tex]
[tex]{H_I}[/tex] is the interaction Hamiltonian.
Then he said Tkl=Tlk, and they're time-dependent because [tex]{H_I}[/tex] is time-dependent. Then [tex]\sum\limits_{l \ne k} {{T_{kl}}} (t){P_l} - {T_{lk}}(t){P_k} = 0[/tex] 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 learned in class, can some one clarify for me?

 

[gtoguy97]

The principle of detailed balance is indeed an axiom in statistical mechanics, and it is based on the assumption of time-reversal symmetry. This means that if a system is in equilibrium, then it will remain in equilibrium regardless of whether time flows forward or backward. In other words, the probability of a transition from state |k> to |l> must be equal to the probability of the reverse transition from state |l> to |k>. This is what your professor was trying to explain with his derivation. However, it is important to note that the detailed balance principle only holds when the system is in equilibrium. It does not necessarily hold for systems that are out of equilibrium.