What is the connection between detailed balance and thermodynamic equilibrium?

[M.B.]

Hey,

I was wondering in what way the detailed balance condition (DBC) has to do with thermodynamic equilibrium.

The DBC is defined as
$$W(i\to j)P(i) = W(j\to i)P(j)$$
with W the rates (probabilities per unit time) to jump from state i to j. P(i) is the equilibrium prob. density to have state i, this could be e.g. a Boltzmann Gibbs distribution in case of canonical equilibrium.
When for a (Markov jump) proces the above equality holds, then the system is time-reversible: it is even likely to follow a path forward in time or backwards in time

My question is:
The above formula is always possible to write down, but how is this related to equilibruim?

 

[0xDEADBEEF]

It is obvious that the probability distribution must not change by the flows. So you see that your equation states that the same amount of thingamajigs are jumping from i to j as from j to i. Hence the distribution stays the same under time evolution. If you know W then you can derive an equilibrium distribution with the DBC.

If the DBC holds than P is an equilibrium distribution. I am not sure if the inverse direction is also true.