Understanding detailed balance (crossposted from stats) I guess I understand the equation of detailed balance, which states that for transition probability $q$ and stationary distribution $\pi$, a Markov Chain satisfies detailed balance if $$q(x|y)\pi(y)=q(y|x)\pi(x),$$
this makes more sense to me if I restate it as: 
$$\frac{q(x|y)}{q(y|x)}= \frac{\pi(x)}{\pi(y)}. $$
Basically, the probability of transition from state $x$ to state $y$ should be proportional to their ratio of probability densities. My question is, how does MCMC fulfilling detailed balance yield the stationary distribution?
 A: Let for purpose of a better understanding
$$\frac{dq(y|x)}{dt}=q(x|y)\pi(y)-q(y|x)\pi(x),$$
the dynamic system represented by a non-stationary Mrkov Chain (commonly called Master Equation), then the detailled ballance is achieved at stationary Markov Chain:
$$\left( \frac{dq(y|x)}{dt} \right)_s=q_s(x|y)\pi_s(y)-q_s(y|x)\pi_s(x)=0$$
and
$$q_s(x|y)\pi_s(y)-q_s(y|x)\pi_s(x)=0$$
Revisiting your case then it is obvious that your distributions are already the stationary distributions with:
$$q_s(x|y)\pi_s(y)=q_s(y|x)\pi_s(x)$$
or:
$$\frac{q_s(x|y)}{q_s(y|x)}= \frac{\pi_s(x)}{\pi_s(y)}$$
What is important that the set up of a non-stationary or stationary Markov Chain requires that you know from some source the (desired) distributions so not to expect to calculate them out of the equation. The equations above just manifest the Chain relations.
Following your comment, undersatnd now what is MCMC meant. In order to apply MCMC you would need to choose from a class of algorithms and apply certain sampling (empiric); see for instance the Metropolis method. The desired distributions, as mentioned above, will then result from your empiric applied to the Chain equation under certain algorithm constraints (this will appear in terms of certain convergence achievable in some finite steps).
In this respect you will need to decide for a method and then follow the cooking receip for building the empiric and then implying your empiric (that includes the desired distribution information) to your Chain.
Some methods are: Metropolisalgorithm, Gibbs-Sampling, Hybrid-Monte-Carlo-Algorithm, Clusteralgorithm, Over-Relaxation-Method.
Hence regarding your question: "How does MCMC fulfilling detailed balance yield the stationary distribution?" One should say this depends and is given by your algorithm of choice.
