In developing methods to perform Monte Carlo simulations one sufficient condition to preserve the stationarity of the target probability distribution is to impose detailed balance i.e.
[Gardiner page 148]
A Markov process satisfies detailed balance if, roughly speaking, in the stationary situation each possible transition balances with the reversed transition.
Where the transition is given by $$(r, v, t)\rightarrow (r', v', t+\tau)$$ and its reversal correspond to the time reversed transition which requires the velocities to be reversed because the motion from r' to r is in the opposite direction from that from r to r'. $$(r', - v', t)\rightarrow (r, - v, t+ \tau).$$
Suppose our transiotions correspond to the movement from a state of the Harmonic oscillator in the phase space $(q,p)$ to another state of the same H.O., $(q',p')$.
Here my question is: how are conservation of Energy and detailed balance linked?
The answer should be:
energy conservation $\Rightarrow$ detailed balance This should be clear because if energy is conserved that means that I am moving on the ellipse in phase space which is composed by all the microstates available to my system.
While the converse should not be always true, i.e. there could be jumps which satisfy detailed balance but violates conservation of energy, but I cannot find an example of them.