Suppose I have a Markov chain (satisfying all conditions of ergodicity) that has a stationary distribution that is easy to sample from. ( Assume that we know the stationary distribution upto a normalization constant e.g the Gibbs distribution).

Is there a way to know how long we should run the markov chain from it's initial distribution so that the resulting distribution of the markov chain after time t is within epsilon of the stationary distribution. ( Assume that the distance between distributions can be measured by your choice of distance functions e.g total variation distance or L-1 norm etc ).

Any pointers in this direction will really be helpful.