  We provide a hybrid method that captures the polynomial speed of convergence
and polynomial speed of mixing for Markov processes. The hybrid method that we
introduce is based on the coupling technique and renewal theory. We propose to
replace some estimates in classical results about the ergodicity of Markov
processes by numerical simulations when the corresponding analytical proof is
difficult. After that, all remaining conclusions can be derived from rigorous
analysis. Then we apply our results to two 1D microscopic heat conduction
models. The mixing rate of these two models are expected to be polynomial but
very difficult to prove. In both examples, our numerical results match the
expected polynomial mixing rate well.
