I bagan to study Markov chains applied to non equilibrium statistical physics, after ergodicity global balance was presented, here are the definitions I have been given:

Ergodicity: A Markov chain is said to be ergodic if there exists a time T so that $W_{ij}^T\neq 0$ for every i,j ($W_{ij}$ is the transition matrix fromm j to i). An equivalent definition is that there exists only one stationary distribution $\pi_i$ so that $\pi_i = \sum_k \pi_k W_{ik}$ and regardless of the initial probability distribution $p_i(t) \rightarrow \pi_i$ for $t \rightarrow \infty$.

Global Balance: $\sum_j (W_{ij}\pi_j-W_{ji}\pi_i)$=0

Are these two equivalent conditions or ergodicity is only sufficient but not necessary to global balance?


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