# What is the meaning of detailed balance in Markov Chains?

I know what it means formally to say that a stochastic matrix and a measure are in detailed balance. ($\lambda_iP_{ij} = \lambda_jP_{ji} \; \forall (i,j)$) but I'm not really sure how to interpret it intuitively, can anyone helpe me?

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## 1 Answer

Imagine that a heap of sand is placed at each site, with a quantity $\lambda_i$ of sand at site $i$. A bell rings. Then each grain of sand at $i$ decides to migrate to $j$ with probability $P_{ij}$, independently of all the others (that is, each individual grain of sand performs a trajectory of the Markov chain and the trajectories are independent). Detailed balance means that as much sand traveled along the edge $(ij)$ from $i$ to $j$ as from $j$ to $i$. Detailed balance implies stationarity, that is, the fact that, once every grain of sand has settled at its new location, each site $i$ has again a quantity $\lambda_i$ of sand. But detailed balance is stronger than stationarity since it means that a film of the movements of the sand looks exactly the same when viewed forwards or backwards.

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