Question

1. Consider an Ehrenfest chain with 6 particles.

(a) Write down the transition matrix and draw the transition diagram. (b) If the chain starts with 3 particles in the left partition, write down the state distribution at the first time step. (c) Find the stationary distribution using the detailed balance condition.

Answer 1

Solution Given that let ㅈt bethe numbe of balls in box! Po at time step t fod oud cose d-6 hen usino e have hen d6 iS . . The toonsition moto σοο 20そ00 The state tdo nsition diaoram fod χ。i5。 CS scanned with CamScanner

5 YC СП The pdobabilities ore meboed as a The stote space of ol,2,d poo his model asing O stoted in port@ ae knoai that Scanned with CamScanner

3 Thust the stote distoibution at the Fiest time.step t리 as 3 Detailed balorce condition stotes that fod a mazhov - chairn oith a statlonasy distoibufho,m fo ou case, T Q dimensional vector and The Detailed balonce condition aill give us 子 equations for the 구 anknowns. using detoiled balance condition ae geé TTI Cs Scanned with CamScanner

5 6 1I 64 ,6, 15, 20 1 15,6 64 Hence The stationary distơibution using detai led bolonce conditio Scanned with CamScannér