Figure 4.

Dependence of the average of diagonal matrix elements $\langle {\text{T}}_{\alpha \alpha }^{\left(e\right)}\rangle$ on the number of exchange attempts per cycle n^e. $\langle {\text{T}}_{\alpha \alpha }^{\left(e\right)}\rangle$ is the average probability that a labeled replica is at the αth thermodynamic state after the exchange process given that it is at the αth thermodynamic state before the exchange process. Plot a shows the dependence of $\langle {\text{T}}_{\alpha \alpha }^{\left(e\right)}\rangle$ on the exchange attempts per RE cycle n_e for one-dimensional MSMRE simulations using different proposal schemes (NNE, nearest neighbor exchange; ISE, independence sampling). Plot b shows the dependence of $\langle {\text{T}}_{\alpha \alpha }^{\left(e\right)}\rangle$ on the number of exchange attempts per RE cycle n_e for the two-dimensional MSMRE simulations using the independence sampling scheme, and the inset shows the dependence of the largest implied time scale (IT) on the number of exchange attempts per RE cycle n_e. The horizontal black lines in plots a and b are the estimates of $\langle {\text{T}}_{\alpha \alpha }^{\left(e\right)}(n_{\text{e}}\to \infty )\rangle$ calculated by eq 10.