MC._init_rate_matrix_pbc¶

MC._init_rate_matrix_pbc(bin_num, diff_bin, df_bin)¶

This function estimates the rate Matrix R for the current free energy and log diffusion profiles over the bins for periodic boundary conditions. The dimension of the matrix is \(n \times n\) with \(n\) as number of the bins. The calculation of the secondary diagonal elements in the rate matrix \(R\) happen with the following equations

\[R_{i+1,i} = \exp \underbrace{\left( \ln \left( \frac{D_{i+\frac{1}{2}}}{\Delta z^2}\right) \right)}_{\mathrm{diff}_\mathrm{bin}} - 0.5(\beta(F(\Delta z_{i+1})-F(\Delta z_{i})\]

\[R_{i,i+1} = \exp \left( \ln \left( \frac{D_{i+\frac{1}{2}}}{\Delta z^2}\right) \right) + 0.5(\beta(F(\Delta z_{i+1})-F(\Delta z_{i})\]

with \(\Delta z\) as the bin width, \(D_{i+\frac{1}{2}}\) as the diffusion between to bins and \(F_i\) as free energy in the bin center. The diagonal elements can be calculated with the secondary elements determine with the equations above.

\[R_{i,i} = -R_{i-1,i}-R_{i+1,i}\]

The corner of the rate matrix is set with:

\[R_{1,1} = - R_{2,1} - R_{N,1}\]

\[R_{N,N} = - R_{N-1,N} - R_{1,N}\]

The periodic boundary conditions are implemeted with

\[R_{1,N} = \exp \left( \ln \left( \frac{D_{N+\frac{1}{2}}}{\Delta z^2}\right) \right) - 0.5(\beta(F(\Delta z_{1})-F(\Delta z_{N}))\]

\[R_{N,1} = \exp \left( \ln \left( \frac{D_{N+\frac{1}{2}}}{\Delta z^2}\right) \right) + 0.5(\beta(F(\Delta z_{1})-F(\Delta z_{N}))\]

Parameters:

bin_numinteger: Number of bins
diff_binlist: Ln diffusion profile over the bins
df_binlist: Free energy profile over the bins

Returns:

rate matrixarray: Rate matrix for the current free energy and ln diffusion profile