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_num : integer
Number of bins
- diff_bin : list
Ln diffusion profile over the bins
- df_bin : list
Free energy profile over the bins
|
Returns: |
- rate matrix : array
Rate matrix for the current free energy and ln diffusion profile
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