deeptime.markov.tools.flux.flux_matrix

deeptime.markov.tools.flux.flux_matrix(T, pi, qminus, qplus, netflux=True)

Compute the TPT flux network for the reaction A–>B.

Parameters:

• T ((M, M) ndarray) – transition matrix

• pi ((M,) ndarray) – Stationary distribution corresponding to T

• qminus ((M,) ndarray) – Backward comittor

• qplus ((M,) ndarray) – Forward committor

• netflux (boolean) – True: net flux matrix will be computed False: gross flux matrix will be computed

Returns:

flux – Matrix of flux values between pairs of states.

Return type:

(M, M) ndarray

See also

committor.forward_committor, committor.backward_committor

Notes

Computation of the flux network relies on transition path theory (TPT) [1]. Here we use discrete transition path theory [2] in the transition matrix formulation [3]. The central object used in transition path theory is the forward and backward comittor function.

The TPT (gross) flux is defined as

$$f_{ij}=\left \{ \begin{array}{rl} \pi_i q_i^{(-)} p_{ij} q_j^{(+)} & i \neq j \\ 0 & i=j \end{array} \right .$$

The TPT net flux is then defined as

$$f_{ij}=\max\{f_{ij} - f_{ji}, 0\} \:\:\:\forall i,j.$$

References

[1]

W. E and E. Vanden-Eijnden. Towards a theory of transition paths. J. Stat. Phys. 123: 503-523 (2006)

[2]

P. Metzner, C. Schuette and E. Vanden-Eijnden. Transition Path Theory for Markov Jump Processes. Multiscale Model Simul 7: 1192-1219 (2009)

[3]

F. Noe, Ch. Schuette, E. Vanden-Eijnden, L. Reich and T. Weikl: Constructing the Full Ensemble of Folding Pathways from Short Off-Equilibrium Simulations. Proc. Natl. Acad. Sci. USA, 106, 19011-19016 (2009)