msmtools.flux.flux_matrix(T, pi, qminus, qplus, netflux=True)

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

  • 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

flux – Matrix of flux values between pairs of states.

Return type:

(M, M) ndarray


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].

See also

committor.forward_committor(), committor.backward_committor()


Computation of the flux network relies on transition path theory (TPT). The central object used in transition path theory is the forward and backward comittor function.

The TPT (gross) flux is defined as

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

The TPT net flux is then defined as

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


[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)