msmtools.analysis.relaxation

msmtools.analysis.relaxation(T, p0, obs, times=1, k=None, ncv=None)

Relaxation experiment.

The relaxation experiment describes the time-evolution of an expectation value starting in a non-equilibrium situation.

Parameters:
  • T ((M, M) ndarray or scipy.sparse matrix) – Transition matrix
  • p0 ((M,) ndarray (optional)) – Initial distribution for a relaxation experiment
  • obs ((M,) ndarray) – Observable, represented as vector on state space
  • times (list of int (optional)) – List of times at which to compute expectation
  • k (int (optional)) – Number of eigenvalues and eigenvectors to use for computation
  • ncv (int (optional)) – The number of Lanczos vectors generated, ncv must be greater than k; it is recommended that ncv > 2*k
Returns:

res – Array of expectation value at given times
Return type:

ndarray

References

[1]Noe, F, S Doose, I Daidone, M Loellmann, M Sauer, J D Chodera and J Smith. 2010. Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. PNAS 108 (12): 4822-4827.

Notes

Relaxation

A relaxation experiment looks at the time dependent expectation value of an observable for a system out of equilibrium

\[\mathbb{E}_{w_{0}}[a(x, t)]=\sum_x w_0(x) a(x, t)=\sum_x w_0(x) \sum_y p^t(x, y) a(y).\]

Examples

>>> import numpy as np
>>> from msmtools.analysis import relaxation

>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> p0 = np.array([1.0, 0.0, 0.0])
>>> a = np.array([1.0, 1.0, 0.0])
>>> times = np.array([1, 5, 10, 20])

>>> rel = relaxation(T, p0, a, times=times)
>>> rel
array([ 1.        ,  0.8407    ,  0.71979377,  0.60624287])