polykin.copolymerization

transitions_multi

transitions_multi(
    f: FloatVectorLike, r: FloatSquareMatrix
) -> FloatSquareMatrix

Calculate the instantaneous transition probabilities for a multicomponent system.

For a multicomponent system, the transition probabilities are given by:

\[ P_{ij} = \frac{r_{ij}^{-1} f_j} {\displaystyle \sum_{k=1}^{N} r_{ik}^{-1} f_k} \]

where \(f_i\) is the molar fraction of monomer \(i\) and \(r_{ij}=k_{ii}/k_{ij}\) is the multicomponent reactivity ratio matrix.

References

• NA Dotson, R Galván, RL Laurence, and M Tirrel. Polymerization process modeling, Wiley, 1996, p. 178.

PARAMETER	DESCRIPTION
f	Vector of instantaneous monomer compositions, \(f_i\). TYPE: FloatVectorLike(N)
r	Matrix of reactivity ratios, \(r_{ij}=k_{ii}/k_{ij}\). TYPE: FloatSquareMatrix(N, N)

RETURNS	DESCRIPTION
FloatSquareMatrix(N, N)	Matrix of transition probabilities, \(P_{ij}\).

See also

• inst_copolymer_multi: instantaneous copolymer composition.
• sequence_multi: instantaneous sequence lengths.
• tuples_multi: instantaneous tuple fractions.

Examples:

>>> from polykin.copolymerization import transitions_multi
>>> import numpy as np

Define reactivity ratio matrix.

>>> r = np.ones((3, 3))
>>> r[0, 1] = 0.2
>>> r[1, 0] = 2.3
>>> r[0, 2] = 3.0
>>> r[2, 0] = 0.9
>>> r[1, 2] = 0.4
>>> r[2, 1] = 1.5

Evaluate transition probabilities.

>>> f = [0.5, 0.3, 0.2]
>>> P = transitions_multi(f, r)
>>> P
array([[0.24193548, 0.72580645, 0.03225806],
       [0.21367521, 0.29487179, 0.49145299],
       [0.58139535, 0.20930233, 0.20930233]])

Source code in src/polykin/copolymerization/multicomponent.py

def transitions_multi(f: FloatVectorLike,
                      r: FloatSquareMatrix
                      ) -> FloatSquareMatrix:
    r"""Calculate the instantaneous transition probabilities for a
    multicomponent system.

    For a multicomponent system, the transition probabilities are given
    by:

    $$ P_{ij} = \frac{r_{ij}^{-1} f_j}
                     {\displaystyle \sum_{k=1}^{N} r_{ik}^{-1} f_k} $$

    where $f_i$ is the molar fraction of monomer $i$ and $r_{ij}=k_{ii}/k_{ij}$
    is the multicomponent reactivity ratio matrix.

    **References**

    *   NA Dotson, R Galván, RL Laurence, and M Tirrel. Polymerization
        process modeling, Wiley, 1996, p. 178.

    Parameters
    ----------
    f : FloatVectorLike (N)
        Vector of instantaneous monomer compositions, $f_i$.
    r : FloatSquareMatrix (N, N)
        Matrix of reactivity ratios, $r_{ij}=k_{ii}/k_{ij}$.

    Returns
    -------
    FloatSquareMatrix (N, N)
        Matrix of transition probabilities, $P_{ij}$.

    See also
    --------
    * [`inst_copolymer_multi`](inst_copolymer_multi.md):
      instantaneous copolymer composition.
    * [`sequence_multi`](sequence_multi.md):
      instantaneous sequence lengths.
    * [`tuples_multi`](tuples_multi.md):
      instantaneous tuple fractions.

    Examples
    --------
    >>> from polykin.copolymerization import transitions_multi
    >>> import numpy as np

    Define reactivity ratio matrix.
    >>> r = np.ones((3, 3))
    >>> r[0, 1] = 0.2
    >>> r[1, 0] = 2.3
    >>> r[0, 2] = 3.0
    >>> r[2, 0] = 0.9
    >>> r[1, 2] = 0.4
    >>> r[2, 1] = 1.5

    Evaluate transition probabilities.
    >>> f = [0.5, 0.3, 0.2]
    >>> P = transitions_multi(f, r)
    >>> P
    array([[0.24193548, 0.72580645, 0.03225806],
           [0.21367521, 0.29487179, 0.49145299],
           [0.58139535, 0.20930233, 0.20930233]])

    """
    f = np.asarray(f)

    # N = len(f)
    # P = np.empty((N, N))
    # for i in range(N):
    #     for j in range(N):
    #         P[i, j] = f[j]/r[i, j] / np.sum(f/r[i, :])
    P = (f/r) / np.sum(f/r, axis=1)[:, np.newaxis]

    return P