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Intrinsic timing in classical master equation dynamics from an extended quadratic format of the evolution law

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Abstract

The paradigm of Markov jump process is widely employed in disparate contexts, like for instance in ecology, epidemiology and chemistry, any time the state-space is discrete and a tracked entity (the system) stochastically jumps from site to site. The classical master equation describes the system’s evolution in terms of site occupation probabilities starting from a given initial condition associated with the initial knowledge about the system. For the master equation of Markov processes admitting stationary occupation probabilities, it is here derived an equivalent quadratic format in an extended space of mutually interrelated variables having physical dimension of inverse of time. The evolution in the probability space is thus mirrored by the evolution in the extended space. This universal format potentially allows one to unveil general traits underlying the master equation dynamics. Here we specifically consider the emergence of an intrinsic rate which, behaving as a state function in the probability space, introduces a timing during the relaxation process. This specific feature has to be taken an empirical discovery which derives from the analysis of numerical calculations; a possible direction towards a formal proof is however proposed. The conjecture made here is that such intrinsic timing is a typical trait (i.e., normally present) of the Markov jump processes.

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Notes

  1. As later discussed in Sect. 2.1, multiple stationary states can be admitted in our analysis. This means that the values \(p_i^\textrm{ss}\) can generally depend on the initial condition.

  2. In the specific context of chemical kinetics, other ways of quadratization have been devised by exploiting smooth transformations of the variables; see for instance Ref. [39].

  3. From Eq. 14 we have that \(z_{(j, \alpha \beta )} = p_j(t)^{-1} dp_j(t)/dt - p_\alpha (t)^{-1} dp_\alpha (t)/dt\). The critical cases are those for which j and/or \(\alpha \) correspond to sites depopulated as \(t \rightarrow \infty \), so that both the occupation probability and its rate of variation tend to zero. Let s be a site of such a kind. We can see that \(p_s(t)^{-1} dp_s(t)/dt\) tends to a finite limit value. Indeed, \(p_s(t)\) must monotonically decay to zero after a certain time \(t^*\), hence \(dp_s(t)/dt < 0\) for \(t \ge t^*\). Then, indicating with \(k^\textrm{exit}_s = \sum _{i \ne s} k_{s \rightarrow i}\) the total depopulation rate of the site s, from Eq. 1 we have that \(dp_s(t)/dt = -k^\textrm{exit}_s p_s(t) + \sum _{i \ne s} p_i(t) k_{i \rightarrow s} > -k^\textrm{exit}_s p_s(t)\). From the two inequalities taken together we obtain \(-k^\textrm{exit}_s< p_s(t)^{-1} dp_s(t)/dt < 0\) for any \(t > t^*\). It follows that also the limit value for \(t \rightarrow \infty \) is bounded. This implies that also the “critical” \(z_{(j, \alpha \beta )}\) have finite values \(z_{(j, \alpha \beta )}^*\).

  4. By noting that \(|V_{(j, \alpha \beta ), (j', \alpha ' \beta ')} |\) is either equal to 0 or to \(h_{(j', \alpha '\beta ')} = p_{\alpha '} k_{\alpha ' \rightarrow \beta '} / p_{j'}\), it follows that \(\sum _{\alpha ', \beta '} |V_{(j, \alpha \beta ), (j', \alpha ' \beta ')}|= c/ p_{j'}\) with the constant factor \(c = \sum _{\alpha ', \beta '} p_{\alpha '} k_{\alpha ' \rightarrow \beta '}\). This factor can be eliminated by exploiting the normalization \(\sum _{j'} p_{j'} = 1\). Ultimately, this leads to \(p_{j'} = \gamma _{j'} / \sum _{j''} \gamma _{j''}\) where \(\gamma _{j'} = \left( \sum _{\alpha ',\beta '} |V_{(j, \alpha \beta ), (j', \alpha ' \beta ')}|\right) ^{-1} \;\;\; \mathrm{for \; any} \; j, \alpha , \beta \). The condition “for any \(j, \alpha , \beta \)” is automatically fulfilled because of the mutual interrelations among the elements of the matrix \(\textbf{V}\). In fact, while the forward transformation \(\textbf{p} \rightarrow \textbf{V}\) specifies unequivocally the matrix \(\textbf{V}\) once the site-site connections are given, the backward transformation is meaningful only if the elements of \(\textbf{V}\) are properly interrelated.

  5. In the ‘hyper-spherical representation’, the system’s state is specified by the variables \((S, {\varvec{\psi }})\) where \({\varvec{\psi }}\) is a state vector normalized as \({\varvec{\psi }}^T {\varvec{\psi }}= 1\) and \(S = \sqrt{\textrm{Tr}\{ \textbf{V}^T \textbf{V}\}} >0\) is the Frobenius norm of the matrix \(\textbf{V}\). The \({\varvec{\psi }}\) components are interpreted as angular-like variables, and S as a radial-like variable. The evolution law has a simple and appealing format which resembles similar laws encountered in the modeling of the evolution of living species in fitness landscapes [54, 55]; see remarks in the Supplementary Material. Of note, the hyper-spherical representation enabled to unveil the existence of orthogonal attracting subspaces in the hyper-spherical space [32, 35].

  6. Computations have been done with double-precision digits. For a cross-check of the accuracy, the time propagation was done also with the implicit ‘Variable Coefficient ODE Solver’ (VODE) [56] for stiff kinetics. The double-precision Fortran77 subroutine ‘DVODE.f’ was employed. The routine is freely available at the URL: https://computing.llnl.gov/projects/odepack/software (last viewed on 28th November 2022). The results were coincident under the graphical resolution of the plots.

  7. Given two arrays \(\textbf{a}\) and \(\textbf{b}\) of equal dimension, the dyadic product is here meant to be the direct product \(\textbf{a} \otimes \textbf{b}\) arranged as square matrix with elements \(a_i b_j\).

  8. Let us consider the scaled quantities \(\alpha _Q^{(n)}(\textbf{p}) := z_Q^{(n)}(\textbf{p})/\max _{Q} \{ |z_Q^{(n)}(\textbf{p}) |\}\). It was found that, for most of the \(\textbf{p}\), the \(\alpha _Q^{(n)}(\textbf{p})\) either go to zero or to some constant value as n increases. In particular, in most cases it was found that \(\alpha _Q^{(n)}(\textbf{p}) \pm 1\). An automatic analysis was conducted for \(n=50\) to inspect on the typicality of this behaviour. The situation \(\alpha _Q^{(n)}(\textbf{p}) \pm 1\) was considered to be occurred if all \(|\alpha _Q^{(n)}(\textbf{p}) |\) were below 0.1 or above 0.9. For \(N = 3\), with jump rates generated at random over a range of 10 orders of magnitude with uniform sampling on the logarithmic scale, and also generating \(\textbf{p}\) at random, the percentage of instances in which such behavior was not observed was of about \(0.7\%\). The percentage increased to about \(1.4 \%\) for \(N = 4\), and to \(2.1 \%\) for \(N = 5\) and \(N = 6\). The number of generated instances was \(10^4\) for \(N = 3, 4\) and \(10^3\) for \(N = 5, 6\). In a relevant fraction of the cases, however, there were indications that \(n = 50\) was presumably too low, and that the expected behavior would have been likely observed going to higher orders.

  9. The following criterion was adopted to assess the convergence to \(\tilde{\textbf{W}}_\infty \). For the elements \((\tilde{\textbf{W}}_{n-1})_{ij} \ne 0\), let us consider the relative variation \(\epsilon _{ij} = |(\tilde{\textbf{W}}_n)_{ij} - (\tilde{\textbf{W}}_{n-1})_{ij} |/(\tilde{\textbf{W}}_{n-1})_{ij}\). The convergence of \(\tilde{\textbf{W}}_n\) to \(\tilde{\textbf{W}}_\infty \) was considered to be achieved (or much likely about to be achieved) if \(\max _{i,j} \{ \epsilon _{ij} \} \le 0.05\) for some \(n \le n_\textrm{max}\) where \(n_\textrm{max}\) is a chosen cutoff. The analysis was done with \(n_\textrm{max} = 10\). Note that this criterion is quite stringent since the elements of \(\tilde{\textbf{W}}_{n-1}\) that possibly go slowly to zero should be neglected to avoid an apparent convergence failure. Moreover, in many cases the convergence could have been likely achieved going to orders much above \(n_\textrm{max}\). Indeed, in the inspected cases the fraction of instances in which the check passed with \(n = n_\textrm{max}\) was of about \(15-20 \%\), which suggests that many instances of missed convergence were merely due to the fact that the applied \(n_\textrm{max}\) was to low.

  10. Calculations have been done also for elements of \(\textbf{V}\) generated with a logarithmic spread. Namely, each element was generated as \(s \times 10^b\) with \(s = \pm 1\) randomly assigned and exponent b drawn from the uniform distribution between \(-10\) and 0. With \(n_\textrm{max} = 10\), it was found that the percentage of convergence was lower than in the case of matrix elements drawn from the uniform distribution. In other calculations, a certain number of matrix elements was randomly set equal to zero to inspect the effect of the sparsity of \(\textbf{V}\). The effect was, again, a decrease of the percentage of convergence.

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Acknowledgements

The author thanks Paolo Nicolini and Alessandro Ceccato for a former long standing and exciting collaboration on closely related topics.

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  1. Department of Chemical Sciences, University of Padova, via Marzolo 1, 35131, Padua, Italy

    Diego Frezzato

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Appendix A: Asymptotic behaviour of \(\tilde{\textbf{W}}_n\) with matrices \(\textbf{V}\) randomly generated

Appendix A: Asymptotic behaviour of \(\tilde{\textbf{W}}_n\) with matrices \(\textbf{V}\) randomly generated

An empirical investigation has been conducted on the behavior of \(\tilde{\textbf{W}}_n\) for matrices \(\textbf{V}\) filled at random. In agreement with the analysis made in Ref. [30], it has been found that the convergence to a stable limit matrix \(\tilde{\textbf{W}}_\infty \) is indeed frequent, although the relationship between convergence and requisites about the entries of \(\textbf{V}\) is still missing.

For instance, if \(\textbf{V}\) is fully filled with entries between -1 and +1 randomly drawn from the uniform distribution, it was estimated that the convergence occurs in at least 15% of the cases for \(d=3\), 8% of the cases for \(d=4\), and 5% for \(d=5\).Footnote 9 The percentage of convergence of course changes if the matrix elements are generated in different ways, and it is found to depend also on the sparsity of \(\textbf{V}\).Footnote 10 The “missed convergence” mostly manifested in an even-odd alternation of the \(\tilde{\textbf{W}}_n\) between two limit forms \(\tilde{\textbf{W}}_\infty \) and \(-\tilde{\textbf{W}}_\infty \). In addition it emerged, at least for the low d cases which could be directly inspected, that \(\tilde{\textbf{W}}_\infty \) typically (but likely not exclusively) tends to a dyadic form in which all rows are given by one row multiplied by a certain factor.

As an example of convergence to \(\tilde{\textbf{W}}_\infty \), let us consider the following matrix randomly generated:

$$\begin{aligned} \textbf{V} = \left[ \begin{array}{ccc} 0.6472 &{} -0.8763 &{} -0.6321 \\ 0.1894 &{} -0.3392 &{} 0.8457 \\ -0.06088 &{} 0.9746 &{} 0.2186 \end{array} \right] \end{aligned}$$
(A1)

Its eigenvalues are \(-0.9763\), \((0.7514 \pm \imath \, 0.2316)\). The matrices \(\tilde{\textbf{W}}_{15}\) and \(\tilde{\textbf{W}}_{14}\) are

$$\begin{aligned} \tilde{\textbf{W}}_{15(14)} = \left[ \begin{array}{ccc} -0.01073 \; (-0.009837) &{} -0.3038 \; (-0.3034) &{} -0.4214 \; (-0.4210) \\ 0.007690 \; (0.007986) &{} 0.1964 \; (0.1970) &{} 0.2734 \; (0.2743) \\ 0.007478 \; (0.007343) &{} 0.1978 \; (0.1982) &{} 0.2741 \; (0.2752) \end{array} \right] \nonumber \\ \end{aligned}$$
(A2)

The convergence to \(\tilde{\textbf{W}}_\infty \) is practically already achieved and, as can be seen, the resulting matrix has a dyadic form: the second and the third rows tend to be identical, while the first row tends to be equal to these rows multiplied by a factor \(\simeq -1.54\). Here is another example:

$$\begin{aligned} \textbf{V} = \left[ \begin{array}{ccc} 0.9049 &{} 0.8256 &{} 0.7216 \\ 0.1398 &{} -0.4014 &{} -0.2607 \\ 0.5747 &{} 0.4002 &{} -0.7200 \end{array} \right] \end{aligned}$$
(A3)

which gives

$$\begin{aligned} \tilde{\textbf{W}}_{15(14)} = \left[ \begin{array}{ccc} 0.5776 \; (0.5782) &{} 0.1559 \; (0.1560) &{} 0.1358 \; (0.1360) \\ 0.007355 \; (0.007320) &{} 0.001968 \; (0.001956) &{} 0.001680 \; (0.001668) \\ 0.3214 \; (0.3206) &{} 0.008650 \; (0.008626) &{} 0.007394 \; (0.007360) \end{array} \right] \nonumber \\ \end{aligned}$$
(A4)

In this case, the eigenvalues of \(\textbf{V}\) are all real and equal to 1.1617, \(-0.8483\), \(-0.5298\). The convergence to \(\tilde{\textbf{W}}_\infty \) is practically achieved and the limit matrix is again a dyadic product in which the second row and the third row are, respectively, about 0.013 and 0.56 times the first row.

Based on the direct observations with d from 3 to 5, the conjecture made here is that the convergence to a fixed matrix \(\tilde{\textbf{W}}_\infty \) is encountered for any dimensions d provided that the matrix \(\textbf{V}\) fulfills some requisites that are however still unknown. In particular, there is no evident relationship between convergence to \(\tilde{\textbf{W}}_\infty \) and eigenvalues of the matrix \(\textbf{V}\): the convergence has been observed either when the eigenvalues are all real or when some are complex-valued, and even regardless of the sign of the real parts. Concerning the convergence rate (versus n) to \(\tilde{\textbf{W}}_\infty \), generally it decreases as d increases and depends on the sparsity of \(\textbf{V}\) in a way not yet systematically characterized.

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Frezzato, D. Intrinsic timing in classical master equation dynamics from an extended quadratic format of the evolution law. J Math Chem 61, 806–834 (2023). https://doi.org/10.1007/s10910-022-01435-7

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