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Journal of Chemical Physics 115, 1716–1733. By Daniel T. Gillespie.

Primary text indexed at Author paper, section II, equations (5)–(9): joint next-reaction density, total propensity, factorization and the direct method. The indexed passage was inspected; the direct PDF URL returned 404 on 2026-11-02. Title, byline and DOI were independently checked against the indexed publisher PDF at moodle2.units.it/pluginfile.php/293921/mod_resource/content/1/Gillespie_2001_Approximate_accelerated_stochastic_simulation_of_chemically_reacting_systems.pdf . The exact algorithm review in section II is cited, rather than the later approximate tau-leaping construction.

Daniel T. Gillespie developed stochastic simulation methods for reacting molecular populations. In the Gillespie algorithm, a reaction propensity function specifies an event rate, and the algorithm samples both the next reaction time and the reaction channel rather than evolving only mean concentrations.^[ref]

If the total propensity is $a_0>0$, the waiting time is exponentially distributed with rate $a_0$, and channel $j$ is selected with probability $a_j/a_0$ at the current state. His tau-leaping approach groups reaction events to accelerate simulation, introducing a controlled approximation rather than retaining the exact event-by-event law.^[ref]

By Daniel T. Gillespie. Annual Review of Physical Chemistry 58, 35–55.

Author text: www.dna.caltech.edu/courses/cs191/paperscs191/gillespie1.pdf

Cited passage: Section 4, printed pp. 44–46, equations (18)–(20), derives the Gaussian approximation to Poisson reaction counts, the chemical Langevin equation and its conditions; printed p. 46 explains the system-size scaling of noise. Byline and journal metadata verified on PDF page 1.