Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method

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As here defined,

ν_{ij}

is the

ν_{ji}

of Refs. 6, 10, 11. The present indexing corresponds to the commonly accepted definition of the stoichiometric matrix.

8.

The Poisson random variable

P (a,τ)

is the integer-valued random variable defined by

Prob {P {(a,τ)=n}=[e}^{−aτ} {(aτ)}^{n}]/n!  (n=0,1,…).

Both the mean and the variance of

P (a,τ)

are equal to

aτ.

P (a,τ)

can be interpreted physically as the number of events that will occur in any finite time τ, given that the probability of an event occurring in any future infinitesimal time

dt

is

adt.

9.

We denote by

N {(m,σ}^{2})

the normal (or Gaussian) random variable with mean m and variance

σ^{2} .

This random variable has the useful property that

N {(m,σ}^{2})=m+σ N (0,1).

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14.

In the thermodynamic limit, the species populations

x_{i}

and the system volume Ω diverge together, and proportionately. It turns out that, in this limit, all propensity functions

a_{j} (x)

diverge linearly with the system size, because the propensity function for an mth-order reaction will contain m factors

x_{i}

along with a factor

Ω^{−(m−1)} .

As a consequence, while the terms under the first summation sign in Eq. (4) are roughly proportional to the system size, the terms under the second summation sign are roughly proportional to the square root of the system size. So, in the thermodynamic limit, the latter terms typically become negligibly small compared to the former terms. Of course, real systems, no matter how large, are necessarily finite, and in situations where the terms in the first summation in Eq. (4) add up to practically zero (for instance at equilibrium), the fluctuating second sum can become important.

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