29 research outputs found
Discreteness Effects in Population Dynamics
We analyse numerically the effects of small population size in the initial
transient regime of a simple example population dynamics. These effects play an
important role for the numerical determination of large deviation functions of
additive observables for stochastic processes. A method commonly used in order
to determine such functions is the so-called cloning algorithm which in its
non-constant population version essentially reduces to the determination of the
growth rate of a population, averaged over many realizations of the dynamics.
However, the averaging of populations is highly dependent not only on the
number of realizations of the population dynamics, and on the initial
population size but also on the cut-off time (or population) considered to stop
their numerical evolution. This may result in an over-influence of discreteness
effects at initial times, caused by small population size. We overcome these
effects by introducing a (realization-dependent) time delay in the evolution of
populations, additional to the discarding of the initial transient regime of
the population growth where these discreteness effects are strong. We show that
the improvement in the estimation of the large deviation function comes
precisely from these two main contributions
Finite-time and finite-size scalings in the evaluation of large-deviation functions: Analytical study using a birth-death process
The Giardin\`a-Kurchan-Peliti algorithm is a numerical procedure that uses
population dynamics in order to calculate large deviation functions associated
to the distribution of time-averaged observables. To study the numerical errors
of this algorithm, we explicitly devise a stochastic birth-death process that
describes the time evolution of the population probability. From this
formulation, we derive that systematic errors of the algorithm decrease
proportionally to the inverse of the population size. Based on this
observation, we propose a simple interpolation technique for the better
estimation of large deviation functions. The approach we present is detailed
explicitly in a two-state model.Comment: 13 pages, 1 figure. First part of pair of companion papers, Part II
being arXiv:1607.0880