  The precision of reaction-diffusion models for mesoscopic physical systems is
limited by fluctuations. To account for this uncertainty, Van Kampen derived a
stochastic Langevin-like reaction-diffusion equation that incorporates
spatio-temporal white noise. The resulting solutions, however, have infinite
standard deviation. Ad hoc modifications that address this issue by introducing
microscopic correlations are inconvenient in many physical contexts of wide
interest. We instead estimate the magnitude of fluctuations by coarse-graining
solutions of the Van Kampen equation at a relevant mesoscopic scale. The
ensuing theory yields fluctuations of finite magnitude. Our approach is
demonstrated for a specific biophysical model--the encoding of positional
information. We discuss the properties of the fluctuations and the role played
by the macroscopic parameters of the underlying reaction-diffusion model. The
analysis and numerical methods developed here can be applied in physical
problems to predict the magnitude of fluctuations. This general approach can
also be extended to other classes of dynamical systems that are described by
partial differential equations.
