  An equation describing the evolution of phenotypic distribution is derived
using methods developed in statistical physics. The equation is solved by using
the singular perturbation method, and assuming that the number of bases in the
genetic sequence is large. Applying the equation to the mutation-selection
model by Eigen provides the critical mutation rate for the error catastrophe.
Phenotypic fluctuation of clones (individuals sharing the same gene) is
introduced into this evolution equation. With this formalism, it is found that
the critical mutation rate is sometimes increased by the phenotypic
fluctuations, i.e., noise can enhance robustness of a fitted state to mutation.
Our formalism is systematic and general, while approximations to derive more
tractable evolution equations are also discussed.
