  We revisit the classical population genetics model of a population evolving
under multiplicative selection, mutation and drift. The number of beneficial
alleles in a multi-locus system can be considered a trait under exponential
selection. Equations of motion are derived for the cumulants of the trait
distribution in the diffusion limit and under the assumption of linkage
equilibrium. Because of the additive nature of cumulants, this reduces to the
problem of determining equations of motion for the expected allele distribution
cumulants at each locus. The cumulant equations form an infinite dimensional
linear system and in an authored appendix Adam Prugel-Bennett provides a closed
form expression for these equations. We derive approximate solutions which are
shown to describe the dynamics well for a broad range of parameters. In
particular, we introduce two approximate analytical solutions: (1) Perturbation
theory is used to solve the dynamics for weak selection and arbitrary mutation
rate. The resulting expansion for the system's eigenvalues reduces to the known
diffusion theory results for the limiting cases with either mutation or
selection absent. (2) For low mutation rates we observe a separation of
time-scales between the slowest mode and the rest which allows us to develop an
approximate analytical solution for the dominant slow mode. The solution is
consistent with the perturbation theory result and provides a good
approximation for much stronger selection intensities.
