  Considerable effort using techniques developed in statistical physics has
been aimed at numerical simulations of agent-based opinion models and analysis
of their results. Such work has elucidated how various rules for interacting
agents can give rise to steady state behaviors in the agent populations that
vary between consensus and fragmentation. At the macroscopic population level,
analysis has been limited due to the lack of an analytically tractable
governing macro-equation for the continuous population state. We use the
integro-differential equation that governs opinion dynamics for the continuous
probability distribution function of agent opinions to develop a novel
nonlinear partial differential equation for the evolution of opinion
distributions. The highly nonlinear equation allows for the generation of a
system of approximations. We consider three initial population distributions
and determine their small-time behavior. Our analysis reveals how the
generation of clusters results from the interplay of diffusion and
anti-diffusion and how initial instabilities arise in different regions of the
population distribution.
