**If the change of frequencies of alleles in a
population is not brought about through selection, it has to be due to other
mechanisms. One is genetic drift (the random sampling of alleles from one
generation to form the next generation), another is migration, where new
individuals come from outside the system and change the frequencies of the
various alleles inside the system. These two mechanisms, along with mutation
which creates new alleles in the first place, are especially interesting to
those working in nonequilibrium statistical mechanics and the theory of
stochastic processes, since their study involves many of the tools and
techniques from these fields. Neutral theories are models of such
selection-free processes. They appear in several different areas: I have been
involved in their study in the fields of population genetics, ecology and
language evolution, where they seem to have a far greater realm of validity
than might naively be expected. Many of the models also seem to be exactly
soluble, or generally amenable to analysis. A paper JSTAT P07018 (2007) based on seminars given by Richard Blythe and myself at the Newton Institute in Cambridge,
gives some more background and information on neutral theories.**

**The concept of neutral theories had its origins in
population genetics and so it is here that the idea is most well developed.
As someone trained in theoretical physics, my preference is for starting from
"microscopic" models, formulated as either Markov chains or master equations
(continuous time Markov chains) and then deriving the corresponding
"macroscopic" model, which in this case are Fokker-Planck equations, frequently
known as Kolmogorov equations in population genetics. Derivations of this
kind are discussed in detail in JSTAT P07018 (2007). It
turns out that, in an application to be discussed below, the case where more
than two alleles of a particular gene exists needed to be studied, and so
Gareth Baxter, Richard Blythe and myself looked at the Fokker-Planck equation describing genetic
drift and mutation when M alleles were present. This is a non-trivial
M-dimensional partial differential equation, which remarkably is
separable, and so exactly soluble. It is interesting to note that when Kimura
first studied these equations in the 1950s, he speculated that "additional
techniques would be required to make the mathematical manipulations manageable"
for an arbitrary number of alleles. The details are given in Math. Biosci. 209, 124-170 (2007),
where many other quantities are calculated exactly, including probabilities of
fixation (all alleles but one have become extinct), mean time to the rth
extinction, probability of a particular sequence of extinctions, and so on.
Care needs to be taken with the boundary conditions of the Fokker-Planck
equation, since the diffusion matrix is state dependent and vanishes at the
boundaries. An alternative method of implementing the boundary conditions,
consisting of working in Fourier space, was worked out with David Waxman, is described in
Jour. Theor. Biol. 247, 849-858 (2007) and gives a
systematic way of approaching these problems.**

**A second field in which neutral theories have
become popular is ecology. The main proponent of a neutral theory of ecology is
Stephen Hubbell, especially through his book The Unified Neutral Theory of
Biodiversity and Biogeography. In Hubbell's neutral theory, genes map
into individuals and alleles into species, thus no one species is any "fitter"
than another, and the time evolution of an ecological community of trophically
similar species is primarily due to random processes of birth, death and
migration. Not surprisingly, this is a controversial claim, however the species
abundance distribution predicted from Hubbell's theory is in surprisingly
good agreement with data. I, together with David Alonso and Ricard Sole were able to calculate this species abundance distribution analytically,
where previously it had only been obtained numerically. We achieved this by
formulating the model as a master equation and mapping it on to a previous
model of ours which we had also solved exactly. The details are given in
Theor. Popul. Biol. 65, 67-73 (2004). A review article,
Trends Ecol. Evol. 21, 451-457 (2006), written together
with David Alonso and
Rampal Etienne discusses the merits of neutral theory in an
ecological context. Neutral theories have a rich mathematical structure which
is explored in two other papers of mine. In the first, Ecol. Lett. 7, 901-910 (2004), the fact that it is a sampling theory is studied
and in the second Jour. Theor. Biol. 248, 522-536 (2007), the consequences of relaxing the zero-sum assumption (i.e. fixed
population size) is investigated.**