**To see why this is so, recall that the time
evolution of the stochastic variables is frequently governed by
either stochastic differential equations (an example is the Langevin equation)
or equations for the time-dependent probability distributions (examples are
the master and Fokker-Planck equations). These equations can usually only be
solved in closed form when the process they are describing is linear, but the
solutions of even nonlinear equations can be formally expressed as a
functional integral. This is the formulation of non-equilibrium systems that
is one of the most attractive, since it enables all of the techniques and
ideas developed to study functional integrals to be brought to bear. For
example, an approximation which holds when the stochastic element (or noise)
is weak is very naturally formulated in this framework. It involves evaluating
the functional integral by steepest descent; the configurations which dominate
the functional integral are known as optimal paths. Furthermore, as
mentioned above, although there is no general formalism analogous to the
partition function for non-equilibrium systems, the generating functionals
that occur in this approach are probably the closest to such a formalism that
we have. In addition, it is possible in this scheme to define non-equilibrium
potentials, and functional integral methods provide powerful tools for
calculating these quantities.**

**In some applications the functional integral
formalism is not the most efficient way of proceeding. For instance, if the
system is described by a master equation and one is interested in the dynamics
when N, the number of constituents, is large, then van Kampen's
system size expansion may be applicable. At leading order (N tending
to infinity) the equations of the corresponding deterministic system are
obtained, and at next-to-leading order a stochastic equation for large N
is found. Crucially, these corrections to the deterministic theory take the
form of linear equations, and so can in principle be solved exactly.
This technique has been used extensively to clarify the appearance of
amplified stochastic cycles. Another application
is to the study of metapopulations. Below I discuss the research I have carried
out on three kinds of stochastic system not discussed in any of the other
sections.**

**In an ecological context, metapopulations are
"populations of populations": groups of local populations in patches which
are simply classified as being either occupied or unoccupied. In other words
the population size of each patch is not the main focus, at least in simple
models. Instead the persistence of the population in a particular patch, and
more generally the persistence of the whole metapopulation, are the main
concern. Patches are linked by immigration and emigration, and local
extinctions will also occur. So the question arises: given local colonisation,
migration and extinction rates, what is the proportion of patches which will
be occupied at any given time? Obviously, stochastic dynamics will be very
important when the number of patches is small, and so a study I carried out
with David Alonso was
particularly concerned with the extinction dynamics of metapopulations. Our
analysis of the model and conclusions are at
Bull. Math. Biol. 64, 913-958 (2002).**

**A few years ago Ricard Sole, David Alonso and I became interested in simple stochastic models of biodiversity.
Imagine an island (the local community) consisting of N individuals
of S possible species (typical values of S might be tens or
hundreds and of N perhaps two or three orders of magnitude bigger).
The number of individuals is fixed, and not all possible species may be
represented on the island. Immigration occurs from a metacommunity, but in
such a way that any immigrants simply displace individuals in the local
community, so that N still remains fixed. In our original model, at
each time step either (i) one individual of species i replaces one of
species j if the score of i against j is positive, or
(ii) an immigrant replaces an arbitrary individual in the local community.
Using a mean-field approximation, it is possible to find an analytic solution
for the stationary probability distribution of this model, and so determine
an analytic form for the species abundance distribution --- which turns out
to reduce to the well-known logseries and lognormal forms in different limits.
The analysis involves the solution of master and Fokker-Planck equations,
which are the central equations of classical non-equilibrium statistical
mechanics. A description of this work can be found at
Phys. Rev. E62, 8466-8484 (2000). A review of
the general ideas behind the approach, as well as a generalisation of the
original model can be found at
Phil. Trans. R. Soc. Lond. B357, 667-681 (2002). The
original model described above can in fact be mapped into Hubbell's model for
local communities which is described in the section on neutral theories. Our model differs from the Hubbell model in two ways: (a)
the Hubbell model has an evolved metacommunity, while the above model has a
uniform metacommunity, (b) there is no equivalent of the matrix which gives a
score to each species i against other species j in the Hubbell
model. In other words, it has no rule which ranks one species of the local
community above another. However, once the mean field approximation has been
made, our model effectively reduces to the Hubbell model. Then, exactly the
same calculations may be carried out, for example, the stationary probability
distribution may be given in closed form.**

**Finally, we give an example of a problem where we
have used a path-integral approach to study a system formulated in terms of
Langevin equations. It concerns the following question: what happens if a
stochastic system is initially in a state which is not stable and more than one
stable state is available for the system to decay into? If this is the case,
the final states will "compete" for occupation, and the system will end up in
one particular one only with a certain probability. This phenomenon, which I
shall refer to as "state selection", occurs in many situations such as fluid
dynamics, lasers, ecology and reaction dynamics in chemistry. I should also
point out, to avoid confusion, that when I use the term "stable", I mean
stable on the time scales of interest. Since these are stochastic systems they
are, of course, metastable and can decay by activation over the barriers
surrounding them. Although techniques based on optimal paths within the the
path integral formalism can be used in this problem, as in the case of noise
activation from one stable state to another, the situation is more subtle.
Nevertheless, Martin Tarlie and I have extended the method to allow us to
calculate the probability of the various states being selected. There are two
distinct cases of interest: type I, where the initial state is near the
separatrix defining the border between the two basins of attraction of two
stable states, and type II, where the initial state is at or near a completely
unstable state (a maximum of the potential, if a potential exists). A general
discussion of the first type may be found at
Phys. Rev. E69, 041106 (2004) and a more
detailed analysis (carried out with Gareth Baxter) may be found at
Phys. Rev. E71, 011106 (2005). An analysis
of the second type of state selection may be found at
Phys. Rev. E64, 026116 (2001). Analytical
expressions for the probability of state selection are given in both cases,
but for type I state selection the result is simpler and, more remarkably,
holds for values of the diffusion constant which are of order one.**