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.