Cumulant expansions and the large deviation rate function

We show that that, by analogy with results from extreme-value theory, the rare events in a Gibbs distribution, as quantified by the exceedance of the distribution, may be expressed in terms of tail joint cumulants that systematically describe rare fluctuations in the tail of the distribution. A joint cumulant expansion based on histogram reweighting is then used to obtain the rate function from large deviation theory, a quantity that characterizes the exponential dependence of the tail of the distribution, for two prototypical systems, namely a 1-D ferromagnet in an external field and an atomistic model of a solid in the isobaric-isothermal ensemble. It is shown that this expansion permits one to capture the behavior of the rate function over a relatively wide range of parameter space from calculations done at one point in this space. Finally, we discuss the extension of this approach to multivariate distributions and suggest strategies for characterizing infrequent events described by the tails of such distributions.