An observation on probability density equations, or, when do simulations reproduce statistics?

Abstract

We are interested in the question 'when does a simulation of a dynamical system reproduce the long time statistics of the original system?'. Specifically we are interested in PDE simulations where the function space of solutions is split into a direct sum of 'resolved' and 'unresolved' modes and one simulates the resolved modes, perhaps with a model accounting for the unresolved modes. This situation arises frequently in applications. We propose a method for estimating the conditional rate of change of the resolved modes from a time series produced by simulation of the full system (both resolved and unresolved modes). The novelty is that we require only moments of the full system and not the conditional measures. We use the same technique to estimate unresolved modes, giving a statistical analogue to inertial, and approximate inertial, manifolds. We can apply our method for estimating the conditional rate of change to understand the sensitivity of solutions to the density transport equation (PDF equation) with respect to the transporting vector field. The sensitivity of the PDF is intimately related to the question of uniqueness of an invariant measure. From a theoretical standpoint it is well known that without strong assumptions the PDF may be sensitive to perturbations in the vector field. We conclude with a numerical example. We attempt to obtain a marginal PDF of the Lorenz equations dynamics by a two-mode model. We show that the dynamical system based on the conditional rate of change fails to capture tile Lorenz dynamics (as is apparent from the topology of 2-d vector fields) and does not reproduce the marginal invariant measure on the resolved subspace. Even so it reproduces a pair of limit cycles of approximately the right size, which is quite remarkable. We conclude that the formalism of conditional probability density functions, in itself, is not sufficient to explain the reproduction of statistics by simulations.