Conditional Expectations and Renormalization

Abstract

Suppose one wants to approximate m components of an n-dimensional system of nonlinear differential equations (m < n) without solving the full system. In general, a smaller system of m equations has right-hand sides which depend on all of the n variables. The simplest approximation is the replacement of those right-hand sides by their conditional expectations given the values of the m variables that are kept. It is assumed that an initial probability density of all the variables is known. This construction is first-order optimal prediction. We here address the problem of actually finding these conditional expectations in the Hamiltonian case. We start from Hald's observation that if the full system is Hamiltonian, then so is the reduced system whose right-hand sides are conditional expectations. The relation between the Hamiltonians of the full system and those of the reduced system is the same as the relation between a renormalized and a bare Hamiltonian in a renormalization group (RNG) transformation. This makes it possible to adapt a small-cell Monte-Carlo RNG method to the calculation of the conditional expectations. For the RNG the construction yields explicit forms of intermediate Hamiltonians in a sequence of renormalized Hamiltonians (the "parameter flow"). A spin system is used to illustrate the ideas.