A Complete Info-Metrics Framework

In this chapter I develop the complete info-metrics framework for inferring problems and theories under all types of uncertainty and missing information. That framework allows for uncertainty in the observed values and about the functional form, as captured by the constraints. Using the derivations of Chapter 8, it also extends the info-metrics framework to include priors. The basic properties of the complete framework are developed as well. Generally speaking, that framework can be viewed as a “meta-theory”—a theory of how to construct theories and consistent models given the available information. This accrues all the benefits of the maximum entropy formalism but additionally accommodates a larger class of problems. The derivations are complemented with a complete visual representation of the info-metrics framework. Theoretical and empirical applications are provided.

Comparison of closure approximations for continuous dislocation dynamics

ABSTRACTWe discuss methods to describe the evolution of dislocation systems in terms of a limited number of continuous field variables while correctly representing the kinematics of systems of flexible and connected lines. We show that a satisfactory continuum representation may be obtained in terms of only four variables. We discuss the consequences of different approximations needed to formulate a closed set of equations for these variables and propose a benchmark problem to assess the performance of the resulting models. We demonstrate that best results are obtained by using the maximum entropy formalism to arrive at an optimal estimate for the dislocation orientation distribution based on its lowest-order angular moments.