Stochastic Geometry of Spatially Structured Birth and Growth Processes. Application to Crystallization Processes

Abstract

• 1 Introduction

• 2 Birth-and-Growth Processes

  • 2.1 The germ process

  • 2.2 Predictability

  • 2.3 The compensator of a marked point process (MPP)

  • 2.4 The geometric process of crystallization

  • 2.5 The nucleation rate

• 3 Elements of Stochastic Geometry

  • 3.1 Hit or miss topology

  • 3.2 Random closed sets (RACS’s) and hitting functional

• 4 Local Densities for Inhomogeneous Random Sets

  • 4.1 An expression for the mean volume density

• 5 The Birth and Growth (Crystallization) Process

  • 5.1 The grain process (growth of crystals)

  • 5.2 The causal cone

• 6 The Hazard Function

• 7 Densities of n-facets of Incomplete Johnson-Mehl Tessellations

  • 7.1 An evolution equation for the n-facets density

• 8 The spherical contact distribution function

  • 8.1 Estimate of the density of d-1-facets via the spherical contact distribution function

  • 8.2 Numerical results

• References