Encompassing over fifty years of research, Professor Gotoh addresses the correlation function of spatial structures and the statistical geometry of random particle assemblies. In this book morphological study is formed into random particle assemblies to which various mathematics are applied such as correlation function, radial distribution function and statistical geometry. This leads to the general comparison between the thermodynamic state such as gases and liquids and the random particle assemblies. Although structures of molecular configurations change at every moment due to thermal vibration, liquids can be regarded as random packing of particles. Similarly, gaseous states correspond to particle dispersion. If physical and chemical properties are taken away from the subject, the remainder is the structure itself. Hence, the structural study is ubiquitous and of fundamental importance. This book will prove useful to chemical engineers working on powder technology as well as mathematicians interested in learning more about the subject.

Key Features

  • Concisely explains various mathematics and tools applied to the subject.
  • Incorporates multiple fields to give a clear view of the application of mathematics in powder technology.


Researchers and students in chemical engineering, materials science and mathematics

Table of Contents


1. Spatial Structure of Random Dispersion of Equal Spheres in One-Dimension

1.1 Discrete System

1.2 Continuous System

2. Spatial Structure of Random Dispersion of Equal Spheres in Two-Dimension

2.1 Outline of Computer Simulation Experiments

2.2 Structure of Random Dispersion

3. Preliminary Mathematics

3.1 Laplace Transform and Inversion Formula

3.2 Fourier Transform and Spectral Density

4. Radial Distribution Function

4.1 RDF Definition

4.2 Ornstein-Zernike Equation

4.3 Solving Procedure

4.4 Analytic Solution by Percus-Yevick Approximation

A. xgn(x)

B. state at contact

4.5 Multi-Sized Particle System

4.6 Binary-Sized Particle System

4.6.1 Influence by the Presence of a Vessel Wall

4.6.2 Pore Size Distribution in Random Assemblies of Equal Spheres

4.6.3 Size Distribution of Aggregates Inherent in Random Dispersion of Equal Spheres

5. Sample Size for Measuring Particle Concentration

5.1 Distribution Functions

5.2 Sample Size of Measurement

6. Introduction to Statistical Thermodynamics

6.1 Quantum States of Steady Thermal Vibration

6.2 Analytical Dynamics and Generalized Coordinates

6.3 Stationary Distribution and Partition Function

6.4 Number Density and Distribution Function

6.5 Equation of State for Gases

7. Structural Comparison of Molecular System and particle Assemblies: Particle Morphology

7.1 Random Packing

7.2 Random Dispersion

7.3 Molecular System and Particle Assemblies

Closing Remarks


No. of pages:
© 2012
Print ISBN:
Electronic ISBN:

About the author

Keishi Gotoh

Affiliations and Expertise

Toyohashi Sozo University and Toyohashi University of Technology, Toyohashi, Japan