On Various Questions in Nonequilibrium Statistical Mechanics Relating to Swarms and Fluid Flow

Russell Kim Standish

Chapters 1 and 5 review the field to set the context of the work. Except where stated, all other chapters report my own original work. Russell Standish

Acknowledgements

I wish to thank Denis Evans for the supervision of the work in this thesis, Brian Robson for his continuing support, Robert Robson for his encouragement and advice, Leo Brewin for his comprehensive knowledge of the MSP operating system, and Bill Alford for his advice on TeX related matters, without which this thesis would have been much more difficult to write. I would also like to thank all the staff and students of the Department of Theoretical Physics, who have over the years provided much stimulating conversation and ideas. Last, but not least, I would like to thank my dearest friend Kim Crichton for her unwavering support throughout this thesis.

Much of the computation described here was performed on the Fujitsu VP100 of the Australian National University Supercomputer Facility.

I gratefully acknowledge the financial assistance of a Commonwealth Postgraduate Research Award.

This thesis has been typeset by the author using LaTeX, a macro package written by Leslie Lamport for Donald Knuth's TeX typesetting system. The source was typed on a MacIntosh, using a macro package written by the author for Matthias Aebi's MEdit text editor. Andrew Trevorrow's OzTeX, a public domain MacIntosh version of LaTeX, was used to preview the document whereas the final version was produced using a Vax. The figures were produced using a picture macro package written by Brendan McKay, the Vax installation of which was done by Bill Alford. Figures to were produced by Arnstein Pritz's FPLoT program, and the plates produced using Uniras.

Publications

Standish RK (1987),

``Nonhydrodynamic Contributions to End Effects in Time of Flight Swarm experiments'', Aust J Phys, 40,519-525

Standish RK and Kumar K (1987),

``End Effects in Time of Flight Experiments'', proc. 5th Intl Swarm Seminar, Birmingham, p104-107

Evans D and Standish RK (1990),

"Non-equilibrium Statistical Mechanics and Molecular Dynamics Calculations" in Computer Modelling of Fluids, Polymers and Solids CRA Catlow et al.(eds.) p125-154

Standish RK (1989)

``Motion of Charged Particles in a Homogeneous Reacting Medium with a One-dimensional Geometry'', Aust J Phys, 42, 223-232

Standish RK and Evans DJ (1990),

``On the Nonlinear Burnett Coefficients'', to be published in Phys. Rev. A.

Abstract:

In this thesis, the theory of nonhydrodynamic phenomena is developed within the framework of the linear Boltzmann equation, and applied to several experimental configurations used in swarm physics. The latter part of the thesis develops fluctuation expressions for the nonlinear Burnett coefficients similar to the Green-Kubo expressions used for linear transport coefficients. A numerical simulation of a simple fluid is used to test these expressions.

• Contents
• List of Figures
• Introduction
  • Continuity Relations
  • Constitutive Relations
  • Burnett Coefficients
  • Boltzmann Equation
  • Chapman-Enskog Solution
  • Nonhydrodynamic Behaviour
• The Linear Boltzmann Equation
  • Time Dependent Transport Coefficients
  • Projection Operator
  • Formal Solution of the Boltzmann Equation
  • The nature of the Spectrum and the Runaway Phenomenon
  • Klein-Kramers Model
• End Effects in Time of Flight Experiments
  • Time of Flight Experiments
  • Lithium ion - Helium Experiment
  • Nonhydrodynamic Contributions to End-Effects in Time-of-Flight Swarm Experiments
  • Model Calculation
  • Peak Arrival Time
  • Pressure Dependencies of Non-hydrodynamic and Higher Order Diffusion Effects
  • Separation of the Different End Effects.
  • Boundaries and Field Inhomogeneities
• Non-hydrodynamic effects in Parallel Plane Steady State Townsend Experiments
  • Steady State Townsend Experiments
  • Conventional Theories
  • Numerical Studies of the Klein-Kramers model
  • Asymptotic behaviour for large times and distances
  • Discussion
• Nonequilibrium Molecular Dynamics Simulation
  • Thermostatted Equations of Motion
  • Formal Solution of Liouville's Equation
  • Nonlinear Response Theory
  • Models of Nonequilibrium Steady States
    • Planar Couette Flow
    • Colour Current
• Nonlinear Burnett Coefficients
  • Equations of Motion
  • Relation between canonical and ensemble
  • Equilibrium Simulation
  • Numerical Evaluation of Burnett Coefficients
• Conclusion
• Perturbation of the eigenfunctions for small k
  • Recursion relations in Component Form
  • Consistency with Biorthonormality
  • Invariance of
• Isotropic Tensors
• System Size Dependence of a Product of Intensive Phase Variables
• Nonlinear Burnett Coefficients Calculation
  • Design Considerations
  • Parameters
  • Equations of Motion
  • The Integrator
  • Periodic Boundary Conditions
  • Forces
  • Miscellany
  • Analysis of Data
  • Putting it all Together
• References
• Mathematical Notation
  • Conventions
    • Vectors and Tensors
    • Differentiation and Integration
    • Other Conventions
  • List of Symbols
• Index
• About this document ...