Introduction

In this chapter we encounter measurements of a type demonstrating some of the unique capabilities of MD. Because the complete trajectories are available, it is no more difficult to measure time-dependent properties, both in and out of equilibrium, than it is to measure thermodynamic and structural properties at equilibrium. Here we concentrate on properties defined in terms of time-dependent correlation functions at the atomic level – the dynamic structure factor and transport coefficients such as the shear viscosity are examples. Most of the analysis is incorporated into the simulation program, but it would of course be possible (though extremely storage intensive) to store the required trajectory data for subsequent processing.

Transport coefficients

Background

Transport coefficients describe the material properties of a fluid within the framework of continuum fluid dynamics. Discrete atoms play no role whatsoever in the continuum picture, but this does not seriously limit the enormous range of practical engineering applications of the continuum approach. The most familiar of the transport coefficients are those applicable to simple fluids; these are the diffusion coefficient, the shear and bulk viscosities and the thermal conductivity. Other transport coefficients appear when dealing with more complex fluids, such as those containing more than one species, or those with novel rheological behavior. In many problems the transport coefficients are assumed to be experimentally determined constants, depending only on the temperature and density of the fluid, which themselves are often assumed constant for a given problem, but in more complex situations transport coefficients can depend on local behavior, an example being the dependence of shear viscosity on the velocity gradient.

I have written this book mainly for students who will need to apply maths in science or engineering courses. It is particularly designed to help the foundation or first year of such a course to run smoothly but it could also be useful to specialist maths students whose particular choice of A-level or pre-university course has meant that there are some gaps in the knowledge required as a basis for their University course. Because it starts by laying the basic groundwork of algebra it will also provide a bridge for students who have not studied maths for some time.

The book is written in such a way that students can use it to sort out any individual difficulties for themselves without needing help from their lecturers.

A message to students

I have made this book as much as possible as though I were talking directly to you about the topics which are in it, sorting out possible difficulties and encouraging your thoughts in return. I want to build up your knowledge and your courage at the same time so that you are able to go forward with confidence in your own ability to handle the techniques which you will need. For this reason, I don't just tell you things, but ask you questions as we go along to give you a chance to think for yourself how the next stage should go.

I have thoroughly revised all the ten chapters in the original edition, both making some changes due to comments from my readers and also checking for errors. I've also added a chapter on vectors which continues naturally from the present chapter on complex numbers.

I wrote the first version of this new chapter as an extension to the book's website (which is now at http://www.mathssurvivalguide.com) building up the pages there gradually. Their content was influenced by emails from visitors, often with particular problems with which they hoped for help. I've now extensively rewritten and rearranged this material. Writing in book form, it was possible to structure the content much more closely than on the Web so that it's easy to see the connections between the different areas and how results can be applied to later problems. The new chapter also has, of course, many practice exercises with complete solutions just as the earlier chapters have.

I'm once again very grateful to Rodie and Tony Sudbery and to David Olive for their helpful suggestions and comments. I must also thank all the people who emailed me, both with comments on the original ten chapters, and also with particular needs in using vectors which I've tried to fulfil here.

I hope that this two-way communication will continue. You can email me from the book's website if you would like to.