In the framework of the phenomenological theory of energy and matter, in Thermodynamics, these properties were measured and interrelated in a logically consistent manner. The resulting theoretical body allowed scientists to write up exact relations between measured properties. However, as long as the molecular or atomic nature of matter was not included in the theory, a complete prediction of the numerical values of these observables was impossible.
To take an example: the entropy difference between two states
of a system was operationally defined as the integral
(assuming a reversible path from to ), but the numerical value
of could not be predicted even for an ideal gas. Similarly, thermodynamic
reasoning shows that the difference between the specific heats
and must exactly equal
, where and
are the expansions coefficient and the isothermal compressibility,
respectively; but a specific value for or cannot
Let us dream for a moment: what if we could calculate, say, the entropy of a piece of matter on the basis of microscopic considerations? Assuming we could find an explicit expression for - entropy as a function of energy and volume - then this would give us access to all the riches of thermodynamic phenomena. All we would have to do is take the inverse function and apply the thermodynamic relations , , etc.
Our project then is to describe the mechanics of an assembly of many ( ) particles with mutual interactions. The only way to do this is by application of statistical methods: a rigorous analysis of the coupled motions of many particles is simply impossible.
It should be noted, however, that with the emergence of fast computing machines it became possible to perform numerical simulations on suitable model systems. It is sometimes sufficient to simulate systems of several hundred particles only: the properties of such small systems differ by no more than a few percent from those of macroscopic samples. In this manner we may check theoretical predictions referring to simplified - and therefore not entirely realistic - model systems. An important example are ``gases'' made up of hard spheres. The properties of such gases may be predicted by theory and ``measured'' in a simulation. The importance of computer simulation for research does not end here. In addition to simple, generic models one may also simulate more realistic and complex systems. In fact, some microscopic properties of such systems are accessible neither to direct experiment nor to theory - but to simulation.
In the context of this course simulations will be used mainly for the visualisation of the statistical-mechanical truths which we derive by mathematical means. Or vice versa: having watched the chaotic buffeting of a few dozen simulated particles we will courageously set out to analyse the essential, regular features hidden in that disorderly motion.
Thus our modest goal will be to identify a microscopic quantity that has all the properties of the thing we call entropy. For a particularly simple model system, the ideal gas, we will even write down an explicit formula for the function . Keeping our eyes open as we take a walk through the neat front yard of ideal gas theory, we may well learn something about other, more complex systems such as crystals, liquids, or photon gases.
The grown-up child's inquiry why the water is wet will have to remain
unanswered for some more time. Even if we reformulate it in appropriate
terms, asking for a
microscopic-statistical explanation for the phenomenon of wetting,
it exceeds the frame of this introductory treatment. - And so much
the better: curiosity, after all, is the well spring of all science.