# 2.3.3 Statistical mechanics

You saw earlier that very strong claims were made for Newtonian mechanics. Many regarded it as a basic framework that would underlie all scientific explanations. It is therefore natural to ask about the relationship between Newtonian mechanics and thermodynamics:

- Do they contradict one another?

- Are they separate aspects of the truth?

- Can thermodynamics be derived from Newtonian mechanics?

These are not easy questions. Thermodynamics was specifically designed to deal with concepts like temperature, heat and entropy, which had no clear Newtonian interpretation. The gulf between the two subjects can be illustrated by taking, say, a glass of water in a state of equilibrium. We now know that this contains an enormous number of molecules (roughly 1024), each feeling electrical forces due to other molecules and moving rapidly around, colliding with other molecules in the liquid and the glass. The Newtonian world-view would require us to keep track of each and every molecule, building up an immensely complicated and detailed description. Of course, this is utterly beyond our powers. Even if it were possible, the results would provide little or no insight. It would be like looking at a painting under a microscope when its true significance is only apparent from a distance of a few metres. Thermodynamics adopts a more practical viewpoint. Rather than tracking each water molecule in detail, it uses just a few well-chosen variables - including energy, volume, pressure, temperature and entropy - to characterise the state of the water as a whole. The amazing thing is that this works. The thermodynamic description is massively incomplete, yet it is sufficient to make useful predictions.

There is a special branch of physics, called statistical mechanics, which attempts to bridge the gap between descriptions on the scale of molecules and thermodynamics. It recognises that our knowledge of a complicated system, such as a glass of water, is inevitably incomplete so we are essentially reduced to making guesses. This may seem to be a terrible weakness, but statistical mechanics actually turns it into an advantage. It replaces precise knowledge of the motion of molecules by probabilities indicating how the molecules are likely to move, on average. It then goes on to estimate the probability of measuring a particular pressure, energy or entropy in the system as a whole. This is rather like the trick pulled by opinion pollsters when they predict the result of a general election without knowing how every individual in the country intends to vote. Pollsters have a mixed reputation, but the calculations of statistical mechanics are much more clear cut. They turn out to provide predictions that are overwhelmingly likely to happen - so much so, that they appear to be laws of Nature. The second law of thermodynamics is a case in point. From the viewpoint of statistical mechanics, the entropy of the Universe is not bound to increase, it is just overwhelmingly likely to do so. Perhaps 'heat death' will not be the end after all. After countless years of dull uniformity, a very unlikely (but possible) new fluctuation may occur with a lower than maximum entropy, and interesting things will start to happen again.

## Boltzmann, entropy and disorder

The statistical interpretation of thermodynamics was pioneered by James Clerk Maxwell (1831-1879) and brought to fruition by the Austrian physicist Ludwig Boltzmann (1844-1906).

Figure17 Ludwig Boltzmann

In 1877 Boltzmann used statistical ideas to gain valuable insight into the meaning of entropy. He realised that entropy could be thought of as a measure of disorder, and that the second law of thermodynamics expressed the fact that disorder tends to increase. You have probably noticed this tendency in everyday life! However, you might also think that you have the power to step in, rearrange things a bit, and restore order. For example, you might decide to tidy up your wardrobe. Would this lead to a decrease in disorder, and hence a decrease in entropy? Actually, it would not. This is because there are inevitable side-effects: whilst sorting out your clothes, you will be breathing, metabolising and warming your surroundings. When everything has been taken into account, the total disorder (as measured by the entropy) will have increased, in spite of the admirable state of order in your wardrobe. The second law of thermodynamics is relentless. The total entropy and the total disorder are overwhelmingly unlikely to decrease.

Boltzmann's contribution was vital, but had a tragic outcome. Towards the end of the nineteenth century several puzzling facts (which eventually led to quantum theory), triggered a reaction against 'materialist' science, and some people even questioned whether atoms exist. Boltzmann, whose work was based on the concept of atoms, found himself cast as their chief defender and the debates became increasingly bitter. Always prone to bouts of depression, Boltzmann came to believe that his life's work had been rejected by the scientific community, although this was far from being true. In 1906, he committed suicide. If despair over rejection, or frustration over being unable to prove his point, were contributing factors the irony would be great indeed. Soon after Boltzmann's death, clinching evidence was found for atoms, and few would ever doubt their existence again.

OpenLearn - The restless Universe
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