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Boltzmann’s Grave

By Daniel Hubbard | August 1, 2009

Ludwig Boltzmann Monument, from Wikipedia

Ludwig Boltzmann Monument, from Wikipedia

Ludwig Boltzmann died on September 5, 1906 and was buried in his hometown of Vienna. His monument is dominated by an imposing bushy-bearded bust that stares down glowering. Above that bust is a single line, a mathematical equation first formulated by Boltzmann-
S = k log W.

Besides the fact that this appears at a grave, why should a genealogist be interested? Ludwig Boltzmann was a physicist and that equation, his equation, has something to say about an entity of great interest to genealogists and historians- time.

We live in a world with three space dimensions. We can move forward and backward (that’s one), left and right without moving forward or backward (that’s two) and up and down without moving forward, backward, left or right (that makes three dimensions). Our universe also has one time dimension. That means that we can move both into the future and into the past, right? No, we can’t move into the past, we can’t even stand still in time. We can only move into the future. The mystery here is, why? Why can’t we move any way we want in what should be a perfectly good dimension?

There are many interrelated ways of looking at time’s arrow. One thought is that the arrow of time is purely psychological, that our minds simply cannot organize the world around us without imposing a sequential flow onto time. Another possibility is that it is an inherent property of time or of matter that causes all material things to rush onwards. There is at least one more angle from which to view the arrow, perhaps the most basic of all, and it can be seen at Boltzmann’s grave. It takes only a few steps to see.

The S in the equation on Boltzmann’s monument stands for entropy, a measure of disorder and chaos. Scientists and engineers became interested in entropy during the 19th century because it told them a great deal about the functioning of the machines of the industrial revolution. Energy can only do useful work if is is ordered, that is, has low entropy. To see the difference between energy and work think of small children. If they are running around, literally bouncing off the walls, they clearly have plenty of energy but they aren’t really doing what an engineer would call useful work. They are instead demonstrating a significant level of chaos.

The concept of entropy isn’t limited to machines and children. It takes work to turn bricks into the highly ordered thing called a house and it takes work to keep the house livable. Abandon the house and those carefully placed and highly ordered bricks will eventually lie spread chaotically over the landscape. It is only a matter of time. Here W from the equation enters the picture. It is the number of different but equivalent arrangements. The larger W is, the larger the entropy. In the case of the house, instead of placing brick 22 between bricks 21 and 23 the bricklayer could have put brick 98 there and placed brick 22 between bricks 97 and 99 and that would have given a different arrangement that is still a house. So it seems like there are a lot of different brick arrangements that can result in a nice house but just think about how many more ways there are to scatter those same bricks on the ground. The difference in the number of arrangements between order and chaos is astronomical.

Before Boltzmann it was observed that entropy always increases or stays the same unless work is done. We can live in a highly ordered world only because the sun is constantly showering us with energy that plants have been harnessing for eons giving us everything from coal to corn flakes. Boltzmann showed that the reason for this entropy increase is that there are many, many more ways for something to be in a chaotic state than an ordered state. Any change is likely to increase the level of chaos simply because there are more ways to achieve disorder than order. Compared to all the possible messes that can exist after you drop an egg, there are very few arrangements that make an unbroken egg. Give a whole egg a jolt and you can easily make a mess. Give a broken egg a jolt and you simply get a different mess. Your chances of causing the egg to reassemble are remote indeed. So, entropy increases simply because it is far, far more likely that a change made to an ordered arrangement leads to less order than to more order.

In this view then, forward in time simply means the direction of increasing entropy. There is always the possibility at least in principle that entropy will spontaneously decrease, eggs can reassemble, houses can arise from ruins and ice cubes can appear floating in a glass of warm water but the probabilities are so incredibly minuscule that these things can be safely said to never happen.

In the end, time flows downhill in a riverbed of probability and the once living and highly ordered Ludwig Boltzmann lies dead, decayed and disordered under the equation that tells us why it is so.

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