David Terr's Website

 

Publications

“Fibonacci Expansions and ‘F-adic’ integers,” The Fibonacci Quarterly, v. 34, 1996

“On the Sums of Digits of Fibonacci Numbers,” The Fibonacci Quarterly, v. 35, 1996..

A Modification of Shanks’ Baby-Step Giant-Step Algorithm,” Mathematics of Computation, v. 69, 2000

Here's a copy of my resume

The Second Law of Thermodynamics

The second law of thermodynamics, as originally stated by Rudolf Clausius in 1865, states that the entropy of an isolated system not in equilibrium tends to increase over time, approaching a maximum value. There is much controversy surrounding this law, much more so than most other physical laws. Part of the reason for this is that unlike all other physical laws, which treat time symmetrically, i.e. they remain true if time is run backward, the second law picks out an arrow of time. Also, it is a statistical law, which means that it's only strictly true for large systems, but can be violated for microscopic systems.

A good example of entropy is a partitioned box containing a gas that is cold on the left side and hot on the right side. When the partition is removed, the left side heats up and the right side cools down until the box reaches equilibrium, i.e. reaches uniform temperature on both sides. While it is doing so, its entropy increases. Statistically, what's going on is that the left (cold) side contains molecules that are moving slower on the average than those on the right (hot) side. Once the partition is removed, some of the fast moving molecules that were originally on the right side move to the left side and some of the slower moving molecules that were originally on the left side move to the right side. This continues until both sides contain on the average the same number of slow and fast moving molecules. But it could theoretically happen that the molecules are moving in just the right way that all the fast moving molecules end up on the right and all the slow molecules on the left, in which case the hot side would become hotter and the cold side colder. However, this is statistically impossible, especially if the box contains many molecules, which any realistic box will.

Maxwell's Demon
In 1877, James Clerk Maxwell devised a thought experiment to attempt to disprove the second law. He imagined a tiny demon inside a partitoned box of gas as described above, but in equilibrium, i.e. both sides starting at the same temperature. The demon guards a little trapdoor between the partitions. When he sees a fast molecule on the left moving to the right or a slow molecule on the right moving to the left, he opens the partition, allowing the particle through, but when he sees a slow molecule on the left moving to the right or a fast molecule on the right moving to the left, he closes the partition. In this way, the gas on the left side cools down and the gas on the right side heats up, thus violating the second law.

Maxwell's demons troubled physicists for a long time. The first inroad to the thought experiement occurred in 1929, when Leo Szilard and Leon Brillouin argued that the demon needs to do work, either by measuring the speed of the molecules or by operating the trapdoor, and that this work requires the expenditure of energy, which causes the entropy of the demon to increase more than the entropy of the gas decreases. More sophistocated thought experiments have since been devised, further verifying the second law.

My Take on the Second Law of Thermodynamics
Although many physicists have long been unhappy with the second law of thermodynamics, some even claiming that it isn't really a law, I believe that it is. The fact that it's a statistical law and not strictly true for microscopic systems doesn't mean it isn't really a law any more than the statistical law of averages not being a law. In other words, if a fair coin is tossed many times, on the average it will land heads half the time and tails half the time, and the proportion of the number of times it lands heads approaches 50% as the number of tosses increases. Entropy is much the same. A large (macroscopic) box of gas will approach and eventually reach equilibrium because of statistics. I would go as far as to argue that other much-cherished physical laws are statistical as well. For instance, it has long been known that the law of conservation of energy (the first law of thermodynamics) isn't strictly true either, in the sense that quantum fluctuations are constantly occurring, borrowing energy from the vacuum for free to create virtual particle-antiparticle pairs and paying back the deficit by having them annihilate each other a short time later in accordance with Heisenberg's uncertainty principle.