This page last updated 4/3/98

I have heard a lot of nonsense about entropy. In this page I will tell you how to build a simple device for testing entropy. You will need a shoe box and a dollars worth of pennies.

When somebody says "That can not happen because of entropy" they are wrong. Entropy does NOT prevent anything. There is no force that embodies the entropic priciple. You have probably heard that the main statement of entropy is that systems always tend to become more disordered. For example a sugar cube dropped into a hot cup of coffee dissolves and the sugar is distributed throughout the coffee. Entropy is why the sugar never comes out of solution and reforms itself into a cube again. In a sugar cube, the sugar molecules are in a highly ordered crystalline state. In solution the sugar is disordered, it is virtually impossible to predict where any given sugar molecule will be. The position of each sugar molecule represents the state of the sugar system. In solution there are an astronomical number of possible states for the system. In a sugar cube the molecules are locked into the crystal and they stay put. Now lets build a machine with a very large number of states.

Every penny can have two states: Heads or Tails. If you have two pennies, there are four states they can be in. Three pennies have 8 states.

  2 Coins        3 Coins
     HH        HHH    THH
     HT        HHT    THT 
     TH        HTH    TTH
     TT        HTT    TTT 
 

It appears that the number of states for N pennys is 2 to the power N. So if we have 100 pennys, the number of states for them is 2^100. This is a very big number.

Now lets get back to the entropy machine. Put $1 of pennys into the shoe box. Put the lid on the box and shake the box vigorously several times. Now open the box and look inside. Are all the pennys indicating Heads? Probably not. The all heads condition is just one possible state for the entropy machine of 2^100 possible states. This is such a remote possibility hat we could repeat this experiment for the rest of our lives and probably never see all the pennys come up heads at the same time. Do not take my word for it, get a shoe box and a bunch of coins and try it. I predict that you will get bored with repeating the experiment before you see the all heads condition.

Let us see how this simple experiment can demonstrate something useful. Some people say "Do not play the state lottery, you can not win." This statement is wrong. A more correct statement is "Do not play the state lottery you will not win." The difference between the two statements is "can not" versus "will not". Perhaps a fine semantic point, but mathematically it is a very big difference. It is not impossible, it is just very unlikely. I live in Florida. The state lottery here consists of choosing six numbers from a group of forty-nine. The against winning are approximately 14,000,000 to 1. This may seem impossible, but yet somebody does it almost every week. Now back to the entropy machine.

Shake the box again. What is the state of the box? It is another 100 digit binary number. What are the odds that the state you produced would occur? The answer is 1:2^100. The exact same odds as the all heads state! You have just witnessed a miracle. In other words, very improbable things happen all the time. Some people that are trying to convince you of some psuedo-scientific crap will point to some bizarre coincidence and ask you to believe some real nonsense. What are the odds that you are reading this web page at this very instant. If that question was asked yesterday it would be highly improbable, but here you are.

Another thing you can demonstrate with the shoebox entropy machine, is something that some people say can not happen, but in fact happens all of the time. This is the spontaneous DECREASE in entropy. Once you have given the box a good shake, you have pretty much maximized the entropy of the system. What happens if you shake it again? There are only two possibilities. The entropy of the system remains the same or the entropy decreases.

In the state of maximum entropy we would expect to find about 50 cents showing heads and fifty cents showing tails. Once you have achieved this state, entropy can not increase any further so it must stay the same or decrease.

I have heard people say that entropy can not decrease, but just look around and there are many examples of a spontaneous decrease in entropy. First there is the solar system. The entropy was higher when it was a cloud of gas. The gravitational collapse to form planets and the sun represents a decrease in entropy. Note that the collapse of the proto-solar system did not cause an increase in the entropy of some larger system as we would find in the classic refrigerator example. The entropy inside the icebox is lower than that of the surrounding kitchen, but in running the motors of the fridge we have heated up the kitchen and the entropy of a larger system was increased in order to lower the entropy around our cold cuts and beer.

The existence of galaxies is also a manifestation of a spontaneous decerase in entropy. The word on the street is that entropy never decreases, and at the level of a coffee cup I have never seen it happen. On the cosmic scale it has happened many times. Maybe if I let my coffee cup sit there for 20 billion years its entropy may also decrease now and again.

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I have recently tried to read "Newton's Principia for the common reader" by S. Chandresekhar. The first thing I learned was that I am not a common reader, at least not in the eyes of Dr. C. It has been a while since I tried to actually do calculus. Dr. C. is a brilliant man. He is the inventor of the Chandresekhar Limit which proves that it takes a star of at least ten solar masses to form a black hole.

Even knowing that the book was not meant for me, I continued trying to read it. There is one thing I did learn that really surprised me, was how Newton derived his laws. When people think of physicists they think of Galileo dropping rocks off the tower of Pisa. This is the empirical approach. Newton didn't do that sort of thing. His work is pure Geometry. And Geometry was his downfall. When Einstein published his work, it did NOT invalidate Newton. For Euclidian geometries Newton works fine. For speeds much less that the speed of light, Einstien and Newton are in total agreement. But Einstein's theorys work in either spherical or hyperbolic geometries. Newton was stuck with Euclid but Einstein knew about Reimann and Lobachevsky.

If I may be so bold as to rag on Dr. C. a bit. Reading his book was a lot like my first calculus course and something like a chemistry course I took. As derivations are made, sometimes the difference between equation one and two is hard for me to follow. To a brilliant person, the fact that eq. 2 follows from eq. 1 may be obvious, but I get nervous when new variables appear out of nowhere. Or maybe they are constants, if the transition is not clear then the new symbol in the equation soup could be either a variable or a constant. I am sure the fault here is my own. Perhaps what I expected was "Newton's Principia for Dummies".

To the readers of this page, I pose a question. Is it common practice for people to integrate or differentiate a function just to see if anything interesting pops out? It seems to me that many learned texts will say something like "Now when we intregrate F(x) we see that...." But they never said what the significance of the integration was. Why did they do it? If you have an answer please e-mail me at "RobertOelrich@worldnet.att.net".

If you think I am showing incredible cheek for daring to criticize Dr. C. please do NOT e-mail, I already know it.