Net Research Project for 4/21/08

Entropy is a well defined thermodynamic quantity (like heat, work, internal energy, etc) that is very useful for describing why certain thermodynamic events occur. The Second Law of Thermodynamics is often stated in terms of entropy. Simply put,

Well, it could stay the same if nothing happened. But for any closed non-reversible process (i.e. any real-world process) entropy always increases. There are several ways to define entropy, but the easiest definition to grasp is "The change in entropy of an object equals the heat flowing in or out of that object divided by the temperature of the object."

One event that is not explain by any other thermodynamic law, but is "intuitively obvious", is that when a hot block is placed in thermal contact with a cold block, the two will eventually come into thermodynamic equilibrium (same temperature). Conservation of Energy will not explain this! The hot block could get hotter and the cold block colder and still conserve energy. So, how does the Second Law prohibit this?

Consider the hot and cold blocks placed in thermal contact within an insulating container. All the heat flowing out of the hot block flows into the cold block. Now look at our definition. The change in entropy of the hot block is negative since dQ is negative (recall the convention that dQ is positive when heat flows into an object) and conversely the change in entropy of the cold block is positive. But since T is larger for the hot block, its negative change in entropy is smaller than the positive change in entropy of the cold block. The net change in entropy for the whole system is positive, i.e. the total entropy increases. If heat flowed the other way, entropy would have decreased and the Second Law would be violated. (Note that a proper calculation would require a bit of calculus since the temperatures of both blocks change as the heat flows. But we can get the gist of the argument by looking at the process for just a short time, wherein the temperatures don't change significantly.)

The Arrow of Time

Consider a shoe box with 2 coins. You placed the coins in the box and shake. The coins will be in one of three possible states - both heads, both tails, or one head and one tail. There is only one way each for the first two states to occur, but two ways for the last one. Therefore, the probability for each of the three states is 1/4, 1/4, and 1/2 respectively. The entropy of the coins in the box is proportional to the probability of the particular state (i.e. the particular number of heads and tails) you find. One head and one tale represents a higher state of entropy than two heads or two tales.

What does the Second Law mean in this case? Well, not much actually. When you shake the box, the natural tendency is for the system to progress to a higher state of entropy - a more probable state. You are most likely to find one head and one tale. But no one would be suprised to find either of the other two states. In the case of two coins, the Second Law is not much of a law. With 10 coins, the bet that the system progresses away from an uneven distribution and towards a more equal distribution of heads and tales becomes considerable more assured. The chances of shaking a box with 5 heads and 5 tales and finding later that it has progressed to a state of all tales or all heads is quite small ... not zero, but small. But with a thousand coins, a million coins ... now you have a statistical law with clout! Check out the Georgia State Second Law of Thermodynamics page and browse the Entropy section. Dave Slaven of Morningside College has put together a really nice Page of Entropy in which entropy is presented in its many guises.

How is this statistical definition connected with the earlier heat definition? The hot block gets cooler and the cool block gets warmer because as the molecules of the blocks interact (i.e. collide at the faces in contact) the more probable states are those in which the energy is distributed more evenly between the molecules. In Saganspeak, with "billions and billions" of molecules, on average, the chance that the collisions transfer more energy to the hot block (at the expense of the cold block) is not zero, but so astronomically small that it is pointless to even consider it as a possibility. Check out the happy molecules page. It is a Java Applet simulation of gas molecules in a box. BE patient and watch for a few minutes until something very odd happens. Then follow the "what's going on?" link. It provides a good description of what might have driven Boltzmann to an early grave.

Entropy Leaks Outside of Physics

The association with order and disorder is probably the most difficult aspect of entropy to understand. It is difficult to quantify order in anything but the most simple systems. Organisms (and their interactions with the environment) are far too complex to quantify with Boltzmann's equation of microstates. But that doesn't stop some people from trying. In the article The Second Law of Thermodynamics, Evolution, and Probability , Frank Steiger debates the creationists' (now called the followers of Intelligent Design) use of entropy and the Second Law to support their views. Most scientists, including those who also consider themselves to be religious, realize that the law of entropy has nothing to say about the existence of God.

Remember, the Universe will eventually reach thermodynamic equilibrium and we'll all be dead. Recent astronomical measurements of the expansion of the Universe indicates this could happen in the next 100 billion years or so! Try not to get too depressed about it.

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