6.1 ENTROPY and PROBABILITY

This section could also be given the mysterious title of “The Meaning of W”, ie the W from the Boltzman Gibbs equation.. [s = k log(W)]. While were at it, lets also point out that (k) is a constant called the Boltzman constant. Its arbitrary and unnecessary however by giving entropy (s) the units of energy per degree or (Joules/deg C) it allows the derivation of the Clausius equation for entropy change due to heat transfer (special case). Note however W (thermodynamic probability) is just a number it has no units.

Now lets play dice, take 3 dice, any pattern you throw will be one of a total of 6^3 = 216 these are the microstates of the system. The probability of any specific patern is 1/216 call it P3. It is essential that all microstates have the same probability. Now a macrostate is just a group of microstates with a common characteristic. We could say the microstates are 'natural' but we choose the macrostate which is therefore intelligent. There are more ways to get some macrostates than others because they contain more microstates. For example there is only one way of getting {666} (ie it does not matter which dice is which) but look at {665} you can get {656} or {566} three ways to do it, but we get those three ways by noting which dice has the 5. Now take {315} essentially three different numbers, there is {351}, {135}, {153}, {531} and {513} ie 6 ways of getting any such group. Our choice of macrostate meant we ignored the order in which they come but required us to identify the dice. We could have chosen just a sum of dots as a macrostate.

The probability of {351} written P(351) = 6 x 1/216 = 6P3 and Pr(665) = 3 x 1/216 = 3P3. So {666} has only 1 micro state, {665} has 3 and {135} has 6. The meaning of 'W' may now be stated as the number of micro states in any given macro state. If we have a box containing the three dice starting with {666} and shake it, the next pattern is 3 times more likely to be {665} and 6 times more likely to be {135}, and for n = any number its 18 times more likely to be {22n} (six macro states) etc. There is a powerful tendency for the arrangement to move towards a macro state with a larger number of micro states because that is more probable. Hence W will tend to increase and so will entropy (s) to a point where W is at its maximum (in this case 6) and tend to stay there which is called equilibrium.

Have a realy nice day..