6.3 Entropy and Probability

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Now consider our three dice in a box in a childs play room. The room is the childs universe but definitely not an isolated system as required by the second law, however the probability outcomes for the box work to illustrate the thermodynamic probability term 'W' in the Boltzman Gibbs equation for absolute entropy. Note we are only interested in the entropy change associated with the process of shaking the dice. That change is the difference between the final and the initial absolute entropies of the two logical states. Note the Botzman Gibbs entropy depends only on the state of the system not how it got that way. For imperfect universes like our play room so long as the 'leaks' are random and small they will not affect the randomness of the process of shaking the dice. Non random energy would be like the child opening the box and deliberately ordering the dice just to trick you. Which is not allowed even in school let alone science.

At an atomic level we observe that heat always travels from hot to cold by any mechanism available.. conduction, convection and radiation. Atoms are very small particles indeed and this is why their random jumbling behaviour is smooth and steady to anything so large as a thermometer. We observe atoms in close proximity share their vibration (heat) energy with those with less (heat) energy. The energy spreads over more atoms and vastly increases the number of ways (micro states) available for it to occupy. So W increases and entropy increases but the real driver of the second law is probability. The effect is to disipate energy. The heat equation for entropy discovered by Rudolf Clasius simply extends the range of possibility for calculation of entropy change using temperature rather than having to count and identify atoms!

Notice the tendancy of heat to move from hot to cold does not require an 'isolated' system to proceed. If we put a source of heat on one end of a metal bar that is a non random input which will maintain an out of equilibrium condition but at no point in the system will we observe heat moving of itself from cold to hot. Putting the heat source and bar in a large insulated box we can measure the increase in entropy of the air surrounding the bar accounting for the entropy debt represented by the non uniform distribution of heat in the bar.. ie the entropy cost of maintaining that improbable state.

Have a specially good day..