| That intuition is still a bit shallow though. I don't mean that in a bad way, some intuition is better than none at all. However if you start to dig you'll find out the terminology goes out of wack. Note that you're describing equilibrium as a unique situation where the number of possible states is at a maximum. Now how can a situation be unique if it has the maximum number of possible states? Clearly the situation is as far from unique as it can be. To resolve the contradiction requires distinguishing between features of the probability distribution and features of a random sample (i.e. a possible state) and also needs an explanation how it even makes sense to view a deterministic physical system (leave quantum mechanics for now) as a random variable. The theory that links everything together is ergodic theory, which has a couple of handy theorems. One is that for a certain kind of dynamical system the average over time and the average over the 'possible states' agree. Such a system can also be assigned an entropy. It even suggests that generally a system will be found around states with a probability close to 2^-entropy (this is not absolutely always true ..but close enough for physicists) Now what does such a system look like? Well we need a state space (easy) and a measure on it which is constant as the system evolves (i.e. we can pick a region in the state space, evolve it and its volume will stay constant). The last part is tricky, but as it turns out classical mechanics gives us the phase space and the canonical volume on it (basically the standard notion of volume) which fit the bill. This gives a probability distribution on the state space and an entropy equal to log(volume in phase space), which matches the definitions in statistical physics but also gives a solid foundation for some of the seemingly arbitrary choices. So there you have it, that's why a system can have a probability distribution attached to it, despite being deterministic, why 'high entropy states' are common, and why physical systems have a uniform distribution (and therefore an entropy which is the log of the number of states). This also explains physicists got away with using a uniform distribution without worrying about which variables they used. By pure 'coincidence' the standard choice of variables that physicists use have this incredibly nice property that makes everything work out. I'm not sure if this is too well known so it might be worth abusing this to 'prove' a perpetuum mobile is possible to stop people using uniform distributions without due deliberation. |