Let's say you have a box full of particles, and each of those particles is independently in one of N states. Each state has a probability of occuring. Let us further suppose that none of these probabilities are the same - if so, then there is one state with the highest probability. If there are then properties E and V which are diagonal and have a 1-1 correspondence to each other amongst all the states, then the most common E and most common V definitely are related to each other by E=MV^2/2.

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Examine the Speed and Energy distributions of the ideal gas. The modal values of these two distributions do not correspond to each other via the E = MV^2/2 relation. Paradox?

resolution:

The modal value of P(v) is the value of v for which the integral from v to v+dv is maximal for any sufficiently small dv. Imagine sliding a small window of width dv over P(v) and finding the interval which contained the most probability density - this interval always includes the modal value of P(v), and that value can be found by taking dv to zero. An identical process applies to the modal energy.

The resolution of the paradox lies in the fact that dv and dE are both the wrong measures to use. The correct measure is number of quantum states, which classically corresponds to phase space. The correct procedure would be to find P in terms of phase space, then to find the values of E and v associated with the modal state. E definitely is the same as mv^2/2 in that case.

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Homework:

Explicitly prove this by showing how the phase space approach derives the correct values of E and V corresponding to using the quantum partition function with the classical limit of particle in a box.

The cool nifty thing that totally made my day was finally getting around to learning the "print" command in gdb. It seems so stupid in retrospect to have been using compiled print statements all these years when I could have examined any variable interactively with a simple debug flag. Now if only gdb could step backwards...

In "Free Will - Even for Robots", John McCarthy describes what he feels are the minimal criteria for a complex system to be considered as possessing free will, namely the ability to state "I can, but I won't".

The decision process of such an entity is divided into two phases, starting with the consideration of all possible actions, i.e. what it can do, and then further searching within that set for what it wants most to do.This manner of breaking down the problem of choice is simply a strategy for building good adaptive goal-seeking systems. It is a strategy that works best when considerations of what can be done and what is wanted are well separated, either in terms of possible perturbations (factors mostly affecting one thing or another) or in terms of time-scale (can possibly changing faster than want). By exploiting this natural factoring, goal-seeking systems are able to adapt faster/more efficiently to changing/different conditions. On top of being structured internally like this, people also try to predict each other's actions by assessing ability (can) and intent (want) separately. Again, it makes sense to do so because the inference and knowledge used to estimate the two vary greatly.

Incompatabilists, of which there are quite a few, hold that free will and determinism are in conflict. These are the people who drag Quantum Mechanics into the discussion. I find determinism to be irrelevant to the issue of free will - a person deciding to do something by flipping a coin does not seem any freer to me. The issue here is that of subjective uncertainty - when flipping a coin, not even the person knows what's going to happen next, and hence that uncertainty is shared amongst all the agents, making it objective. A person with free will is one whose actions are unpredictable to others and yet can in retrospect be proven to have been fully decided on by the person internally. The question is, how does one furnish such a proof?