How can entropy increase in a deterministic universe

[Stephen Tashi]

But I wasn't talking about time, I was talking about uncertainty. If all you know is that you're in some blue state, then there is a 1/9900 chance that you're in any particular blue state. That's subjective probability.

Being a Bayesian by preference, I approve of subjective probability.

I just meant that it is improbable (using a subjective notion of probability) that the next state will be lower entropy, if all we know about the current state is that it is a blue state. The statement about the fraction of time spent in blue versus yellow states may also be true, but we would have to make more complicated assumptions about the nature of the transition relation to make that conclusion.

In order to make statement about the subjective probability that a system makes a transition from one color state to another color at randomly selected time I think we must stipulate that the system spends an equal time "during" each microstate transistion and "on" each microstate, if we are modeling a "rest" interval between transitions.

What's the realistic situation for a laboratory experiment? I can conceive of preparing a system so it has a definite state and continuously measuring its state at evenly spaced discrete intervals over a 1 hr time interval. That would allow us to say what fraction "of the time" the system made transitions of a certain type ( yellow to blue etc).

I'm assuming that the color is a macroscopically observable property of the state. The point of using an example where the two colors corresponded to different numbers of states is just that otherwise, the entropy would always be constant. Entropy is only a useful concept when it differs from state to state.

That's a very important observation!

I'm not sure what you mean by "sub-microstate". Do you mean that the microstate itself may actually be a macrostate, with even more microscopic details?

Yes.

My general line of thinking is this: When the physical state of a system is described by a vector of values $(x_1,x_2,x_3,...x_n)$ where each $x_i$ may take values in a continuous range of real numbers, then approximating the vector as a discrete microstate presumably involves defining how a set of these vectors is to be regarded as the same microstate. For example, if we want microstates to be "boxes" we could define a microstate $m_k$ to be the set $(x_1,x_2,x_3,...,x_n): a_{k1} < x_1 \le b_{k1}, a_{k2} < x_2 \le b_{k2}, ... ,a_{kn} < x_n \le b_{kn}$.

If we wish to make direction connection between "fraction of the number of states that are so-and-do" and "fraction of the time the system is in a state that is so-and-do" then we cannot pick the boundaries $a_{k_1},a_{k2},...,b_{k1},b_{k2},...,b_{kn}$ arbitrarily. We need to pick them so that the dynamical law that governs the system's trajectory through the states $(x_1,x_2,...,x_n)$ implies that the system spends approximately the same time in each microstate $m_k$.