I thought I recalled that “macrostate” was only used for the “microcanonical ensemble” (fancy phrase for a uniform-over-all-microstates-with-same-(E,N,V) probability distribution), but in fact it’s a little ambiguous.
Wikipedia says
Treatments on statistical mechanics[2][3] define a macrostate as follows: a particular set of values of energy, the number of particles, and the volume of an isolated thermodynamic system is said to specify a particular macrostate of it.
which implies microcanonical ensemble (the other are parametrized by things other than (E, N, V) triples), but then later it talks about both the canonical and microcanonical ensemble.
I think a lot of our confusion comes from way physicists equivocate between macrostates as a set of microstates (with the probability distribution) unspecified) and as a probability distribution. Wiki’s “definition” is ambiguous: a particular (E, N, V) triple specifies both a set of microstates (with those values) and a distribution (uniform over that set).
In contrast, the canonical ensemble is a probability distribution defined by a triple (T,N,V), with each microstate having probability proportional to exp(- E / kT) if it has particle number N and volume V, otherwise probability zero. I’m not sure what “a macrostate specified by (T,N,V)” should mean here: either the set of microstates with (N, V) (and any E), or the non-uniform distribution I just described.
(By the way: note that when T is being used here, it doesn’t mean the average energy, kinetic or otherwise. kT isn’t the actual energy of anything, it’s just the slope of the exponential decay of probability with respect to energy. A consequence of this definition is that the expected kinetic energy in some contexts is proportional to temperature, but this expectation is for a probability distribution over many microstates that may have more or less kinetic energy than that. Another consequence is that for large systems, the average kinetic energy of particles in the actual true microstate is very likely to be very close to (some multiple of) kT, but this is because of the law of large numbers and is not true for small systems. Note that there’s two different senses of “average” here.)
I agree that equal probabilities / uniform distributions are not a fundamental part of anything here and are just a useful special case to consider.
Honestly, I’m confused about this now.
I thought I recalled that “macrostate” was only used for the “microcanonical ensemble” (fancy phrase for a uniform-over-all-microstates-with-same-(E,N,V) probability distribution), but in fact it’s a little ambiguous.
Wikipedia says
which implies microcanonical ensemble (the other are parametrized by things other than (E, N, V) triples), but then later it talks about both the canonical and microcanonical ensemble.
I think a lot of our confusion comes from way physicists equivocate between macrostates as a set of microstates (with the probability distribution) unspecified) and as a probability distribution. Wiki’s “definition” is ambiguous: a particular (E, N, V) triple specifies both a set of microstates (with those values) and a distribution (uniform over that set).
In contrast, the canonical ensemble is a probability distribution defined by a triple (T,N,V), with each microstate having probability proportional to exp(- E / kT) if it has particle number N and volume V, otherwise probability zero. I’m not sure what “a macrostate specified by (T,N,V)” should mean here: either the set of microstates with (N, V) (and any E), or the non-uniform distribution I just described.
(By the way: note that when T is being used here, it doesn’t mean the average energy, kinetic or otherwise. kT isn’t the actual energy of anything, it’s just the slope of the exponential decay of probability with respect to energy. A consequence of this definition is that the expected kinetic energy in some contexts is proportional to temperature, but this expectation is for a probability distribution over many microstates that may have more or less kinetic energy than that. Another consequence is that for large systems, the average kinetic energy of particles in the actual true microstate is very likely to be very close to (some multiple of) kT, but this is because of the law of large numbers and is not true for small systems. Note that there’s two different senses of “average” here.)
I agree that equal probabilities / uniform distributions are not a fundamental part of anything here and are just a useful special case to consider.