# Can a distribution with sharper energy maximum than the exp-function give an equivalent theory?

Because for many particles the distribution $\varrho\sim\mathrm e^{-\beta\ H}$ has an extremely sharp maximum, the expectation values of the canonical ensemble agrees with that of the microcanonical ensemble. The exp-function has many desirable computational properties and emerges in the derivation from the mircocanonical ensable when coupled to an external bath e.g. from the $\log$ of involved in the entropy $S$ as defined for the microcanonical ensemble.

My question is if another distribution would also give an equivalent theory, one neighter $\mathrm{const}$ vs. $0$ as in the microcanonical case, but steeper than $e^{-\beta\ H}$ of the canoncial ensemble.

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You know the sharp maximum is a consequence of the size of a thermodynamic system and not the choice of expressing things in terms of the log of the number of states instead of the number of states? – NowIGetToLearnWhatAHeadIs Sep 16 '13 at 17:58
@NowIGetToLearnWhatAHeadIs: Yeah. Plus the $\log$ is not arbitrary. It makes $\mathrm d S$ additive w.r.t. joining systems. There question is, though, can one compute expectation values in the canonical ensemble with a function which is sharp around the same point as the exp-function, and even sharper than that. – NikolajK Sep 16 '13 at 18:42

I've been thinking a lot about this sort of problem recently, after reading Gibbs' Elementary Principles in Statistical Mechanics. I'd recommend you read it, as Gibbs talks about many subtle points regarding the different ensembles, and no other text that I've seen goes into so much detail.

From what I can tell, the canonical ensemble is a fundamental ensemble in its own right. While it is often "derived" in textbooks by using a microcanonical ensemble of a system combined with heat bath, it is not really necessary to assume the microcanonical ensemble. What I mean is, you can use a variety of ways to define the heat bath's ensemble, as there are many ways to get a sharply peaked function. As long as the total ensemble has reasonable macroscopic thermodynamic qualities, the canonical ensemble will pop out in the end (or grand canonical, if you allow for particles to leave) for any small part of the total system.

The origin of the universality is essentially that the heat bath's density of states tends to grow as energy^(huge power) and the typical heat bath energy of concern is a large number. Therefore the logarithm of the heat bath's density of states can be taylor expanded as a linear function, over a wide range of energies. In the thermodynamic limit (imagine taking your heat bath, replicating it 10000000x, and then merging the replicas into one giant heat bath with only the total energy conserved), the log(density of states) becomes arbitrarily linear in energy, and the probability distribution of energies for the heat bath converges to a relatively narrow gaussian (this occurs regardless of the original heat bath ensemble, by central limit theorem). Once you attach the system of concern to this giant heat bath, you invariably get the canonical ensemble for that system.

(I have been a bit vague and non-rigorous here, but I hope you get the gist.)

TLDR version: It is the linearity in log(heat bath density of states) that leads to the canonical ensemble for a system attached to the heat bath; the choice of ensemble for the heat bath is not essential.

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Thanks for the response! As the microcanonical ensemble gives constant value (0 or 1, before normalization), the proof always drops the symbol $\varrho$ soon for the phase space volume in question as such, and so it's difficult for me to extend the derivation and keep the $\varrho$ from scratch. But I also fail to see how the $\log$ which you expand comes into the picture - why would $S\propto \log vol$ be reasonable if you don't start with the microcanonical ensemble? – NikolajK Oct 16 '13 at 17:48
Yes, that's one nice thing about the microcanonical way, that ϱ has such a simple form. I'd add a more careful derivation showing what I am alluding to, but I've got a bit of a cold and my brain power seems to be elsewhere at the moment :). ------- Anyway, it's not really about entropy, but more about the statistics of combining systems. If I recall correctly, the density of states and probability distribution of a combined system are each given by a convolution of the respective functions from the parts. – Nanite Oct 16 '13 at 20:03