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I need to consider a couple of examples of systems which have energies that are intensive variables - not extensive. I'be been thinking about this and I am not coming up with anything. My understanding is that extensive variables (at least wrt usual energies) scales with mass or length (system size). It also seems that some 'energies' depend upon the model used, such as how strong the interactions are in neighbors of atoms or dipoles, etc., or whether one is considering chemical potential or not, etc.

Any good suggestions?

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The total internal energy of a system is completely out of the question as an answer, of course. I would even go as far as saying that it is the quintessential and most important extensive variable of a system. Therefore, most things that have to do with 'energies' within thermodynamics will also be extensive variables. I'm not sure of what is being expected from you as an answer, but as a tentative guess from a condensed matter bloke, without prior knowledge about the particular context in which this question was formulated, I would think of 'energy per something' quantities which are characteristic signatures of the thermodynamic state of a physical system.

That would be the case of the bond energy per atom in a condensate (i.e., solid or liquid) system or the mean thermal energy per degree of freedom which is $\frac{1}{2}k_BT$ from the equipartition theorem (and an intensive quantity, as temperature is also intensive).

Other possible answers that come to mind would be the Fermi energy of the gas of electrons in a neutral solid, or the characteristic gap energies of an insulating, semiconducting or superconducting material, although these are very solid state physics-centered answers.

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I don't know a system for which its energy would be an intensive variable. However, there are systems whose energy does not scale linearly with volume and some parts of energy do not scale linearly with mass.

A simple example would be gravitating mass. For example consider a uniform gravitating sphere. It's gravitational energy according to Wikipedia is given by

$$U = \frac{3GM^2}{5r}$$

As you see this expression is linear neither in mass nor in volume.

You might be interested in non-extensive thermodynamics. Probably they did find some exotic system with energy being intensive variable. Unfortunately I don't know a good introduction to the subject, look through the links in the SklogWiki page a I linked. Or ask a reference question.

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