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I am looking for a good book doing classical mechanics and statistical mechanics in terms of the Liouville operator. I have not found a lot on this subject and even books like Mathematical Methods of Classical Mechanics by V. l. Arnold don't seem to cover it. Especially the definition of the operator without the imaginary unit so just

$$L \Gamma = \{H,\Gamma\}$$

seems to be rare.

Really good online sources would be equally appreciated. If I don't have too mathematically versed to understand it, that would be nice as well.

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Some of the mathematical aspects of the Liouville operator can be found in the second book by Reed and Simon, in section X.14 (it is not a comprehensive account, but it gives the basic ideas and proofs). In the notes at the end of chapter X, in the part dedicated to section X.14, there is also a quite extensive bibliography that may be useful.

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    $\begingroup$ [insert arbitrary topic in functional analysis related to physics] can be found in Reed & Simon. $\endgroup$ – Danu May 13 '15 at 10:02
  • $\begingroup$ @Danu :-D I agree, and I was very tempted to write "as always" after can be found... $\endgroup$ – yuggib May 13 '15 at 10:03
  • $\begingroup$ Maybe I'll take a look but i had those in my hands before, they are quite a bit too advanced for my skills i am afraid. $\endgroup$ – pindakaas May 13 '15 at 10:18
  • $\begingroup$ @pindakaas I understand; however on that particular topic the exposition is not so difficult (at least in my opinion). $\endgroup$ – yuggib May 13 '15 at 10:29
  • $\begingroup$ Oh man I took a look at it it is really hard to gauge what is even going on for me since i really don't know any of the prerequisites. They proove that $iL$ is a generator for a transformation and thus i think its self adjointness follows. I really have no idea how that works... Still thanks for the Tip. $\endgroup$ – pindakaas May 13 '15 at 14:25

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