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In classical mechanics the concept of energy is very simple. If I have a bunch of particles $r_1$...$r_n$. Then the total energy is:

$$E=\frac{1}{2}m(\dot r_1^2+...\dot r_n^2)+U(r_1...r_n)$$

Now in thermodynamics; I read from callen's book that Energy is a function that dependa on volume ($V$); number of particles ($N$) and entropy($S$). That is:

$$E=f(S;V;N)$$

What is the connection between these two Es? How can I go from $r_1..r_n;r_1'...r_n'$ to $S;V;N$. Given that the potential energy function is known (suppose lennard jones).

Main purpose of this post:

I want to find the connection between classical mechanics and thermodynamics; without discretizing the phase space and going through the concept of entropy.

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    $\begingroup$ Where did you get the first equation from? It looks like some form of kinetic energy in which case that would not be the total energy of a system. $\endgroup$
    – Bob D
    Commented Mar 28 at 20:59
  • $\begingroup$ U is the potential energy. $\endgroup$
    – Robin
    Commented Mar 28 at 21:00
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    $\begingroup$ The general form of potential energy has nothing to do with kinetic energy. $\endgroup$
    – Bob D
    Commented Mar 28 at 21:32
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    $\begingroup$ You are going to want to look into a little something called "statistical mechanics." See, for example, this website: en.wikipedia.org/wiki/Statistical_mechanics $\endgroup$
    – hft
    Commented Mar 29 at 0:23
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    $\begingroup$ Voting to reopen. This is a perfectly clear question. It is only some of the comments above have created any confusion. $\endgroup$
    – gandalf61
    Commented Mar 29 at 11:52

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