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In thermodynamics, in the canonical ensemble, it is said that the state of the system with the lowest free energy will be the equilibrium one.

However, I don't understand how we can defined the free energy of a "state". Indeed, given a system, let us describe a microscopic configuration simply by "$\phi$". Usually in thermodynamics we use $(q,p)$, just to specify what I am talking about. Then, in the canonical ensemble the Free energy is defined as :

$$ F=- k_B T \ln(Z) $$ Where (roughly)

$$ Z = \int d\phi e^{-\beta E(\phi)} $$

Where $E(\phi)$ is the energy of a microscopic configuration $\phi$. Given that, I do not understand, given two different states $\phi_1$ and $\phi_2$, how could I compare their "free energy". Indeed we integrate over all possible states to define the free energy, and thus I do not see how to define it for individual states. Is there a notion of "compartimentalized" free energy, where in some sense we restrict the integral defining $Z$ just to "subset" of states ?

In several resources of thermodynamics, this discussion is not brought up, even though it seems to be a very important point.

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    $\begingroup$ Source of the statement? Check the definition of what is meant by state in that reference (e.g. micro state, macro state...?) $\endgroup$
    – Tobias Fünke
    Commented Jul 26, 2022 at 18:02
  • $\begingroup$ It is usually stated in most thermodynamics course, I can produce a specific one if needed. Otherwise, the wikipedia page for the Helmotlz free energy : At constant temperature, the Helmholtz free energy is minimized at equilibrium. $\endgroup$
    – Frotaur
    Commented Jul 26, 2022 at 18:05

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In thermodynamics and statistical physics we distinguish between microstates and macrostates. A macrostate is defined by the average values of the (macroscopic) thermodynamic variables, such as pressure, volume, temperature etc. The microstate of a system is defined by the exact configuration of each microscopic constituent of the system, e.g. by the position and momentum of each particle in a gas.

Both macro and micro states are simply referred to a 'states' or 'the state of the system' as it is normally obvious from context which is meant. Occasionally, however this can lead to ambiguity. In your question you use the word state to mean microstate, however free energy is a property of a macrostate, leading to your confusion.

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  • $\begingroup$ I don't understand how the definition I gave of Z depends on the macrostate. To me, it is completely agnostic to the macrostate, since we integrate over all possible microstates of the system, and hence also macrostates. Or is my understanding incorrect ? $\endgroup$
    – Frotaur
    Commented Jul 26, 2022 at 18:14
  • $\begingroup$ The partition function will depend on a number of fixed external parameters, for example the temperature or volume of the system, which can be seen as determining the macrostate of the system $\endgroup$
    – By Symmetry
    Commented Jul 26, 2022 at 18:18
  • $\begingroup$ I somehow see what you mean, but if two macrostates have different temperatures they do not belong in the same canonical ensemble, as their temperatures are different, right ? In my case (I was thinking about the Hawking-Page transition), the two "states" have the same temperature. So I guess if I have two macrostates which can be distinguished by some macro parameter $A$, I could define $Z$ as the integral of $\phi$ restricted to the microstates that have macro parameter $A$. This seems to be correct, I will have to think about it a little bit. $\endgroup$
    – Frotaur
    Commented Jul 26, 2022 at 18:32
  • $\begingroup$ "if two macrostates have different temperatures they do not belong in the same canonical ensemble" - In the sense that a system can only have one temperature, sure I guess? But it is not like we have to go right back to first principles every time we increase the temperature os our sample by half a degree. Thinking of $Z$ as a function of temperature, volume or whatever is fine, and indeed I have seen treatments that would argue it is correct to view $Z$ as a function of the constrained in your ensemble $\endgroup$
    – By Symmetry
    Commented Jul 26, 2022 at 18:44
  • $\begingroup$ "So I guess if I have two macrostates which can be distinguished by some macro parameter $A$..." a better way to approach this is to say that $Z$ is a function of the parameters held fixed in your ensemble (say for example temperature and volume). Having found $Z$ we can then find other thermodynamic variables (say pressure) in terms of those fixed parameters. We can then, if we so choose, invert those equations to find our intial fixed parameters in terms of the other thermodynamic variables. We can then define our macrostate in terms of these variables of our choice. $\endgroup$
    – By Symmetry
    Commented Jul 26, 2022 at 18:47

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