Timeline for How is free energy defined for a single state?
Current License: CC BY-SA 4.0
9 events
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| Jul 26, 2022 at 20:41 | comment | added | Frotaur | Yes, but depending on which parameters are being held fixed, the quantity to be minimized changes... Thus you cannot do this with any parameter that you want. If let the amount of particles to fluctuate for instance it will be the minimum of the Gibbs free energy instead. I will read on thermodynamics as it has been a bit since I studied it, but this seems still very hazy | |
| Jul 26, 2022 at 19:00 | comment | added | By Symmetry | The variables that are not being held fixed will find values that minimize the free energy. Look at it this way, if I have a gas in an insulating box with fixed volume $V$, the gas can find a pressure, temperature and entropy that it likes, but it cannot magic energy in or out of existence and the volume of the box will still be $V$. By minimizing the free energy the fixed parameters determine the values of all others and so determine the macrostate | |
| Jul 26, 2022 at 18:49 | comment | added | Frotaur | I am just saying that because the fact that the free energy must be minimized is only valid for a system at fixed temperature T, right ? If we consider a system which can change temperature T, then we are not in the canonical ensemble, so all these claims about free energies become incorrect. Or I am mixing things up ? | |
| Jul 26, 2022 at 18:47 | comment | added | By Symmetry | "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. | |
| Jul 26, 2022 at 18:44 | comment | added | By Symmetry | "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 | |
| Jul 26, 2022 at 18:32 | comment | added | Frotaur | 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. | |
| Jul 26, 2022 at 18:18 | comment | added | By Symmetry | 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 | |
| Jul 26, 2022 at 18:14 | comment | added | Frotaur | 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 ? | |
| Jul 26, 2022 at 18:10 | history | answered | By Symmetry | CC BY-SA 4.0 |