Extrinsic thermodynamic variable confusion

[BruceW]

Hi all,
Something has been troubling me. To begin with, I have never been certain about this concept of 'extrinsic' thermodynamic variables. I mean, they don't have to be linear with system size, right? They just need to increase with system size? And also, I have a specific 'example problem' where I am not really sure what should be extrinsic. I am talking about the canonical ensemble here, so let's use $H$ as the total energy of the system, which is a random variable. So for example,
$\langle H \rangle = \frac{1}{Z} \int \ E \ g(E) \ \exp (- \beta E) \ dE$
Right, so now you know what kind of notation I'm using, let's get to my problem. OK, from the fluctuation-dissipation theorem, we have:
$\langle H^2 \rangle - (\langle H \rangle )^2 = k_B T^2 C_v$
Where $k_B$ is boltzmann's constant and $T$ is temperature and $C_v$ is the heat capacity. So now, I know that heat capacity is an extrinsic variable, which means the variance of energy (on the left-hand side) is extrinsic variable too. Now, if we assume that heat capacity increases linearly with system size, and if we assume the average total energy increases linearly with system size, then $C_v/\langle H\rangle$ should be an intrinsic variable, right? Now, if we use the above equation and divide by the average total energy, we get:
$$\frac{\langle H^2 \rangle - (\langle H \rangle )^2}{\langle H\rangle } = k_B T^2 \frac{C_v}{\langle H\rangle }$$
On the right-hand side we have an intrinsic variable, so that means the left-hand side is also an intrinsic variable... But it has units of energy ?! This seems totally weird to me, that something has units of energy, yet is an intrinsic variable.

Also something that is bugging me, is that I assumed that heat capacity and average total energy both increase linearly with system size. But this is not necessarily going to be true. It is true in the case of independent particles, or molecules, or whatever. But when they are not independent, it will be more complicated, generally. So we cannot even say if $C_v/\langle H\rangle$ is intrinsic or not! And so we wouldn't know (generally) if the "variance of energy / average energy" is an intrinsic variable or an extrinsic variable. Does this mean that it will depend on which system we are talking about?! So the definition of thermodynamic variables into intrinsic and extrinsic depends on which specific system we are considering?!

Anyway, thanks for reading my uhh... rant about how the concept does not make sense to me. If anyone has advice/solution, that would be cool

 

[mikeph]

I don't know nearly enough to follow your particular example, but what is your average total energy? Averaged with respect to what time? If it's an average over space then surely this is an intrinsic property. Would would doubling the system size double the average energy?

Or have I misunderstood?

 

[BruceW]

It is averaged over the canonical ensemble. (not over time, and not over space). You can interpret the canonical ensemble of a system like this: ok so the system is in contact with a much larger 'reservoir' or 'heat bath' and the system can exchange heat freely with that reservoir, which is kept at a specific temperature. In other words, the energy of the system is not kept constant, but we assume equilibrium, so on average there is no change with time. And then you can say the total energy of the system has a certain probability to take on each possible value. And the average over these possibilities is the average total energy.

And if (for example) we had a lattice of N independent harmonic oscillators, then the average energy of the entire system is simply N times the average energy of one of the harmonic oscillators. So in this case, the system size is exactly proportional to the average energy. But if the harmonic oscillators affected each other in some way, then system size might not be proportional to average energy.

 

[atyy]

The term "extensive" means proportional to N as N goes to infinity, not for finite N, because of the interacting system problem you mentioned.

I'm not sure about the other part of your question.

 

[sardar]

.Hello,

I understand your confusion about extrinsic thermodynamic variables. Let me try to clarify a few things.

Firstly, extrinsic thermodynamic variables are not necessarily linear with system size. They simply refer to variables that are affected by the size of the system. For example, heat capacity, which is a measure of how much heat is needed to raise the temperature of a system, is affected by the size of the system. A larger system will have a higher heat capacity compared to a smaller system.

Secondly, in your example, $C_v$ is indeed an extrinsic variable, but it is not necessarily linear with system size. It depends on the properties of the system itself. As you mentioned, in the case of independent particles, it may be linear, but in more complex systems, it may not be.

As for the issue of the units of energy for an intrinsic variable, this is not uncommon in thermodynamics. For example, entropy, which is an intrinsic variable, also has units of energy/temperature. This is because it is a measure of the amount of energy that is unavailable for work in a system.

Finally, you are correct in saying that the definition of intrinsic and extrinsic variables can depend on the specific system being considered. This is because different systems can have different properties and behaviors, and therefore the variables that are intrinsic or extrinsic to them can also differ.

I hope this helps to clarify some of your confusion. If you have any further questions, please feel free to ask.