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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:
[tex]\frac{\langle H^2 \rangle - (\langle H \rangle )^2}{\langle H\rangle } = k_B T^2 \frac{C_v}{\langle H\rangle }[/tex]
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
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:
[tex]\frac{\langle H^2 \rangle - (\langle H \rangle )^2}{\langle H\rangle } = k_B T^2 \frac{C_v}{\langle H\rangle }[/tex]
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
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