I’m going to open up with a technical point: it is important, not only in general but particularly in thermodynamics, to specify what quantities are being held fixed when taking partial derivatives. For example, you use this relation early on:
dSdQ=1T.
This is a relationship at constant volume. Specifically, the somewhat standard notation would be
∂U∂S∣∣V=T,
where U is the internal energy. The change in internal energy at constant volume is equal to the heat transfer, so it reduces to the relationship you used.
That brings us to the lemma you wanted to use:
dXdt∝dSdX.
To get what you wanted, it has to actually be the derivative with constant volume on the right, but then there’s a problem: it doesn’t succeed in giving you the time derivative of V since ∂S∂V∣∣V=0.
Let’s assume that problem with the lemma can somehow be fixed, though, for sake of discussion. There’s another issue, which is that if the proportionality depends on thermodynamic variables, then you can have basically any relationship. For example, your heat equation:
dQ1dt∝dS1dQ1−dS2dQ2=1T1−1T2.
If these proportionalities were T21 and T22, it would actually give Newton’s law of cooling. For an ideal gas, the equation of state U=32NkT means that the change in internal energy (which is just the heat transfer at constant volume) ought to be directly proportional to the temperature change, with no dependence on other thermodynamic variables (besides N) in the proportionality.
Now we have your formula for the derivative of temperature:
dTdt∝−d2SdQ2T2(1T−1T∞).
As a side note, I’m not sure how this is a heat capacity; it doesn’t match any of the heat capacity formulas I remember. But the appearance of T2 is notable; it makes it look a lot closer to Newton’s law of cooling, and comparing it to the earlier equation for heat shows how the proposed proportionalities from the first lemma contain dependence on other thermodynamic variables. But you changed from T1 and T2 to T and T∞ before this, so it’s worth remembering that there should be a symmetric relationship between the two subsystems. Multiplying both of the inverse temperature terms by a single temperature produces an asymmetry in the time derivatives for the two subsystems.
This asymmetry in the temperature dependence would predict that one subsystem will heat faster than the other subsystem cools, which would tend to violate energy conservation. If we just imagine an ideal gas in two separate with an identical number of particles in each container, any temperature increase in one gas has to be exactly compensated by an identical magnitude temperature decrease in the other gas, since the internal energy is just proportional to temperature.
So I argue that this proposed law does not hold up.
Ah, so I’m working at a level of generality that applies to all sorts of dynamical systems, including ones with no well-defined volume. As long as there’s a conserved quantity Q, we can define the entropy S(Q) as the log of the number of states with that value of Q. This is a univariate function of Q, and temperature can be defined as the multiplicative inverse of the derivative dSdQ.
if the proportionality depends on thermodynamic variables
By
dQ1dt∝1T1−1T2
I mean
dQ1dt(t)=a(1T1(t)−1T2(t))
for some constant a that doesn’t vary with time. So it’s incompatible with Newton’s law.
This asymmetry in the temperature dependence would predict that one subsystem will heat faster than the other subsystem cools
Oh, the asymmetric formula relies on the assumption I made that subsystem 2 is so much bigger than subsystem 1 that its temperature doesn’t change appreciably during the cooling process. I wasn’t clear about that, sorry.
Ah, so I’m working at a level of generality that applies to all sorts of dynamical systems, including ones with no well-defined volume. As long as there’s a conserved quantity Q, we can define the entropy S(Q) as the log of the number of states with that value of Q. This is a univariate function of Q, and temperature can be defined as the multiplicative inverse of the derivative dSdQ.
You still in general to specify which macroscopic variables are being held fixed when taking partial derivatives. Taking a derivative with volume held constant is different from one with pressure held constant, etc. It’s not a universal fact that all such derivatives give temperature. The fact that we’re talking about a thermodynamic system with some macroscopic quantities requires us to specify this, and we have various types of energy functions, related by Legendre transformations, defined based off which conjugate pairs of thermodynamic quantities they are functions.
By
dQ1dt∝1T1−1T2
I mean
dQ1dt(t)=a(1T1(t)−1T2(t))
for some constant a that doesn’t vary with time. So it’s incompatible with Newton’s law.
And I don’t believe this proportionality holds, given what I demonstrated between the forms for what you get when applying this ansatz with T versus with Q. Can you demonstrate, for example, that the two different proportionalities you get between T and Q are consistent in the case of an ideal gas law, given that the two should differ only by a constant independent of thermodynamic quantities in that case?
Oh, the asymmetric formula relies on the assumption I made that subsystem 2 is so much bigger than subsystem 1 that its temperature doesn’t change appreciably during the cooling process. I wasn’t clear about that, sorry.
Since it seems like the non-idealized symmetric form would multiply one term by T21 and the other term by T22, can you explain why the non-idealized version doesn’t just reduce to something like Newton’s law of cooling, then?
Here is some further discussion on issues with the 1T law.
For an ideal gas, the root mean square velocity vrms is proportional to √T. Scaling temperature up by a factor of 4 scales up all the velocities by a factor of 2, for example. This applies not just to the rms velocity but to the entire velocity distribution. The punchline is, looking at a video of a hot ideal gas is not distinguishable from looking at a sped-up video of a cold ideal gas, keeping the volume fixed.
Continuing this scaling investigation, for a gas with collisions, slowing down the playback of a video has the effect of increasing the time between collisions, and as discussed, slowing down the video should look like lowering the temperature. And given a hard sphere-like collision of two particles, scaling up the velocities of the particles involved also scales up the energy exchanged in the collision. So, just from kinetic theory, we see that the rate of heat transfer between two gases must increase if the temperature of both gases were increased by the same proportionality. This is what Newton’s law of cooling says, and it is the opposite of what your proposed law says.
Here is a further oddity: your law predicts that an infinitely hot heat bath has a bounded rate of heat exchange with any system with a finite, non-zero temperature, which similar to the above, doesn’t agree with how would understand it from the kinetic theory of gases.
Scaling temperature up by a factor of 4 scales up all the velocities by a factor of 2 [...] slowing down the playback of a video has the effect of increasing the time between collisions [....]
Oh, good point! But hm, scaling up temperature by 4x should increase velocities by 2x and energy transfer per collision by 4x. And it should increase the rate of collisions per time by 2x. So the rate of energy transfer per time should increase 8x. But that violates Newton’s law as well. What am I missing here?
Material properties such as thermal conductivity can depend on temperature. The actual calculation of thermal conductivity of various materials is very much outside of my area, but Schroeder’s “An Introduction to Thermal Physics” has a somewhat similar derivation showing the thermal conductivity of an ideal gas being proportional to √T based off the rms velocity and mean free path (which can be related to average time between collisions).
I’m going to open up with a technical point: it is important, not only in general but particularly in thermodynamics, to specify what quantities are being held fixed when taking partial derivatives. For example, you use this relation early on:
dSdQ=1T.
This is a relationship at constant volume. Specifically, the somewhat standard notation would be
∂U∂S∣∣V=T,
where U is the internal energy. The change in internal energy at constant volume is equal to the heat transfer, so it reduces to the relationship you used.
That brings us to the lemma you wanted to use:
dXdt∝dSdX.
To get what you wanted, it has to actually be the derivative with constant volume on the right, but then there’s a problem: it doesn’t succeed in giving you the time derivative of V since ∂S∂V∣∣V=0.
Let’s assume that problem with the lemma can somehow be fixed, though, for sake of discussion. There’s another issue, which is that if the proportionality depends on thermodynamic variables, then you can have basically any relationship. For example, your heat equation:
dQ1dt∝dS1dQ1−dS2dQ2=1T1−1T2.
If these proportionalities were T21 and T22, it would actually give Newton’s law of cooling. For an ideal gas, the equation of state U=32NkT means that the change in internal energy (which is just the heat transfer at constant volume) ought to be directly proportional to the temperature change, with no dependence on other thermodynamic variables (besides N) in the proportionality.
Now we have your formula for the derivative of temperature:
dTdt∝−d2SdQ2T2(1T−1T∞).
As a side note, I’m not sure how this is a heat capacity; it doesn’t match any of the heat capacity formulas I remember. But the appearance of T2 is notable; it makes it look a lot closer to Newton’s law of cooling, and comparing it to the earlier equation for heat shows how the proposed proportionalities from the first lemma contain dependence on other thermodynamic variables. But you changed from T1 and T2 to T and T∞ before this, so it’s worth remembering that there should be a symmetric relationship between the two subsystems. Multiplying both of the inverse temperature terms by a single temperature produces an asymmetry in the time derivatives for the two subsystems.
This asymmetry in the temperature dependence would predict that one subsystem will heat faster than the other subsystem cools, which would tend to violate energy conservation. If we just imagine an ideal gas in two separate with an identical number of particles in each container, any temperature increase in one gas has to be exactly compensated by an identical magnitude temperature decrease in the other gas, since the internal energy is just proportional to temperature.
So I argue that this proposed law does not hold up.
Ah, so I’m working at a level of generality that applies to all sorts of dynamical systems, including ones with no well-defined volume. As long as there’s a conserved quantity Q, we can define the entropy S(Q) as the log of the number of states with that value of Q. This is a univariate function of Q, and temperature can be defined as the multiplicative inverse of the derivative dSdQ.
By
dQ1dt∝1T1−1T2
I mean
dQ1dt(t)=a(1T1(t)−1T2(t))
for some constant a that doesn’t vary with time. So it’s incompatible with Newton’s law.
Oh, the asymmetric formula relies on the assumption I made that subsystem 2 is so much bigger than subsystem 1 that its temperature doesn’t change appreciably during the cooling process. I wasn’t clear about that, sorry.
Q in thermodynamics is not a conserved quantity, otherwise, heat engines couldn’t work! It’s not a function of microstates either.
See http://www.av8n.com/physics/thermo/path-cycle.html for details, or pages 240-242 of Kittel & Kroemer.
You still in general to specify which macroscopic variables are being held fixed when taking partial derivatives. Taking a derivative with volume held constant is different from one with pressure held constant, etc. It’s not a universal fact that all such derivatives give temperature. The fact that we’re talking about a thermodynamic system with some macroscopic quantities requires us to specify this, and we have various types of energy functions, related by Legendre transformations, defined based off which conjugate pairs of thermodynamic quantities they are functions.
And I don’t believe this proportionality holds, given what I demonstrated between the forms for what you get when applying this ansatz with T versus with Q. Can you demonstrate, for example, that the two different proportionalities you get between T and Q are consistent in the case of an ideal gas law, given that the two should differ only by a constant independent of thermodynamic quantities in that case?
Since it seems like the non-idealized symmetric form would multiply one term by T21 and the other term by T22, can you explain why the non-idealized version doesn’t just reduce to something like Newton’s law of cooling, then?
Here is some further discussion on issues with the 1T law.
For an ideal gas, the root mean square velocity vrms is proportional to √T. Scaling temperature up by a factor of 4 scales up all the velocities by a factor of 2, for example. This applies not just to the rms velocity but to the entire velocity distribution. The punchline is, looking at a video of a hot ideal gas is not distinguishable from looking at a sped-up video of a cold ideal gas, keeping the volume fixed.
Continuing this scaling investigation, for a gas with collisions, slowing down the playback of a video has the effect of increasing the time between collisions, and as discussed, slowing down the video should look like lowering the temperature. And given a hard sphere-like collision of two particles, scaling up the velocities of the particles involved also scales up the energy exchanged in the collision. So, just from kinetic theory, we see that the rate of heat transfer between two gases must increase if the temperature of both gases were increased by the same proportionality. This is what Newton’s law of cooling says, and it is the opposite of what your proposed law says.
Here is a further oddity: your law predicts that an infinitely hot heat bath has a bounded rate of heat exchange with any system with a finite, non-zero temperature, which similar to the above, doesn’t agree with how would understand it from the kinetic theory of gases.
Oh, good point! But hm, scaling up temperature by 4x should increase velocities by 2x and energy transfer per collision by 4x. And it should increase the rate of collisions per time by 2x. So the rate of energy transfer per time should increase 8x. But that violates Newton’s law as well. What am I missing here?
Material properties such as thermal conductivity can depend on temperature. The actual calculation of thermal conductivity of various materials is very much outside of my area, but Schroeder’s “An Introduction to Thermal Physics” has a somewhat similar derivation showing the thermal conductivity of an ideal gas being proportional to √T based off the rms velocity and mean free path (which can be related to average time between collisions).