A high-school physics textbook will tell you that, according to Newton’s law of cooling, a hot thing will cool down according to
dQdt∝T∞−T
where Q is heat, t is time, T is temperature, and T∞ is the temperature of the environment. But this is slightly wrong! It turns out that the correct cooling rate, which I’ll call Lambert’s law, is
dQdt∝1T−1T∞.
(Newton’s law approximates this pretty well.) Let’s derive this from first principles, and try to test it empirically.
Epistemic status: Logical deductions from a speculative physical model. I don’t actually know nonequilibrium thermodynamics.
Rates of change
First, a lemma: I claim that if X is any macroscopic variable, then the rate of change of X is proportional to the derivative of the entropy with respect to X:
dXdt∝dSdX
Why? Suppose our system changes in discrete timesteps, and X can only change in steps of size ΔX. And suppose the probability of a transition X↦X+ΔX is proportional to the number of microstates Ω(X+ΔX). Then you can calculate the expected change in X per timestep, and from there prove the lemma.[1]
The Lambert cooling law
Now let our system comprise two subsystems. There’s a fixed amount of heat split between the subsystems. Apply our lemma, taking X to be Q1, the amount of heat in subsystem 1. We find that[2]
dQ1dt∝dS1dQ1−dS2dQ2=1T1−1T2
Here the inverse of temperature, or coldness, shows up. The coldness of a system is defined to be the derivative of entropy with respect to energy. (The only energy here is heat.)
Now if subsystem 2 is very big, its temperature doesn’t change much. So let’s rename Q1 to Q, T1 to T; and T2 to T∞, because that’s the temperature subsystem 1 will have in the limit:
dQdt∝1T−1T∞
We can use this to get an expression for dT/dt involving the second derivative of entropy, which is related to the heat capacity:
dTdt∝−d2SdQ2T2(1T−1T∞)
However, textbooks tend to assume that the heat capacity is temperature-independent, in which case heat is roughly proportional to temperature. So we have
dTdt∝1T−1T∞
This differential equation has a neat solution after a change of variables. Let
τ=(TT∞−1)eT/T∞−1
Then the solution is exponential decay:
τ=τ0e−rt
where τ0 is the value of τ at t=0, and r is a constant. If you want to express the solution in terms of T, you’ll need the Lambert W function.
Experimental support?
Newton’s law is a linear approximation of the Lambert law. If T is close to T∞ (and both are far from absolute zero), then the two laws are hard to distinguish.
I tried to distinguish them anyway, by measuring a hot baking sheet with a cooking thermometer. A best-fit Newton law predicts the data pretty well. The best-fit Lambert law is quite close to Newton (within 3 degrees Kelvin), but is a *worse* fit than Newton. I think my experimental setup has too much error to distinguish the two laws.
I’d guess a proper experiment would have to control convection, prevent the environment from heating up, and explicitly account for changing heat capacity. Cooling down to liquid nitrogen temperatures would make Lambert easier to distinguish from Newton, but that might be harder to work with.
Conclusion
If this cooling law is described somewhere in the nonequilibrium thermodynamics literature, I’d love to see a reference to it. Also, I’d love to know more about the assumption I started with: That a macroscopic variable must change gradually, and that the transition probabilities are proportional to the number of microstates; it seems relevant to dynamics in the same way that ergodicity is relevant to equilibrium.
[Retracted] Newton’s law of cooling from first principles
This post is wrong. See this comment.
A high-school physics textbook will tell you that, according to Newton’s law of cooling, a hot thing will cool down according to
dQdt∝T∞−T
where Q is heat, t is time, T is temperature, and T∞ is the temperature of the environment. But this is slightly wrong! It turns out that the correct cooling rate, which I’ll call Lambert’s law, is
dQdt∝1T−1T∞.
(Newton’s law approximates this pretty well.) Let’s derive this from first principles, and try to test it empirically.
Epistemic status: Logical deductions from a speculative physical model. I don’t actually know nonequilibrium thermodynamics.
Rates of change
First, a lemma: I claim that if X is any macroscopic variable, then the rate of change of X is proportional to the derivative of the entropy with respect to X:
dXdt∝dSdX
Why? Suppose our system changes in discrete timesteps, and X can only change in steps of size ΔX. And suppose the probability of a transition X↦X+ΔX is proportional to the number of microstates Ω(X+ΔX). Then you can calculate the expected change in X per timestep, and from there prove the lemma.[1]
The Lambert cooling law
Now let our system comprise two subsystems. There’s a fixed amount of heat split between the subsystems. Apply our lemma, taking X to be Q1, the amount of heat in subsystem 1. We find that[2]
dQ1dt∝dS1dQ1−dS2dQ2=1T1−1T2
Here the inverse of temperature, or coldness, shows up. The coldness of a system is defined to be the derivative of entropy with respect to energy. (The only energy here is heat.)
Now if subsystem 2 is very big, its temperature doesn’t change much. So let’s rename Q1 to Q, T1 to T; and T2 to T∞, because that’s the temperature subsystem 1 will have in the limit:
dQdt∝1T−1T∞
We can use this to get an expression for dT/dt involving the second derivative of entropy, which is related to the heat capacity:
dTdt∝−d2SdQ2T2(1T−1T∞)
However, textbooks tend to assume that the heat capacity is temperature-independent, in which case heat is roughly proportional to temperature. So we have
dTdt∝1T−1T∞
This differential equation has a neat solution after a change of variables. Let
τ=(TT∞−1)eT/T∞−1
Then the solution is exponential decay:
τ=τ0e−rt
where τ0 is the value of τ at t=0, and r is a constant. If you want to express the solution in terms of T, you’ll need the Lambert W function.
Experimental support?
Newton’s law is a linear approximation of the Lambert law. If T is close to T∞ (and both are far from absolute zero), then the two laws are hard to distinguish.
I tried to distinguish them anyway, by measuring a hot baking sheet with a cooking thermometer. A best-fit Newton law predicts the data pretty well. The best-fit Lambert law is quite close to Newton (within 3 degrees Kelvin), but is a *worse* fit than Newton. I think my experimental setup has too much error to distinguish the two laws.
I’d guess a proper experiment would have to control convection, prevent the environment from heating up, and explicitly account for changing heat capacity. Cooling down to liquid nitrogen temperatures would make Lambert easier to distinguish from Newton, but that might be harder to work with.
Conclusion
If this cooling law is described somewhere in the nonequilibrium thermodynamics literature, I’d love to see a reference to it. Also, I’d love to know more about the assumption I started with: That a macroscopic variable must change gradually, and that the transition probabilities are proportional to the number of microstates; it seems relevant to dynamics in the same way that ergodicity is relevant to equilibrium.
Hints: Remember that S=logΩ. Also, to first order, the denominator Ω(X−ΔX)+Ω(X)+Ω(X+ΔX) is approximately 3Ω(X).
Hints: The entropy of the whole system is the sum of the entropies of the subsystems, S=S1+S2. And dQ2dQ1=−1.