Kind of an extension to this question:
If you heat up an object, and put it in contact with a colder object, in an ideal insulated box, the heat from one will transfer to the other through thermal conduction and they will eventually reach an equilibrium temperature at the midpoint, correct?
Now if you have a hot resistor (electrical component) and a cold resistor, and connect them by their leads, so that they make a circuit:
there will be the same conduction and radiation heat transfers. But also, the hotter resistor will have a larger noise current, right? So will there additionally be a transfer of electrical energy from one resistor to the other? Would completing the circuit allow them to reach equilibrium temperature faster than if they were just touching through an insulator with the same thermal conductivity?
Best Answer
A resistor at a temperature T has a fluctuating voltage. This is a consequence of the fluctuation dissipation theorem which you can use to calculate the spectrum of the voltage. The wikipedia article on the Fluctuation Dissipation Theorem has a section on resistor thermal noise. When measured over a bandwidth $\Delta\nu$, the average squared voltage is: $$\langle V^2\rangle = 4Rk_BT\Delta\nu$$ where $k_B= 1.38\times 10^{-23}$J/K is Boltzmann's constant.
Suppose we have a resistor $R$ maintained at a temperature T, and hooked up to a resistor $R_0$ initially at absolute zero. The thermal noise of the warm resistor will be as given above. When you apply a voltage $V$ to a resistor $R_0$, the dissipation (in watts) will be given by $IV = V^2R_0$, so the watts applied to the resistor initially at absolute zero, over a bandwidth $\Delta\nu$ will be $$\langle V^2\rangle = 4R_0Rk_BT\Delta\nu.$$
As that resistor warms up to temperature $T_0$, it will apply a fluctuating voltage on the warm resistor. Following the above, but with the two resistors swapped, the power applied to the warm resistor by the colder resistor at temperature $T_0$ will be: $$\langle V^2\rangle = 4RR_0k_BT_0\Delta\nu.$$
The system will be in balance when the above two powers are equal. This happens algebraically when $T=T_0$.
A possible source of paradoxical confusion is that the above calculation was done over a limited bandwidth range. But the calculation does not depend on frequency; instead the power transmitted is simply proportional to the range of bandwidths.
For the usual physical system, we consider frequencies that run from 0 to infinity. Thus the total bandwidth is infinite. This suggests that the power flow in the above should be infinite. This paradox is avoided by noting that physical resistors have a limited bandwidth. There is always a parasitic capacitance so that the bandwidth is limited on the high side. Thus the power transfer rate depends on how ideal your resistors are.
As an example calculation, suppose that a resistor has a maximum frequency of 100 GHz $= 10^{11}$ Hz, a (room) temperature of 300K, and a resistance of 1000 ohms. Then the power transfer rate is: $$ 4\times 1000 \times 1.38\times 10^{-23} 10^{11} \times 300 = 1.66\;\;\textrm{uWatts}$$ Given the heat capacity of the resistor, you can compute the relaxation time with which the colder resistor exponentially approaches an equal temperature.