I have a question which is inspired by considering the light field coming off an
ID: 1375569 • Letter: I
Question
I have a question which is inspired by considering the light field coming off an incandescent lightbulb. As a blackbody radiation field, the light is in thermal equilibrium at temperature T, which implies that each normal mode has a mean energy given by Planck's law, and a random phase. Thus, if I were to look, microscopically, at the electric field, I would see a fairly complicated random function that can only really be considered constant at timescales ???/kBT (at which the corresponding modes have next to no amplitude and therefore do not affect the electric field's time dependence).
This illustrates a general aspect of thermal and thermodynamic equilibrium: they are only relevant concepts when the systems involved are looked at on timescales far longer than their relevant dynamics.
My question, then, is this: are examples of slowly-varying systems (where by "slowly" I mean on the timescales of seconds, or preferably longer) that can be considered to be in thermal equilibrium on timescales longer than that known?
Explanation / Answer
Good question! When I started writing this answer I couldn't think of an example, but then I realised that Brownian motion fits the bill, as I'll explain below. So please forgive the somewhat tangential introduction:
To a reasonable degree of approximation, temperature can be thought of as energy per degree of freedom. I say approximation because in quantum systems the number of degrees of freedom can depend on the temperature as well, and then it gets complicated, but let's imagine for a moment that we live in a classical world where that doesn't happen.
The range of temperatures we observe in everyday life ranges up to a few thousands of Kelvins. (You can get much higher temperatures in some experiments, but only for very brief periods of time.) Multiplying by Boltzmann's constant, this corresponds to a maximum of around 10?19 Joules per degree of freedom in any system that you're likely to be able to sit around and watch for a long period of time.
The rate of motion this corresponds to depends on the mass associated with the degree of freedom. For example, if I surround a 1kg pendulum with air at 300K then both the gas molecules and the pendulum will have an average energy of the order 10?21 J. For the gas molecules this corresponds to a speed of about 500ms?1, whereas the pendulum will have an average velocity of about 10?21???????10?11ms?1 once the system reaches equilibrium. However, the amplitude of its movement will be too small to measure, because the force restoring it to its equilibrium position will easily overwhelm its thermal movement.
However, for a mass of intermediate size, such an amount of energy gives rise to motion that is both slow enough to be perceived by the human eye, and of sufficient magnitude to be seen through a microscope. It was first observed by Brown in the case of pollen grains suspended in water. The grains jiggle around all over the place, but when looked at over long time scales they're in thermal equilibrium - their motion will continue in the same way forever. Unlike a pendulum the grains have no force restoring them to their original position, so they will wonder all over the container they're in.
The time scale can be increased by increasing the mass of the grains. However, as the grains get larger it will be harder and harder to eliminate forces that make them want to do things like sticking to the sides of the container. Still, I guess that with the right setup, and precise enough measurements, you could probably observe Brownian motion taking place on as long a time scale as you want.
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