torsdag 29 juli 2010

Mathematics of Blackbody Radiation, Without Backradiation

Blackbody in the form of a grand piano as a system of vibrating strings of different eigen-frequencies, including a "cut-off" damping mechanism transforming high overtones into heat.

Computational Blackbody Radiation presents an analysis of the following model of blackbody radiation:
  • U_tt - U_xx - R U_ttt - D^2 U_xxt = F
  • E_t = int (F^2 - RU_tt^2) dx
where U(x,t) is the amplitude of a vibrating string at position x at time t, with the subindices indicating differentiation. The string is subject to radiative forcing of intensity F^2 (incoming waves) and responds by emitting radiation (outgoing waves) with intensity RU_tt^2. The difference between incoming and outgoing radiation acts as a heat source to the internal energy
  • E = vibrational energy + heat
  • vibrational energy = int (U_t^2 + U_x^2) dx.
Finally, - D^2U_xxt represents a dissipative mechanism grinding waves of high frequency into heat. The coefficient D = H/T with H a (small) spatial mesh scale, represents a cut-off length or smallest coordination length, which is the smallest wave length which can be carried by the vibrating string as coordinated wave motion. Further E ~T^2 with T temperature.

As discussed in Computational Blackbody Radiation the dissipation can take different forms with more or less sharp cut-off. We here consider the simplest case including essentials.

The cut-off length D decreases with increasing temperature T: The hotter the more fine details can be represented and emitted. The colder, the "dumber" is the string.

There is a corresponding cut-off frequency T/H conforming with Wien's Displacement Law.

We assume that R is small expressing that the emission is a small perturbation on top of carrier wave of temperature T ~ 1/R. Incoming waves trigger resonances in the system from which waves are re-emitted. The effect is that an incoming blackbody spectrum below cut-off
can be fully absorbed and re-emitted as the same blackbody spectrum.

A blackbody spectrum is characterized by equal temperature of all frequencies (below cut-off).

The model can be seen as a collection of (atomic) resonators with a range of frequencies from small to medium to large, the motion of which is sustained by incoming waves (with blackbody spectrum). The resonators absorb an incoming blackbody spectrum and react by
  • re-emitting a blackbody spectrum below cut-off
  • transforming incoming spectrum above cut-off into heat (as part of internal energy).
The transformation of high frequency input comes from an inability of the system to correctly absorb certain input, because the required coordination length is too small for the
available precision, and the system therefore distorts incoming high-frequency waves into incoherent high-frequency motion kept as internal (heat) energy, which is not radiated. The heating is similar to blushing from an inability to properly respond to a (sharp/nasty) remark.
The model shows two basic features of blackbody radiation:
  • low incoming frequencies are re-emitted without causing heating.
  • high incoming frequencies are transformed into heat (=incoherent high frequencies).
The net result is that a warm blackbody can heat a colder blackbody, through incoming frequencies above cut-off. But a cold blackbody cannot heat a warmer, because incoming frequencies below cut-off will be re-emitted without heating effect.

Since "backradiation" refers to the latter case, the model indicates that "backradiation" is not physical.

The question is now to what extent the model captures real physics? Is a blackbody an analog computer performing some form of analog computation of finite precision, when absorbing and emitting radiation acting as a system of resonators with finite coordination length?

It is like a system of crickets able to emit variable pitch sound with the top pitch
increasing with "temperature" or "excitation level". Or the frequency of "the wave" in the stadium increasing as the excitation of the public increases. Or like politicians delivering increasingly high pitch coordinated messages as the campaign temperature increases towards election.

Note that the above model is deterministic with the quanta statistics of Planck's classical model (created in an "act of despair" from an apparent collapse of classical mechanics), being replaced by finite precision computation (as an "act of resurrection of the hope" of classical mechanics).

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