In the The Emperor's New Mind book. While author explains why classical physics fails to explain our universe, says:

The root cause of these is that two kinds of physical object must coexist: particles, each described by a small finite number (six) of parameters (three positions and three momenta); and fields, requiring an infinite number of parameters. This dichotomy is not really physically consistent. For a system with both particles and fields to be in equilibrium (i.e. 'fully settled down'), all energy gets taken from the particles and put into the fields. This is a result of the phenomenon called 'equipartition of energy': at equilibrium, the energy is spread evenly among all the degrees of freedom of the system. Since the fields have infinitely many degrees of freedom, the poor particles get left with none at all. http://www.amazon.com/The-Emperors-New-Mind-Concerning/dp/0192861980

My questions: What does "energy gets taken from the particles and put into the fields" mean? How does this relate with classical physics failure?


This stems from the equiparition of energy. The idea is simple enough: suppose you have a system with $n$ degrees of freedom, e.g. $n$ different vibrational modes, then interactions in the system will spread the energy of the system over all the $n$ modes.

If you take a simple particle with no internal structure then this particle has only three degrees of freedom, i.e. motion in the $x$, $y$ and $z$ directions.

If you take a field, e.g. a gravitational field, then to specify the field you have to specify the field strength at every possible position in space, and there are an infinite number of points in space. That means the field has an infinite number of degrees of freedom.

So if you consider a particle interacting with a field, in the combined system the field has infinitely more degrees of freedom than the particle does. So if you spread the available energy evenly over all the degrees the field will get all the energy leaving the particle with none.

But I don't think Penrose intends this to be a rigorous argument, and indeed it isn't. What the equipartition theorem says is that at a temperature $T$ every degree of freedom will get an energy $\tfrac{1}{2}kT$. So the particle energy would be $\tfrac{3}{2}kT$ just as for a free particle, but the field would get an infinite energy. The problem is then explaining why fields don't actually have infinite energy. In fact exactly this prpoblem cropped up in the context of black body radiation, where the classical theory predicts that the energy of the electromagnetc field should go infinity as the wavelength goes to zero. This problem is resolved by quantum mechanics.

  • $\begingroup$ Thank you. You said:"So the particle energy would be 3/2kT just as for a free particle, but the field would get an infinite energy." Get from where? For example 2 charged particle create an electric field. The source of the energy is the particles. Right? But Penrose said:" all energy gets taken from the particles and put into the fields" and you said:"but the field would get an infinite energy". I think what your saying is giving heat energy from outside. So the source is outside. Right? Is your example and Penrose's example different according to the source of the energy? $\endgroup$ – user50322 Aug 9 '14 at 13:34
  • $\begingroup$ Yes, if the temperature is T that assumes the particle/field is in equilibrium with some external heat bath at temperature T, and the energy would come from this heat bath. In effect the field would have an infinite heat capacity. If the particle/field were isolated and contained only a finite amount of energy then the temperature of the system would be infinitesimally small (because of the infinite heat capacity). $\endgroup$ – John Rennie Aug 9 '14 at 13:38
  • $\begingroup$ So author intended to say: "Fields must have particle like property." Right? $\endgroup$ – user50322 Aug 9 '14 at 14:48
  • $\begingroup$ Penrose's point is that you need quantum mechanics to properly describe fields, because in a purely classical theory inconsistencies like infinities must inevitably arise. $\endgroup$ – John Rennie Aug 9 '14 at 14:52

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