Abstract
A new explanation of the spectrum of black-body radiation is presented based on finite precision computation
instead of statistics.
All these fifty years of conscious brooding have brought me no nearer to the answer to the question, ‘What are light quanta?’ Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken…The quanta really are a hopeless mess. (Einstein 1954)
The whole procedure was an act of despair because a theoretical interpretation (of black-body radiation) had to be found at any price, no matter how high that might be…I was ready to sacrifice any of my previous convictions about physics...(Planck 1900)
Black Hole of Classical Physics
The starting point of modern physics was that classical physics failed to explain black body radiation:
A black body absorbs all incoming light of all frequencies, but only emits low frequencies with a
cut-off of high frequency depending on temperature. This is illustrated in the following figures showing
that the cut-off depends on the temperature and occurs for higher frequencies/shorter wave-lengths for higher temperature:
which is a graphical representation of
showing the exponential cut-off factor.
This is why the Earth absorbing light from the Sun of all frequencies from low to high, does not itself glow like the Sun, but only radiates low frequency infrared light:
The trouble with classical wave mechanics facing the physicists in the late 19th century, was that it seemed to be fully reversible, and thus whatever was absorbed ultimately had to be emitted. Classical physics
predicted that the Earth would shine like the Sun, with no cut-off of high frequencies as indicated in the following figure:
predicted that the Earth would shine like the Sun, with no cut-off of high frequencies as indicated in the following figure:
But the Earth does not glow like the Sun and the question to be answered was (and is):
- Why is formally reversible wave mechanics, in reality irreversible with cut-off of high frequencies?
To answer this question Planck used a form of statistical mechanics, where the high-frequency cut-off was explained as an effect of smaller probability of high-frequency quanta. But Planck was not happy with his resolution, which he believed was only a mathematical trick without physical meaning, and Planck became a revolutionary against his will.
Finite Precision Computation as Explanation of Cut-Off.
The above question is the question of irreversiblity in a formally reversible Hamiltonian system, which is answered by the new version of the second law of thermodynamics based on finite precision computation instead of statistics. The answer is the same because wave mechanics of absorption and emission of light is an example of a formally reversible Hamiltonian system. The answer is developed in computational black-body radiation [1]and in concentrate reads:
- a black-body can be seen as a network of interacting atomic oscillators which are excited by incoming waves and can emit waves by coordinated oscillation
- irreversibility arises from finite precision computation and not from statistics
- cut-off of high frequencies occurs because the required coordination of atomic oscillations cannot be met by finite precision computation
- physics is a form of analog computation of finite precision
- the cut-off frequency increases with temperature because the amplitude of oscillation increases which increases relative precision
- a black-body transforms coordinated high-frequency input into low-frequency infrared heat radiation
- cut-off of high frequencies is a well-known phenomenon in computational wave propagation arising from finite precision and not from statistics.
A Classical Model of Black-Body Radiation
Consider the following wave equation for the deflection U of a vibrating string of temperature T subject to a distributed forcing F and radiating energy to the surrounding medium:
U_tt – U_xx - R U_ttt - H^2U_txx= F for 0 < x < 1, t > 0
T_t = integral (F2 – R u_tt u_tt ) dx for t > 0,
where R > 0 is a coefficient of radiation, H = h/T is an effective mesh size representing the cut-off wave length, where h is a nominal mesh size representing Planck’s constant, and the subscript denotes differentiation with respect to time t or space x.
This model is a variant of the model used by Planck[2]in deriving his radiation law, with the additional term H2Utxx modeling diffusion from computation causing computational dissipation H2UtxUtx entering the energy balance of incoming energy from forcing and outgoing radiating energy, generating heat increasing the temperature.
Computational results showing that this model generates Planck’s radiation law will be presented shortly. The conclusion is that Planck’s radiation law can be derived from a computational version of a classical wave equation model, without reference to any ad hoc statistics of quanta.
The World As Computation 