Physics as Analog Computation

· blackbody radiation

Let us present an example illustrating our main idea of The World as Computation with physics viewed as different forms of analog computation with finite time step, which can be simulated by digital computation with finite time step.

Time-Stepping the Blackbody Equation

To solve the blackbody model equation

  • U_{tt} - U_{xx} - \gamma U_{ttt} = f

where \gamma is a small positive coefficient of radiative damping, we meet the problem of time-stepping an equation of the principal form

  • V -\gamma V_{t} = F,

which has an unstable unphysical component with large exponential growth of the form  \exp(\frac{t}{\gamma}). This unphysical component can be filtered out by time-stepping with a suitably chosen time step k according to Backward Euler:

  • V(t+k) -\frac{\gamma}{k}(V(t+k)- V(t))=F(t+k)

that is

  • V(t+k) = -\frac{\frac{\gamma}{k}}{1-\frac{\gamma}{k}}V(t) + \frac{1}{1-\frac{\gamma}{k}}F(t),

which eliminates the unstable component (assuming \frac{\gamma}{k}<1) if

  • \frac{\frac{\gamma}{k}}{1-\frac{\gamma}{k}}<1

that is if k >2\gamma.

Here  the physical solution is captured by time-stepping with a time step which is not too small, in contrast to the more usual case where the accuracy gets better with decreasing time step.

Notice that the coefficient  \gamma of the third time derivative is small, which allows the time step restriction k>2\gamma to represent physics.

Note that Forward Euler

  • V(t) -\frac{\gamma}{k}(V(t+k)- V(t))=F(t) with
  • V(t+k) = (\frac{k}{\gamma}+1)V(t)-\frac{k}{\gamma}F(t)

is unstable since the exponential growth from the factor \frac{k}{\gamma} is present.

This fits with the familiar experience of numerical analysis that Backward Euler as an implicit method is closer to physics than Forward Euler, which can be criticized as a naive non-physical method.

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