A simulation can calculate a century in a minute or spend a day computing a fraction of a second. Its progress bar therefore tells us something about a computer task, but not directly about the duration experienced by the system being modelled.
Three quantities deserve separate names: the number of calculation steps, the time represented inside the model, and the elapsed time spent running the calculation.
Three kinds of progress
The iteration count might be n = 0, 1, 2 and so on. A numerical scheme may associate those steps with model times tₙ = nh, where h is a chosen time step. The computer's wall-clock time records how long the computation takes to execute.
Changing the processor can change the third quantity without changing the model's result. Changing h can alter the numerical result even when the simulated duration remains fixed. Changing the physical model changes the question being calculated.
The distinctions are easy to lose in an animation. A screen refresh may display one model step, several steps, or an interpolated frame between steps. Watching a smooth motion does not reveal how the underlying calculation is organized.
One model, several calculations
Consider a dimensionless quantity x that decays according to dx/dt = −kx, with x(0) = 1 and k = 1 s⁻¹. The exact solution at one second is e⁻¹, approximately 0.367879.
A simple forward Euler calculation replaces the differential equation by an update:
xₙ₊₁ = xₙ(1 − kh).
Using different step sizes to reach the same model time of one second gives:
| Step h | Updates | Computed x(1 s) | Absolute error |
|---|---|---|---|
| 0.50 s | 2 | 0.250000 | 0.117879 |
| 0.25 s | 4 | 0.316406 | 0.051473 |
| 0.10 s | 10 | 0.348678 | 0.019201 |
For this example, the smaller steps improve the approximation. We have not discovered that the physical decay contains two, four or ten fundamental moments. We changed the numerical resolution used to approximate a continuous model.
Nor is taking more steps automatically enough in every problem. A calculation needs an appropriate method, sensible parameter choices and a check that the result behaves as expected when numerical choices change.
A discrete model makes a different claim
Now suppose the update rule itself is proposed as fundamental. In that case, n labels the model's native sequence, rather than an approximation chosen to solve a previously specified differential equation.
Even then, the rule must explain how a physical duration is assigned to its steps. One update is a unit of ordering. To call it one second requires a relationship to an operational clock. A model might obtain that relationship through recurring processes or through a calibrated correspondence with measurements.
This is the point developed in Duration and order. A sequence alone does not determine the durations between its entries. The additional structure matters whether the sequence is displayed on a computer or proposed as a description of nature.
What a simulation can establish
A carefully constructed simulation can expose consequences of explicit assumptions. It can reveal a counterexample, demonstrate numerical convergence in a specified setting, or show how an observable changes when a parameter varies.
It cannot establish that its assumptions describe the world merely by producing an attractive animation. Agreement with an observation is more informative when the observable, comparison procedure and adjustable choices are made clear.
In the decay example, an elementary validation is available because the exact solution is known. In a harder problem, useful checks may involve conservation laws, limiting cases or independent implementations. The appropriate check depends on the claim being made.
Simulation and inner generation
A computer program can illustrate a distinction between constructing a record and reading the completed result. That can help explain the inner/outer-time distinction in the Duality of Time programme.
The analogy still needs care. A program has its own processor, memory and execution order. Those are parts of the illustration; they are not automatically features of the proposed physical ontology. Ask which structural relationship the simulation is meant to demonstrate and which implementation details are incidental.
For a concrete next step, revisit the predictive memory laboratory. Identify the model state, the displayed output, the update count and the prediction being tested. Then ask which of those would remain unchanged if the same calculation ran on a faster machine.
Sources and further reading
- Locality begins with neighbours: an example in which the definition of one update determines a propagation bound.
- Claims and evidence: connecting a demonstration to the particular claim it supports.
The decay calculation is an original worked example using the forward Euler formula. Its exact solution and displayed errors are calculated directly; the numbers are not experimental measurements.