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Lowly engineer here. I am struggling - I think like many - to develop intuition on DEs.

From looking at the history and applications of DEs, general themes that come to mind are, conservation, energy (eg. isochrone problem), causality, feedback (control systems), etc.

However, I can't seem to understand why they are considered such good descriptions of reality. DEs and systems dynamics seems to be almost synonymous with math modelling.

There was an interesting post on hackernews recently. Some guy worked with Feynman on a company that was trying to achieve parallel processing with a million processors. They gave Feynman the problem of analysing "the router" or "connection machine" that would allow processors to share information.

Feynmann apparently solved this problem using differential equations.

I thought this was rather interesting and potentially quite illuminating on DEs in general.

The article doesn't go into any of the math and I couldn't find any papers on it. So I thought I'd ask here.

Does anyone have any guesses/insight as to how Feynmann converted this discrete logic problem into a DE problem? What might that model have looked like in a general sense? Perhaps you can write out some expressions?

From looking at the history and applications of DEs, general themes that come to mind are, conservation, energy (eg. isochrone problem), causality, feedback (control systems), etc.

However, I can't seem to understand why they are considered such good descriptions of reality. DEs and systems dynamics seems to be almost synonymous with math modelling.

There was an interesting post on hackernews recently. Some guy worked with Feynman on a company that was trying to achieve parallel processing with a million processors. They gave Feynman the problem of analysing "the router" or "connection machine" that would allow processors to share information.

*The router of the Connection Machine was the part of the hardware that allowed the processors to communicate. It was a complicated device; by comparison, the processors themselves were simple. Connecting a separate communication wire between each pair of processors was impractical since a million processors would require $10^{12]$ wires. Instead, we planned to connect the processors in a 20-dimensional hypercube so that each processor would only need to talk to 20 others directly. Because many processors had to communicate simultaneously, many messages would contend for the same wires. The router's job was to find a free path through this 20-dimensional traffic jam or, if it couldn't, to hold onto the message in a buffer until a path became free. Our question to Richard Feynman was whether we had allowed enough buffers for the router to operate efficiently.*Feynmann apparently solved this problem using differential equations.

I thought this was rather interesting and potentially quite illuminating on DEs in general.

*By the end of that summer of 1983, Richard had completed his analysis of the behavior of the router, and much to our surprise and amusement, he presented his answer in the form of a set of partial differential equations. To a physicist this may seem natural, but to a computer designer, treating a set of boolean circuits as a continuous, differentiable system is a bit strange. Feynman's router equations were in terms of variables representing continuous quantities such as "the average number of 1 bits in a message address." I was much more accustomed to seeing analysis in terms of inductive proof and case analysis than taking the derivative of "the number of 1's" with respect to time. Our discrete analysis said we needed seven buffers per chip; Feynman's equations suggested that we only needed five. We decided to play it safe and ignore Feynman.*The article doesn't go into any of the math and I couldn't find any papers on it. So I thought I'd ask here.

Does anyone have any guesses/insight as to how Feynmann converted this discrete logic problem into a DE problem? What might that model have looked like in a general sense? Perhaps you can write out some expressions?

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