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I've recently become interested in DDEs, but I don't know much about them. A DDE has the form

$$\begin{align*}\dot{x}(t) = f(t, x(t - \tau)) && \tau > 0\end{align*}$$

My understanding from the readings I've done is that they are used to approximate systems with delay when the system is otherwise too difficult to model in any other way.

However, intuitively, I feel that that a system's true behavior cannot possibly be modeled via DDEs, because such a behavior would seem to require an infinite amount of memory.

Are there any physical, real-world phenomena that are actually described by DDEs (neglecting quantum effects)? In other words, are DDEs simply a tool for approximating behaviors whose true natures are too difficult to describe, or do they actually describe what happens in the real world?

Qmechanic
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user541686
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    This is a bit too broad for this site as you're effectively asking us to review the area for you. Michael Atiyah got interested in the area recently, and there is a review by him here that you might be interested to read. As far as I know the area remains at the periphery of modern physics. – John Rennie May 14 '15 at 06:41
  • @JohnRennie: Thanks for the link! I'm looking at it right now, but I don't understand why this should require anyone to review the area... all it would take to answer this question is either a simple "no", or a "yes, here's the DDE that governs [some physical effect]"... it should be a pretty short answer either way. – user541686 May 14 '15 at 06:55
  • @Mehrdad - A yes or no answer may be sufficient (? + !!!) for your purpose, but isn't good for the long time health of the site. – 299792458 May 14 '15 at 08:36
  • Everything in physics is an approximation, why shouldn't this be? The above equation is too general to be identified with any particular physics, but delays are highly physical scenarios which appear in basically all areas that are governed by some form of wave equation. I can think of at least a couple of applications for the case where f is a linear function. So that's an affirmative. It's a possible physical model and no more of a simplification than anything else. – CuriousOne May 14 '15 at 08:42
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    @CuriousOne: Don't get too philosophical about what I mean when I say "approximation". I mean it in the obvious and common sense. For example, for the purposes of my question, when writing down the equation of the dynamics of a swinging pendulum, the original nonlinear equation would be considered exact, whereas the simple harmonic motion equation is (obviously) considered an approximation. Hence the true dynamics are nonlinear, but the approximation is linear. Similarly, linear friction is obviously an approximation. My question is, do DDEs ever describe true dynamics, or only approximations? – user541686 May 14 '15 at 08:57
  • There is nothing philosophical about approximation in physics. Everything in physics is just an approximation and that's why we are spending so much time on error analysis. In that sense your question completely misses the point. – CuriousOne May 14 '15 at 09:05
  • I am merely pointing out the obvious: physics is not mathematics and I gave you an affirmative answer, anyway. Delay equations do play an important role in physics and I am sure an average physicist can come up with more than one problem where they would be a nice approximation. Sorry if the latter statement violates your sense of perfection. – CuriousOne May 14 '15 at 09:13
  • Related: https://physics.stackexchange.com/q/27143/2451 – Qmechanic Aug 29 '20 at 14:25

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Delay equations are not necessarily approximations. They often appear if a projection is made, see http://www.physicsoverflow.org/17968/how-to-handle-nonmarkovian-dynamics-in-open-quantum-system

Urgje
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