Amara wrote:
>Lee,
>
>I'll venture a guess that you and I defining "differential equation"
>differently.
>
>My definition of "differential equation":
>
> function => evolve(function)
> where "evolve" is over time or over space or both
>
>Differential equations can describe almost every phenomena that
>I can think of.
Yes. This is broader than what I meant by "differential equation".
Into your definition, it looks to me as if the number theory
Collatz function (defined on each positive integer)
x => x/2 (if x is even)
x => 3x + 1 (if x is odd)
is also a "differential equation". Is it, (in your book)?
I, on the other hand, had characterized differential equations as
always expressing infinitesimal constraints, and stated that they
were useful only in this "arena" (a very poor word choice).
You then gave a nice list of "examples that coupled partial
>differential equations can model:
>
> fluids, clouds, chemical waves, populations, markets,
> cerebral cell assemblies, ferromagnetism, the immune system,
> avalanches, interstellar medium, shocks, bird flocks,
> urban growth, and snow crystals
But isn't it true that, to take populations for example, one
proceeds by assuming continuous values, rather than discrete?
It's not surprizing that this approach can yield a great deal,
(though some of your examples are new to me).
You then go on to summarize in a very nice way (and in a way
that I'm sure I don't fully appreciate) some interesting results
concerning nonlinear dissipative systems and so forth, that I
don't quite understand the relevance of in this discussion you
and I are having.
>>From a computational perspective, the myriads of intermediate
>>results necessary in the numerical solution of differential
>>equations are merely approximations to the mathematical truth.
>
>One can always choose the time at which to compute the functional
>evolution, so I still don't follow why you are emphasizing
>"intermediate results". One can study the system to determine the
>phase transitions, in order to know at which time important
>structures appear.
Well, what I was referring to was the kind of calculation that
needs to be done in, say, the three-body problem, where one must
calculate numerical values for differentials, add the results to
the variables in question, and recalculate the differentials and
so forth. The Lorenz equations, as you know, are the prime example,
(but the Rossler quations are easier to type :-)
x' = -y - z (where ' denotes derivative with respect to t)
y' = x + ay
z' = b + (x - c)z
It's in this sense that "intermediate" values are needed; one cannot
simply calculate what x, y, and z are at time t=20, for example.
As I think you said, there are closed form solutions for exceedingly
few differential equations, so computer generated solutions always
take some kind of path like this. (I was using Greg Egan's phrase
"intermediate results" mostly to try to read him in the most charitable
way.) But here is exactly what I meant by the foregoing paragraph:
Take the Rossler equations above, for a concrete example. Given
particular initial values, they define a mathematical truth. There
exist exact real numbers that correspond to values of t, but we can
never do more than approximate these real values (unlike the case
with the Collatz iteration, say). That's all I meant by "the numerical
solution of differential equations are merely approximations to the
mathematical truth." Is this still problematic?
>>But in iterative calculations of this other kind, e.g., Life, chess,
>>Turing machines, and the disjointed details of our lives on Earth,
>>the necessary intermediate results **are** the "mathematical" truths
>>themselves.
>
>Hmm. Are you meaning "emergent order" ?? [no]
>Differential equations (albeit VERY coupled and nonlinear and dissipative
>and far-from-equilibrium, and hard to compute on today's computers) can
>model most of these systems.
Am I reading you right? You believe that chess can be modeled
by differential equations? And Turing machines, i.e., computer
programs? And Conway's Life? It's simply not the case that
differential equations can model these in any useful way, unless
(again) you're using the term "differential equation" in an
amazingly broad way. (I.e., are Conway's Rules further examples
of what you'd call "differential equations"???)
Now of course when I write about "our lives on Earth", that's so
broad that of course some aspects are captured extremely well by
differential equations---by many aspects, I submit, cannot be
so captured.
Lee
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