From: Lee Corbin <lcorbin@ricochet.net>, Fri, 11 May 2001
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.
The subject of this thread says "reversible" and since I jumped into
this topic in the middle without reading all of the earlier posts, I
don't know if we are still assuming "reversible". For the record, I'm
not.
>We have since the days of the Bernoullis been formulating differential
>equations to describe a lot of physical phenomena. Whether it's heat
>flow in solids (Fourier's big breakthrough) or cables hanging in
>catenaries, using the language of mathematics to express infinitesimal
>constraints has been extremely productive. But it is only in this
>arena, important as it is, that differential equations are of any use.
I don't agree. Here are some examples that coupled partial
differential equations can model:
(These are macroscopic phenomena)
fluids
clouds
chemical waves
populations
markets
cerebral cell assemblies
ferromagnetism
the immune system
avalanches
interstellar medium
shocks
bird flocks
urban growth
snow crystals
The microscopic view of these complex systems can be described by
the differential ("evolution") equation, where each component
depends on the space and time of the components:
x' = F(x,lambda)
where F in a nonlinear function F of the vector x, depending on an
external control parameter, lambda. The emergence of order, or phase
transitions, can occur a couple of ways: reversible structures in
thermal equilibrium (growth of snow crystals for example), or
irreversible structures far from equilibrium (most of the interesting
phenomena...)
During the phase transitions of nonlinear dissipative systems, the
old structures become unstable and break down by changing control
parameters. On the microscopic level, the stable modes of the old
states are dominated by the unstable modes. Hermann Hakan calls this
the 'slaving principle' because the stable modes are 'enslaved' by
the unstable modes at a certain threshold. Variables are eliminated
which results in an enormous reduction of the degrees of freedom,
and a new structure emerges because the remaining unstable modes
serve as "order parameters" for the macroscopic behavior of the
system. The evolution of the macroscopic parameters are described by
differential equations. In contrast to the properties of the
elements of a system at the microscopic level (for example, atoms,
molecules), the order parameters denote macroscopic features of the
whole system. A qualitative metaphoric example would be a laser:
where the the slowly varying amplitudes of modes serve as the order
parameters because they start to enslave the atomic system.
>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.
>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" ??
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.
Amara
one useful reference:
Mainzer, K. _Thinking in Complexity_, Springer Verlag, 1994,
pg. 67-68.
********************************************************************
Amara Graps email: amara@amara.com
Computational Physics vita: finger agraps@shell5.ba.best.com
Multiplex Answers URL: http://www.amara.com/
********************************************************************
"Whenever I see an adult on a bicycle, I do not despair for the
future of the human race." -- H. G. Wells
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