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(FG 1)
Supported by: DFG: Priority Program ``Analysis, Modellbildung und Simulation von Mehrskalenproblemen'' (Analysis, modeling and simulation of multiscale problems)
Description:
This is a joint project of Dreyer/Sprekels. In the last period two questions were posed and answered:
Regarding the first question, we have shown that the answer is affirmative only for a special class of microscopic initial conditions, namely those that can be prepared macroscopically.
Regarding the second question, we have studied simple cases, where the microscopic observables only depend on time but not on the particle index. A well-known example is the oscillator motion of the atomic chain, which can be found in [1].
Let be the microscopic time.
denotes a
microscopic observable that depends continuously on the microscopic
variable
with
. The
macroscopic time is defined by
where
is a small positive parameter, and the macroscopic limit can be
established for
. To this end we define
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(1) |
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(2) |
A second simple example serves to illustrate the suitable mathematics of the
macroscopic limit. We consider the following Newtonian system that
contains the small parameter :
The macroscopic time is defined by , so that the force f
varies only on the macroscopic scale, while the variable Q oscillates on
the microscopic scale. Fig. 1 shows a typical example:
There is a rapid oscillation for small
from the macroscopic
viewpoint.
For fixed and fixed macroscopic time tfinal there
exists, within the time interval
, a solution
of
(3) that we rescale according to
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(4) |
The most important observable is the energy e, which is given by
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(5) |
The functions and
converge to Young measures
and
, respectively. Precisely we write
and
although
and
are no functions but
Young measures. The Young measures
and
are
completely determined by the following properties
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(6) |
References:
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