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Collaborator: H. Stephan
Cooperation with: G. Wachutka (Technische Universität München), E.Ya. Khruslov (B. Verkin Institute for Low Temperature Physics and Engineering, Kharkov, Ukraine)
Supported by: DFG: ``Energiemodelle für heterogene Halbleiterstrukturen'' (Energy models for heterogeneous semiconductor structures)
Description:
Driftdiffusion equations provide a powerful description of microscopic particle transport on the phenomenological level. However, the relation between the phenomenological description and the underlying microscopic transport phenomena, in general, is not clearcut. Often only special solutions (the stationary solution), global properties (the free energy) or functionals of the solution (moments) are known. Therefore it is interesting to consider example problems which are nontrivial and the underlying microscopic processes are well known. We demonstrate this in two examples.
Ex. 1: A powerful thermodynamically consistent method for
deriving driftdiffusion equations
has been developed in the last years by H. Gajewski
and others (see [1, 2, 3]).
Looking for a phenomenological
evolution equation for a concentration u(x, t) with
u(x, t) 0,
u(x, t) 1
(
x _{n}, t 0),
we consider at first a microscopic picture. Let
u(x, t) be the solution of a KolmogorovChapman equation
Now we are going to derive a phenomenological equation for u by Gajewski's method, assuming that a free energy of type (3), with some reference concentration u^{*}, is given. This is a typical situation in applications.
We calculate the stationary state u_{s}(x)
by Lagrange's method, varying the
functional (u) under the constraint
u(x, t)dx = 1.
This leads to the functional
L(u) = (u)  u(x)dx  1.
Thus, u_{s}(x) is the solution to the EulerLagrange equation
= u_{s}(x) = ()u^{*}(x) 
For deriving an evolution equation, we assume that
the Lagrange multiplier
is the chemical potential depending
on x and t, and
u(x, t) = ((x, t))u^{*}(x) 
1, u(t) = 1,(t) =  1, =  1, = 0 
u(t) = ,(t) =  , =  , 0 , 
We have to choose and
in such a manner that
the equations (4) and (1)
describe the same physical problem. As usual,
is taken as a gradient
=
with an n x n matrix
(x).
There is good reason, see [3] and the references there,
to take
similar to the inverse Hessian of
= u^{*} 
Ex. 2: An other typical situation is the following: We want to derive
an equation for u(x, t) of type (4) for the diffusion
of particles in a spacehomogeneous medium.
However, for an exact description of the problem
we have to take into
account more state parameters than x, e.g., the velocity v, too.
Let
v and
x .
Let us assume that the
trajectory in phase space is Markovian with
the probability density W(v, x, t). Then the KolmogorovChapman
equation describing the time
evolution of W(v, x, t) has the form
Ultimately, we are only interested in
the concentration (space density)
u(x, t) = W(v, x, t)dv.
A typical method to calculate u
is to derive a system of equations for the vmoments
of W and close this system in a heuristic way.
In the case of the general Brownian motion, governed by equation
(6) and
(f )(v) = avf + bf + 
u(x, t) = 1  e^{at}u(x, t) + 
In the limit
t we get the equation
u(x, t)  =  u(x, t) + 
In the limit
t 0 we get
the secondorder hyperbolic equation
u(x, t) = u(x, t) , 
References:



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