## Numerical Evaluation of Diffusion Coefficients for Strongly Interacting
Systems^{*}

Tapio Ala-Nissila

Helsinki Institute of Physics and Laboratory of Physics,
Helsinki University of Technology, Espoo, Finland

and

Department of Physics, Brown University, Providence RI 02912

I will review recent progress in developing methodology for the
evaluation of tracer and collective diffusion coefficients
through numerical simulations for cases, where
many-particle interaction effects are dominant.
The first approach is called
the Dynamical Mean Field (DMF) theory, and it expresses the
diffusion coefficients in a form where they are proportional to
the local jump rate [1]. This makes it numerically very
efficient to study surface diffusion, and local jump rates may also
be measured experimentally. The main approximation
in the DMF theory is the neglect of memory effects.
We have
tested the DMF theory for a variety of strongly interacting systems, and
find that it gives a very good approximation for the temperature
and concentration dependence of collective diffusion. In the
case of tracer diffusion where memory effects are more pronounced,
the DMF theory is less accurate quantitatively [1,2].

To improve on the DMF theory, we have developed a memory
expansion for the diffusion coefficients [3]. This theory is formally an
exact decomposition of the relevant autocorrelation functions
in terms of successive terms containing all the memory effects.
These terms are particularly easy to evaluate through computer
simulations. In addition to diffusion coefficients, the method
can be generalized to study other transport coefficients.
The theory has been tested for strongly interacting systems
in the case of surface diffusion, and
it makes it possible to obtain very accurate values of the
diffusion coefficients with up to about two orders of magnitude reduction
in computer time as compared to the standard methods. Through
the method it is also possible to study the behavior of the
memory functions corresponding to the generalized Langevin equations
for tracer and collective diffusion, respectively.

- [1]
- T. Hjelt, I. Vattulainen, J. Merikoski, T. Ala-Nissila,
and S. C. Ying, Surface Sci. (Lett.)
**380** (2/3), L501 (1997).
- [2]
- I. Vattulainen, J. Merikoski, T. Ala-Nissila,
and S. C. Ying, Phys. Rev. Lett.
**79**, 257 (1997);
Phys. Rev. B **57**, 1896 (1998).
- [3]
- S. C. Ying, I. Vattulainen, J. Merikoski, T. Hjelt,
and T. Ala-Nissila, Phys. Rev. B
**58**, 2170 (1998).

^{*}Work done in collaboration with T. Hjelt, J. Merikoski,
I. Vattulainen, and S. C. Ying.

**Acknowledgment and Disclaimer**