Dynamics of interacting system

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Collective diffusion

Collective diffusion is a dynamic process that involves many particles and due to their cooperative movements leads to various interesting effects. Coefficient of the collective diffusion is a parameter that is a part of the equation describing local density diffusion process nearby the equilibrium state:
${\frac{ \partial \rho }{ \partial t }} = {\nabla_r D(\rho)\nabla_r (\rho)}$

Diffusion coefficient as a function of a particle density and other parameters of the system describes dynamics of the diffusion processes in the system. Calculation of this parameter in the general case is not an easy task. Lately a new and very effective method has been worked out in our group. It is variational approach to the problem. According to this approach, calculation of the diffusion coefficient consists of the definition of the transition matrix in the given system, and definition of the proper variational eigenvector. Next, the parameters are found that minimalize the following formula that describes diffusion parameter:
$\lambda (k) \approx \frac{ \Phi^{\dagger}_{var}(k)\cdot {\mathbb M}(k) \cdot [P^{eq} \Phi_{var}(k)]}{ \Phi^\dagger_{var}(k) \cdot [P^{eq} \Phi_{var}(k)]}$ $\lambda(k) \rightarrow D k^2$

Finally diffusion coefficient as a function of density, temperature and other physical condition is found. In this way we are able to describe various interesting and sometimes unexpected system behaviors. We show how the diffusion coefficient depends on the interaction strength between particles and on the interaction dependent barrier height. Interesting case of fractal like structure of the diffusion coefficient for a system with devil staircase phase diagram caused by long range interactions within medium. Diffusion on non-homogeneous potential has been calculated and two dimensional systems are recently studied.

We show that an exponent of a power-like time domain growth is determined not only by the conservation or nonconservation of the order parameter, but also by the asymmetry of single particle jumps. Domains that have an anisotropic pattern, such as (2X1), have a tendency to grow faster in certain direction than they do in others. The rate of expansion in different directions depends on the barriers for single particle jumps. As a result, dynamical behavior of systems which start in the same configurations and eventually reach the same equilibrium states is completely different. We show how differences in microscopic dynamics in a one dimensional Potts model lead to different rates of domain growth. We observe a similar effect for a two-dimensional (2X1)ordering by changing the way in which a barrier for a jump depends on the number of neighboring particles. We show examples of the domain power growth which are characterized by different exponents.

Figure

Examples of the domain pattern

Collective diffusion published:

Low-Temperature Ultrafast Mobility in Systems with Long-Range Rapulsive Interactions: Pb/Si(111).

Microscopic approach to the collective diffusion in the interacting lattice gas.

Chemical diffusion in a interacting lattice gas: Analytic theory and simple applications.

Collective diffusion in a interacting one-dimensional lattice gas: Arbitrary interactions, activation energy, and nonequilibrium diffusion.

Kinetic lattice gas model of collective diffusion in a one-dimensional system with long-range repulsive interactions.

Ritz variational principle for collective diffusion in an adsorbate on a non-homogeneous substrate.

Collective diffusion presentations:

Collective diffusion of the interacting surface gas.

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