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zero). Equation (1) is solved with the appropriate boundary conditions, which reflect the hard-sphere nature of the file:

= \big( D\partial_{x_{j+1}} P(\mathbf{x},t\mid \mathbf{x_0})\big)_{x_{j+1}=x_j}; \qquad j=-M,\ldots,M-1, |}}

and with the appropriate initial condition:

In a simple file, the initial density is fixed, namely,, where is a parameter that represents a microscopic length. The PDFs' coordinates must obey the order: .

Heterogeneous files

In such

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