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Markovian Random Walks on Square Lattice with Constant Non-Symmetric Diffusion Coefficients
Within the rigor typical for physical models a new type non-symmetric diffusion problem is considered and the corresponding Brownian motion implementing such diffusion processes is constructed. As a particular example, random walks with internal causality on a square lattice are studied in detail. By construction, one elementary step of a random walker on the lattice may consist of its two succeeding jumps to the nearest neighboring nodes along the x- and then y-axis or the y- and then x-axis ordered, e.g., clock-wise. It is essential that the second fragment of elementary step is caused by the first one, meaning that the second fragment can arise only if the first one has been implemented, but not vice versa. In particular, if for some reasons the second fragment is blocked, the first one may be not affected, whereas if the first fragment is blocked, the second one cannot be implemented in any case. As demonstrated, on time scales much larger then the duration of one elementary step these random walks are characterized by a diffusion matrix with non-zero anti-symmetric component, which is also justified by numerical simulation.