Random walk analysis of the collective cosine formulation of correlation effects in atom transport in a concentrated alloy

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Abstract
The phenomenological coefficients Lij, which are defined in non-equilibrium thermodynamics and which characterize atom transport near thermodynamic equilibrium, can be expanded in terms of so-called collective cosines. A typical such quantity is (cos θji (n)) which is defined as the average of the cosine of the angle between the direction of an initial jump of an atom of species j and a final jump of an atom of species i when there are exactly n jumps of atoms of species i following the initial j atom jump and n = 1,2,3,…New exact relations between the four quantities (cos θji (n)), for i,j = A,B and arbitrary n are derived for a binary random alloy of two atomic species (A,B) with transport by a very small concentration of vacancies. A new simplified formula for the off-diagonal coefficient Lij,j ≠ i, is also derived in terms of these collective cosines. The approximate calculation of the collective cosines by enumeration of random walks is examined; results for n = 1,2 and 3 are in very good agreement with earlier Monte Carlo simulation results for jump frequency ratios of the two atom components of 0·1 and 0·01.