Variational Principle for Mass Transport

A variation principle for mass transport in solids is derived that recasts transport coefficients as minima of local thermodynamic average quantities. The result is independent of diffusion mechanisms and applies to amorphous and crystalline systems. This unifies different computational approaches for diffusion and provides a framework for the creation of new approximation methods with error estimation. It gives a different physical interpretation of the Green function. Finally, the variational principle quantifies the accuracy of competing approaches for a nontrivial diffusion problem.

Figure 1

Diffusivity of solute atom $B$ and solvent $A$ in a random alloy on a square lattice, scaled by vacancy concentration. The vacancy exchange rate with atom $A$ is one, while we consider different relative rates for vacancy-$B$ exchange and different solute concentrations from zero to one. The largest deviation is for the case where ${\nu }_B=0$, which requires longer range correlation to capture the percolation limit near $c_B\approx 0.5$; the “percolation” simulations are averages of 256 Green function calculations on a $256\times 256$ periodic cell.