Variational Principle for Mass Transport

Abstract

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 mechanism, 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.

General Canonical Variational Principle and Noether Theorem, Their New Classical and Quantum Physics, Solution to Crisis Deducing All Physics Laws in Phase Space

“…Eqs. (26), (27) and (28) are the group G's infinitesimal generating functions, ε σ are independent infinitesimal parameters related to ω.…”

“…A lot of references, e.g., [23][24][25][26][27][28] have very well investigated various variational principles and their useful actual applications, and lots of good works, e.g., [29], [30], [31] have very well researched on Noether's Theorem and the useful pratical applications in different branches of physics.…”

Elastodiffusion and cluster mobilities using kinetic Monte Carlo simulations: Fast first-passage algorithms for reversible diffusion processes

“…Path factorization amounts to constructing the triangular matrices involved in LU or LDL T decomposition as well as their inverses, as shown in Appendix B. It is shown that the repeated application of updating rules (32) and (33) to obtain τ (N ) and θ (N ) from τ (0) and θ (0) amounts to directly applying the inverted triangular factors on τ (0) and θ (0) . Noticeably, transitions to new transient states are removed within path factorization in the same way as matrix elements are canceled through Gaussian elimination.…”

“…They are obtained by inverting a matrix, which amounts to summing over all paths between two consecutive vacancy-solute exchanges. Green functions used in atomic transport theory [31,32] also appear as pseudo-inverses of transition rate matrices and may also be interpreted as geometric sums of path probabilities.…”

“…From the matrix products above and property L + n L − n = I, we deduce that L + L − = I, hence L − corresponds to the inverse of L + . The decomposition of L − and U − into product of triangular elemental matrices are used in the updating rule (32) to compute τ (N ) , the vector of mean first-passage times. The sequential procedure (32) amounts to applying vector b = B 1 on matrices L − , D −1 and U − , successively:…”