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General thermodynamic approach for diffusion on a lattice
Phys. Rev. E 114, 024152 – Published 25 August, 2026
DOI: https://doi.org/10.1103/p8qx-8ym2
Abstract
This work presents a general thermodynamic approach to describe particle diffusion on a lattice, a model used to study transport processes in solids and on surfaces. By treating each lattice site as an open thermodynamic system, the effects of microscopic particle interactions are represented through the chemical potential. A fundamental relationship between the Onsager matrix and its ideal-system counterpart (, where interactions are neglected) using the determinant of the covariance matrix is demonstrated. This framework allows for the calculation of transport coefficients using the combination of their ideal values and thermodynamic properties. The general result is successfully applied to reproduce the Darken equation for substitutional diffusion in solids and to derive the nondiagonal diffusion matrix of the Zhdanov model for surface diffusion of Langmuir particles. In the last case, analytical predictions are further validated through numerical simulations across various interaction potentials.
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References (29)
- A. Paul, T. Laurila, V. Vuorinen, and S. V. Divinski, Thermodynamics, Diffusion and the Kirkendall Effect in Solids (Springer, Heidelberg, 2014).
- H. Mehrer, Diffusion in Solids (Springer, Berlin, 2007).
- P. Shewmon, Diffusion in Solids (Springer, Cham, 2016).
- T. Ala-Nissila, R. Ferrando, and S. C. Ying, Collective and single particle diffusion on surfaces, Adv. Phys. 51, 949 (2002).
- R. Gomer, Diffusion of adsorbates on metal surfaces, Rep. Prog. Phys. 53, 917 (1990).
- G. Antczak and G. Ehrlich, Surface Diffusion, Metals, Metal Atoms, and Clusters (Cambridge University Press, Cambridge, UK, 2010).
- T. T. Tsong, Mechanisms of surface diffusion, Prog. Surf. Sci. 67, 235 (2001).
- L. S. Darken, Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. AIME 175, 184 (1948).
- S. Yu. Krylov, J. J. M. Beenakker, and M. C. Tringides, On the theory of surface diffusion: Kinetic versus lattice gas approach, Surf. Sci. 420, 233 (1999).
- V. P. Zhdanov, Diffusion of coadsorbed particles, Surf. Sci. 194, 1 (1988).
- V. P. Zhdanov, Manifestation of Onsager's off-diagonal fluxes in diffusion of coadsorbed particles, Surf. Sci. 704, 121740 (2021).
- M. Tammaro and J. W. Evans, Chemical diffusivity and wave propagation in surface reactions: Lattice-gas model mimicking CO-oxidation with high CO-mobility, J. Chem. Phys. 108, 762 (1998).
- A. N. Gorban, H. P. Sargsyan, and H. A. Wahab, Quasichemical models of multicomponent nonlinear diffusion, Math. Model. Nat. Phenom. 6, 184 (2011).
- V. Skakauskas and P. Katauskis, Modelling of the “surface explosion” of the reaction over supported catalysts, J. Math. Chem. 58, 1531 (2020).
- M. A. Di Muro and M. Hoyuelos, Application of the Widom insertion formula to transition rates in a lattice, Phys. Rev. E 104, 044104 (2021).
- M. S. Green, Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids, J. Chem. Phys. 22, 398 (1954).
- R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
- H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (John Wiley & Sons, Singapore, 1985).
- M. Hoyuelos, M. A. Di Muro, and P. Gimenez, Physical meaning of non-extensive term in Massieu functions, Phys. Rev. E 111, 024120 (2025).
- S.-I. Amari, Information geometry, Contemp. Math. 203, 81 (1997).
- S.-I. Amari, Information Geometry and Its Applications (Springer, Japan, 2016).
- F. Barbaresco, Geometric theory of heat from Souriau Lie groups thermodynamics and Koszul Hessian geometry: Applications in information geometry for exponential families, Entropy 18, 386 (2016).
- F. Nielsen, An elementary introduction to information geometry, Entropy 22, 1100 (2020).
- G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory, Rev. Mod. Phys. 67, 605 (1995).
- K. Binder, How Monte Carlo simulations can clarify complex problems in statistical physics, Int. J. Mod. Phys. B 15, 1193 (2001).
- L. Zhang, Y.-C. Liu, and Q. Wang, Molecular dynamics simulation of self- and mutual diffusion coefficients for confined mixtures, J. Chem. Phys. 123, 144701 (2005).
- M. A. Hartmann, R. Weinkamer, P. Fratzl, J. Svoboda, and F. D. Fischer, Onsager's coefficients and diffusion laws—a Monte Carlo study, Philos. Mag. 85, 1243 (2005).
- T. M. Liggett, Interacting Particle Systems (Springer, New York, NY, 1985).
- M. Mińkowski and M. A. Załuska-Kotur, Collective diffusion of dense adsorbate at surfaces of arbitrary geometry, J. Stat. Mech. (2018) 053208.
