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Stochastic growth equations and reparametrization invariance

Matteo Marsili, Amos Maritan, Flavio Toigo, and Jayanth R. Banavar

Matteo Marsili

  • Department of Physics, Shuster Laboratory, The University of Manchester, M13 9PL Manchester, United Kingdom

Amos Maritan

  • Istituto Nazionale Fisica della Materia and International School for Advanced Studies (S.I.S.S.A.) via Beirut 2-4, Trieste 34014 and sezione INFN di Trieste, Italy

Flavio Toigo

  • Istituto Nazionale Fisica della Materia and Dipartimento di Fisica, Padova University, via Marzolo 8, 35100 Padova, Italy

Jayanth R. Banavar

  • Department of Physics and Center for Material Physics, 104 Davey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802

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Rev. Mod. Phys. 68, 963 – Published 1 October, 1996

DOI: https://doi.org/10.1103/RevModPhys.68.963

Abstract

This article reviews the role of reparametrization invariance (the invariance of the properties of a system with respect to the choice of the co-ordinate system used to describe it) in deriving stochastic equations that describe the growth of surfaces. By imposing reparametrization invariance on a system, the authors identify the physical origin of many of the terms in its growth equations. Both continuum-growth equations for interfaces and equations for the coarse-grained evolution of discrete-lattice models are derived with this method. A detailed analysis of the discrete-lattice case and its small-gradient expansion provides a physical basis for terms found in commonly studied growth equations. The reparametrization-invariant formulation of growth processes also has the advantage of allowing one to model shadowing effects that are lost in the no-overhang approximation and to conserve underlying symmetries of the system that are lost in a small-gradient expansion. Finally, a knowledge of the full equation of motion, beyond the lowest-order gradient expansion, may be relevant in problems where the usual perturbative renormalization methods fail. [S0034-6861(96)00104-3]

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Rev. Mod. Phys. 68, 963– Published 1 October, 1996

    © 1996 American Physical Society

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