Stochastic Dynamics of Interface Growth Processes
Summary
Interface growth processes describe the evolution of a boundary separating distinct phases under the influence of random fluctuations and deterministic mechanisms. These phenomena appear in contexts as diverse as thin-film deposition, erosion fronts, microbial colony expansion and fluid invasion in porous media. The stochastic character of growth emerges from discrete deposition events, thermal noise or environmental heterogeneity, leading to irregular surface morphologies. Theoretical treatments employ continuum stochastic partial differential equations alongside discrete lattice models to capture key features of height fluctuations, correlation functions and scaling behaviour. Central to this field is the notion of self-affine roughness, quantified by scaling exponents that relate spatial extent to height variations and temporal dynamics. The Kardar–Parisi–Zhang (KPZ) equation stands as a prototypical model, incorporating local smoothing, lateral growth and additive noise, and serving as the archetype of a broad universality class. Exact solutions in one dimension, renormalisation-inspired scaling arguments and high-precision simulations have together established a coherent picture of fluctuation statistics. Attention now turns to multidimensional extensions, nonlocal interactions and the coupling between deterministic instabilities and stochastic forcing, with a view to controlling pattern formation in advanced material coatings, biological systems and environmental applications.
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Stochastic Dynamics of Interface Growth Processes publication trend
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Technical terms
Kardar–Parisi–Zhang (KPZ) equation: A nonlinear stochastic partial differential equation describing the evolution of a growing interface subject to local smoothing, lateral growth and random fluctuations.
Roughness exponent: A dimensionless parameter characterising how the standard deviation of interface height scales with system size.
Universality class: A grouping of growth processes sharing the same set of critical exponents and scaling functions, regardless of microscopic details.
Dynamic scaling ansatz: A hypothesis that relates spatial and temporal correlation functions through power-law scaling, thereby predicting how interface roughness evolves over time.
References
- Surface Evolution of Polymer Films Grown by Vapor Deposition: Growth of Local and Global Slopes of Interfaces. Polymers (2024).
- Analytical and numerical study of diffusion propelled surface growth phenomena. Partial Differential Equations in Applied Mathematics (2024).
- A CLASS OF GROWTH MODELS RESCALING TO KPZ. Forum of Mathematics Pi (2018).
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