Celebrating the 40th Anniversary of the Kardar–Parisi–Zhang (KPZ) Equation
A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".
Deadline for manuscript submissions: 31 December 2026 | Viewed by 256
Special Issue Editors
Interests: game theories in AI epoch–human ; interface growth dynamics; complex systems; infophysics; econophysics
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Special Issue Information
Dear Colleagues,
Since its formulation in 1986, the Kardar–Parisi–Zhang (KPZ) equation has emerged as a cornerstone paradigm in non-equilibrium statistical physics, unifying the study of kinetic interface roughening across diverse natural and artificial systems. From bacterial colony growth and forest fire boundaries to semiconductor thin-film deposition and ecological movement patterns, the KPZ equation’s ability to capture the interplay of surface relaxation, nonlinear lateral growth, and stochastic noise has made it a universal framework for understanding complex scaling behaviors. As we mark its 40th anniversary in 2026, this Special Issue aims to showcase the latest advances, unresolved challenges, and cross-disciplinary expansions of KPZ theory, honoring its enduring impact on physics, chemistry, biology, materials science, and beyond.
Over four decades, research on the KPZ equation has achieved remarkable breakthroughs. The resolution of its mathematical well-posedness in 2013 laid the foundation for rigorous analytical investigations, while variational approaches have opened up new avenues for studying non-equilibrium dynamics irrespective of substrate dimensionality. Key insights include the identification of non-equilibrium potential (NEP) as a Lyapunov-like functional, the discovery that memory effects in asymptotic statistics depend on initial conditions, and the extension of KPZ universality to related models such as the Golubović–Bruinsma (GB) model. Stochastic thermodynamics has further enriched our understanding, linking NEP to entropy production and enabling the derivation of fluctuation theorems and thermodynamic uncertainty relations. Cross-disciplinary applications have expanded the equation’s reach, from modeling anomalous diffusion in animal movement to explaining formations of spatial patterns in ecosystems and granular aggregates. Despite these strides, the following critical challenges persist: analytical solutions remain elusive substrates, the precise dependence of significant exponents on noise correlations and boundary conditions requires deeper investigation, and bridging theoretical predictions with experimental observations in complex real-world systems remains a pressing goal. Additionally, the integration of KPZ dynamics with other non-equilibrium phenomena, such as synchronization in oscillator networks or active matter dynamics, presents promising frontiers for exploration.
This Special Issue invites state-of-the-art research that advances KPZ theory and its applications across scientific disciplines. We welcome contributions addressing, but not limited to, the following topics:
- Theoretical advances in KPZ universality, including critical exponents, scaling behaviors, and large-deviation statistics;
- Novel analytical and numerical methods for solving KPZ and KPZ-like equations, such as variational approaches, path-integral formulations, and discretization schemes preserving key symmetries;
- Non-equilibrium thermodynamics of KPZ systems, including entropy production, fluctuation–dissipation relations, and memory effects;
- Extensions and generalizations of the KPZ equation, such as the GB model, coupled KPZ systems, and KPZ dynamics in correlated or heterogeneous environments;
- Cross-disciplinary applications, including interface growth in materials science, ecological movement patterns, bacterial colony expansion, and thin-film deposition;
- Experimental validation and observation of KPZ universality in natural or engineered systems;
- Connections between KPZ dynamics and other fields, such as directed polymers in random media, Burgers turbulence, and active matter.
We particularly encourage submissions that push the boundaries of KPZ research—whether by resolving long-standing theoretical puzzles, developing innovative computational tools, or uncovering unexpected applications in emerging fields. This Special Issue aims to provide a comprehensive insight into the current state of KPZ research, honoring its 40-year legacy while charting future directions for the next generation of non-equilibrium physics, as well as possible inspiration for GenAI.
Prof. Dr. Yi-Cheng Zhang
Prof. Dr. Zi-Ke Zhang
Dr. Fei Jing
Guest Editors
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Keywords
- non-equilibrium potential (NEP)
- stochastic thermodynamics
- memory effects
- golubović–bruinsma (GB) model
- scaling behavior
- noise regularization
- large deviation statistics
- interface growth dynamics
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