Spatial Correlations Achieve Exponential Scaling of Diffusion, Restoring Zwanzig’s Prediction

Understanding how particles move through complex, uneven energy landscapes is fundamental to many areas of science, from chemical reactions to the behaviour of biological molecules and disordered materials. Biman Bagchi, along with colleagues from the Indian Institute of Science and others, investigates the long-standing question of how ‘roughness’ in these landscapes affects the speed of diffusion. Previous work suggested that roughness always slows movement in a predictable way, a theory originally proposed by Zwanzig, but simulations revealed this prediction breaks down when disorder is not evenly distributed. This team now demonstrates that introducing spatial correlations, a concept borrowed from astrophysics describing turbulence, effectively smooths out the landscape, suppressing problematic traps and restoring the accuracy of Zwanzig’s original theory, offering a new understanding of diffusion in complex systems and providing a framework to explain why earlier predictions failed.

Spatial Correlations Slow Diffusion in Rough Landscapes

This research details the impact of spatial correlations on diffusion within rugged energy landscapes, building upon the foundational work of Zwanzig and extending it with insights from simulations and theoretical analysis. The study addresses limitations in earlier theoretical work, demonstrating that Zwanzig’s mean-field theory, predicting a simple exponential relationship between landscape roughness and diffusion, fails when disorder is uncorrelated. The core finding is that spatial correlations fundamentally alter the landscape, mitigating the formation of rare, deep three-site traps that drastically slow down diffusion in uncorrelated landscapes. Gaussian spatial correlations, originally developed in astrophysics to describe turbulent density fluctuations, are used as a mathematical model for this smoothing effect, quantified by a correlation length (λ).

These correlations reduce the variance of energy differences between neighboring points and ensure energy changes are not independent, preventing extreme local minima and suppressing deep asymmetric traps. Brownian dynamics simulations confirm that Zwanzig’s prediction overestimates the diffusion coefficient in uncorrelated landscapes, while correlated landscapes with finite spatial correlation length closely match the predicted diffusion. Numerical examples illustrate a dramatic reduction in escape times from traps in correlated landscapes compared to uncorrelated ones, highlighting the importance of spatial structure in accurately modeling complex systems. This work highlights that understanding transport in disordered landscapes requires considering not just the amount of disorder, but also its spatial structure. Correlated roughness is common in biological systems, such as protein sliding on DNA and enzyme diffusion, and materials science, including polymers, glasses, and proteins. By smoothing the potential and suppressing pathological traps, correlations restore the validity of Zwanzig’s original theory and provide a more realistic framework for understanding transport in a wide range of physical and biological systems.

Correlated Landscapes Enhance Molecular Diffusion

Scientists investigated diffusion within rugged energy landscapes, relevant to diverse systems including ion transport in glasses and protein dynamics, by developing a novel approach to understanding how landscape roughness affects molecular movement. Researchers demonstrated that the emergence of rare, deep three-site traps dominates transport in uncorrelated landscapes, significantly reducing the diffusion constant. To overcome this, the team pioneered the introduction of Gaussian spatial correlations into the landscape roughness, inspired by techniques used in astrophysics to model turbulent density fluctuations. This method constructs landscapes where the energy at one location is statistically related to nearby locations, effectively smoothing the landscape and suppressing the formation of pathological traps.

The research clarifies that Zwanzig’s local averaging technique can be understood as a Gaussian cumulant expansion, and its validity breaks down with uncorrelated disorder. The team’s work demonstrates how Gaussian spatial correlations reshape roughness increments, eliminating asymmetric multi-site traps and restoring expected mean-field behavior. Numerical examples revealed a dramatic reduction in escape times, with correlated landscapes exhibiting significantly faster diffusion compared to uncorrelated ones. This approach provides a more realistic framework for understanding transport in systems with finite correlation lengths, such as DNA sequences and disordered solids.

Spatial Correlations Rescue Diffusion in Rough Landscapes

Scientists investigated diffusion within rugged energy landscapes, crucial to understanding processes in diverse systems including biomolecular dynamics and disordered solids. The research centers on refining a classic theory predicting how landscape roughness impacts the rate of movement, and demonstrates how spatial correlations fundamentally alter this relationship. Initial work revealed that a simple mean-field theory breaks down when disorder is uncorrelated, due to the prevalence of rare, deep three-site traps. The team demonstrated that introducing Gaussian spatial correlations, originally developed to model turbulence in astrophysics, effectively suppresses these problematic traps and restores the original exponential scaling predicted by the mean-field theory.

The work clarifies why Zwanzig’s local averaging approach fails with uncorrelated disorder, and how spatial correlations reinstate its validity. Measurements confirm that the introduction of Gaussian spatial correlations dramatically reduces escape times from traps, showcasing a significant improvement in transport efficiency. The researchers established that the correlated model, incorporating a finite correlation length, more accurately reflects the molecular connectivity and intermolecular forces present in real physical systems like DNA sequences and polymer chains.

Spatial Correlation Rescues Rough Landscape Dynamics

This research clarifies how the spatial arrangement of disorder impacts movement across rugged energy landscapes, a phenomenon relevant to diverse fields including chemical physics and biomolecular dynamics. Scientists demonstrated that traditional theories predicting movement in these landscapes break down when disorder is randomly distributed, due to the formation of rare, deep traps that impede progress. However, by introducing spatial correlations, borrowing concepts originally developed to model turbulence in astrophysics, the team showed that even a small degree of correlation significantly reduces these problematic traps and restores the accuracy of simpler, established theories. The key achievement lies in establishing that the structure of roughness, not just its magnitude, governs diffusion in disordered systems.

Gaussian spatial correlations effectively smooth the energy landscape, reducing extreme asymmetry and allowing for more predictable movement. This smoothing effect arises because correlations link successive energy variations, preventing the formation of the deep, isolated traps that plague uncorrelated landscapes. This approach provides a valuable framework for understanding how spatial structure influences transport in complex systems, with implications for understanding protein movement on DNA and the behaviour of polymers and glasses.