Realising Continuous Functions Achieves Hydrodynamic Limit with Weierstrass-type Result

The behaviour of complex systems often hinges on understanding how microscopic interactions translate into macroscopic properties, a challenge that has long occupied physicists and mathematicians. Gabriel S. Nahum, working independently, now demonstrates a surprising degree of flexibility in these systems, proving that any positive, continuous, and bounded function can serve as the governing coefficient within a specific type of evolutionary equation. This result, achieved through the construction of a tailored process and utilising entropy methods, establishes a powerful connection between abstract mathematical functions and the dynamics of interacting particle systems. The findings significantly broaden the scope of models capable of describing real-world phenomena, offering new avenues for exploring emergent behaviour in diverse fields. This work represents a substantial step towards a more complete theoretical framework for understanding non-equilibrium statistical mechanics.

This work studies the local density of particles in a discrete Markovian system, which converges to the solution of a particular differential equation after rescaling in space and time. From a probabilistic viewpoint, this represents a law of large numbers, bridging the gap between discrete and continuum models. The research investigates the limiting behaviour of this system as the number of particles approaches infinity, focusing on the characterisation of the points at which this limit is achieved. This work contributes a novel understanding of the local density behaviour in discrete Markovian systems, providing a mathematical framework for analysing the emergence of macroscopic properties from microscopic interactions.

Constructing Gradient Systems and Diffusivity Calculations

Gradient systems represent a mathematically tractable class of interacting particle systems, distinguished by their local structure and simplifying hydrodynamical analysis. These models are defined by currents expressible as discrete gradients of a potential, enabling more readily calculated equilibrium and non-equilibrium fluctuations than in non-gradient systems. Crucially, gradient systems allow for explicit computation of diffusivity, offering insights into the static versus dynamic behaviour of diffusion. This research addresses the fundamental question of constructing microscopic dynamics that produce a desired diffusion equation, focusing on bulk dynamics within a one-dimensional discrete torus.

The study builds upon the Symmetric Simple Exclusion Process, a classical example exhibiting linear diffusivity, and extends to non-linear diffusivities such as those found in the Porous Media Model, where diffusion coefficients take the form D(ρ) = ρm. The Interpolating Model successfully demonstrated a microscopic, continuous description of the transition between slow and fast diffusion for values of m within the range of (−1, 1), demonstrating the possibility of continuous transitions in diffusion behaviour. This approach seeks to expand the understanding of how versatile gradient systems can be in replicating diverse diffusive behaviours.

Diffusivity from Entropy and Interacting Particles

Scientists have demonstrated that any positive, continuous, and bounded function can be realised as the coefficient of an evolution equation linked to a gradient interacting particle system. This breakthrough relies on the construction of a specifically designed system and the application of the entropy method, offering a novel approach to modelling diffusive processes. The research team explicitly computed the diffusivity, a feat typically achieved using the more complex Green Kubo formula, and measured a dynamical term that vanishes only when the model adheres to a gradient type, indicating purely static behaviour. Further investigations explored diffusivities beyond linear models, deriving polynomial diffusivities in the form of the Porous Media Model, allowing diffusion coefficients of the form D(ρ) = ρm, for positive integer values of m, and extending this to non-integer values of m through the development of the Interpolating Model.

This advancement builds upon earlier research deriving diffusivities from continuous processes involving coupled oscillators, offering a more streamlined approach. Crucially, the team established the need for a ‘replacement lemma’ to account for the non-uniform continuity of the microscopic potential associated with the gradient property. The model is a nearest-neighbour, non-cooperative, symmetric exclusion process of gradient type with non-local constraints, where hopping rates depend on a “mesoscopic” part of the configuration.

Functions as Coefficients in Particle Systems

This research demonstrates that any positive, continuous, and bounded function can be represented as a coefficient within an evolution equation linked to a gradient interacting particle system. The authors achieved this through the construction of a specific model and application of the entropy method, notably introducing a new ‘replacement lemma’ to address challenges arising from the microscopic potential’s non-uniform continuity. This work builds upon existing models, offering a distinct approach and detailed proof for certain key estimates. The significance of this lies in establishing a connection between abstract mathematical functions and the behaviour of interacting particle systems, potentially offering new avenues for modelling complex phenomena.

While the current model is restricted to positive diffusivities, a condition necessary to ensure irreducibility, the authors suggest its potential as a bridge between non-cooperative and cooperative models, particularly if extended to functions that can attain zero values. Future research will focus on exploring the extension to functions attaining zero, potentially utilising non-gradient tools, and investigating the possibility of deriving fractional hydrodynamic equations for long-range models, requiring a new mathematical basis beyond existing frameworks. The behaviour of complex systems often hinges on understanding how microscopic interactions translate into macroscopic properties, a challenge that has long occupied physicists and mathematicians. Nahum, working independently, now demonstrates a surprising degree of flexibility in these systems, proving that any positive, continuous, and bounded function can serve as the governing coefficient within a specific type of evolutionary equation.

This result, achieved through the construction of a tailored process and utilising entropy methods, establishes a powerful connection between abstract mathematical functions and the dynamics of interacting particle systems. The findings significantly broaden the scope of models capable of describing real-world phenomena, offering new avenues for exploring emergent behaviour in diverse fields. This work represents a substantial step towards a more complete theoretical framework for understanding non-equilibrium statistical mechanics.