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Fractional diffusion equations imply non-Gaussian distributions that generalise the standard diffusive process. Recent advances in fractional calculus lead to a class of new fractional operators defined by non-singular memory kernels,…
We study the diffusion of an ensemble of overdamped particles sliding over a tilted random poten- tial (produced by the interaction of a particle with a random polymer) with long-range correlations. We found that the diffusion properties of…
Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible…
Established theoretical studies of diffusion in rugged (or rough) potential surfaces have largely focused on quenched energy landscapes. Here we study diffusion on a rugged energy landscape in the presence of dynamic disorder, a situation…
We study the effects of spatially inhomogeneous diffusion on the non-equilibrium phase transition in the contact process. The directed-percolation critical point in the contact process is known to be stable against the addition of a…
By event-driven molecular dynamics simulations we investigate magneto-transport in a two-dimensional model with randomly distributed scatterers close to the field-induced localization transition. This transition is generated by percolating…
Generative models have enjoyed widespread success in a variety of applications. However, they encounter inherent mathematical limitations in modeling distributions where samples are constrained by equalities, as is frequently the setting in…
The Hubbard model, despite its simplicity, is remarkably successful at describing numerous many-body phenomena. However, due to the small class of problems which can be solved exactly, there has been substantial interest in quantum…
Using two models of opinion dynamics, the $q$-voter model with independence and the $q$-voter model with anticonformity, we discuss how the change of disorder from annealed to quenched affects phase transitions on networks. To derive phase…
We study how to extend the use of the diffusion model to answer the causal question from the observational data under the existence of unmeasured confounders. In Pearl's framework of using a Directed Acyclic Graph (DAG) to capture the…
In this paper Gaussian models of retarded and accelerated anomalous diffusion are considered. Stochastic differential equations of fractional order driven by single or multiple fractional Gaussian noise terms are introduced to describe…
We revisit the diffusion properties and the mean drift induced by an external field of a random walk process in a class of branched structures, as the comb lattice and the linear chains of plaquettes. A simple treatment based on scaling…
Diffusion models are a class of generative models that serve to establish a stochastic transport map between an empirically observed, yet unknown, target distribution and a known prior. Despite their remarkable success in real-world…
Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by…
Subdiffusion on a fractal comb is considered. A mechanism of subdiffusion with a transport exponent different from 1/2 is suggested. It is shown that the transport exponent is determined by the fractal geometry of the comb.
Diffusive transport in many complex systems features a crossover between anomalous diffusion at short times and normal diffusion at long times. This behavior can be mathematically modeled by cutting off (tempering) beyond a mesoscopic…
We investigate the dynamics of a quantum particle in disordered tight-binding models in one and two dimensions which are exceptions to the common wisdom on Anderson localization, in the sense that the localization length diverges at some…
This paper explores the efficacy of diffusion-based generative models as neural operators for partial differential equations (PDEs). Neural operators are neural networks that learn a mapping from the parameter space to the solution space of…
We use numerical simulations to study the behavior of 2D frictionless disk systems under cyclic shear as a function of reversal amplitude \gamma_r. Our studies focus on mean bulk and disk dynamics. These measurements suggest a crossover…
In this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a…