Related papers: From Baking a Cake to Solving the Schrodinger Equa…
Einstein's kinetic theory of the Brownian motion, based upon light water molecules continuously bombarding a heavy pollen, provided an explanation of diffusion from the Newtonian mechanics. Since the discovery of quantum mechanics it has…
We suggest an explanation of typical incubation times statistical features based on the universal behavior of exit times for diffusion models. We give a mathematically rigorous proof of the characteristic right skewness of the incubation…
Diffusion models have achieved remarkable success in generating samples from unknown data distributions. Most popular stochastic differential equation-based diffusion models perturb the target distribution by adding Gaussian noise,…
We prove a global well-posedness result for defocusing nonlinear Schrodinger equations with time dependent potential. We then focus on time dependent harmonic potentials. This aspect is motivated by Physics (Bose--Einstein condensation),…
We study numerically quantum diffusion of a particle on small-world networks by integrating the time-dependent Schr\"odinger equation with a localized initial state. The participation ratio, which corresponds to the number of visited sites…
For the solution $q(t)=(q_n(t))_{n\in\mathbb Z}$ to one-dimensional discrete Schr\"odinger equation $${\rm i}\dot{q}_n=-(q_{n+1}+q_{n-1})+ V(\theta+n\omega) q_n, \quad n\in\mathbb Z,$$ with $\omega\in\mathbb R^d$ Diophantine, and $V$ a…
We study the homogenization of a Schrodinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating…
The separating time for two probability measures on a filtered space is an extended stopping time which captures the phase transition between equivalence and singularity. More specifically, two probability measures are equivalent before…
Of primary interest in this paper is the numerical approximation of a time dependent fractional, in space, diffusion equation where the domain is assumed to be nonhomogeneous, having different axial diffusion coefficients. This work is…
We consider the problem of simulating diffusion bridges, which are diffusion processes that are conditioned to initialize and terminate at two given states. The simulation of diffusion bridges has applications in diverse scientific fields…
We present an analytical closed form expression, which gives a good approximate propagator for diffusion on the sphere. Our formula is the spherical counterpart of the Gaussian propagator for diffusion on the plane. While the analytical…
Single particle tracking has become a standard tool to investigate diffusive properties, especially in small systems such as biological cells. Usually the resulting time series are analyzed in terms of time averages over individual…
In this paper we prove the following: (1) The basic error of time-dependent perturbation theory is using the sum of first finite order of perturbed solutions to substitute the exact solution in the divergent interval of the series for…
Denoising diffusion models are a novel class of generative models that have recently become extremely popular in machine learning. In this paper, we describe how such ideas can also be used to sample from posterior distributions and, more…
As well known, the generalized Langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle. Here, we focus attention on some new aspects of the…
Various degenerate diffusion equations exhibit a waiting time phenomenon: Dependening on the "flatness" of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount…
The global-in-time existence of bounded weak solutions to the Maxwell-Stefan-Fourier equations in Fick-Onsager form is proved. The model consists of the mass balance equations for the partial mass densities and and the energy balance…
When frying potato snacks, it is typically observed that the dough, which is submerged in hot oil, after some critical time increases its buoyancy and floats to the surface. The lift-off time is a useful metric in ensuring that the snacks…
One of the greatest scientific achievements of physics in the 20th century is the discovery of quantum mechanics. The Schrodinger equation is the most fundamental equation in quantum mechanics describing the time-based evolution of the…
In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order $1 \le \alpha \le 2$ is revisited. This equation interpolates between the diffusion and the wave equations that behave quite…