An official website of the United States government
Motion by mean curvature and Dyson Brownian Motion
We construct Dyson Brownian motion for β ∈ (0, ∞] by adapting the extrinsic construc- tion of Brownian motion on Riemannian manifolds to the geometry of group orbits within the space of Hermitian matrices. When β is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits evolve by motion by (minus one half times) mean curvature.
more »
« less
- Award ID(s):
- 2107205
- PAR ID:
- 10549939
- Publisher / Repository:
- Project Euclid
- Date Published:
- Journal Name:
- Electronic Communications in Probability
- Volume:
- 28
- ISSN:
- 1083-589X
- Subject(s) / Keyword(s):
- Dyson Brownian motion mean curvature Riemannian submersion.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.more » « less
-
Brownian Motion, with some persistence in the direction of motion, typically known as active Brownian Motion, has been observed in many significant chemical and biological transport processes. Here, we present a model of drifted Brownian Motion that considers a nonlinear stochastic drift with constant or fluctuating diffusivity. The interplay between nonlinearity and structural heterogeneity of the environment can explain three essential features of active transport. These features, which are commonly observed in experiments and molecular dynamics simulations, include transient superdiffusion, ephemeral non-Gaussian displacement distribution, and non-monotonic evolution of non-Gaussian parameter. Our results compare qualitatively well with experiments of self-propelled particles in simple hydrogen peroxide solutions and molecular dynamics simulations of self-propelled particles in more complex settings such as viscoelastic polymeric media.more » « less
-
Abstract The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot’s fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking.more » « less
-
Biophysical and condensed matter systems connection is of great importance nowadays due to the need for a new approach in microelectronic biodevices, biocomputers or biochips advanced development. Considering that the living and nonliving systems’ submicroparticles are identical, we can establish the biunivocally correspondent relation between these two particle systems, as a biomimetic correlation based on Brownian motion fractal nature similarities, as the integrative property. In our research, we used the experimental results of bacterial motion under the influence of energetic impulses, like music, and also some biomolecule motion data. Our goal is to define the relation between biophysical and physical particle systems, by introducing mathematical analytical forms and applying Brownian motion fractal nature characterization and fractal interpolation. This work is an advanced research in the field of new solutions for high-level microelectronic integrations, which include submicrobiosystems like part of even organic microelectronic considerations, together with some physical systems of particles in solid-state solutions as a nonorganic part. Our research is based on Brownian motion minimal joint properties within the integrated biophysical systems in the wholeness of nature.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1214/23-ECP540
