Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime

We show that the uncertainty in distance and time measurements found by the heuristic combination of quantum mechanics and general relativity is reproduced in a purely classical and flat multi-fractal spacetime whose geometry changes with the probed scale (dimensional flow) and has non-zero imaginary dimension, corresponding to a discrete scale invariance at short distances. Thus, dimensional flow can manifest itself as an intrinsic measurement uncertainty and, conversely, measurement-uncertainty estimates are generally valid because they rely on this universal property of quantum geometries. These general results affect multi-fractional theories, a recent proposal related to quantum gravity, in two ways: they can fix two parameters previously left free (in particular, the value of the spacetime dimension at short scales) and point towards a reinterpretation of the ultraviolet structure of geometry as a stochastic foam or fuzziness. This is also confirmed by a correspondence we establish between Nottale scale relativity and the stochastic geometry of multi-fractional models.Comment: 25 pages. v2: minor typos corrected, references adde

Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error bounds and Algorithms

Biochemical reactions can happen on different time scales and also the abundance of species in these reactions can be very different from each other. Classical approaches, such as deterministic or stochastic approach, fail to account for or to exploit this multi-scale nature, respectively. In this paper, we propose a jump-diffusion approximation for multi-scale Markov jump processes that couples the two modeling approaches. An error bound of the proposed approximation is derived and used to partition the reactions into fast and slow sets, where the fast set is simulated by a stochastic differential equation and the slow set is modeled by a discrete chain. The error bound leads to a very efficient dynamic partitioning algorithm which has been implemented for several multi-scale reaction systems. The gain in computational efficiency is illustrated by a realistically sized model of a signal transduction cascade coupled to a gene expression dynamics.Comment: 32 pages, 7 figure

Data assimilation in a multi-scale model

Data assimilation for multi-scale models is an important contemporary research topic. Especially the role of unresolved scales and model error in data assimilation needs to be systematically addressed. Here we examine these issues using the Ensemble Kalman filter (EnKF) with the two-level Lorenz-96 model as a conceptual prototype model of the multi-scale climate system. We use stochastic parameterization schemes to mitigate the model errors from the unresolved scales. Our results indicate that a third-order autoregressive process performs better than a first-order autoregressive process in the stochastic parameterization schemes, especially for the system with a large time-scale separation.Model errors can also arise from imprecise model parameters. We find that the accuracy of the analysis (an optimal estimate of a model state) is linearly correlated to the forcing error in the Lorenz-96 model. Furthermore, we propose novel observation strategies to deal with the fact that the dimension of the observations is much smaller than the model states. We also propose a new analog method to increase the size of the ensemble when its size is too small

Knot in Cen A: Stochastic Magnetic Field for Diffusive Synchrotron Radiation?

The emission of relativistic electrons moving in the random and small-scale magnetic field is presented by diffusive synchrotron radiation (DSR). In this Letter, we revisit the perturbative treatment of DSR. We propose that random and small-scale magnetic field might be generated by the turbulence. As an example, multi-band radiation of the knot in Cen A comes from the electrons with energy $\gamma_e\sim 10^3-10^4$ in the magnetic field of $10^{-3}G$. The multi-band spectrum of DSR is well determined by the feature of stochastic magnetic field. These results put strong constraint to the models of particle acceleration.Comment: accepted by ApJL, comments are welcom

Multi-Scale Stochastic Simulation for Diffusive Molecular Communication

Recently, hybrid models have emerged that combine microscopic and mesoscopic regimes in a single stochastic reaction-diffusion simulation. Microscopic simulations track every individual molecule and are generally more accurate. Mesoscopic simulations partition the environment into subvolumes, track when molecules move between adjacent subvolumes, and are generally more computationally efficient. In this paper, we present the foundation of a multi-scale stochastic simulator from the perspective of molecular communication, for both mesoscopic and hybrid models, where we emphasize simulation accuracy at the receiver and efficiency in regions that are far from the communication link. Our multi-scale models use subvolumes of different sizes, between which we derive the diffusion event transition rate. Simulation results compare the accuracy and efficiency of traditional approaches with that of a regular hybrid method and with those of our proposed multi-scale methods.Comment: 7 pages, 2 tables, 6 figures. Will be presented at the 2015 IEEE International Conference on Communications (ICC) in June 201