Planck space

The Omori-Utsu law shows the temporal power-law-like decrease of the frequency of earthquake aftershocks and, interestingly, is found in a variety of complex systems/phenomena exhibiting catastrophes. Now, it may be interpreted as a characteristic response of such systems to large events. Here, hierarchical dynamics with the fast and slow degrees of freedom is studied on the basis of the Fokker-Planck theory for the load-state distribution to formulate the law as a relaxation process, in which diffusion coefficient in the space of the load state is treated as a fluctuating slow variable...

We study a switching heroin epidemic model in this paper, in which the switching of supply of heroin occurs due to the flowering period and fruiting period of opium poppy plants. Precisely, we give three equations to represent the dynamics of the susceptible, the dynamics of the untreated drug addicts and the dynamics of the drug addicts under treatment, respectively, within a local population, and the coefficients of each equation are functions of Markov chains taking values in a finite state space. The first concern is to prove the existence and uniqueness of a global positive solution to the switching model...

We studied the translocation of polyelectrolyte (PE) chains driven by an electric field through a pore by means of molecular dynamics simulations of a coarse-grained HP model mimicking high salt conditions. Charged monomers were considered as polar (P) and neutral monomers as hydrophobic (H). We considered PE sequences that had equally spaced charges along the hydrophobic backbone. Hydrophobic PEs were in the globular form in which H-type and P-type monomers were partially segregated and they unfolded in order to translocate through the narrow channel under the electric field...

We demonstrate a free-space optical communication link with an optical transmitter that harvests naturally occurring Planck radiation from a warm body and modulates the emitted intensity. The transmitter exploits an electro-thermo-optic effect in a multilayer graphene device that electrically controls the surface emissivity of the device resulting in control of the intensity of the emitted Planck radiation. We design an amplitude-modulated optical communication scheme and provide a link budget for communications data rate and range based on our experimental electro-optic characterization of the transmitter...

It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic...

For a theoretical analysis of mass transfer processes in electromembrane systems, the Nernst-Planck and Poisson equations (NPP) are generally used. In the case of 1D direct-current-mode modelling, a fixed potential (for example, zero) is set on one of the boundaries of the considered region, and on the other-a condition connecting the spatial derivative of the potential and the given current density. Therefore, in the approach based on the system of NPP equations, the accuracy of the solution is significantly affected by the accuracy of calculating the concentration and potential fields at this boundary...

Physics-informed neural networks (PiNNs) recently emerged as a powerful solver for a large class of partial differential equations (PDEs) under various initial and boundary conditions. In this paper, we propose trapz-PiNNs, physics-informed neural networks incorporated with a modified trapezoidal rule recently developed for accurately evaluating fractional Laplacian and solve the space-fractional Fokker-Planck equations in 2D and 3D. We describe the modified trapezoidal rule in detail and verify the second-order accuracy...

We study a generalization of the random walk (RW) based on a deformed translation of the unitary step, inherited by the q algebra, a mathematical structure underlying nonextensive statistics. The RW with deformed step implies an associated deformed random walk (DRW) provided with a deformed Pascal triangle along with an inhomogeneous diffusion. The paths of the RW in deformed space are divergent, while those corresponding to the DRW converge to a fixed point. Standard random walk is recovered for q→1 and a suppression of randomness is manifested for the DRW with -1<γ_{q}<1 and γ_{q}=1-q...

We outline formal and physical similarities between the quantum dynamics of open systems and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, which, for suitable timescales, becomes analogous to the Fokker-Planck equation describing classical advection and diffusion. This correspondence allows, in principle, to surmise a finite resolution, other than the Planck scale, for the quantized state space of the open system, particularly meaningful when the latter underlies chaotic classical dynamics...

We investigate, by simulations and analytic theory, the sensitivity of nonequilibrium relaxation to interaction potential and dimensionality by using Boltzmann's H function H(t). We evaluate H(t) for three different intermolecular potentials in all three dimensions and find that the well-known H theorem is valid and that the H function exhibits rather strong sensitivity to all these factors. The relaxation of H(t) is long in one dimension, but short in three dimensions, longer for the Lennard-Jones potential than for the hard spheres...

The electric AC response of electrolytic cells with DC bias is analyzed solving numerically the Poisson-Nernst-Planck equations and avoiding the commonly used infinite solution approximation. The results show the presence of an additional low-frequency dispersion process associated with the finite spacing of the electrodes. Moreover, we find that the condition of fixed ionic content inside the electrolytic cell has a strong bearing on both the steady-state and the frequency response. For example: the characteristic frequency of the high-frequency dispersion decreases when the DC potential increases and/or the electrode spacing decreases in the closed cell case, while it remains essentially insensitive on these changes for open cells...

Memory-limited partially observable stochastic control (ML-POSC) is the stochastic optimal control problem under incomplete information and memory limitation. To obtain the optimal control function of ML-POSC, a system of the forward Fokker-Planck (FP) equation and the backward Hamilton-Jacobi-Bellman (HJB) equation needs to be solved. In this work, we first show that the system of HJB-FP equations can be interpreted via Pontryagin's minimum principle on the probability density function space. Based on this interpretation, we then propose the forward-backward sweep method (FBSM) for ML-POSC...

The basic building blocks of the electrophysiology of cardiomyocytes are ion channels integrated in the cell membranes. Close to the ion channels there are very strong electrical and chemical gradients. However, these gradients extend for only a few nano-meters and are therefore commonly ignored in mathematical models. The full complexity of the dynamics is modelled by the Poisson-Nernst-Planck (PNP) equations but these equations must be solved using temporal and spatial scales of nano-seconds and nano-meters...

Quantum electrodynamic fields possess fluctuations corresponding to transient particle-antiparticle dipoles, which can be characterized by a nonvanishing polarizability density. Here, we extend a recently proposed quantum scaling law to describe the volumetric and radial polarizability density of a quantum field corresponding to electrons and positrons and derive the Casimir self-interaction energy (SIE) density of the field, E[over ¯]_{SIE}, in terms of the fine-structure constant. The proposed model obeys the cosmological equation of state w=-1 and the magnitude of the calculated E[over ¯]_{SIE} lies in between the two recent measurements of the cosmological constant Λ obtained by the Planck Mission and the Hubble Space Telescope...

Inducing transport in electrolyte-filled nanopores with dc fields has led to influential applications ranging from nanosensors to DNA sequencing. Here we use the Poisson-Nernst-Planck and Navier-Stokes equations to show that unbiased ac fields can induce comparable directional flows in gated conical nanopores. This flow exclusively occurs at intermediate driving frequencies and hinges on the resonance of two competing timescales, representing space charge development at the ends and in the interior of the pore...

Electron diffusion by whistler-mode chorus waves is one of the key processes controlling the dynamics of relativistic electron fluxes in the Earth's radiation belts. It is responsible for the acceleration of sub-relativistic electrons injected from the plasma sheet to relativistic energies as well as for their precipitation and loss into the atmosphere. Based on analytical estimates of chorus wave-driven quasi-linear electron energy and pitch-angle diffusion rates, we provide analytical steady-state solutions to the corresponding Fokker-Planck equation for the relativistic electron distribution and flux...

The Fokker-Planck (FP) approach for the description of vibrational kinetics is extended in order to include multiquanta transitions and time dependent solutions. Due to the importance of vibrational ladder climbing for the optimization of plasma-assisted nitrogen fixation, nitrogen is used as a test case with a comprehensive set of elementary processes affecting the vibrational distribution function (VDF). The inclusion of the vibrational energy equation is shown to be the best way to model transient conditions in a plasma reactor using the FP approach...

Fluidic devices exhibiting ion current rectification (ICR), or ionic diodes, are of broad interest for applications including desalination, energy harvesting, and sensing, among others. For such applications a large conductance is desirable, which can be achieved by simultaneously using thin membranes and wide pores. In this paper we demonstrate ICR in micrometer sized conical channels in a thin silicon membrane with pore diameters comparable to the membrane thickness but both much larger than the electrolyte screening length...

We consider graviton scattering in maximal supergravity on anti-de Sitter space (AdS) in d+1 dimensions for d=3, 4, and 6 with no extra compact spacetime factor. Holography suggests that this theory is dual to an exotic maximally supersymmetric conformal field theory (CFT) in d dimensions whose only light single trace operator is the stress tensor. This contrasts with more standard cases like type IIB string theory on AdS_{5}×S^{5} dual to N=4 super-Yang-Mills, where the CFT has light single trace operators for each Kaluza-Klein mode on S^{5}...

A minimum length scale of the order of Planck length is a feature of many models of quantum gravity that seek to unify quantum mechanics and gravitation. Recently, Perivolaropoulos in his seminal work (Perivolaropoulos in Phys. Rev. D 95:103523, 2017) predicted the simultaneous existence of minimal and maximal length measurements of quantum gravity. More recently, we have shown that both measurable lengths can be obtained from position-dependent noncommutativity (Lawson in J. Phys. A Math.Theor. 53:115303, 2020)...