Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Long-Range Electron Transport Donor-Acceptor in Nonlinear Lattices

We study here several simple models of the electron transfer (ET) in a one-dimensional nonlinear lattice between a donor and an acceptor and propose a new fast mechanism of electron surfing on soliton-like excitations along the lattice. The nonlinear lattice is modeled as a classical one-dimensional Morse chain and the dynamics of the electrons are considered in the tight-binding approximation. This model is applied to the processes along a covalent bridge connecting donors and acceptors. First, it is shown that the electron forms bound states with the solitonic excitations in the lattice. These so-called solectrons may move with supersonic speed. In a heated system, the electron transfer between a donor and an acceptor is modeled as a diffusion-like process. We study in detail the role of thermal factors on the electron transfer. Then, we develop a simple model based on the classical Smoluchowski–Chandrasekhar picture of diffusion-controlled reactions as stochastic processes with emitters and absorbers. Acceptors are modeled by an absorbing boundary. Finally, we compare the new ET mechanisms described here with known ET data. We conclude that electron surfing on solitons could be a special fast way for ET over quite long distances

On the temperature dependence of fast electron transport in crystal lattices

Building upon the findings of Muto et al. [Phys. Lett. A 136, 33 (1989)] and Marchesoni and Lucheroni [Phys. Rev. E 44, 5303 (1991)] about the growth of the number of (anharmonic) lattice solitons with increasing temperature and using a recent transport theory developed by the present authors [A.P. Chetverikov, W. Ebeling, G. Röpke, M.G. Velarde, Eur. Phys. J. B 87, 153 (2014)] here we provide the fractional power law of the temperature dependence of resistivity in a rather general model for one-dimensional crystal lattices as, e.g., conducting polymers. We also show that the determining factor for the transport is the possibility of forming electron-soliton bound states (in short solectrons) with a most significant contribution arising from the (bosonic) bound state of two electrons to a soliton (in short bisolectrons)

High electrical conductivity in nonlinear model lattice crystals mediated by thermal excitation of solectrons

The quantum statistics of electrons interacting with nonlinear excitations of a classical heated nonlinear lattice of atoms is studied. By using tight-binding approximation, Wigner momentum distributions and computer simulations we show the existence of quite fast and nearly loss-free motions of charges along crystallographic axes and estimate the range of values of transport coefficients. Using minimization of free energy we estimate the density of mobile bound states between electrons and lattice solitons (so-called solectrons). We calculate the momenta of Wigner velocity distributions and from Kubo relations the diffusivity and the electrical conductivity using the relaxation time approximation. We show that thermally excited solectrons in nonlinear media may lead to a significant transport enhancement. Our estimates and computer simulations demonstrate the existence of a temperature window, where solectrons are relatively stable and contribute strongly to transport. The electrical conductivity may be enhanced up to two orders of magnitude

Control of electron and electron–hole pair dynamics on nonlinear lattice bilayers by strong solitons

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in: Chetverikov, A. P., Ebeling, W., Schöll, E., & Velarde, M. G. (2021). Control of electron and electron–hole pair dynamics on nonlinear lattice bilayers by strong solitons. In Chaos: An Interdisciplinary Journal of Nonlinear Science (Vol. 31, Issue 8, p. 083123). AIP Publishing. https://doi.org/10.1063/5.0057084 and may be found at https://doi.org/10.1063/5.0057084.We consider the dynamics of electrons and holes moving in two-dimensional lattice layers and bilayers. As an example, we study triangular lattices with units interacting via anharmonic Morse potentials and investigate the dynamics of excess electrons and electron–hole pairs according to the Schrödinger equation in the tight binding approximation. We show that when single-site lattice solitons or M-solitons are excited in one of the layers, those lattice deformations are capable of trapping excess electrons or electron–hole pairs, thus forming quasiparticle compounds moving approximately with the velocity of the solitons. We study the temporal and spatial nonlinear dynamical evolution of localized excitations on coupled triangular double layers. Furthermore, we find that the motion of electrons or electron–hole pairs on a bilayer is slaved by solitons. By case studies of the dynamics of charges bound to solitons, we demonstrate that the slaving effect may be exploited for controlling the motion of the electrons and holes in lattice layers, including also bosonic electron–hole–soliton compounds in lattice bilayers, which represent a novel form of quasiparticles.DFG, 163436311, Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept

Nonlinear excitations and bound states of electrons, holes and solitons in bilayers of triangular lattices

We study the temporal and spatial nonlinear dynamical evolution of a coupled triangular lattice crystal bilayer system where in one layer one excess free electron is injected while an excess positive charge, a hole, is created in the other. The atoms of each of the backbone lattices interact with anharmonic (short range) Morse potentials whereas the charges interact via (long range) Coulomb potentials. Computer simulations are provided of the possibilities offered by varying interlayer separation, strength of the Coulomb force between the charges and the diverse dynamical role played by excited solitons supersonically moving along crystallographic axes in one of the layers. Optimal conditions are identified for the occurrence of electron–hole pairs and for the more significant case of a boson-like electron–hole–soliton coupled compound, a new form of quasiparticle moving along the coupled bilayer system with no need of applying an external electric field

Head-on and head-off collisions of discrete breathers in two-dimensional anharmonic crystal lattices

Collisions of discrete breathers (DB) moving toward each other along neighboring close packed atomic rows in a 2D crystal lattice are investigated by molecular dynamics computer simulations. It is shown that a DB can draw energy from the other and emerge from the collision with amplitude much greater than its initial amplitude