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

Turbulence of second sound waves in superfluid He II

We communicate the results of numerical studies of acoustic turbulence in a system of slightly dissipating, nonlinear second sound waves in superfluid He II. It is shown that at sufficiently high amplitude of the external driving force the power-like energy distribution over frequency is formed in the system of second sound waves. This distribution is attributed to formation of the acoustic turbulence regime in the system. The interval of frequencies in which the distribution has a power-like form is expanded to high frequencies with increasing the amplitude of the driving force. The distribution of the energy inside this interval is close to Eω~ 2. It is shown that the distribution of energy Eω depends on the value of the nonlinearity coefficient of the second sound, but does not depend on the sign of the coefficient, i.e., the coherent structures (shock waves) do not contribute to the statistical properties of the turbulent state

Nonlinear second sound waves and acoustic turbulence in superfluid He-4.

The preliminary results of an investigation of nonlinear second sound waves in a high quality resonator filled with superfluid 4He are presented and discussed. It is found that, for a sufficiently strong periodic driving force, a cascade of second sound waves is formed at multiple harmonics of the driving frequency over the extremely wide frequency range 1–100 kHz. It can be described by a power law A ω =const.×ω −m , where the scaling index m≈1. These observation can be attributed to the formation of a Kolmogorov-like turbulent cascade in the system of second sound waves, accompanied by directed energy flux through the frequency scales. It manifests itself as a limiting of the amplitude of a standing wave, a distortion of the shape of the initially harmonic waves, and a reduction of the effective quality factor Q of the resonator

Turbulence of Second Sound Waves in Superfluid 4He: Effect of Low-Frequency Resonant Perturbations.

We report the results of investigations of acoustic turbulence in a system of nonlinear second sound waves in a high-quality resonator filled with superfluid 4He. It was observed that subharmonics of a periodic driving force applied to the system may be generated via a parametric instability. We find that application of an additional low-frequency pumping to the turbulent system results in the generation of waves at combination frequencies of the driving forces and also leads to substantial changes in the energy spectrum of the acoustic oscillations

Rogue waves in superfluid helium.

Rogue waves have been observed in superfluid helium. The experimental system consists of high intensity second sound (temperature-entropy) waves within a resonant cavity. Under steady state conditions, with a constant oscillatory driving force at the resonant frequency, the waves are turbulent and there are fluxes of energy towards both high and low frequencies. Rogue waves appear under the nonequilibrium conditions that prevail shortly after the drive has been switched on, prior to establishment of the steady state. The experiment is described briefly, relevant results are presented and discussed theoretically in terms of nonlinear wave interactions, and possible connections to rogue waves on the ocean are considered