Lie-Poisson Neural Networks (LPNets): Data-based computing of Hamiltonian systems with symmetries.

An accurate data-based prediction of the long-term evolution of Hamiltonian systems requires a network that preserves the appropriate structure under each time step. Every Hamiltonian system contains two essential ingredients: the Poisson bracket and the Hamiltonian. Hamiltonian systems with symmetries, whose paradigm examples are the Lie-Poisson systems, have been shown to describe a broad category of physical phenomena, from satellite motion to underwater vehicles, fluids, geophysical applications, complex fluids, and plasma physics. The Poisson bracket in these systems comes from the symmetries, while the Hamiltonian comes from the underlying physics. We view the symmetry of the system as primary, hence the Lie-Poisson bracket is known exactly, whereas the Hamiltonian is regarded as coming from physics and is considered not known, or known approximately. Using this approach, we develop a network based on transformations that exactly preserve the Poisson bracket and the special functions of the Lie-Poisson systems (Casimirs) to machine precision. We present two flavors of such systems: one, where the parameters of transformations are computed from data using a dense neural network (LPNets), and another, where the composition of transformations is used as building blocks (G-LPNets). We also show how to adapt these methods to a larger class of Poisson brackets. We apply the resulting methods to several examples, such as rigid body (satellite) motion, underwater vehicles, a particle in a magnetic field, and others. The methods developed in this paper are important for the construction of accurate data-based methods for simulating the long-term dynamics of physical systems.

A nanostructured surface increases friction exponentially at the solid-gas interface.

According to Stokes' law, a moving solid surface experiences viscous drag that is linearly related to its velocity and the viscosity of the medium. The viscous interactions result in dissipation that is known to scale as the square root of the kinematic viscosity times the density of the gas. We observed that when an oscillating surface is modified with nanostructures, the experimentally measured dissipation shows an exponential dependence on kinematic viscosity. The surface nanostructures alter solid-gas interplay greatly, amplifying the dissipation response exponentially for even minute variations in viscosity. Nanostructured resonator thus allows discrimination of otherwise narrow range of gaseous viscosity making dissipation an ideal parameter for analysis of a gaseous media. We attribute the observed exponential enhancement to the stochastic nature of interactions of many coupled nanostructures with the gas media.

Recording oscillations of sub-micron size cantilevers by extreme ultraviolet Fourier transform holography.

We recorded the fast oscillation of sub-micron cantilevers using time-resolved extreme ultraviolet (EUV) Fourier transform holography. A tabletop capillary discharge EUV laser with a wavelength of 46.9 nm provided a large flux of coherent illumination that was split using a Fresnel zone plate to generate the object and the reference beams. The reference wave was produced by the first order focus while a central opening in the zone plate provided a direct illumination of the cantilevers. Single-shot holograms allowed for the composition of a movie featuring the fast oscillation. Three-dimensional displacements of the object were determined as well by numerical back-propagation, or "refocusing" of the electromagnetic fields during the reconstruction of a single hologram.

Dynamics of elastic rods in perfect friction contact.

One of the most challenging and basic problems in elastic rod dynamics is a description of rods in contact that prevents any unphysical self-intersections. Most previous works addressed this issue through the introduction of short-range potentials. We study the dynamics of elastic rods with perfect rolling contact which is physically relevant for rods with a rough surface. Such dynamics cannot be described by the introduction of any kind of potential. We show that, surprisingly, the presence of rolling contact in rod dynamics leads to highly complex behavior even for the evolution of small disturbances.

Ordered and disordered dynamics in monolayers of rolling particles.

We consider the ordered and disordered dynamics for monolayers of rolling self-interacting particles modeling water molecules. The rolling constraint represents a simplified model of a strong, but rapidly decaying bond with the surface. We show the existence and nonlinear stability of ordered lattice states, as well as disturbance propagation through and chaotic vibrations of these states. We study the dynamics of disordered gas states and show that there is a surprising and universal linear connection between distributions of angular and linear velocity, allowing definition of temperature.

Manipulation of single atoms by atomic force microscopy as a resonance effect.

Extraction and deposition of single atoms using an atomic force microscope tip is a promising technique for building nanostructures. Previous theoretical models for this technique, based on adiabatic atom motion in either classical or quantum mechanics settings, encountered an apparent difficulty in explaining atom extraction and deposition for the same experimental conditions. We resolve that difficulty by showing that both extraction and deposition of atoms can be formulated in terms of pure classical mechanics as a resonance effect, arising from a combination of interatomic forces and vibrations of individual atoms.

Relaxation dynamics of nucleosomal DNA.

Recent experimental and theoretical evidence demonstrates that proteins and water in the hydration layer can follow complex stretched exponential or power law relaxation dynamics. Here, we report on a 50 ns all atom molecular dynamics (MD) simulation of the yeast nucleosome, where the interactions between DNA, histones, surrounding water and ions are explicitly included. DNA interacts with the histone core in 14 locations, approximately every 10.4 base pairs. We demonstrate that all sites of interaction exhibit anomalously slow power law relaxation, extending up to 10 ns, while fast exponential relaxation dynamics of hundreds of picoseconds applies to DNA regions outside these locations. The appearance of 1/f(alpha) noise or pink noise in DNA dynamics is ubiquitous. For histone-bound nucleotide dynamics alpha --> 1 and is a signature of complexity of the protein-DNA interactions. For control purposes two additional DNA simulations free of protein are conducted. Both utilize the same sequence of DNA, as found the in the nucleosome. In one simulation the initial conformation of the double helix is a straight B-form. In the other, the initial conformation is super helical. Neither of these simulations exhibits the variation of alpha as a function of position, the measure of power law for dynamical behavior, which we observe in the nucleosome simulation. The unique correspondence (high alpha to DNA-histone interaction sites, low alpha to free DNA sites), suggests that alpha may be an important and new quantification of protein-DNA interactions for future experiments.

Meandering fluid streams in the presence of flow-rate fluctuations.

We report that meandering of a rivulet flowing down a nonerodible, partially wetting incline is triggered by flow-rate fluctuations and sustained by external noise forcing. In our experiments, the former is provided by an electronically controlled valve, and the latter is due to fluid droplets left on the surface by previous meanderings. We observe power-law behavior of the averaged spectrum of the deviations of the stream from its center line, which rules out the existence of a preferred wavelength in ongoing meandering. We derive a simple theoretical model of rivulet meandering from first principles, incorporating stream dynamics and external noise forcing. The model provides an accurate statistical description of the stream deviation from a nonmeandering path.

Instabilities, bifurcations, and multiple solutions in expanding channel flows.

We present an experimental realization of the classical Jeffery-Hamel flows inside a wedge-shaped channel. We compare the measured velocity fields with the predictions of Jeffery-Hamel theory. A detailed experimental study of bifurcation diagrams for the solutions reveals the absolute stability of the pure outflow solution and an interesting hysteretic structure for bifurcations. We also observe a multiple-vortex flow regime predicted earlier numerically and analytically.

Aggregation of finite-size particles with variable mobility.

New model equations are derived for dynamics of aggregation of finite-size particles. The differences from standard Debye-Hückel and Keller-Segel models are that the mobility of particles depends on the configuration of their neighbors and linear diffusion acts on locally averaged particle density. The evolution of collapsed states in these models reduces exactly to finite-dimensional dynamics of interacting particle clumps. Simulations show these collapsed (clumped) states emerge from smooth initial conditions, even in one spatial dimension. Extensions to two and three dimensions are also discussed.

Braiding patterns on an inclined plane.

A jet of fluid flowing down a partially wetting, inclined plane usually meanders but--by maintaining a constant flow rate--meandering can be suppressed, leading to the emergence of a beautiful braided structure. Here we show that this flow pattern can be explained by the interplay between surface tension, which tends to narrow the jet, and fluid inertia, which drives the jet to widen. These observations dispel misconceptions about the relationship between braiding and meandering that have persisted for over 20 years.

Turbulent wake solutions of the Prandtl alpha equations.

A derivation of the Navier-Stokes alpha equations for spatially dependent alpha is presented. It is shown that an extra term in the equation is necessary to ensure the conservation of momentum. The Prandtl form of these variable alpha equations are determined for both planar and axisymmetric pressure-gradient-driven boundary-layer flows correcting previous work on the subject. The Prandtl equations are then solved analytically for four different asymptotic wake flows: the classical planar wake, the classical axisymmetric wake, the planar dragless wake, and the axisymmetric dragless wake. Least-squares fits of the theoretical solutions with available turbulent mean-flow velocity data for classical planar and axisymmetric wakes are given. We point out that the dissipation coefficient does not have to be equal to the kinematic viscosity, and its numerical value may be estimated from experimental data.