Non-conservative Interactions Research Articles

Evaluation and optimization of quartz resonant-frequency retuned fork force sensors with high Q factors, and the associated electric circuits, for non-contact atomic force microscopy.

High-Q factor retuned fork (RTF) force sensors made from quartz tuning forks, and the electric circuits for the sensors, were evaluated and optimized to improve the performance of non-contact atomic force microscopy (nc-AFM) performed under ultrahigh vacuum (UHV) conditions. To exploit the high Q factor of the RTF sensor, the oscillation of the RTF sensor was excited at its resonant frequency, using a stray capacitance compensation circuit to cancel the excitation signal leaked through the stray capacitor of the sensor. To improve the signal-to-noise (S/N) ratio in the detected signal, a small capacitor was inserted before the input of an operational (OP) amplifier placed in an UHV chamber, which reduced the output noise from the amplifier. A low-noise, wideband OP amplifier produced a superior S/N ratio, compared with a precision OP amplifier. The thermal vibrational density spectra of the RTF sensors were evaluated using the circuit. The RTF sensor with an effective spring constant value as low as 1000 N/m provided a lower minimum detection limit for force differentiation. A nc-AFM image of a Si(111)-7 × 7 surface was produced with atomic resolution using the RTF sensor in a constant frequency shift mode; tunneling current and energy dissipation images with atomic resolution were also simultaneously produced. The high-Q factor RTF sensor showed potential for the high sensitivity of energy dissipation as small as 1 meV/cycle and the high-resolution analysis of non-conservative force interactions.

“SLIMPLECTIC” INTEGRATORS: VARIATIONAL INTEGRATORS FOR GENERAL NONCONSERVATIVE SYSTEMS

Symplectic integrators are widely used for long-term integration of conservative astrophysical problems due to their ability to preserve the constants of motion; however, they cannot in general be applied in the presence of nonconservative interactions. In this Letter, we develop the "slimplectic" integrator, a new type of numerical integrator that shares many of the benefits of traditional symplectic integrators yet is applicable to general nonconservative systems. We utilize a fixed time-step variational integrator formalism applied to the principle of stationary nonconservative action developed in Galley, 2013; Galley, Tsang & Stein, 2014. As a result, the generalized momenta and energy (Noether current) evolutions are well-tracked. We discuss several example systems, including damped harmonic oscillators, Poynting-Robertson drag, and gravitational radiation reaction, by utilizing our new publicly available code to demonstrate the slimplectic integrator algorithm. Slimplectic integrators are well-suited for integrations of systems where nonconservative effects play an important role in the long-term dynamical evolution. As such they are particularly appropriate for cosmological or celestial N-body dynamics problems where nonconservative interactions, e.g. gas interactions or dissipative tides, can play an important role.

Structure Formation of Sulfur-Based Composite: The Model

In material science the simultaneous application of theoretical examination, experimental and numerical studies are often required. This is especially true for modern composite materials with extra inter-boundary nanoscale layers. Thickness of layers is usually about tens of nanometers, while diameters of particles of filler are about several hundreds of nanometers. Thus, during the theoretical study and numerical experiments the size and properties of inter-boundary layer must be taken into account. The proper choice of the model is the key factor for the adequate results of simulation. In the present work we have derived such a model. The system under investigation – disperse-filled composite material with inter-boundary layers of different properties – is represented by particle system; these classes of models can be characterized by high generality. Initial equation for the law of motion is sequentially extended with terms which account for different phenomena – conservative binary interaction, non-conservative interaction with environment, interaction with planar boundaries and non-conservative particle-particle interaction via inter-boundary layer. The reduction of the law of motion to the system of ordinary differential equations had opened the possibility for utilization of the vast majority of numerical algorithms for the prediction of the structural properties of nanomodified sulfur-based composite.

The Role of Nonconservative Interactions in the Asymptotic Limit of Thermostatted Kinetic Models

This paper is concerned with the asymptotic analysis of space-velocity dependent thermostatted kinetic frameworks which include conservative, nonconservative and stochastic operators. The mathematical frameworks are integro-partial differential equations that can be proposed for the modeling of most phenomena occurring in biological and chemical systems. Specifically the paper focuses on the derivation of macroscopic equations obtained by performing a low-field and a high-field scaling into the thermostatted kinetic framework and considering the related convergence when the scaling parameter goes to zero. In the low-field limit, the macroscopic equations show diffusion with respect to both the space variable and a scalar variable that is introduced for the modeling of the strategy of the particle system. In the high-field limit, the macroscopic equations show hyperbolic behavior. The asymptotic analysis is also generalized to systems decomposed in various functional subsystems.

Multi-scale coarse-graining of non-conservative interactions in molecular liquids

A new bottom-up procedure for constructing non-conservative (dissipative and stochastic) interactions for dissipative particle dynamics (DPD) models is described and applied to perform hierarchical coarse-graining of a polar molecular liquid (nitromethane). The distant-dependent radial and shear frictions in functional-free form are derived consistently with a chosen form for conservative interactions by matching two-body force-velocity and three-body velocity-velocity correlations along the microscopic trajectories of the centroids of Voronoi cells (clusters), which represent the dissipative particles within the DPD description. The Voronoi tessellation is achieved by application of the K-means clustering algorithm at regular time intervals. Consistently with a notion of many-body DPD, the conservative interactions are determined through the multi-scale coarse-graining (MS-CG) method, which naturally implements a pairwise decomposition of the microscopic free energy. A hierarchy of MS-CG/DPD models starting with one molecule per Voronoi cell and up to 64 molecules per cell is derived. The radial contribution to the friction appears to be dominant for all models. As the Voronoi cell sizes increase, the dissipative forces rapidly become confined to the first coordination shell. For Voronoi cells of two and more molecules the time dependence of the velocity autocorrelation function becomes monotonic and well reproduced by the respective MS-CG/DPD models. A comparative analysis of force and velocity correlations in the atomistic and CG ensembles indicates Markovian behavior with as low as two molecules per dissipative particle. The models with one and two molecules per Voronoi cell yield transport properties (diffusion and shear viscosity) that are in good agreement with the atomistic data. The coarser models produce slower dynamics that can be appreciably attributed to unaccounted dissipation introduced by regular Voronoi re-partitioning as well as by larger numerical errors in mapping out the dissipative forces. The framework presented herein can be used to develop computational models of real liquids which are capable of bridging the atomistic and mesoscopic scales.

How do mutative events modify moments evolution in thermostatted kinetic models?

This short communication aims at developing a thermostatted kinetic framework which includes conservative and nonconservative interactions. Specifically nonconservative interactions refer to proliferative/destructive and mutative events. The thermostatted kinetic framework is a set of autonomous partial integro-differential equations with quadratic nonlinearity. How the moments evolution is modified by mutative interactions is explored in the present communication. Applications refer to the cancer-immune system competition.

A multiscale quasicontinuum method for dissipative lattice models and discrete networks

Lattice models and discrete networks naturally describe mechanical phenomena at the mesoscale of fibrous materials. A disadvantage of lattice models is their computational cost. The quasicontinuum (QC) method is a suitable multiscale approach that reduces the computational cost of lattice models and allows the incorporation of local lattice defects in large-scale problems. So far, all QC methods are formulated for conservative (mostly atomistic) lattice models. Lattice models of fibrous materials however, often require non-conservative interactions. In this paper, a QC formulation is derived based on the virtual-power of a non-conservative lattice model. By using the virtual-power statement instead of force-equilibrium, errors in the governing equations of the force-based QC formulations are avoided. Nevertheless, the non-conservative interaction forces can still be directly inserted in the virtual-power QC framework. The summation rules for energy-based QC methods can still be used in the proposed framework as shown by two multiscale examples.

About an H-theorem for systems with non-conservative interactions

We present some arguments in favor of an H-theorem for a generalization of the Boltzmann equation including non-conservative interactions and a linear Fokker–Planck-like thermostatting term. Such a non-linear equation describing the evolution of the single particle probability Pi(t) of being in state i at time t is a suitable model for granular gases and is referred to here as the Boltzmann–Fokker–Planck (BFP) equation. The conjectured H-functional, which appears to be non-increasing, is HC(t) = ∑iPi(t)lnPi(t)/Πi with Πi = limt→∞Pi(t), in analogy with the H-functional of Markov processes. The extension to continuous states is straightforward. A simple proof can be given for the elastic BFP equation. A semi-analytical proof is also offered for the BFP equation for so-called inelastic Maxwell molecules. Other evidence is obtained by solving particular BFP cases through numerical integration or through ‘particle schemes’ such as the direct simulation Monte Carlo.

Bifurcation and nonlinear analysis of nonconservative interaction between rotor and blade row

It is well known that nonconventional modal interaction between the collective blade motion and rotor lateral whirling can cause instability at a specific speed range. The bifurcation and nonlinear analysis of this interaction are studied in this research for a rotating bladed disk with long blades. In order to obtain a qualitative and general conclusion, a simple model of rotor and blades is considered in which the disk is supposed to be rigid. The blades are considered as inverted pendulums. The bearings and blade stiffness are assumed to be nonlinear and symmetric. The nonlinear stiffness of the blades and bearings includes cubic Duffing terms. A multiple scale method is employed for obtaining the bifurcation equation. The frequency coalescence phenomenon is studied through the bifurcation equation. The effect of rotor and blade damping as well as rotating speed is investigated. The results highlight some interesting phenomena related to the eigenvalue interaction and the nonlinearity in the system. An approximation for the final asynchronous limit cycle is also obtained and compared with the numerical results.

Controllability in Hybrid Kinetic Equations Modeling Nonequilibrium Multicellular Systems

This paper is concerned with the derivation of hybrid kinetic partial integrodifferential equations that can be proposed for the mathematical modeling of multicellular systems subjected to external force fields and characterized by nonconservative interactions. In order to prevent an uncontrolled time evolution of the moments of the solution, a control operator is introduced which is based on the Gaussian thermostat. Specifically, the analysis shows that the moments are solution of a Riccati-type differential equation.

Modeling Complex Systems with Particles Refuge by Thermostatted Kinetic Theory Methods

This paper is concerned with the mathematical modeling of complex systems characterized by particles refuge. Specifically the paper focuses on the derivation and moments analysis of thermostatted kinetic frameworks with conservative and nonconservative interactions for closed and open complex systems at nonequilibrium. Applications and future research perspectives are discussed in the last section of the paper.

Drive-amplitude-modulation atomic force microscopy: From vacuum to liquids

We introduce drive-amplitude-modulation atomic force microscopy as a dynamic mode with outstanding performance in all environments from vacuum to liquids. As with frequency modulation, the new mode follows a feedback scheme with two nested loops: The first keeps the cantilever oscillation amplitude constant by regulating the driving force, and the second uses the driving force as the feedback variable for topography. Additionally, a phase-locked loop can be used as a parallel feedback allowing separation of the conservative and nonconservative interactions. We describe the basis of this mode and present some examples of its performance in three different environments. Drive-amplutide modulation is a very stable, intuitive and easy to use mode that is free of the feedback instability associated with the noncontact-to-contact transition that occurs in the frequency-modulation mode.

Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states

We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation (BTE) in d-dimensions for variable hard sphere (VHS) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computation is reduced to a separate integral over the unit sphere S d - 1 . The conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or elastic models of slowing down process. Our simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation, both for elastic and inelastic VHS interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in Bobylev et al. [A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, Journal of Statistical Physics 111 (2003) 403–417] and generalized to a wide range of related models in Bobylev et al. [A.V. Bobylev, C. Cercignani, I.M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Communication in Mathematical Physics, in press. URL: <http://arxiv.org/abs/math-ph/0608035>]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods [T.V. Noije, M. Ernst, Velocity distributions in homogeneously cooling and heated granular fluids, Granular Matter 1(57) (1998); M.H. Ernst, R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, Journal of Statistical Physics 109 (2002) 407–432; S.J. Moon, M.D. Shattuck, J. Swift, Velocity distributions and correlations in homogeneously heated granular media, Physical Review E 64 (2001) 031303; I.M. Gamba, S. Rjasanow, W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Mathematical and Computer Modelling 42 (2005) 683–700] and rigorously proven in Gamba et al. [I.M. Gamba, V. Panferov, C. Villani, On the Boltzmann equation for diffusively excited granular media, Communications in Mathematical Physics 246 (2004) 503–541(39)] and [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, Journal of Statistical Physics 116 (2004) 1651–1682].

Imaging Beyond Topography

As a relatively new scanning probe technique, Dynamic Force Microscopy [1] has proved to be a powerful tool, allowing for imaging the topography of a sample surface with true atomic resolution. Besides topographic imaging, the simultaneously recorded damping signal is related to non-conservative interaction between tip and sample. For a molecular system, we show the tip-induced switching of the functionalized groups of the organic molecule leading to an enhanced damping signal.

Generalized Landau–Lifshitz–Gilbert equation for uniformly magnetized bodies

We consider generalized Landau–Lifshitz–Gilbert (LLG) deterministic dynamics in uniformly magnetized bodies. The dynamics take place on the unit sphere Σ , and are characterized by a vector field v tangential to Σ . By using Helmholtz decomposition on Σ , it is proven that v is uniquely defined by two potentials χ and ψ . Potential χ can be identified with the free energy of the system, while ψ describes non-conservative interactions of the system with the environment. The presence of ψ modifies the usual energy balance of LLG dynamics. Instead of purely relaxation dynamics we may have steady injection of energy through non-conservative interactions. The implications of the new form of the energy balance are discussed in detail.

A Model of Mechanical Polishing in the Presence of a Lubricant

Modeling of the friction and wear at the contact of rough surfaces in the presence of a lubricant is important for the development of modern technologies. This is a complex problem involving a consistent description of the elastic straining of contacting bodies and of the flow of viscous liquid between them. It is demonstrated that this problem can be significantly simplified in the most interesting case, in which the lubricant thickness is very small and the decisive contribution to both the contact interaction and the friction force is due to the interactions between a finite number of inhomogeneities approaching one another to a certain distance, this distance being much smaller than the average distance between the two bodies. The lubricant dynamics can be modeled in terms of the nonconservative interactions between the particles, which depend on the distance and relative velocity. The proposed approach is used to describe the process of mechanical polishing in the presence of abrasive particles suspended in the lubricant layer. This hydrodynamic polishing process results in the formation of a surface relief with a statistically equilibrium roughness that exhibits a fractal character.

Dynamic AFM using the FM technique with constant excitation amplitude

Dynamic atomic force microscopy using the frequency modulation technique is investigated for the case that the excitation amplitude is kept constant. This mode of operation has very unique properties. A computer simulation is used to investigate the distance dependence of the measurement signals frequency and amplitude using various conservative and non-conservative interaction forces. It is shown how the two measurement channels are interlinked and influenced by both conservative and dissipative interactions. Further, discontinuous frequency shift versus distance curves as reported in the literature were not obtained.

Long-range nonconservative interactions of moving neutral atoms with surface plasmons: A possibility of experimental verification

We are discussing a possibility of experimental measurement of the long-range nonconservative interactions of a neutral beam of helium atoms on an aluminum surface in the velocity range 0.1–2.5 a.u. General formulas for the lateral force with account of plasmon excitation effects and in the local Drude approximation to the surface response are given. A measurable positive energy gain of helium atoms caused by surface plasmon coupling is predicted.

Global dynamics in the simplest models of three-dimensional water-wave patterns

We explore the idea that periodic or chaotic finite motions corresponding to attractors in the simplest models of resonant wave interactions might shed light on the problem of pattern formation. First we identify those dynamical regimes of interest which imply certain specific relations between physically observable variables, e.g. between amplitudes and phases of Fourier harmonics comprising the pattern. To be of relevance to reality, the regimes must be robust. The issue of structural stability of low-dimensional dynamical models is central to our work. We show that the classical model of three-wave resonant interactions in a non-conservative medium is structurally unstable with respect to small cubic interactions. The structural instability is found to be due to the presence of certain extremely sensitive points in the unperturbed system attractors. The model describing the horse-shoe pattern formation due to non-conservative quintet interactions [11] is also analyzed and a rich family of attractors is mapped. The absence of such sensitive points in the found attractors thus indicates the robustness of the regimes of interest. Applicability of these models to the problem of 3-D water wave patterns is discussed. Our general conclusion is that extreme caution is necessary in applying the dynamical system approach, based upon low-dimensional models, to the problem of water wave pattern formation.

Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas

A kinetic approach is presented for the analysis of a gas mixture with two kinds of nonconservative interactions. In a bimolecular chemical reaction, mass transfer and energy of chemical link arise, and in inelastic mechanical encounters, molecules get excited or de‐excited due to their quantized structure. Molecules undergo transitions between energy levels also by absorption and emission of photons of the self‐consistent radiation field. From the kinetic Boltzmann‐type equations, the problem of equilibria and of their stability is addressed. A detailed balance principle is proved and a Lyapunov functional is constructed; mass action law and Planck's law of radiation are recovered.