Trans-Scale Mechanics: Looking For The Missing Links Between Continuum And Micro/Nanoscopic Reality

Problems involving coupled multiple space and time scales offer a real challenge for conventional frameworks of either particle or continuum mechanics. In this paper, four cases studies (shear band formation in bulk metallic glasses, spallation resulting from stress wave, interaction between a probe tip and sample, the simulation of nanoindentation with molecular statistical thermodynamics) are provided to illustrate the three levels of trans-scale problems (problems due to various physical mechanisms at macro-level, problems due to micro-structural evolution at macro/micro-level, problems due to the coupling of atoms/molecules and a finite size body at micro/nano-level) and their formulations. Accordingly, non-equilibrium statistical mechanics, coupled trans-scale equations and simultaneous solutions, and trans-scale algorithms based on atomic/molecular interaction are suggested as the three possible modes of trans-scale mechanics.

Nonequilibrium statistical mechanics of one-dimensional turbulence

The method of statistical mechanics is applied to the study of the one-dimensional model of turbulence proposed in an earlier paper. The closure problem is solved by the variational approach which has been developed for the three-dimensional case, yielding two integral equations for two unknown functions. By solving the two integral equations, the Kolmogorov k−5/3 law is derived and the (one-dimensional) Kolmogorov constant Ko is evaluated, obtaining Ko=0.55, which is in good agreement with the result of numerical experiments on one-dimensional turbulence.

Nonequilibrium statistical mechanics of two-dimensional turbulence

The vorticity dynamics of two-dimensional turbulence are investigated analytically, applying the method of Qian (1983). The vorticity equation and its Fourier transform are presented; a set of modal parameters and a modal dynamic equation are derived; and the corresponding Liouville equation for the probability distribution in phase space is solved using a Langevin/Fokker-Planck approach to obtain integral equations for the enstrophy and for the dynamic damping coefficient eta. The equilibrium spectrum for inviscid flow is found to be a stationary solution of the enstrophy equation, and the inertial-range spectrum is determined by introducing a localization factor in the two integral equations and evaluating the localized versions numerically.

Statistical-theory of multiphoton excitation of polyatomic-molecules based on the wigner function

This paper is aimed at establishing a statistical theory of rotational and vibrational excitation of polyatomic molecules by an intense IR laser. Starting from the Wigner function of quantum statistical mechanics, we treat the rotational motion in the classical approximation; the vibrational modes are classified into active ones which are coupled directly with the laser and the background modes which are not coupled with the laser. The reduced Wigner function, i.e., the Wigner function integrated over all background coordinates should satisfy an integro-differential equation. We introduce the idea of ``viscous damping'' to handle the interaction between the active modes and the background. The damping coefficient can be calculated with the aid of the well-known Schwartz–Slawsky–Herzfeld theory. The resulting equation is solved by the method of moment equations. There is only one adjustable parameter in our scheme; it is introduced due to the lack of precise knowledge about the molecular potential. The theory developed in this paper explains satisfactorily the recent absorption experiments of SF6 irradiated by a short pulse CO2 laser, which are in sharp contradiction with the prevailing quasi-continuum theory. We also refined the density of energy levels which is responsible for the muliphoton excitation of polyatomic molecules.

Non-modal instabilities of two-dimensional disturbances in plane Couette flow of a power-law fluid

Instabilities of fluid flows have traditionally been investigated by normal mode analysis, i.e. by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem. However, the results of eigenvalue analysis agree poorly in many cases with experiments, especially for shear flows. In this paper we study the instabilities of two-dimensional Couette flow of a polymeric fluid in the framework of non-modal stability theory rather than normal mode analysis. A power-law model is used to describe the polymeric liquid. We focus on the response to external excitations and initial conditions by examining the pseudospectra structures and the transient energy growths. For both Newtonian and non-Newtonian flows, the results show that there can be a rather large transient growth even though the linear operator of Couette flow has no unstable eigenvalue. The effects of non-Newtonian viscosity on the transient behaviors are examined in this study. The results show that the "shear-thinning/shear-thickening" effect increases/decreases the amplitude of responses to external excitations and initial conditions. (C) 2010 Elsevier B.V. All rights reserved.

Cross-correlation between WMAP and 2MASS: non-Gaussianity induced by the SZ effect

We study the non-Gaussianity induced by the Sunyaev-Zel'dovich (SZ) effect in cosmic microwave background (CMB) fluctuation maps. If a CMB map is contaminated by the SZ effect of galaxies or galaxy clusters, the CMB maps should have similar non-Gaussian features to the galaxy and cluster fields. Using the WMAP data and 2MASS galaxy catalogue, we show that the non-Gaussianity of the 2MASS galaxies is imprinted on WMAP maps. The signature of non-Gaussianity can be seen with the fourth-order cross-correlation between the wavelet variables of the WMAP maps and 2MASS clusters. The intensity of the fourth-order non-Gaussian features is found to be consistent with the contamination of the SZ effect of 2MASS galaxies. We also show that this non-Gaussianity can not be seen by the high-order autocorrelation of the WMAP. This is because the SZ signals in the autocorrelations of the WMAP data generally are weaker than the WMAP-2MASS cross-correlations by a factor f(2), which is the ratio between the powers of the SZ-effect map and the CMB fluctuations on the scale considered. Therefore, the ratio of high-order autocorrelations of CMB maps to cross-correlations of the CMB maps and galaxy field would be effective to constrain the powers of the SZ effect on various scales.

Energy landscape theory, funnels, specificity, and optimal criterion of biomolecular binding

We study the nature of biomolecular binding. We found that in general there exists several thermodynamic phases: a native binding phase, a non-native phase, and a glass or local trapping phase. The quantitative optimal criterion for the binding specificity is found to be the maximization of the ratio of the binding transition temperature versus the trapping transition temperature, or equivalently the ratio of the energy gap of binding between the native state and the average non-native states versus the dispersion or variance of the non-native states. This leads to a funneled binding energy landscape.

A thermal-elastic analysis on laser-induced reverse bulging and plugging in circular brass foil

A new failure mode is observed in circular brass foils induced by laser beam. The new failure is based on the following experimental facts : (1) the peripheries of the circular brass foils are fixed and the surfaces of the foils are radiated by laser beam ; (2) the laser beam used is considered to be non-Gaussian spatially, actually an approximately uniform distribution limited in a certain size spot ; (3) the pulse on time of laser beam should be 250 μs, i.e. so called long duration pulse laser. The failure process consists of three stages ; i.e. thermal bulging, localized shear deformation and perforation by plugging. The word reverse in reverse bulging and plugging mode means that bulging and plugging occur in the direction of incident laser beam. To study the newly-discovered type of failure quantitatively, analytical solutions for the axisymmetric temperature field and deflection curve are derived. The calculated results show that the newly discovered failure mode is attributed to the spatial structure effect of laser beam indeed.

Multiscale Coupling in Complex Mechanical Systems

Multiscale coupling attracts broad interests from mechanics, physics and chemistry to biology. The diversity and coupling of physics at different scales are two essential features of multiscale problems in far-from-equilibrium systems. The two features present fundamental difficulties and are great challenges to multiscale modeling and simulation. The theory of dynamical system and statistical mechanics provide fundamental tools for the multiscale coupling problems. The paper presents some closed multiscale formulations, e.g., the mapping closure approximation, multiscale large-eddy simulation and statistical mesoscopic damage mechanics, for two typical multiscale coupling problems in mechanics, that is, turbulence in fluids and failure in solids. It is pointed that developing a tractable, closed nonequilibrium statistical theory may be an effective approach to deal with the multiscale coupling problems. Some common characteristics of the statistical theory are discussed.

Multiscale Coupling: Challenges and Opportunities

Multiscale coupling is ubiquitous in nature and attracts broad interests of scientists from mathematicians, physicists, machinists, chemists to biologists. However, much less attention has been paid to its intrinsic implication. In this paper, multiscale coupling is introduced by studying two typical examples in classic mechanics: fluid turbulence and solid failure. The nature of multiscale coupling in the two examples lies in their physical diversities and strong coupling over wide-range scales. The theories of dynamical system and statistical mechanics provide fundamental methods for the multiscale coupling problems. The diverse multiscale couplings call for unified approaches and might expedite new concepts, theories and disciplines.

Turbulent passive scalar field of a small prandtl number

The statistical-mechanics theory of the passive scalar field convected by turbulence, developed in an earlier paper [Phys. Fluids 28, 1299 (1985)], is extended to the case of a small molecular Prandtl number. The set of governing integral equations is solved by the equation-error method. The resultant scalar-variance spectrum for the inertial range is F(k)~x−5/3/[1+1.21x1.67(1+0.353x2.32)], where x is the wavenumber scaled by Corrsin's dissipation wavenumber. This result reduces to the − (5)/(3) law in the inertial-convective range. It also approximately reduces to the − (17)/(3) law in the inertial-diffusive range, but the proportionality constant differs from Batchelor's by a factor of 3.6.

A passive scalar field convected by turbulence

Classical statistical mechanics is applied to the study of a passive scalar field convected by isotropic turbulence. A complete set of independent real parameters and dynamic equations are worked out to describe the dynamic state of the passive scalar field. The corresponding Liouville equation is solved by a perturbation method based upon a Langevin–Fokker–Planck model. The closure problem is treated by a variational approach reported in earlier papers. Two integral equations are obtained for two unknown functions: the scalar variance spectrum F(k) and the effective damping coefficient (k). The appearance of the energy spectrum of the velocity field in the two integral equations represents the coupling of the scalar field with the velocity field. As an application of the theory, the two integral equations are solved to derive the inertial-convective-range spectrum, obtaining F(k)=0.61 −1/3 k−5/3. Here is the dissipation rate of the scalar variance and is the dissipation rate of the energy of the velocity field. This theoretical value of the scalar Kolmogorov constant, 0.61, is in good agreement with experiments.

Universal equilibrium range of turbulence

An attempt is made to determine the form of F(x), the dimensionless function of universal nature which occurs in the energy spectrum for the universal equilibrium range of fully developed turbulence, by the method of statistical mechanics without introducing any parameter of semiempirical nature. Then, the validity of the variational approach to the closure problem of turbulence theory is tested by applying it to the study of the universal equilbrium range of turbulence.

Variational approach to the closure problem of turbulence theory

A new method is proposed to solve the closure problem of turbulence theory and to drive the Kolmogorov law in an Eulerian framework. Instead of using complex Fourier components of velocity field as modal parameters, a complete set of independent real parameters and dynamic equations are worked out to describe the dynamic states of a turbulence. Classical statistical mechanics is used to study the statistical behavior of the turbulence. An approximate stationary solution of the Liouville equation is obtained by a perturbation method based on a Langevin-Fokker-Planck (LFP) model. The dynamic damping coefficient eta of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution; this leads to a convergent integral equation for eta to replace the divergent response equation of Kraichnan's direct-interaction (DI) approximation, thereby solving the closure problem without appealing to a Lagrangian formulation. The Kolmogorov constant Ko is evaluated numerically, obtaining Ko = 1.2, which is compatible with the experimental data given by Gibson and Schwartz, (1963).

Transverse mode formation in longitudinally pumped miniature slab lasers

The formation of transverse modes in longitudinally pumped miniature slab lasers is investigated theoretically and experimentally. The longitudinally non-uniform gain-guiding is studied by expanding the electric field into the Hermite-Gaussian functions that satisfy boundary conditions of the resonator. Non-Gaussian transversal beam profiles in the near field are found and the beam diameter is reduced when the pump spot becomes smaller. The experimental observation agrees with the theoretical calculation.