Scale-dependent Theory Research Articles

Scale-dependent theory of the disordered electron liquid

We review the scaling theory of disordered itinerant electrons with e−e interactions. We first show how to adjust the microscopic Fermi-liquid theory to the presence of disorder. Then we describe the non-linear sigma model (NLSM) with interactions (Finkel’stein’s model). This model is closely connected to the Fermi liquid, but is more generally applicable, since it can also be viewed as a minimal effective functional describing disordered interacting electrons. Our discussion emphasizes the general structure of the theory, and the connection of the scaling parameters to conservation laws. We then move on to discuss the metal–insulator transition (MIT) in the disordered electron liquid in two and three dimensions. Generally speaking, this MIT is a non-trivial example of a quantum phase transition. The NLSM approach allows to identify the dynamical exponent connecting the spatial and energy scales, which is central for the description of the kinetic and thermodynamic behavior of the system in the critical region of the MIT in three dimensions. In two dimensions, the system can be discussed in terms of a flow in the disorder-interaction phase plane, which is controlled by a fixed point. We demonstrate that the two-parameter RG-equations accurately describe electrons in Si-MOSFETs including the observed non-monotonic behavior of the resistance and its strong drop at low temperatures. The theory can also be applied to systems with an attractive interaction in the Cooper channel, where it describes the suppression of superconductivity in disordered amorphous films. We extend our discussion to heat transport in the two-dimensional electron liquid. Similar to the electric conductivity, the thermal conductivity also acquires logarithmic corrections induced by the interplay of the electron interaction and disorder. The resulting thermal conductivity can be calculated in the NLSM formalism after introducing so-called gravitational potentials. It may be concluded that the combined effect of disorder and e−e interactions determines all aspects of the physics of itinerant electrons in the presence of disorder.

Achieving Effective Renormalization Scale and Scheme Independence via the Principle of Observable Effective Matching

In this work, we explicate a new approach for eliminating renormalization scale and scheme (RSS) dependence in observables. We develop this approach by matching RSS-dependent observables (such as cross-sections and decay rates) to a theory which is independent of both these forms of dependencies. We term the fundamental basis behind this approach as the principle of observable effective matching (POEM), which entails matching of a scale- and scheme-dependent observable with the fully physical scale (PS) and dynamical scale-dependent theory at loop orders at which RSS independence is guaranteed. This is aimed toward achieving so-called “effective” RSS-independent expressions as the resulting dynamical dependence is derived from a particular order in RSS-dependent perturbation theory. With this matching at a PS at which the coupling (and masses) is experimentally determined at this scale, we obtain an “effective theoretical observable (ETO)”, a finite-order RSS-independent version of the RSS-dependent observable. We illustrate our approach with a study of the cross-section ratio Re+e− for e+e− → hadrons, which is demonstrated to achieve scale and scheme independence utilizing the three- and four-loop order MS̄ scheme expression in QCD perturbation theory via matching at both one-loop and two-loop orders for obtaining the ETO. With two-loop matching, we obtain an ETO prediction of 311Re+e−eff=1.052431−0.0006+0.0006 at Q = 31.6GeV, which is in excellent agreement with the experimental value of 311Re+e−exp=1.0527−0.005+0.005. Given its new conceptual basis, ease of use, and performance, we contend that POEM be explored in its application for obtaining ETOs for predicting RSS-independent observables across domains of high-energy theory and phenomenology as well as other areas of fundamental and applied physics, such as cosmology and statistical and condensed matter physics.

On Stochastic Heating and Its Phase-space Signatures in Low-beta Kinetic Turbulence

Abstract We revisit the theory of stochastic heating of ions and investigate its phase-space signatures in kinetic turbulence of relevance to low-β portions of the solar wind. In particular, we retain a full scale-dependent approach in our treatment, and we explicitly consider the case in which electric-field fluctuations can be described by a generalized Ohm’s law that includes Hall and thermoelectric effects. These two electric-field terms provide the dominant contributions to stochastic ion heating when the ion-Larmor scale is much smaller than the ion skin depth, ρ i ≪ d i, which is the case at β ≪ 1. Employing well-known spectral scaling laws for Alfvén-wave and kinetic-Alfvén-wave turbulent fluctuations, we obtain scaling relations characterizing the field-perpendicular particle-energization rate and energy diffusion coefficient associated with stochastic heating in these two regimes. Phase-space signatures of ion heating are then investigated using three-dimensional hybrid-kinetic simulations of continuously driven Alfvénic turbulence at low β (namely, β i = β e = 0.3 and β i = β e = 1/9). In these simulations, energization of ions parallel to the magnetic field is subdominant compared to its perpendicular counterpart (Q ∥,i ≪ Q ⊥,i), and the fraction of turbulent energy that goes into ion heating is ≈75% at β i = 0.3 and ≈40% at β i ≃ 0.1. The phase-space signatures of ion energization are consistent with Landau-resonant collisionless damping and a (β-dependent) combination of ion-cyclotron and stochastic heating. We demonstrate good agreement between our scale-dependent theory and various signatures associated with the stochastic portion of the heating. We discuss briefly the effect of intermittency on stochastic heating and the implications of our work for the interpretation of stochastic heating in solar-wind spacecraft data.

A micro/nano-scale Timoshenko-Ehrenfest beam model for bending, buckling and vibration analyses based on doublet mechanics theory

A micro-nano-scale Timoshenko-Ehrenfest beam model is investigated using doublet mechanics theory in the present study. The governing equations and all possible boundary conditions are obtained based on doublet mechanics model. The static bending, buckling and vibration problems of Timoshenko microbeams are examined in detail. Deflection, rotation, critical buckling loads and natural frequencies predicted by the present doublet mechanics model are obtained for simply supported micro-scale Timoshenko beams by the Navier solution method. The obtained results are compared to other classical and non-classical continuum theories. To illustrate the present doublet mechanics model, the influences of thickness to length scale parameter ratio of the considered material and slenderness ratio on static bending, buckling and vibration problems are investigated. It is shown that there are two frequency spectrums in the vibration of nanobeams similar to macro Timoshenko beams. It is interesting to note that acceptable physical frequencies (mode numbers) have an upper bound due to Van Hove singularity depending on geometrical and material properties of the beam. That fact is observed first time in the open literature by using scale dependent theories.

Nonlinear frequency behaviour of magneto-electromechanical mass nanosensors using vibrating MEE nanoplates with multiple nanoparticles

Nanomechanical mass sensors are promising candidates for low-cost, rapid and sensitive detection of ultrasmall particles such as biomolecules, bacteria and pollutant objects. This advanced mass sensors function based on the change in the vibration behaviour due to the nanoparticle attachment. A nonlinear continuum formulation is developed in this analysis for the large-amplitude vibrations of magneto-electromechanical mass nanosensors. The strain gradient influence and the effect of the nonlocality of stress components are incorporated into the nonlinear analysis via a higher-order scale-dependent theory of elasticity. The nanomechanical sensor makes use of vibrating magneto-electro-elastic (MEE) nanoplates with several possible locations for trapping nanoparticles. The nonlinear equations of the nanosensor are obtained and solved using Hamilton’s principle and perturbation technique, respectively. A verification study is performed by using available finite element results from the literature for graphene-based mechanical nanosensors. The nonlinear frequency shifts are obtained for different scale-dependent theories and various types of compatibility equations. The influences of the mass, position and number of nanoparticles on the frequency shift are studied considering the effects of stress nonlocality, strain gradient, mass moment of inertia, and geometrical nonlinearity.

Vibration of layered nanobeams with periodic nanostructures

Experimental studies show that softening or hardening behaviors of micro/nanostructures depend on the microstructure of the considered material. Some scale dependent theories like nonlocal elasticity, strain gradient models, and modified couple stress theory predict only softening or hardening behavior. Unlike these theories, doublet mechanics predicts both softening and hardening behaviors in some nanomaterial structures. In the present study, vibration of layered nanobeams with periodic nanostructures is investigated using doublet mechanics theory. After deriving the governing equations, Navier-type solution is applied in the vibration analysis of layered nanobeams. Parametric results are presented for different material and geometrical properties.

Nonlinear buckling and free vibration of curved CNTs by doublet mechanics

In this manuscript, static and dynamic behaviors of geometrically imperfect carbon nanotubes (CNTs) subject to different types of end conditions are investigated. The Doublet Mechanics (DM) theory, which is length scale dependent theory, is used in the analysis. The Euler-Bernoulli kinematic and nonlinear mid-plane stretching effect are considered through analysis. The governing equation of imperfect CNTs is a sixth order nonlinear integro-partial-differential equation. The buckling problem is discretized via the differential-integral-quadrature method (DIQM) and then it is solved using Newton's method. The equation of linear vibration problem is discretized using DIQM and then solved as a linear eigenvalue problem to get natural frequencies and corresponding mode shapes. The DIQM results are compared with analytical ones available in the literature and excellent agreement is obtained. The numerical results are depicted to illustrate the influence of length scale parameter, imperfection amplitude and shear foundation constant on critical buckling load, post-buckling configuration and linear vibration behavior. The current model is effective in designing of NEMS, nano-sensor and nano-actuator manufactured by CNTs.

Local dynamic analysis of imperfect fluid-conveying nanotubes with large deformations incorporating nonlinear damping

An attempt is made in this article to analyse the large-amplitude local dynamics of nanofluid-conveying nanotubes with geometrical imperfections. Each element of the nanotube can have displacements along both longitudinal and transverse directions. A nonlinear damping model is also taken into account utilising the Kelvin–Voigt approach. The stress nonlocality and strain gradient influences are modelled using an advanced scale-dependent theory. Moreover, the Beskok–Karniadakis approach is applied for relative motions at the nanotube wall. To present the coupled motion equations of the coupled nanotube, Hamilton’s approach is used. Moreover, to develop a reliable solution procedure, Galerkin’s method along with continuation technique is utilised. The effects of nonlinear damping, geometrical imperfection, being at nanoscales, fluid velocity and relative motion at the wall on the large-amplitude local dynamics are investigated.

Wavelets and Renormalization Group in Quantum Field Theory Problems

Using continuous wavelet transform it is possible to construct a regularization procedure for scale-dependent quantum field theory models, which is complementary to functional renormalization group method in the sense that it sums up the fluctuations of larger scales in order to get the effective action at small observation scale (M.V.Altaisky, Phys. Rev. D93(2016) 105043. The standard RG results for $\phi^4$ model are reproduced. The fixed points of the scale-dependent theory are studied in one loop approximation.

Axial Wave Reflection and Transmission in Stepped Nanorods Using Doublet Mechanics Theory

A numerical investigation of the reflection and transmission of axial waves at stepped nanorods is presented. The scale dependent doublet mechanics theory is used in the analysis. The main difference of the doublet mechanics from other scale dependent models (stress gradient, strain gradient and couple stress theories) is its direct dependence to the micro/nano structure of the solid. Scale parameter is directly related to atomic structure of the material in doublet mechanics theory and it is assumed as carbon-carbon bond length in the present study. However, identification of scale parameters in other scale dependent theories is difficult compared to doublet mechanics theory. Governing equations of stepped nanorods are derived in the framework of doublet mechanics using the Hamilton Principle. The numerical results predicted by doublet mechanics are shown and compared with the classical elasticity.

Wave propagation in double walled carbon nanotubes by using doublet mechanics theory

Flexural and axial wave propagation in double walled carbon nanotubes embedded in an elastic medium and axial wave propagation in single walled carbon nanotubes are investigated. A length scale dependent theory which is called doublet mechanics is used in the analysis. Governing equations are obtained by using Hamilton principle. Doublet mechanics results are compared with classical elasticity and other size dependent continuum theories such as strain gradient theory, nonlocal theory and lattice dynamics. In addition, experimental wave frequencies of graphite are compared with the doublet mechanics theory. It is obtained that doublet mechanics gives accurate results for flexural and axial wave propagation in nanotubes. Thus, doublet mechanics can be used for the design of electro-mechanical nano-devices such as nanomotors, nanosensors and oscillators.

Vibration and buckling analysis of nanotubes (nanofibers) embedded in an elastic medium using Doublet Mechanics

In the present study, vibration and buckling of nanotubes (nanofibers) embedded in an elastic medium are studied. A length scale-dependent theory called Doublet Mechanics (DM) is used in the formulation. In this theory, discrete microstructure of solids is considered in the formulation and using a bottom-up approach macro level strains and stresses are obtained from microlevel strains and stresses. Taylor series expansion of the microlevel displacement is used in the definition of the micro strains. The number of terms in the Taylor series describes the microstructure of the considered solids. In this study, nanotube fibers are assumed as an Euler–Bernoulli beam embedded in an elastic medium. Simply supported and clamped boundary conditions are considered at the edges of the beams. Free vibration frequencies and critical buckling loads are obtained and compared with the classical elasticity results. It is shown that scale-dependent DM can be used at the nanolength scale.