Gradient Expansion Research Articles

Research and analysis of manipulator control method based on deep learning

Aiming at the stability and accuracy of grasping objects, this paper studies the adaptive neural network control and learning of manipulator control system with unknown system dynamics. First, a stable adaptive neural network (NN) controller is designed, and the unknown closed-loop system dynamics of the manipulator is approximated by using radial basis function (RBF) neural network. The LSTM long and short memory algorithm is introduced, and the RBF based hybrid neural network model is constructed. The LSTM algorithm is used to design the input gate, forgetting gate and output gate structures to suppress the gradient expansion problem during coordinate data training, and accurate trajectory correction instructions are given. Lyapunov stability theorem is used to analyze stability. Partial persistent excitation (PE) conditions of some internal signals in the closed-loop system are satisfied in the control process of tracking the circular reference trajectory. Under the PE condition, the proposed adaptive NN controller can accurately identify the dynamic uncertainties of the manipulator in the stability control process. Then, a new NN learning control method is proposed, which effectively uses the knowledge learned without re adapting to the unknown manipulator control system dynamics to achieve closed-loop stability and improve control performance. The effectiveness of the proposed method is studied through simulation.Keyboard

Derivative Formulas and Gradient of Functions with Non-Independent Variables

Stochastic characterizations of functions subject to constraints result in treating them as functions with non-independent variables. By using the distribution function or copula of the input variables that comply with such constraints, we derive two types of partial derivatives of functions with non-independent variables (i.e., actual and dependent derivatives) and argue in favor of the latter. Dependent partial derivatives of functions with non-independent variables rely on the dependent Jacobian matrix of non-independent variables, which is also used to define a tensor metric. The differential geometric framework allows us to derive the gradient, Hessian, and Taylor-type expansions of functions with non-independent variables.

An update on adiabatic modes in cosmology and δN formalism

In this paper, we generalize the Weinberg's procedure to determine the comoving curvature perturbation ℛ to non-attractor inflationary regimes. We show that both modes of ℛ are related to a symmetry of the perturbative equations in the Newtonian gauge. As a byproduct, we clarify that adiabaticity does not generally imply constancy of ℛ, not even in the k ⟶ 0 limit. We then show that there exist non-equivalent definitions of δN that would reproduce ℛ or the uniform density curvature perturbation ζ at linear order. We have then shown that the perturbative δN definition in terms of difference between the number of e-foldings of different gauges, can be extended non-perturbatively at leading order in gradient expansion. Nevertheless, the computer friendly definition in terms of the difference of e-foldings obtained from the evolution of a local FRW Universe, respectively with perturbed and un-perturbed initial conditions, might only give information about the linear order curvature perturbations, contrary to what is stated in the literature.

Stability studies of first-order spin-hydrodynamic frameworks

We study the stability of first-order dissipative spin-hydrodynamic frameworks. We considered two different first-order dissipative spin-hydrodynamic frameworks. The first one considers the spin-chemical potential (${\ensuremath{\omega}}^{\ensuremath{\alpha}\ensuremath{\beta}}$) to be first order [$\mathcal{O}(\ensuremath{\partial})$] in the hydrodynamic gradient expansion. The hydrodynamic gradient ordering of the spin-chemical potential is a debatable issue within the frameworks of spin hydrodynamics. Therefore, as a second choice, we also consider the spin-hydrodynamic equations with ${\ensuremath{\omega}}^{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{\sim}\mathcal{O}(1)$. We find that, for both frameworks, at the level of linear perturbations some spin modes can be unstable. To remove these generic instabilities, we consider the Frenkel condition. We argue that the Frenkel condition helps get rid of the unstable solutions in both cases but with a physical drawback for the case where ${\ensuremath{\omega}}^{\ensuremath{\mu}\ensuremath{\nu}}\ensuremath{\sim}\mathcal{O}(\ensuremath{\partial})$.

Discussion on the transport processes in electrons with non-Maxwellian energy distribution function in partially-ionized plasmas

In a previous work (Alvarez Laguna et al 2022 Phys. Plasmas 29 083507), we have developed a non-linear moment model for electrons that self-consistently captures non-Maxwellian electron energy distribution function effects. The model does not rely in the local approximation and the transport coefficients are calculated by expanding the distribution function into Hermite polynomials and by taking moments of the Boltzmann equation, including the collision operator for elastic and inelastic collisions with arbitrary cross sections. This model captures the classical Fick’s, Fourier’s, and Ohm’s law as well as Soret, Dufour, and Peltier effects. In addition, novel non-local transport phenomena appear as a result of spatial gradients of the kurtosis of the distribution function. In this paper, we discuss on the transport effects by analyzing two collisional models: constant collision frequency and constant cross section. We estimate the order of magnitude of the transport processes in non-equilibrium electrons by analyzing the Langmuir probe measurements of a low-pressure argon inductively-coupled discharge. The results show that, under these conditions, the transport produced by the spatial gradients in the kurtosis of the distribution function produces a heat-flux contribution that is of the same order of magnitude as the Fourier and Dufour’s effects. These transport effects are beyond the local field or the electron gradient expansions, commonly used in the low-temperature plasma modeling.

Quantum partonic transport in QCD matter

We study gradient corrections to the transport equation for energetic light partons in dense QCD environments. In the diffusion limit, the transport dynamics is solely controlled by small-angle elastic scatterings, leading to transverse momentum broadening with respect to the parton's initial direction. Such a parton propagation is usually considered in the limit of transversely homogeneous matter. The transport processes admit a classical description and the transverse spatial dependence of the medium properties emerges only through the jet quenching parameter. In this work, we show that a gradient expansion of the all-order evolution equation for the partonic Wigner function leads to an evolution equation in the Boltzmann-diffusion form only up to the leading order in transverse gradients. At the second order in gradients, the quantum corrections associated with nonlocal interactions give rise to a novel transport that can be implemented in Monte Carlo simulations. In addition, using our results, we compute the gradient corrections to the jet quenching parameter in inhomogeneous matter.

Gravitational chiral anomaly and anomalous transport for fields with spin 3/2

In a fluid with vorticity and acceleration, an axial current arises in the third order of gradient expansion, the corresponding phenomenon can be called the Kinematical vortical effect (KVE). It has recently been shown that the KVE while existing in the absence of gravitational fields, is nonetheless associated with effects in curved space-time, namely with the gravitational chiral quantum anomaly. In this paper, we find the KVE transport coefficients using the Zubarev quantum-statistical density operator for the Rarita-Schwinger-Adler theory, which includes massless fields with spins 3/2 and 1/2, and demonstrate the relationship with the gravitational anomaly. A prediction is made about the possible form of the transport coefficients for massless fields with arbitrary spin.

Reconstruction of spectra and an algorithm based on the theorems of Darboux and Puiseux

Assuming only a known dispersion relation of a single mode in the spectrum of a meromorphic two-point function (in the complex frequency plane at fixed wavevector) in some quantum field theory, we investigate when and how the reconstruction of the complete spectrum of physical excitations is possible. In particular, we develop a constructive algorithm based on the theorems of Darboux and Puiseux that allows for such a reconstruction of all modes connected by level-crossings. For concreteness, we focus on theories in which the known mode is a gapless excitation described by the hydrodynamic gradient expansion, known at least to some (preferably high) order. We first apply the algorithm to a simple algebraic example and then to the transverse momentum excitations in the holographic theory that describes a stack of M2 branes and includes momentum diffusion as its gapless excitation.

Pseudomagnetic fields, particle-hole asymmetry, and microscopic effective continuum Hamiltonians of twisted bilayer graphene

Using the method developed in the companion paper, we construct the effective continuum theories for two different microscopic tight binding models of the twisted bilayer graphene at the twist angle of $1.05^\circ$, one Slater-Koster based and the other ab-initio Wannier based. The energy spectra obtained from the continuum theory -- either for rigid twist or including lattice relaxation -- are found to be in nearly perfect agreement with the spectra from the tight binding models when the gradient expansion is carried out to second order, demonstrating the validity of the method. We also analyze the properties of the Bloch states of the resulting narrow bands, finding non-negligible particle-hole symmetry breaking near the $\Gamma$ point in our continuum theory constructed for the ab-initio based microscopic model due to a term in the continuum theory that was previously overlooked. This reveals the difference with all existing continuum models where the particle-hole symmetry of the narrow band Hilbert space is nearly perfect.

Scanning drift tube measurements and kinetic studies of electron transport in CO

We present scanning drift tube measurements of electron swarm transport coefficients in CO as a function of the reduced electric field at room temperature under time-of-flight conditions. The measurements are compared to modeling results and other available experimental data on swarm transport over the broad range of from 2 Td to 1603 Td. The modeling results are obtained in Monte Carlo simulations and by solving the electron Boltzmann equation using a multi-term approach and the density gradient expansion procedure. We find generally good agreement between the measured and calculated transport coefficients. We propose a strategy to improve the cross-section set used to explain certain discrepancies at lower values. Measurements and calculations of electron transport coefficients under hydrodynamic conditions are complemented by Monte Carlo simulations of electron transport in an idealized steady-state Townsend (SST) setup. The ionization coefficient is calculated as a function of from the spatial density profiles of the electrons and compared to the corresponding values evaluated from the knowledge of the effective ionization frequency, drift velocity and longitudinal diffusion coefficient. Contrary to the traditional views, according to which the spatial relaxation of the mean energy and other transport properties for electrons in molecular gases is most commonly monotonic or quasi-monotonic, we find a ‘window’ of where the SST transport properties of the electrons exhibit oscillatory behavior as they relax towards the equilibrium state far downstream from the electron emitting boundary.

Non-Abelian chiral kinetic equations in the Cartan-Weyl basis

<sec>Non-Abelian gauge field is the fundamental element of the standard model. Non-Abelian chiral kinetic theory can be used to describe how the chiral fermions in standard model transport in a non-equilibrium system. </sec><sec>In our previous work, we decomposed the non-Abelian chiral kinetic equations into color singlet and multiplet in the <inline-formula><tex-math id="M1">\begin{document}$SU(N)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M1.png"/></alternatives></inline-formula> color space. In this formalism, the chiral kinetic equations preserve the gauge symmetry in a very apparent way. However, sometimes we need to describe the microscopic process of the specific color degree, e.g. the color connection in the hadronization stage. In order to describe such a process, it will be more convenient to decompose the non-Abelian chiral kinetic equations in the Cartan-Weyl basis. </sec><sec>In this work, we choose the matrix elements of the Wigner function in fundamental representation of color space as the direct variables and decompose the gauge field or strength tensor field in the Cartan-Weyl basis. By using the covariant gradient expansion, we decompose the non-Abelian chiral kinetic equations into the coupled kinetic equations for diagonal distribution function and non-diagonal distribution function up to the first order. When only diagonal elements exist in the gauge field with non-diagonal elements and diagonal elements decoupled, the non-Ableian chiral kinetic equation will be reduced to the form in the Abelian case. When the non-diagonal elements of the gauge field are present, the kinetic equations are totally tangled between diagonal distribution function and non-diagonal distribution function. Especially, the <inline-formula><tex-math id="M2">\begin{document}$0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M2.png"/></alternatives></inline-formula>th-order non-diagonal distribution function could induce the <inline-formula><tex-math id="M3">\begin{document}$1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M3.png"/></alternatives></inline-formula>st-order diagonal Wigner function, and the <inline-formula><tex-math id="M4">\begin{document}$0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M4.png"/></alternatives></inline-formula>th-order diagonal distribution function could also induce the <inline-formula><tex-math id="M5">\begin{document}$1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M5.png"/></alternatives></inline-formula>st-order non-diagonal Wigner function. </sec>

Resummation and cancellation of the VIA source in electroweak baryogenesis

We re-derive the vev-insertion approximation (VIA) source in electroweak baryogenesis. In contrast to the original derivation, we rely solely on 1-particle-irreducible self-energy diagrams. We solve the Green’s function equations both perturbatively and resummed over all vev-insertions. The VIA source corresponds to the leading order contribution in the gradient expansion of the Kadanoff-Baym (KB) equations. We find that it vanishes both for bosons and fermions, both in the perturbative and in the resummed approach. The non-existence of the source is a result of a cancellation between different terms in the KB equations, and persists after resumming the masses.

An Effective Field Theory for large oscillons

We consider oscillons — localized, quasiperiodic, and extremely long-living classical solutions in models with real scalar fields. We develop their effective description in the limit of large size at finite field strength. Namely, we note that nonlinear long-range field configurations can be described by an effective complex field ψ(t,x) which is related to the original fields by a canonical transformation. The action for ψ has the form of a systematic gradient expansion. At every order of the expansion, such an effective theory has a global U(1) symmetry and hence a family of stationary nontopological solitons — oscillons. The decay of the latter objects is a nonperturbative process from the viewpoint of the effective theory. Our approach gives an intuitive understanding of oscillons in full nonlinearity and explains their longevity. Importantly, it also provides reliable selection criteria for models with long-lived oscillons. This technique is more precise in the nonrelativistic limit, in the notable cases of nonlinear, extremely long-lived, and large objects, and also in lower spatial dimensions. We test the effective theory by performing explicit numerical simulations of a (d + 1)-dimensional scalar field with a plateau potential.

SU(2) gauge theory of the pseudogap phase in the two-dimensional Hubbard model

We present a SU(2) gauge theory of fluctuating magnetic order in the two-dimensional Hubbard model. The theory is based on a fractionalization of electrons in fermionic chargons and bosonic spinons. The chargons undergo N\'eel or spiral magnetic order below a density-dependent transition temperature ${T}^{*}$. Fluctuations of the spin orientation are described by a nonlinear sigma model obtained from a gradient expansion of the spinon action. The spin stiffnesses are computed from a renormalization group improved random phase approximation. Our approximations are designed for moderate, not for strong, Hubbard interactions. The stiffnesses are strongly doping dependent with discontinuities at half-filling and a pronounced electron-hole asymmetry. The spinon fluctuations prevent magnetic long-range order of the electrons at any finite temperature. The phase with magnetic chargon order shares characteristic features with the pseudogap regime in high-${T}_{c}$ cuprates: a strong reduction of charge carrier density, a spin gap, and Fermi arcs. A substantial fraction of the pseudogap regime exhibits electronic nematicity.

Gradient corrections to the local-density approximation in the one-dimensional Bose gas

The local density approximation (LDA) is the central technical tool in the modeling of quantum gases in trapping potentials. It consists in treating the gas as an assembly of independent mesoscopic fluid cells at equilibrium with a local chemical potential, and it is justified when the correlation length is larger than the size of the cells. The LDA is often regarded as a crude approximation, particularly in the ground state of the one-dimensional (1D) Bose gas, { where the correlation length is "therefore said to be" infinite (in the sense that correlation functions decay as a power law).} Here we take another look at the LDA. The local density $\rho(x)$ is viewed as a functional of the trapping potential $V(x)$, to which one applies a gradient expansion. The zeroth order in that expansion is the LDA. The first-order correction in the gradient expansion vanishes due to reflection symmetry. At second order, there are two corrections proportional to $d^2V/dx^2$ and $(dV/dx)^2$, and we propose a method to determine the corresponding coefficients by a perturbative calculation in the Lieb-Liniger model. This leads to an expression for the coefficients in terms of matrix elements of the density operator, which can in principle be evaluated numerically for an arbitrary coupling constant; here we show how to efficiently evaluate the coefficient associated to the curvature of the potential $d^2V/dx^2$, which dominates the deviation to LDA near local minima or maxima of the trapping potential. Both coefficients are evaluated analytically in the limits of infinite repulsion (hard-core bosons) and small repulsion (quasi-condensate).} The corrected LDA density profiles are compared to DMRG calculations, with significant improvement compared to zeroth-order LDA.

Coarse graining in time with the functional renormalization group: Relaxation in Brownian motion.

We apply the functional renormalization group (fRG) to study relaxation in a stochastic process governed by an overdamped Langevin equationwith one degree of freedom, exploiting the connection with supersymmetric quantum mechanics in imaginary time. After reviewing the functional integral formulation of the system and its underlying symmetries, including the resulting Ward-Takahashi identities for arbitrary initial conditions, we compute the effective action Γ from the fRG, approximated in terms of the leading and subleading terms in the gradient expansion: the local potential approximation and wave-function renormalization, respectively. This is achieved by coarse graining the thermal fluctuations in time resulting in, e.g., an effective potential incorporating fluctuations at all timescales. We then use the resulting effective equationsof motion to describe the decay of the covariance and the relaxation of the average position and variance toward their equilibrium values at different temperatures. We use as examples a simple polynomial potential, an unequal Lennard-Jones type potential, and a more complex potential with multiple trapping wells and barriers. We find that these are all handled well, with the accuracy of the approximations improving as the relaxation's spectral representation shifts to lower eigenvalues, in line with expectations about the validity of the gradient expansion. The spectral representation's range also correlates with temperature, leading to the conclusion that the gradient expansion works better for higher temperatures than lower ones. This paper demonstrates the ability of the fRG to expedite the computation of statistical objects in otherwise long-timescale simulations, acting as a first step to more complicated systems.

Relativistic Hydrodynamics: A Singulant Perspective

There is growing evidence that the hydrodynamic gradient expansion is factorially divergent. We advocate for using Dingle's singulants as a way to gain analytic control over its large-order behaviour for nonlinear flows. Within our approach, singulants can be viewed as new emergent degrees of freedom which reorganise the large-order gradient expansion. We work out the physics of singulants for longitudinal flows, where they obey simple evolution equations which we compute in M\"uller-Israel-Stewart-like models, holography and kinetic theory. These equations determine the dynamics of the large-order behaviour of the hydrodynamic expansion, which we confirm with explicit numerical calculations. One of our key findings is a duality between singulant dynamics and a certain linear response theory problem. Finally, we discuss the role of singulants in optimal truncation of the hydrodynamic gradient expansion. A by-product of our analysis is a new M\"uller-Israel-Stewart-like model, where the qualitative behaviour of singulants shares more similarities with holography than models considered hitherto.

Novel treatments for the bioconvective radiative Ellis nanofluids wedge flow with viscous dissipation and an activation energy

This paper aims to examine time-dependent magnetohydrodynamic Ellis nanofluids flow over a wedge in the presence of the activation energy, thermal radiation and motile oxytactic microorganisms. New techniques are presented to explore this situation that is depending on expressing the stress tensor as an expansion in the velocity gradients for particularly values of the power-index α(α=0,0.5,1,2). Due the importance of the dynamic viscosity in Ellis's case, the viscous dissipation impacts are not neglected and a stagnation-point is taken place in the flow area. At the wedge surface, the Newtonian heating and passively controlling of the nanoparticles are considered. The transformed governing equations are solved using the Finite Difference Method (FDM) with Blottner algorithm and the obtained results are validated by applying Runge-Kutta method. The main outcomes disclosed that for all values of α, the growing in the material parameter A gives a great enhancement in values of RexCfx. Additionally, the increase in the unsteadiness parameter A1 enhances the forced convection process and hence (NnxRex1/2) , RexCfx and (NuxRex1/2) have higher values.

Comment on “Curved-space Dirac description of elastically deformed monolayer graphene is generally incorrect”

In a recent paper [M. M. Roberts and T. Wiseman, Phys. Rev. B 105, 195412 (2022)], based on methods typical of the condensed-matter literature (the real-space gradient expansion applied to the tight-binding model), conclude that the curved-space Dirac description of the low-energy conductivity electrons of monolayer graphene is incorrect when only strain is present (elastic monolayer graphene). In this Comment we point out that, in a much earlier paper [A. Iorio and P. Pais, Phys. Rev. D 92, 125005 (2015)], basing our analysis on methods typical of quantum field theory in curved space-time, we had concluded the same. Unfortunately, the authors of [M. M. Roberts and T. Wiseman, Phys. Rev. B 105, 195412 (2022)] have missed that result (even though it is very clearly stated in various places, starting from the Abstract), to the extent of citing our paper as one where strain is actually used for that wrong purpose. Since the paper in point is written with the declared intent of bringing clarity into the debate, we believe that this Comment of ours is very much due, precisely to give it another try to achieve the latter purpose.

Higher Sensory Sensitivity is Linked to Greater Expansion Amongst Functional Connectivity Gradients

Insofar as the autistic-like phenotype presents in the general population, it consists of partially dissociable traits, such as social and sensory issues. Here, we investigate individual differences in cortical organisation related to autistic-like traits. Connectome gradient decomposition based on resting state fMRI data reliably reveals a principal gradient spanning from unimodal to transmodal regions, reflecting the transition from perception to abstract cognition. In our non-clinical sample, this gradient's expansion, indicating less integration between visual and default mode networks, correlates with subjective sensory sensitivity (measured using the Glasgow Sensory Questionnaire, GSQ), but not other autistic-like traits (measured using the Autism Spectrum Quotient, AQ). This novel brain-based correlate of the GSQ demonstrates sensory issues can be disentangled from the wider autistic-like phenotype.