Quantum theory of pure liquid metals as two-component systems

With the ultimate aims of clarifying the interpretation and the utility of effective ion-ion interactions in liquid metals, and of understanding the unusual isotopic mass dependence of the shear viscosity of liquid metal Li, a fully quantum statistical mechanical theory is developed from the many-body Hamiltonian of the conduction electron-positive ion assembly.We have set up quantum equations of motion which are analogs of classical continuity and conservation equations by expanding the equation for the Wigner distribution function about its diagonal. The most important of these equations for our present purposes relates the time derivative of the current density j(r, t) to the flux of current and to density-density correlation functions for electrons, electron-ions, and ions.This theory is then applied to neutron scattering by liquid metals. While the theory is sufficiently general in principle to treat electron-ion interaction of arbitrary strength, it is shown that when the interacion is weak, the usual results are recovered, along with the effective ion-ion interaction. In this latter connection, it is also demonstrated how the effective Ornstein-Zernike function C(q) in a liquid metal is related to bare ion and bare electron direct correlation functions and to the bare electron partial structure factor. Combining C(q) with one of the classical equations of liquid structure such as Born-Green or Percus-Yevick then relates the effective ion-ion interaction to the partial correlation functions of the bare ions and electrons.It is further shown how gradient expansions of the correlation functions lead to equations of motion for the density, current, and energy density which are simply the hydrodynamic equations of the present quantum theory of two-component systems. It is pointed out that the analog of the Navier-Stokes equation for the two-component system may be used to identify the quantity $\text{4}\text{3}\text{η + ζ}$ for the liquid metal, η and ζ being respectively the shear and bulk viscosities. Finally, it is demonstrated that $\text{4}\text{3}\text{η + ζ}$ depends explicitly on functional derivatives of the nonequilibrium pair distribution functions of ion-ion, electron-ion, and electron-electron correlations.     

Kinetic theory of reacting systems

In this paper transport processes of reacting systems are investigated, based on the Boltzmann equations. The Boltzmann equations are solved by means of Grad's moment method to thirteen moments and some formal results are obtained for transport properties. It is shown that the rate coefficients are quadratic functions of hydrodynamic fluxes and are in the form

where

are the scalar moments associated with the reaction and q, J, Π are heat flux, material flux and traceless symmetric stress tensor. k⁽⁰⁾_i is the usual local equilibrium formula for reaction rate constant. Iterative solutions for the equations of change for

, q, J and Π are obtained from which transport coefficients are calculated for the reacting system. It is shown that the solutions, when specialized to nonreacting mixtures, lead to results for the transport coefficients which are exactly in agreement with the Chapman-Enskog theory results. The modifications of the transport coefficients due to reactions are obtained from the iterative solutions and the bracket integrals necessary for their calculations are explicitly given in an appendix.     

Longitudinal relaxation spectra of the transverse Ising model close to Tc

The correlative effects of the nature of the interaction and of the method of calculation on the shape of the longitudinal relaxation function (LRF) for the transverse Ising model are analysed. The LRF is calculated in two ways: (i) its continued fraction representation within the three pole approximation (TPA); and (ii) the resolution of kinetic equations derived for the correlation functions beyond the random phase approximation (RPA). The effects of the nature of the interaction on the LRF spectral characteristics are investigated using an interaction made of three variable contributions: uniaxial dipolar, isotropic infinite range and anisotropic nearest-neighbour interactions. Contrary to the TPA, the kinetic-equation-method (KEM) leads to LRF's exhibiting a three peak structure for every q-value except q = 0 (q = 0_⊥ if the interaction is of dipolar nature) whatever the interaction. The approximations underlying both methods are specified and discussed. Comments on recent neutron scattering experiments on Li Tb_pY_1-pF₄ by Youngblood et al. are made.     

A mode coupling theory of higher order time correlation functions

Kawasaki's mode coupling theory [Ann. Phys. 61 (1970) 1] is used to compute time correlation functions of the form 〈A_k0(t₀) … A_kn(t_n)〉, where A_k(t) represents some slowly varying quantity. The Gaussian and Bare Vertex approximations are made, thus yielding extremely simple expressions for these higher order correlation functions. These do not contain any bare transport coefficients and suggest relatively simple tests by which the theory could be checked. Examples relating to light scattering in nonequilibrium systems and the hydrodynamics of simple fluids are presented.     

Tsallis' quantum q-fields

We generalize several well known quantum equations to a Tsallis' q-scenario, and provide a quantum version of some classical fields associated with them in the recent literature. We refer to the q-Schro¨dinger, q-KleinGordon, q-Dirac, and q-Proca equations advanced in, respectively, Phys. Rev. Lett. 106, 140601(2011), EPL 118,61004(2017) and references therein. We also introduce here equations corresponding to q-Yang-Mills fields, both in the Abelian and non-Abelian instances. We show how to define the q-quantum field theories corresponding to the above equations, introduce the pertinent actions, and obtain equations of motion via the minimum action principle.These q-fields are meaningful at very high energies(Te V scale) for q = 1.15, high energies(Ge V scale) for q = 1.001,and low energies(Me V scale) for q =1.000001 [Nucl. Phys. A 955(2016) 16 and references therein].(See the ALICE experiment at the LHC). Surprisingly enough, these q-fields are simultaneously q-exponential functions of the usual linear fields' logarithms.     

Microscopic theory of hydrodynamic modes in a planar ferromagnet

The hydrodynamic equations for an easy-plane (planar) ferromagnet are derived by analogy with the hydrodynamics of liquid helium. The relaxation of the magnitude of an order parameter is taken into account.The derivation is based on microscopic dynamics and the nonequilibrium statistical operator method. The hydrodynamic equations are generally nonlocal in space and in time and hold for all T<T_c.Transport coefficients are expressed in terms of time correlation functions and their symmetry properties are investigated.     

Energy, momentum, and force in classical electrodynamics: Application to negative-index media

The classical theory of electromagnetism is based on Maxwell's macroscopic equations, an energy postulate, a momentum postulate, and a generalized form of the Lorentz law of force. These seven postulates constitute the foundation of a complete and consistent theory, thus eliminating the need for physical models of polarization P and magnetization M — these being the distinguishing features of Maxwell's macroscopic equations. In the proposed formulation, P(r, t) and M(r, t) are arbitrary functions of space and time, their physical properties being embedded in the seven postulates of the theory. The postulates are self-consistent, comply with special relativity, and satisfy the laws of conservation of energy, linear momentum, and angular momentum. The Abraham momentum density p_EM(r,t) = E(r,t) × H(r,t) / c² emerges as the universal electromagnetic momentum that does not depend on whether the field is propagating or evanescent, and whether or not the host media are homogeneous, transparent, isotropic, linear, dispersive, magnetic, hysteretic, negative-index, etc. Any variation with time of the total electromagnetic momentum of a closed system results in a force exerted on the material media within the system in accordance with the generalized Lorentz law.     

A classical model of EPR experiment with quantum mechanical correlations and bell inequalities

A simple model of a classical break-up process is given in which the correlation E(a,b) of the components A and B of the spins of the two subsystems along directions a and b gives precisely the quantum mechanical result $\text{?cos(}\text{a·b}\text{)}$. The model is “local”, but the normalization procedure of correlation functions in terms of “hidden variables” is different from that used in deriving Bell's inequalities. A discretization procedure of the classical spins is then given which reproduces fully the dichotomous quantum mechanical results both for probabilities and for correlation functions. This procedure illustrates particularly clearly the difference between quantum and classical spins and provides a possible intuitive picture for the notion of the “reduction of the wave function”.     

Toroidal hydromagnetics

The complete set of hydromagnetic equations is transformed into Poisson equations and equations of motion for flux densities and their associated variables. The toroidal components of the vector potential A and of the momentum density $\text{a}\text{≠}\text{πv}$ are represented by the po loidal flux densities Ψ and Ψ, respectively, for which the equations of motion are derived. The poloidal components A_⊥ and a_⊥ are represen ed by the potentials atΦ, U and φ, u, for which we obtain Poisson equations in the poloidal plane. Thus one has to solve two Dirichlet and two von Neumann problems at every time step. The source terms of the four Poisson equations define the remaining four variables, namely, $\text{Λ = ▽ ·}\text{A}$,$\text{Ω=(▽×}\text{A}\text{)ζ/R}$, $\text{λ=?·}\text{a}$, and $\text{ω=(?×}\text{a}\text{)ζ/R}$, for which equations of motion are also derived. In the limit of small toroidicity ? we look fo r a selfconsistent scaling of the equations with v_⊥～ε. But the curl of v_⊥×B in Faraday's law creates a toroidal plasma component of B which is one order of magnitude larger than in the case of a low β equilibrium; therefore, the motion becomes fully three-dimensional. Finally, an artificial pressure law is needed to balance the lowest order of the Lorentz force. The conclusion is then that the scaling laws previously used are not applicable for toroidal geometry, and that the effort to obtain numerical solutions is not dramatically higher than without using any scaling law.     

Nature of electric and magnetic dipoles gleaned from the Poynting theorem and the Lorentz force law of classical electrodynamics

Starting with the most general form of Maxwell's macroscopic equations in which the free charge and free current densities, ρ_free and J_free, as well as the densities of polarization and magnetization, P and M, are arbitrary functions of space and time, we compare and contrast two versions of the Poynting vector, namely, S = μ_o^− 1E × B and S = E × H. Here E is the electric field, H is the magnetic field, B is the magnetic induction, and μ_o is the permeability of free space. We argue that the identification of one or the other of these Poynting vectors with the rate of flow of electromagnetic energy is intimately tied to the nature of magnetic dipoles and the way in which these dipoles exchange energy with the electromagnetic field. In addition, the manifest nature of both electric and magnetic dipoles in their interactions with the electromagnetic field has consequences for the Lorentz law of force. If the conventional identification of magnetic dipoles with Amperian current loops is extended beyond Maxwell's macroscopic equations to the domain where energy, force, torque, momentum, and angular momentum are active participants, it will be shown that “hidden energy” and “hidden momentum” become inescapable consequences of such identification with Amperian current loops. Hidden energy and hidden momentum can be avoided, however, if we adopt S = E × H as the true Poynting vector, and also accept a generalized version of the Lorentz force law. We conclude that the identification of magnetic dipoles with Amperian current loops, while certainly acceptable within the confines of Maxwell's macroscopic equations, is inadequate and leads to complications when considering energy, force, torque, momentum, and angular momentum in electromagnetic systems that involve the interaction of fields and matter.     

Projection operators in classical kinetic theory

Zwanzig's projected kinetic equation is rederived by a perturbation method. A choice of projection is proposed which, in conjunction with appropriate initial-value conditions, yields kinetic equations for the two time distribution functions of phase subsets for a system in equilibrium. These equations are generalizations of the Fokker-Planck equations in which the dissipative terms are non-Markoffian.

It is shown that exact equations for the van Hove self and distinct correlation functions are particular cases of these equations.     

Dipolar interaction in the transverse Ising model of H-bonded ferroelectrics

Numerical solutions are presented for the set of approximated kinetic equations previously obtained to describe the dynamical behaviour of the pseudospin correlation functions of the transverse Ising model in H-bonded ferroelectrics. Stability of the soft mode and spectral characteristics of the correlation functions as T → ∞ are discussed in terms of an interaction made of both nearest-neighbour and dipole-dipole contributions. This infinite-temperature calculation shows how shift (towards zero frequency) and width of the soft mode vary with the interdipolar strength of the interaction. The asymmetry of the imaginary part of susceptibility _X″zz(ω) exhibited by KDP in its paraelectric phase is also shown to be due to the dipolar character of the interaction.     

Dynamics of Many-Body Correlation Green's Functions Ⅱ. Equal Time Limit

Under equal time limit,it is shown that the dynamic equations of the n-body correlation Green's functions G(n)c leads to the set of equations for time evolution of the n-body correlations c_n and the many-body correlation Green's functions and their equations of motion are independent of the time order.     

A statistical-mechanical theory of self-diffusion in glass-forming liquids

A statistical-mechanical theory of self-diffusion in glass-forming liquids is presented. A non-Markov linear Langevin equation is derived from a Newton equation by employing the Tokuyama-Mori projection operator method. The memory function is explicitly written in terms of the force-force correlation functions. The equations for the mean-square displacement, the mean-fourth displacement, and the non-Gaussian parameter are then formally derived. The present theory is applied to the glass transitions in the glass-forming liquids to discuss the crossover phenomena in the dynamics of a single particle from a short-time ballistic motion to a long-time self-diffusion process via a β (caging) stage. The effects of the renormalized friction coefficient on self-diffusion are thus explored with the aid of analyses of the simulation results by the mean-field theory proposed recently by the present author. It is thus shown that the relaxation time of the renormalized memory function is given by the β-relaxation time. It is also shown that for times longer than the β-relaxation time the dynamics of a single particle is identical to that discussed in the suspensions.     

Critical fluctuations in continuous non-equilibrium systems

We treat a continuously extended non-equilibrium system described by a set of space and time dependent variablesq. Theq's are assumed to obey linearized equations of motion with fluctuating forces which areδ-correlated in space and time. We calculate the two-time and two-space-point correlation functions.     

Stochastic equations arising from test particle problems: II. Application to the harmonic lattice and the electron plasma

To compare the different kinetic equations derived in a previous paper for weakly coupled systems, the results are applied to coupled harmonic oscillators and a one-component plasma in a magnetic field. Using the harmonic interaction as an example, it is demonstrated that reasonable results can be inferred only from the kinetic equation, which is characterized by an additional time average. Applying this equation to an electron plasma in a magnetic field B, the Balescu-Lenard equation is recovered for B = 0, but a modification is obtained for B ≠. For strong fields the diffusion coefficient is discussed.     

Electron Green's function in the planar t − J model

The electron Green's functions G(k, ω) within the t ? J model and in the regime of intermediate doping is studied analytically using equations of motion for projected fermionic operators and the decoupling of the self energy into the single-particle and spin fluctuations. It is shown that the assumption of marginal spin dynamics at T = 0 leads to an anomalous quasiparticle damping. Numerical results show also a pronounced asymmetry between the hole (ω < 0) and the electron (ω > 0) part of the spectral function, whereby hole-like quasiparticles are generally overdamped.     

Long-wavelength excitations in a Bose gas at zero temperature

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.