Quantum Hydrodynamics Approach to The Research of Quantum Effects and Vorticity Evolution in Spin Quantum Plasmas

In this paper we explicate a method of magneto quantum hydrodynamics (MQHD) for the study of the quantum evolution of a system of spinning fermions in an external electromagnetic field. The fundamental equations of microscopic quantum hydrodynamics (the momentum balance equation and the magnetic moment density equation) were derived from the many-particle microscopic Schroodinger equation with $Spin-Spin and Coulomb modified Hamiltonian. Using the developed approach the extended vorticity evolution equation for the quantum spinning plasma were derived. The effects of new spin forces and spin-spin interaction contributions on the motion of fermions, evolution of the magnetic moment density and vorticity generation were predicted. The influence of the intrinsic spin of electrons in the nonlinear whistler mode turbulence was investigated. This results can be used for the theoretical studies of spinning many-particle systems, especially dense quantum plasmas in compact astrophysical objects, plasmas in semiconductors and micro-mechanical systems, in quantum x-ray free-electron lasers.


I. INTRODUCTION
The spinning quantum fluid plasma is becoming of increasing current interest 1 -7 . Hydrodynamics equations of a spinning fluid for the Pauli equation with the quantum particle angular momentum spin was presented since the pioneering works by Takabayashi and Vigier 8 -11 . The vector representation of non-relativistic spinning particle leads to appearance of new quantum effects had been separated as non-linear terms which arises from the inhomogeneity of spin distribution. The extension of the interpretation to developed approach had been carried in the 12 - 16 .
The quantum effects in plasma can be represented by three main quantum corrections. The first is a quantum force, the multiparticle quantum Bohm or Madelung potential, proportional to powers of and produced by density fluctuations 17 , 18 . The second is associated with the quantum particle angular momentum spin by the possible inhomogeneity of the external and spin magnetic fields. In the momentum balance equation this force appears through the magnetization energy 1 . And the latter force associated only with the spin magnetic moment of the particle 8 .
The most interesting and defining features of a quantum spinning plasmas can be derived from the vorticity equation. It had been derived by 19 that the vorticity, constructed from spin field of a quantum spinning plasma, combines with the classical generalized vorticity to yield a new grand generalized vorticity that obeys the standard vortex dynamics. Astrophysics is also a rapidly growing field of research. It is important that the consequences of turbulent plasma movement in the solar photosphere lead to the generation of vorticity, while magnetic vortices are produced by magnetic tension. For example, magnetohydrodynamics (MHD) simulations of magnetoconvection have been used to analyze the generation of small-scale vortex motions in the solar photosphere. Using the vorticity equation, combined with G-band radiative diagnostics, it has been shown that two different types of photospheric vorticity, magnetic and non-magnetic, are generated in the domain 20 . The presence of vortex motions for the astrophysics had been developed in 21 -22 .
The extracting coherent vortices out of turbulent flows had been applied to simulations of resistive driftwave turbulence in magnetized plasma 23 . The quasihydrodynamic and quasi-adiabatic regimes had been investigated.
The formation and dynamics of dark solitons and vortices in quantum electron plasmas had been studied in 24 . A pair of equations comprising the nonlinear Schrdinger and Poisson system of equations, which conserves the number of electrons as well as their momentum and energy had been used. It had been shown that the gradient free-energy contained in equilibrium spin vorticity can cause electromagnetic modes, in particular the light wave 25 .
The collective electron angular momentum spin effects in spinning quantum plasmas can be investigated using insights from quantum kinetic theory or some effective theory. We propose a method of quantum hydrodynamics that allows one to obtain a description of the collective effects in magnetized quantum plasmas in terms of functions in physical space. The fermion model was developed in Refs. 1 , 2 , 26 and 27 . The waves in the magnetized plasma with the spin had been studied in 26 exploring of new quantum hydrodynamics method of the generation wave in the plasma. The new formalism given in this references had been used in this article for studying of vorticity evolution in the magnetized plasma with the spin. A quantum mechanics description for systems of N interacting spinning particles is based upon the many-particle Schrdinger equation (MPSE) that specifies a wave function in a 3N-dimensional configuration space. As wave processes, processes of information transfer, and other spin transport processes occur in 3D physical space, it becomes necessary to turn to a mathematical method of physically observable values that are determined in a 3D physical space. To do this, we should derive the fundamental equations that determine the dynamics of functions of three variables, starting from MPSE. This problem has been solved with the creation of a manyparticle quantum hydrodynamics (MPQHD) method.
In this article for studying of vorticity and spin vortex effects we generalize and use the method of the manyparticle quantum hydrodynamics MQHD approach. We derive the fundamental balance equation, the magnetic moment evolution equation and new vorticity dynamics equation and the magnetic vortex evolution equation for the magnetized quantum plasmas.

II. FUNDAMENTAL EQUATIONS OF THE FERMION QUANTUM HYDRODYNAMICS
In this section we derive the system of magneto quantum hydrodynamics (MQHD) equations for charged and neutral particles from the many-particle microscopic Schrodinger equation where R = ( r 1 , ..., r N ). We consider a system of N interacting fermions with equal masses m j , charged and proper magnetic moments in an external electromagnetic field. A state of the system of N fermions is determined by a wave function in the 3N-dimensional configuration space, which is a rank − N spinor The Hamiltonian has the form where µ j = gµ B /2, µ jB -is the electron or positron magnetic moment and µ jB = q j /2m j c -is the Bohr magneton, q j stands for the charge of electrons q e = −e or for the charge of positrons q p = e, and -is the Planck constant, g ≃ 2.0023193. The covariant derivative operator isD where A ext , ϕ j,ext -are the vector and scalar potentials of external electromagnetic field. Green's functions of the Coulomb and Spin − Spin interaction are The first step in the construction of MQHD apparatus is to determine the concentration of particles in the neighborhood of r in a physical space. If we define the concentration of particles as quantum average of the concentration operator in the coordinate representation Differentiation of ρ( r, t) with respect to time and applying of the Schrodinger equation with Hamiltonian 3 leads to continuity equation where the current density takes a form of +ψ + s (R, t)D α j ψ s (R, t)), A momentum balance equation can be derived by differentiating current density 8 with respect to time represents the momentum current density tensor. Momentum balance equation 9 contains the particle magnetic moment density 1 The Coulomb and Spin − Spin interactions between the particles are represented in Eq. 9 by the terms where is the two-particle probability density for the occurrence of two particle in the neighborhoods of the points r and r ′ normalized by N (N − 1), and two-particle tensor of the magnetic moment density Differentiation of M α with respect to time and applying of the Schrodinger equation with Hamiltonian 3 leads to magnetization equation. The equation representing the non-relativistic evolution of spin − 1/2 motion takes a form of where the tensor of the magnetic moment flux density is The spinning quantum magnetohydrodynamics should explain the vorticity evolution. The main idea of this paper was to create a hydrodynamics foundation for the vortex dynamic in the context of spinning quantum plasma. We use the MPQHD approach to receive the equations for the particle vorticity density, obeying the standard vortex dynamics. We determine the vorticity density vector of particles in the neighborhood of r in a physical space as where we construct the vorticity density in term of the wave function, we denote the macroscopic vorticity density as Ω = ∇ × j, as will be shown below. The classical generalized vorticity density Ω can be defined as the curl of the current density. But in article 19 the ordinary vorticity of the plasma is proportional to the curl of the flow velocity of the fermions (vorticity have the dimensions of the magnetic field).On the other hand vorticity can be defined as the curl of velocity ω = ∇ × υ 20 . Our idea is that we use the definition 16 and MQHD method based upon the many-particle Schrodinger equation to derive the generalized dynamical equation for classical vorticity ω = ∇ × υ similar to 20 and 21 (see below), but contained the information about interactions inside the fluid.

A. Velocity field
The velocity of j − th particle υ j is determined by equation The quantity υ j (R, t) describe the current of probability connected with the motion of j−th particle, in general case υ j (R, t) depend on coordinate of all particles of the system R, where R is the totality of 3N coordinate of N particles of the system R = ( r 1 , ..., r N ).
The S(R, t) value in the formula 17 represents the phase of the wave function and as the electron has spin, the wave function is now be expressed in the form where ϕ, normalized such that ϕ + ϕ = 1, is the new spinor, defined in the local frame of reference with the origin at the point r. The spinor gives the spin part of the wave function.
We substituted the wave function in the definition of the basic hydrodynamical quantities. Using that the velocity field υ is the velocity of the local center of mass and determined by equation the vorticity density field 16 and the momentum current density tensor 10 have the new form of is the well known kinetic pressure tensor. Value u α j ( r, R, t) is a quantum equivalent of the thermal speed and u α j ( r, R, t) = υ α j (R, t) − υ( r, t). The tensor Λ αβ is proportional to 2 , has a purely quantum origin and can therefore be interpreted as an additional quantum pressure The quantum tensor 25 is the quantity which can be rewritten in terms of concentration ρ in the approximation of noninteracting particles, using the definition using simple manipulation with the expression 24 we may replace for the large system of noninteracting particles, this tensor is It should be explained that the tensor 25 arises as a consequence of the quantum Madelung potential and can be interpreted as an additional quantum pressure.
The tensor Υ αβ s appears in the theory as a result of representations rotating electrons as an assembly of bodies continuously distributed in space. In the context of quantum hydrodynamics the force due to a new spin stress inside the fluid takes the form of This new force emerges from the inhomogeneity of spin distribution and must be considered in the equation of motion, being the order of 2 .
On the other hand, after the presentation of the wave function in the exponential form the tensor of the magnetic moment flux density takes the form of where In the context of quantum hydrodynamics we have the additional spin torque ). (29) B. Energy evolution equation The energy density taking into account the Coulomb and Spin-Spin interactions is given by 1 Differentiation of 30 with respect to time and application of the Schrodinger equation with Hamiltonian 3 leads to the energy balance equation where A(r, t) -is the density of internal force and Q(r, t) -is the internal energy flux density. The internal energy flux density is given by The force density of internal forces in (31) consists of Coulomb force density A cl (r, t) and Spin-Spin force density A s−s (r, t) Using the fact that the velocity field is the velocity of the local center of mass and is determined by 19 and using the definitions 26 and 29 the energy evolution equation reads Let us discuss the physical significance of the terms on the right-hand side of the system of MQHD equation 34. The third and fifth terms on the right-hand side in Eq. 34 describe a quantum force produced by density fluctuations, which has its origin in the so-called Madelung potential. The forth term represents the well known pressure tensor influence. The sixth term characterizes the energy density generation by the spin stress and the seventh term on the left-hand side of 34 describes the magnetic moment density torque influence.
The relative energy density the of spinning fermions in an external electromagnetic field takes the form The first term on the right-hand side of the expression 35 describes the quantum equivalent of the ther-mal speed contribution, the second term characterizes the quantum Madelung potential contribution and the third term presents the internal spin potential influence. The forth and fifth terms describe a force field that represents interactions between particles, namely the Coulomb interaction of charges and spinspin interactions.
Note that to simplify the problem we consider that the thermal spin-interactions are neglected and microscopic spin s α j = s α is equal to macroscopic average s α . Taken in the approximation of self-consistent field, from 9, 14 we have the set of MQHD equation for the electrons and positrons (p=e, i): continuity equation, momentum balance equation, magnetic moment density equation take the form Lets discuss the physical significance of terms on the right side of the system of MQHD equations obtained above 36 -38. The first and second terms in Eq. 37describe the well known interaction with the external electromagnetic field, where the first term represents the effect of the external electric field on the charge density and the second term is the Lorentz force field. The fourth term is a quantum force produced by density fluctuations, which has its origin in the so-called Madelung potential. The fifth term appears in the equation of motion 37 through the magnetization energy and depends on the spin or magnetic moment density of particles. The sixth term represents the self-force or magnetic moment density stress inside the electron or positron fluid. This spin self-force appears even in the absence of the electromagnetic fields and arises from the inhomogeneity of the magnetic moment density distribution. Other terms in 37 describe a force field that represents interactions between particles, namely the Coulomb interaction of charges and Spin − Spin interactions.
The second term in the equation of magnetic moment density motion 38 represents the effect additional magnetic moment density torque on the magnetic moment density evolution and tends to align spins parallel. It's important that the second term has a similar form respectively to the the contribution of exchange interaction in ferromagnetic media for isotropic cubic ferromagnetic.
Using the definition 16 and the Madelung decomposition 17 with the momentum balance dynamical equation 37 the hydrodynamics classical vorticity dynamical equa- The vorticity evolution equation 39 shows the different physical factors associated with the generation of vorticity. The second term on the right side of 39 is proportional to the gas pressure and is responsible for the hydrodynamic baroclinic vorticity generation of the classical vortex field. The third term represents the magnetic baroclinic vorticity and is associated with the anisotropic magnetic pressure effect. The fourth term contains information about the vorticity generated by the magnetic tension. The sixth term is associated with the magnetic vorticity generation, even in the absence of the magnetic field. The sixth and seventh terms characterize the effect of Coulumb and Spin − Spin interactions in the vorticity evolution. Equation 39 contains the normal electron or positron current density j ep = q p ρ p υ p , and the magnetic moment density M p = ρ p µ p . The vorticity evolution equation 39 is a generalization of classical vorticity equation which had been presented in works 19 , 20 , 28 and 29 . At first, Eq. 39 combines the erstwhile generalized classical vorticity, but in contrast to 20 and 29 contains the information about interactions inside the quantum vortical fluid and have been derived using the MQHD method.
Note, that for a 3D system of particles the momentum balance equation 37, the magnetic density equation 38 and the vorticity evolution equation 39 may be written down in terms of magnetic intensity of the field that is created by charges q p and spins s p of the particle system and (42) where ω p = ω p + qp mpc B -is the generalized vorticity and the effective magnetic field B ef f = B + B in includes the total magnetic field and internal magnetic field B in The total magnetic field B consists of the field generated by the charge and the field generated by the spins. Amp'eres law including the magnetization spin current j m = 2µ/ ∇ × (ρ s) takes the form of We must note that the spin stress term have notably interesting nature, exist even in absence of magnetic field, have only quantum foundation and arising out from the spin part of the wave function. Equation 42 was rewritten by separating the magnetic and non-magnetic terms 8 . effective magnetic field and the last three terms produced by the magnetization vortex generation. Lets to rewrite the spin-vortex evolution equation from ??, using the vector Ξ = ∇ × s This is new equation was obtained with the method of magneto quantum hydrodynamics (MQHD) for the study of the quantum evolution of a system of spinning fermions.

C. The Whistler Mode Turbulence in Magnetized Plasmas
The nonlinear turbulent processes associated with electromagnetic waves in spinning plasmas have attracted interest. Nonlinear whistler mode turbulence has been studied in a magnetized plasma 30 -33 . The authors had focused on low-frequency (in comparison with the electron gyro-frequency) nonlinearly interacting electron whistlers and nonlinearly interacting Hallmagnetohydrodynamic (H-MHD) fluctuations in 30 . In this section we investigate the electron whistler wave properties based on extended 2D magnetohydrodynamic equations. However, we understand that the electron spin effect on the whistler wave dispersion typically requires a strong external magnetic field.
Two-dimensional turbulence has been studied in a magnetized plasma involving incompressible electrons and immobile ions. We consider that the electrons carry currents, while the immobile ions provide a neutralizing background to a quasi-neutral spinning plasma. Using the fact that the electron fluid velocity is associated not only with the rotational magnetic field but also with the magnetization spin current j M = 2ρ 0 µ e ∇ × s/ which is determined by the spin vector s, we have from the Amp'eres law where µ e = −ge /4m e c, m e -is the electron mass, ρ 0 -is the electron density. We take into account that the electron density is constant and the electron continuity equation 36 shows a divergence-less electron fluid velocity ∇ υ e = 0. All physical quantity is presented in the form of sum of equilibrium part and small perturbations f ρ e ( r, t) = ρ 0 .., υ e ( r, t) = υ 1 ( r, t) + .., ω e ( r, t) = ω 1 ( r, t) + .., where B 0 -is the external uniform magnetic field directed along the axis y, s 0 -is the unperturbed spin vector.
In this case if we assume that linear excitations f 1 are proportional to exp(−iωt + i k x), where ω -is the wave frequency and k 2 = k 2 x +k 2 y . The three-dimensional equation 42 closed by 41 transformed into two dimensional by the regarding variation in the z-direction as ignorable or ∂/∂z = 0 and used the separation for the total magnetic field into two scalar variables B 1 = z × ∇ψ + b z 30 .
We will assume propagation of the waves along an external magnetic field B 0 or k = k y . A linearized set of equations 41 and 42 in this case gives us the dispersion equation where the length and time scales are normalized respectively d e = c/ω pe and ω c = eB 0 /m e c, d e -is the electron skin depth or inertial length scale, ω 2 pe = 4πe 2 ρ 0 /m eis the electron plasma frequency ω c -is the electron cyclotron frequency and c -is the speed of light. The other physical quantities are normalized as -is the spin-precession frequency which includes the internal magnetic field influence and ω µ = g 2 |s 0 |/4m e d 2 eis a frequency that involves a spin correction due to the plasma magnetization current and appears even in the absence of the external magnetic field B 0 , -is the reduced Planck constant. We use that an unperturbed spin state s 0 = − /2 tanh(µ B B 0 /k B T e ) antiparallel to the background magnetic field. This function appears as the solution of the spin evolution equation for spin quantum plasmas where the spin inertia and the spin thermal coupling terms are neglected 4 , 34 . The temperature T e is the Fermi electron temperature T F = 2 (3π 2 ρ 0 ) 2/3 /2m e k B , where k B is the Boltzmann constant. A situation magnetization effects might be important in a regime of very strong magnetic field in which the external field strength approaches or exceeds the quantum critical magnetic field B 0 ∼ 4.4138 × 10 13 G and highly dense plasmas ρ 0 ∼ 10 30 1/sm 3 . But it can be assumed that the internal magnetic field inside the fluid which is dependent on the gradient of the spin distribution 43, can tend to align neighboring spins parallel.
The effect of the frequency that involves the spin correction due to the plasma magnetization current is small ω µ < ω c the cubic expression 48 may be expanded to yield formulae ω k in the following form and The relation 49 expresses the dispersion of lowfrequency whistler waves in the spinning quantum plasma using the model based on the 2D electromagnetic turbulence equation 42. The solution 50 expresses the dispersion of waves that emerge as a result of spin dynamics. The spectrum is divided by the electron inertial skin depth into two regions, short scale kd e > 1, ω k ∼ 1 and long scale region kd e < 1, ω k ∼ k 2 .

III. CONCLUSIONS
In this paper we analyzed vorticity excitations caused by the magnetic moment density dynamics in systems of charged 1/2 − spin particles. MQHD equations are a consequence of MPSE in which particles interaction is directly taken into account. In our work we consider the Coulomb and Spin − Spin interactions. The system of MQHD equations we have constructed comprises equations of continuity, of the momentum balance, of the energy evolution equation, of the magnetic moment density evolution, and of the vorticity density dynamics. In our studies of wave processes we have used a self-consistent field approximation of the MQHD equations.
The equations we are interested in, determining the system dynamics, are the hydrodynamic equations for the spinning plasma. This equations (37 and 38) have an additional quantum contribution proportional to 2 and spin corrections, additional M agnetic M oment Stress and M agnetic M oment T orque which have been derived in the absence of (thermal) fluctuation of the spin about the macroscopic average. But in such a situation (thermal) effects on the spin might be important. The main objective of this paper was to construct an appropriate a new generalized vorticity equation 42 for spin quantum plasmas that contains the magnetic, non-magnetic terms and the spin dependent forces being non potential. The turbulent processes in plasmas had been investigated using the vorticity equation 19 , 20 , 21 . We had derived the vortex dynamic formulation of spinning non -relativistic quantum plasma, using the method of magneto quantum hydrodynamics (MQHD). We had generalized the classical vorticity equation for a spinning quantum fluid plasma and derived the vorticity equation 39 in which particles interactions ( Coulomb and Spin − Spin) is directly taken into account. Important that the quantum Madelung potential do not contribute to the vorticity evolution.
Using MQHD equations we analyzed elementary excitations in various physical systems in a linear approximation. We had studied the influence of the intrinsic spin of electrons in the nonlinear whistler mode turbulence. Dispersion branches characterize a new waves, one of which propagates below the electron cyclotron frequency 49, one above the spin-precession frequency due to the spin perturbations 50. This result had been derived for the incompressible electrons in the model based on the two-dimensional vorticity equation. The spin effects are seen to be substantial in the very strong magnetic field, dense plasmas and the graphical representation of the waves is similar to that found in The investigation of this approach leads to interesting spin effects dense quantum plasmas in compact astrophysical objects, plasmas in semiconductors and micro-mechanical systems, in quantum x-ray freeelectron lasers.