Physica A

Hydrodynamic Excitation Spectrum and Time Correlation Functions for the Multicomponent Mixtures of Magnetic and Nonmagnetic Particles

O.F.Batsevych^b, I.M.Mryglod^a, Yu.K.Rudavskii^b, M.V.Tokarchuk^a

Introduction
Equations of Molecular Hydrodynamics
Hydrodynamic Excitation Spectrum
Time-Dependent Correlation Functions
Simple fluid
Mixture of magnetic and nonmagnetic particles in paramagnetic state
The dynamical structure factors for three-component mixture of nonmagnetic particles
Bibliography

Introduction

The study of the dynamics of magnetic liquids and their mixtures is of great importance due to the diversity of physical features of such systems and big amount of problems that should be solved. Beside this, the magnetic liquids, mixtures of magnetic and nonmagnetic atoms in the external fields of mechanical or electromagnetic origin, have already taken their significant place in chemical, electronic and other modern technologies. The recent experimental investigations [1] have already proved the existence of feromagnetic phase in overcooled (?supercooled) liquid alloy Co₈₀-Pd₂₀ similar to that of alloy Au-Co [2,3,4] with Heizenberg exchange interaction, studied previously. This revives investigations in the area of physics of magnetic liquids. We must note that this subject is actual despite of whether exist there feromagnetic state in magnetic liquids or not, because the investigation of dynamics of systems with translational and spin degrees of freedom in external magnetic fields is of significant importance. One face with problems mentioned above while describing spin relaxation in liquids [5,6,7], investigating the dynamics of ferocolloid systems [8,9,10,11,12], dynamical properties of polar liquids [13,14,15,16,17,18], surface absorption dynamics [18,19], etc. The existence of two interacting subsystems in magnetic liquid give rize to different new phenomena which are absent in simple liquids as well as in solid magnetics. Certainly, their nature depends on parameters of interaction betwen particles for liquid and magnetic subsystems separately and their influence on each other. The statistical hydrodynamics of Heizenberg magnetic liquid was presented in [20,21] with the help of Zubarev's method of nonequilibrium statistical operator [22,23]. The investigations of time-dependent correlation functions and collective modes spectrum were performed in [24,25], where the analytical expressions of dynamical structure factors ``density-density'' $S(k, \omega)$ and ``spin density-spin density'' $S_m(k,\omega)$ were given, which is important for explaining of neutron scattering experiments in presence of external magnetic field. The expression for the Landau-Placheck coefficients, characterizing corresponding integral intensities of central and lateral peaks in functions $S(k, \omega)$ and $S_m(k,\omega)$ were found. The forecasting of qualitative changes in magnetic structure factor $S_m(k,\omega)$ in presence of magnetic field is one of sequences which should be tested experimentally. In this case the nonzero contribution from the sound mode appears, which can be observed as lateral Brillouen peak in $S_m(k,\omega)$ . The investigation of transport coefficients (viscosity, heatconductivity, magnetic diffusion, etc.), and time correlation functions describing the fluctuations of mass, momentum, energy for the liquid mixture of magnetic and nonmagnetic particles is of exceptional interest and value. It gives us possibility of deep insight into the systems with coupled classical and quantum subsystems. Studying the collective modes spectrum, partial magnetic and dynamical structure factors reveals the peculiarities of propagation of sound, head, interdiffusion, spin waves. Performing these investigations withing the frame of rigorous statistical approach allows to avoid groundless phenomenological assumptions. The statistical hydrodynamics for a mixture of magnetic and nonmagnetic particles in nonhomogeneous external magnetic field have been presented in [20,21]. With the help of the method of nonequilibrium statistical operator [22,23], the generalized hydrodynamics equations valid for describing both strong and weak nonequilibrium states were derived. To describe the weak nonequilibrium processes the linearized equations of molecular hydrodynamics and equations for time correlations functions were found, and corresponding memory functions, connected with generalized viscosity, thermoconductivity, cross-spin diffusion etc. transport coefficients analyzed. The goal of this paper is to study the hydrodynamic region of wave number k and frequency $\omega$ , using the results obtained in [20,21]. The peculiarity of approach proposed is that generalizing the case of binary magnetic mixture (chapter 2), the consideration and results are presented for more general case of multicomponent mixtures which include arbitrary number of magnetic components. In chapter 3 the hydrodynamic modes spectrum is found. All time dependent correlation functions are analyzed in chapter 4, the analytical expressions for weight coefficients describing contributions of each mode to time correlation functions are given. Next chapters are devoted to analysis of particular cases of mixtures. In 6 and 7 there are analyzed binary magnetic mixture and three-component nonmagnetic mixture, respectively. The hydrodynamic modes spectrum, expressions for dynamical structure factors and magnetic structure factor (for binary magnetic mixture) are written via the thermodynamical parameters and transport coefficients.

Equations of Molecular Hydrodynamics

Following papers [28,26,27] where hydrodynamics processes of magnetic fluids and mixtures of magnetic and nonmagnetic particles in weak nonequilibrium state were investigated, let us begin with consideration of binary liquid mixture of magnetic and nonmagnetic particles in hydrodynamic limit. The system is described by Hamiltonian [28] with Heisenberg-like interaction between spins. Using Zubarev'z method of nonequilibrium statistical operator [22,23], the equations of molecular hydrodynamics and equations for time correlation function can be derived for such a system [26,27]. For the small deviations of the system from the equilibrium state we can write Laplace-transformed equations of generalized hydrodynamics in the matrix form [26] as follows:

$\begin{displaymath} \left\{\mbox{\rm i}\omega{\cdot}\tilde 1-\mbox{\rm i}\tilde... ...ega)\right\}\langle \Delta \hat Y_i(k)\rangle ^\omega=0. \end{displaymath}$

(2.1)

Laplace transforms of time correlation functions $\hat F(k,z)$ satisfy the equation

$\begin{displaymath} \left\{z{\cdot}\tilde1-\mbox{\rm i}\tilde\Omega(k)+\tilde\Phi(k,z)\right\}\,\tilde F(k,z)=\tilde F_0(k), \end{displaymath}$

(2.2)

where $\mbox{\rm i}\tilde\Omega(k)$ and $\tilde\Phi(k,z)$ are the frequency matrix and the matrix of memory function defined as follows:

$\displaystyle \mbox{\rm i}\tilde\Omega(k)$

=

$\displaystyle (\mbox{\rm i}\hat L{\cdot}{\hat Y}(k),\hat Y(-k)) \ (\hat Y(k),\hat Y^+(-k))^{-1},$

(2.3)

$\displaystyle \tilde\Phi(k,z)$

=

$\displaystyle \left((1-{\cal P})\,\mbox{\rm i}\hat L{\cdot}\hat Y, \frac{1}{z+(... ...ox{\rm i}\hat L{\cdot}\hat Y^+\right) \left(\hat Y(k),\hat Y^+(-k)\right)^{-1},$

(2.4)

where $\mbox{\rm i}\hat L$ is a Liouville operator of the system and ${\cal P}$ is a projection Mori operator which obeys:

$\begin{displaymath} {\cal P}\cdot\tilde A=\left(\tilde A,\hat Y(-k)\right)\left(\hat Y(k),\hat Y^+(-k)\right)^{-1}\hat Y(k), \end{displaymath}$

(2.5)

$(\hat A,\hat B)$ means correlation function:

$\begin{displaymath} (\hat A,\hat B)=\int_0^1\langle \hat A\rho_0^\tau\hat B\rho_0^{-\tau}\rangle \,d\tau , \end{displaymath}$

(2.6)

where $\langle \dots\rangle$ is an average with respect to equilibrium Gibs distribution $\rho_0$ and $\hat Y(k)$ is the vector-column of parameters of abbreviated description of the system. As the parameters of abbreviated description the Fourier components of conservative quantities should be chosen. After performing the ortogonalization procedure for the binary mixture of magnetic and nonmagnetic particles [28] they can be written as:

$\begin{displaymath} \hat Y(k) = \{ \hat n_1( k), {\hat n_2}(k), \hat p(k), \hat s(k), \hat h(k) \} , \end{displaymath}$

(2.7)

and are the Fourier components of the partial density of particle number of nonmagnetic (species 1) and magnetic (species 2) particles, total momentum, projected magnetization and enthalpy, respectively which depend on the wave number k. Matrix of static correlation functions $\tilde F_0(k) = \left(\hat Y_i(k),\hat Y^+_i(k)\right)$ in the right-hand side of (2.2) has block-diagonal structure [28] for the case of orthogonalized parameters of abbreviated description (2.7). Using the symmetry properties of the correlation functions with respect to time inversion and space symmetry operations, it can be shown [27] that in the system posed in external magnetic field and describing by five parameters of abbreviated description (2.7) matrix $\mbox{\rm i}\tilde\Omega$ will have crosslike structure: only those elements which have index corresponding to momentum, will be nonzero. It can be easily shown that the same structure of $\mbox{\rm i}\tilde\Omega$ remains for more general case of multicomponent system. That is why we will consider more general case of m+2 parameters of abbreviated description, where m=m₁+m₂. Here m₁ is the number of components of fluid subsystem (number of species), which consists of parameters $\{\hat n_1(k),\dots,\hat n_{m_1}(k)\}$ ; m₂ is the number of conserved variables, which correspond to spin subsystem (these could be the partial magnetic momentums of species of magnetic particles $\{\hat s_1(k),\dots,\hat s_{m_2}(k)\}$ , in case they commutate with Hamiltonian, or other conserved quantities constructed on the basis of spin variables); the rest two parameters are the density of total momentum $\hat p(k)$ and the enthalpy $\hat h(k)$ . It is easy to verify, that due to the conservation of parameters of abbreviated description the frequency matrix (2.3) in hydrodynamic approximation is linear with respect to k:

$\begin{displaymath} \mbox{\rm i}\tilde\Omega=\mbox{\rm i}k{\cdot}v_s{\cdot}\tilde\nu =\delta{\cdot}v_s{\cdot}\tilde\nu, \end{displaymath}$

(2.8)

the matrix of memory functions is quadratic, respectively:

$\begin{displaymath} \tilde\Phi=-k^2{\cdot}v_s{\cdot}\tilde\varphi =\delta^2{\cdot}v_s{\cdot}\tilde\varphi , \end{displaymath}$

(2.9)

Here and further we will use notification $\delta\equiv \mbox{\rm i}k$ and suppose $\delta$ to be a small parameter. The contribution to the frequency matrix, proportional to $\delta^2$ is equal to zero. The coefficient v_s in (2.8), (2.9) is chosen as follows:

$\begin{displaymath} v_s^2= {1\over2} \mathop{\mbox{\rm Sp}}\nolimits\,\left[\left({\mbox{\rm i}\tilde\Omega\over\delta}\right)^2\right], \end{displaymath}$

(2.10)

from which follows that the trace of square of matrix $\tilde\nu$ is equal to 2:

$\begin{displaymath} \mathop{\mbox{\rm Sp}}\nolimits\,\tilde\nu^2=2. \end{displaymath}$

(2.11)

As it will be shown further, coefficient v_s gives us the propagation velocity of sound waves, and elements of matrices $\tilde\nu$ and $\tilde\varphi$ introduced in (2.8), (2.9) with its mediation are dimensionless, that allows to simplify calculations significantly. Using the Markovian approximation, which is asymptotically exact in hydrodynamic limit, we conclude, that full hydrodynamic matrix $\tilde T_H\equiv \mbox{\rm i}\tilde\Omega-\tilde\Phi$ does not depend on z. That is why analyzing equations (2.1), (2.2) in the hydrodynamic region for the small k and z, we can reduce our consideration to the hydrodynamic matrix $\tilde T_\delta$ :

$\begin{displaymath} \tilde T_\delta(z)=\tilde\nu+\delta{\cdot}\tilde\varphi , \end{displaymath}$

(2.12)

remembering that:

$\begin{displaymath} \tilde T_H= \mbox{\rm i}\tilde\Omega - \tilde\Phi= \delta{\cdot}v_s{\cdot}\tilde T_\delta. \end{displaymath}$

(2.13)

Let in the set of m+2 parameters of abbreviated description the momentum variable have number $\pi$ $(\hat Y_\pi(k)=\hat p(k))$ , then using definition of $\tilde\nu$ (2.8) and symmetry properties mentioned above, $\tilde\nu$ for such a system will read [28]:

$\begin{displaymath} \tilde\nu=\left( \begin{array}{ccccccc} 0&\dots&0&\nu_{1,\... ... 0&\dots&0&\nu_{m+2,\pi}&0&\dots&0\\ \end{array}\right). \end{displaymath}$

(2.14)

Due to the same reasons matrix $\tilde\varphi$ will have the opposite to $\tilde\nu$ structure:

$\begin{displaymath} \tilde\varphi =\left( \begin{array}{ccccccc} \varphi _{1,... ...2,\pi+1}&\dots &\varphi _{m+2,m+2}\\ \end{array}\right). \end{displaymath}$

(2.15)

It is easy to verify, that the matrix $\tilde\nu$ (2.14) satisfies:

$\begin{displaymath} \tilde\nu^3=\tilde\nu, \end{displaymath}$

(2.16)

Hydrodynamic Excitation Spectrum

To solve equations (2.1) and (2.2) we must find the eigenvalues {z_i} of hydrodynamic operator $\tilde T_\delta$ (2.12), which are proportional to the eigenvalues {Z_i} of full hydrodynamic operator $\tilde T_H$ of equation (2.1) (see (2.13)):

$\begin{displaymath} Z_i=\delta\,v_s{\cdot}z_i. \end{displaymath}$

(3.1)

Eigenvalues z_i could be found as series over $\delta$ :

$\begin{displaymath} z_i = \lambda_i + \delta{\cdot}{D}_i + \delta^2{\cdot}\gamma_i + \dots, \end{displaymath}$

(3.2)

where term $\delta^2{\cdot}\gamma_i$ in (3.2) could be found only taking into account the second and higher powers of $\delta$ in hydrodynamic matrix $\tilde T_\delta$ , that is why using expressions (2.8) and (2.9), we must restrict ourselves to the zeroth and linear approximations of z_i with respect to $\delta$ in resulting expressions. $\lambda_i$ and D_i can be found from:

$\displaystyle \det$

$\textstyle \tilde B(\lambda,D)$

=0,

(3.3)

$\textstyle \tilde B(\lambda,D)$

$\displaystyle \equiv\tilde\nu-\lambda+\delta{\cdot}(\tilde\varphi -D),$

(3.4)

where $\lambda$ and D are the corresponding diagonal matrices. Zeroes of characteristic polynom $\det(\tilde\nu-\lambda)= (\lambda^2-\frac12\mathop{\mbox{\rm Sp}}\nolimits\,\tilde\nu^2)(-\lambda)^m$ of matrix $\tilde\nu$ yield three different solutions for the eigenvalues (3.2) in zero approximation:

$\begin{displaymath} \lambda_+\equiv\lambda_{m+1}=+1,\quad\lambda_-\equiv\lambda_{m+2}=-1, \quad\lambda_0\equiv\lambda_1=\dots=\lambda_m=0, \end{displaymath}$

(3.5)

where $\lambda_+$ , $\lambda_-$ are the simple, and $\lambda_0$ is m time degenerated root. Eigenvalues $\lambda_+$ , $\lambda_-$ correspond to eigenvalues $\pm\delta\,v_s$ of full hydrodynamic operator $\tilde T_H$ (see (3.1)), so we conclude, that $\lambda_+$ , $\lambda_-$ describe propagation of sound waves in the system $\cite {MF2}$ ; $\lambda_0$ describe dissipative modes which correspond to dissipative processes. To find D_i, let us develop determinant of matrix $\tilde B(\lambda,D)$ (3.4) into the series with accuracy of linear terms over $\delta$ :

$\displaystyle \det\tilde B(\lambda,D)$

$\textstyle \simeq$

$\displaystyle \det(\tilde\nu-\lambda)+ \delta\cdot\sum\limits _{i,j}(\varphi _{... ...\lambda,D)\over\partial (\delta{\cdot}\varphi _{ij})} \right\vert _{\delta=0} =$

(3.6)

=

$\displaystyle \det(\tilde\nu-\lambda)+\delta\cdot\sum\limits _{i,j}(\varphi _{ij}-D\delta_{ij}) A_{ij}(\lambda) =0,$

where $\delta_{ij}$ - Kroneker symbol, $A_{ij}(\lambda)=\left.\mbox{Ad}_{ij} \!\left(\tilde B(\lambda,D)\right)\right\vert _{\delta=0}$ - an algebraic adjunct of the matrix $\tilde B$ to the element B_ij for $\delta=0$ . It is easy to see, that $\left.\tilde B(\lambda,D)\right\vert _{\delta=0} =\tilde\nu-\lambda$ , and from the general matrix theory it follows, that [29]:

$\begin{displaymath} A_{ij}(\lambda) = (\tilde\nu-\lambda)^{-1}_{ji}\cdot\det(\tilde\nu-\lambda). \end{displaymath}$

(3.7)

It is easy to find the inverse matrix to the matrix $(\tilde\nu-\lambda)$ , and we will have:

$\begin{displaymath} A_{ij}(\lambda)= (-\lambda)^{m-1}{\cdot} \big(\lambda^2-1+\lambda\tilde\nu+\lambda\tilde\nu^2\big)_{ji}. \end{displaymath}$

(3.8)

Equation (3.6) now will read:

$\begin{displaymath} \det\tilde B(\lambda,D)\simeq\det(\tilde\nu-\lambda)+ (-... ...hi -D) (\lambda^2-1+\lambda\tilde\nu+\tilde\nu^2)\right]. \end{displaymath}$

(3.9)

Substituting eigenvalue $\lambda$ in (3.9) by $\lambda_\pm$ , equation for finding $D_\pm$ will read:

$\begin{displaymath} \mathop{\mbox{\rm Sp}}\nolimits\,\left[(\tilde\varphi -D_\pm) (\lambda_\pm^2-1+\lambda_\pm\tilde\nu+\tilde\nu^2)\right]=0. \end{displaymath}$

(3.10)

Which yields $D_\pm= {\left(\mathop{\mbox{\rm Sp}}\nolimits\,(\tilde\varphi \tilde\nu^2)\pm... ...\nolimits\,(\tilde\nu^2)\pm\mathop{\mbox{\rm Sp}}\nolimits\,(\tilde\nu)\right)}$ , and taking into account that $\mathop{\mbox{\rm Sp}}\nolimits\,(\tilde\varphi \tilde\nu)=\mathop{\mbox{\rm Sp}}\nolimits\,(\tilde\nu)=0$ , finally:

$\begin{displaymath} D_*\equiv D_+=D_-={\mathop{\mbox{\rm Sp}}\nolimits\,(\tilde... ...\nu^2)\over\mathop{\mbox{\rm Sp}}\nolimits\,(\tilde\nu^2)}, \end{displaymath}$

(3.11)

The most easy way to find solutions for the dissipative modes {D_i, $i=1,\dots,m\}$ is to use equation (3.3) directly. For $\lambda=\lambda_0=0$ we have:

$\begin{displaymath} \det\tilde B(0,D)=\det\left(\tilde\nu+\delta{\cdot}(\tilde\varphi -D)\right)=0. \end{displaymath}$

(3.12)

Let us decompose $\det\tilde B(0,D)$ into the terms with respect to the elements of the $\pi$ -th row:

$\begin{displaymath} \det\tilde B(0,D)= \sum\limits _{i(\neq\pi)}\nu_{\pi i}{\... ...i} + \delta{\cdot}(\varphi _{\pi\pi}-D){\cdot}A_{\pi\pi}. \end{displaymath}$

(3.13)

The multiplier $A_{\pi i}=\mbox{Ad}_{\pi i}\tilde B(0,D)$ is an algebraic adjunct of the element $B_{\pi i}$ (which is equal to $\nu_{\pi i}$ ) of matrix $\tilde B(0,D)$ , and it can be treated as a determinant of some matrix $\tilde B'$ of dimension (m+1) x (m+1), obtained from the matrix $\tilde B(0,D)$ , of dimension (m+2) x (m+2), by deleting in it $\pi$ -th row and i-th column. In the matrix $\tilde B'$ all columns, except of $\pi$ -th are proportional to $\delta$ , that is why the multiplier $\delta^m$ can be put out from within the determinant:

$\begin{displaymath}A_{\pi i}=\mbox{Ad}_{\pi i}(\tilde\nu + \delta\cdot(\tilde\va... ...ox{Ad}_{\pi i}(\tilde\nu + \tilde\varphi -D), \quad i\neq\pi,\end{displaymath}$

Also it is easy to see that $A_{\pi\pi} = \delta^{m+1} \cdot \mbox{Ad}_{\pi\pi}(\tilde\varphi -D)$ , and finally we can write down (3.13) as follows:

$\begin{displaymath} \det\tilde B(0,D)= \delta^m\cdot \sum\limits _{i(\neq\pi)}... ...}-D){\cdot}\mbox{Ad}_{\pi\pi}\!\left(\tilde\varphi -D\right). \end{displaymath}$

(3.14)

Coefficient near the multiplier $\delta^m$ gives us an algebraic equation of m-th order from which m diffusive coefficients of dissipative modes can be found. It can be written in an abbreviated form as follows:

$\begin{displaymath} \det(\tilde\ae({D}))=0, \end{displaymath}$

(3.15)

where

$\begin{displaymath} \tilde\ae({D})=\left.(\tilde\nu+\tilde\varphi -{D}{\cdot}\t... ...vert _{ \mbox{\scriptsize element}_{\pi,\pi}({D})\equiv0}. \end{displaymath}$

(3.16)

For example for the binary magnetic mixture with the parameters of abbreviated description (2.7) we have:

$\begin{displaymath} \tilde\ae({D})=\left(\begin{array}{ccccc} \varphi _{n_1,n_... ...arphi _{h,s}& \varphi _{h,h}-{D}\\ \end{array}\right). \end{displaymath}$

(3.17)

Equation (3.15) can be given some geometrical interpretation. To do so let us introduce matrix $\tilde\theta$ of dimension (m+1) x (m+1) obtained from the matrix $\tilde\varphi$ by deleting $\pi$ -th row and $\pi$ -th column; (m+1) - dimensional vector-column

$\begin{displaymath}\vec\nu=(\nu_{1,\pi};\nu_{2,\pi};\dots;\nu_{\pi-1,\pi};\nu_{\pi+1,\pi};\dots;\nu_{m+2,\pi})^т,\end{displaymath}$

which is the $\pi$ -th row of the matrix $\tilde\nu$ (2.14) excluding element $\nu_{\pi,\pi}(=0)$ ; and (m+1) - dimensional vector-row

$\begin{displaymath}\vec\nu'=(\nu_{\pi,1};\dots;\nu_{\pi,\pi-1};\nu_{\pi,\pi+1};\dots;\nu_{\pi,m+2}), \end{displaymath}$

which consists of nonzero components of $\pi$ -th row of matrix $\tilde\nu$ . Let us introduce x-dependent ``metric tensor'': $\tilde g(x)=(\tilde\theta-x{\cdot}\tilde1)^{-1}$ , then decomposing $\det(\tilde\ae({D}))$ with respect to elements of $\pi$ -th column and $\pi$ -th row, we can prove, that expression (3.15) is equivalent to the next one:

$\begin{displaymath} \vec\nu'{\cdot}\tilde g({D}){\cdot}\vec\nu=0. \end{displaymath}$

(3.18)

As we see, the problem is equivalent to that of finding such values {D_i} for which $\vec\nu$ and $\vec\nu'$ are orthogonal with respect to metric given by tensor $\tilde g({D}_i)$ . Now we can write eigenvalues (3.2) of hydrodynamic matrix $\tilde T_\delta$ as follows:

$\displaystyle z_i=0+\delta{\cdot}{D}_i+\delta^2{\cdot}\gamma_i,\quad\quad i=1,\dots,m,$

(3.19)

$\displaystyle z_+=z_{m+1}=1+\delta{\cdot}{D}_*+\delta^2{\cdot} \gamma_+,$

(3.20)

$\displaystyle z_-=z_{m+2}=-1+\delta{\cdot}{D}_*+\delta^2{\cdot}\gamma_- ,$

(3.21)

Where values of index $i=\{1,\dots,m\}$ correspond to dissipative modes, and i=m+1, m+2 - to sound modes, D_* is given by equation (3.11), D_i - by (3.15).

Time-Dependent Correlation Functions

Receiving of time correlation functions $\tilde F(k,z)$ from (2.2) is equivalent to finding of matrix $\tilde M(z)=(z{\cdot}\tilde1-\tilde T_H(k))^{-1}$ , which is a function of the hydrodynamic operator T_H(k), really:

$\begin{displaymath} \tilde F(k,z)=(z-\tilde T_H(k))^{-1}\tilde F_0(k)=\tilde M(z){\cdot}\tilde F_0(k). \end{displaymath}$

(4.1)

In general case, an arbitrary function of matrix $\tilde A$ which has eigenvalues {z_i}, can be written as follows (see [29]):

$\begin{displaymath} f(\tilde A)=\sum\limits _{i=1}^{n}f(z_i)\,\tilde G^i, \end{displaymath}$

(4.2)

if $\tilde A$ is a matrix of the simple structure. Coefficients $\tilde G^i$ in (4.2) are so-called weight coefficients of the matrix $\tilde A$ . Matrix $\tilde T_\delta$ which we are interested in is a matrix of the simple structure for an arbitrary $\delta$ , because for $\delta\neq0$ all eigenvalues are different (3.19) - (3.21), and for $\delta=0$ the minimal ?annulating polynom $\eta(x)=x(x-1)(x+1)$ of matrix $\tilde\nu$ ( $=\lim\limits_{\delta\to0}\tilde T_\delta$ ) includes monom with degenerated eigenvalue $(\lambda=0)$ in the first power, that is the ??need and sufficient condition of simplicity. The fact, that $\eta(x)$ is the minimal ?annulating polynom of matrix $\tilde\nu$ (it means that $\eta(\tilde\nu)=0$ ) follows from the equality (2.16). So we can write for $\tilde\nu$ and $\tilde T_\delta$ and for the arbitrary analytical function f(x):

$\displaystyle f(\tilde\nu)=f(1)\,\tilde G_\nu^++f(-1)\,\tilde G_\nu^-+f(0)\, \tilde G_\nu^0 ,$

(4.3)

$\displaystyle f(\tilde T_\delta)=\sum\limits _{i=1}^{m+2}f(z_i)\,\tilde G_\delta^i.$

(4.4)

As it will be shown further, in the limit $\delta\to0$ the weight coefficients $\tilde G_\delta^i$ yield m+2 different weight coefficients $\{\tilde g_0^i$ , $i=1,\dots,m\}$ , that is why we will formally distinguish matrices $\tilde\nu$ and $\tilde T_0=\lim\limits_{\delta\to 0}\tilde T_\delta$ because their system of weight coefficients $\{\tilde G_\nu^\pm$ , $\tilde G_\nu^0\}$ and $\{\tilde g_0^i$ , $i=1,\dots,m\}$ , correspondingly, do not coinside. Equality (4.2) for the matrix $\tilde T_\delta$ can be written down with the accuracy of the second order over the small parameter $\delta$ as follows:

$\begin{displaymath} f(\tilde\nu+\delta\,\tilde\varphi )=\sum\limits _{i=1}^{m+2... ...i+\delta{\cdot}\tilde g_1^i+\delta^2{\cdot}\tilde g_2^i), \end{displaymath}$

(4.5)

where $\tilde g_0^i$ , $\tilde g_1^i$ , $\tilde g_2^i$ are the zeroth, the first and the second approximations, respectively of the weight coefficients $\tilde G^i_\delta$ (4.4) with respect to the small parameter $\delta$ :

$\begin{displaymath} \tilde G_\delta^i=\tilde g_0^i+\delta{\cdot}\tilde g_1^i+\delta^2{\cdot}\tilde g_2^i+\dots. \end{displaymath}$

(4.6)

It is easy to convince that weight coefficients $\tilde G^i_\delta$ satisfy the equalities:

$\begin{displaymath} \tilde G_\delta^i{\cdot}\tilde G_\delta^j=\tilde G_\delta^i{\cdot}\delta_{ij}, \end{displaymath}$

(4.7)

where $\delta_{ij}$ - the Cronnekker symbol. Putting (4.6) into (4.7), for the approximations $\tilde g_0^i$ , $\tilde g_1^i$ , $\dots$ we will have the chain of equalities:

$\displaystyle \tilde g_0^i{\cdot}\tilde g_0^j=\tilde g_0^i{\cdot}\delta_{ij},$

(4.8)

$\displaystyle \tilde g_0^i{\cdot}\tilde g_1^j + \tilde g_1^i{\cdot}\tilde g_0^j = \tilde g_1^i{\cdot}\delta_{ij}.$

(4.9)

$\displaystyle \dots\dots\dots$

After developing into the series the left-hand side of (4.5) with respect to $\delta$ , we can write:

$\begin{displaymath} f(\tilde\nu+\delta\,\tilde\varphi )=\sum\limits _{k\geq0}\f... ...}\frac{f^{(k)}(0)}{k!}\{\tilde\nu^{k-2},\tilde\varphi ^2\}, \end{displaymath}$

(4.10)

where

$\displaystyle \{\tilde\nu^s,\tilde\varphi \}\equiv\sum\limits _{k=0}^{s}\tilde\... ...phi +\tilde\nu^{s-1}\tilde\varphi \tilde\nu+\dots +\tilde\varphi \tilde\nu^{s},$

(4.11)

$\displaystyle \{\tilde\nu^s,\tilde\varphi ^2\}\equiv\sum\limits _{k=0}^{s}\sum\... ...l=0}^{s-k}\tilde\nu^k \tilde\varphi \tilde\nu^l\tilde\varphi \tilde\nu^{s-k-l}.$

(4.12)

The first term in the right-hand side of (4.10) can be written as $f(\tilde\nu)$ . Developing into the series both sides of (4.5) and equating to zero coefficients near the powers of $\delta$ , we will have three equalities:

$\displaystyle f(\tilde\nu)=\sum\limits _{i=1}^{m+2}f(\lambda_i){\cdot}\tilde g_0^i,$

(4.13)

$\displaystyle \sum\limits _{k\geq1}\frac{f^{(k)}(0)}{k!}\{\tilde\nu^{k-1},\tild... ...eft[f(\lambda_i){\cdot}\tilde g_1^i+f'(\lambda_i)D_i{\cdot}\tilde g_0^i\right],$

(4.14)

$\displaystyle \sum\limits _{k\geq2}\frac{f^{(k)}(0)}{k!}\{\tilde\nu^{k-2},\tild... ...bda_i)\gamma_i+\frac{f''(\lambda_i)}{2!}D_i^2\right){\cdot}\tilde g_0^i\right].$

(4.15)

From the previous consideration we can conclude, that calculating of time correlation functions is based on finding of the weight coefficients (4.6). Finding the weight coefficients is based on the extracting their approximations $(\tilde g_0^i, \tilde g_1^i, \dots)$ from the equations (4.15)-(4.15) where they are involved in. Let us analyze (4.15). Due to the fact, that $\lambda_i$ adopts three values $\lambda_\pm=\pm1$ , $\lambda_0=0$ , the equation (4.15) can be written as:

$\begin{displaymath} f(\tilde\nu)=f(1)\,\tilde g_0^++f(-1)\,\tilde g_0^-+f(0)\,\tilde P, \end{displaymath}$

(4.16)

where

$\begin{displaymath} \tilde P=\sum\limits _{i=1}^m\tilde g_0^i. \end{displaymath}$

(4.17)

Let us consider three functions f₊(x)=x(x+1), f_-(x)=x(x-1), f₀(x)=(x-1)(x+1) subsequently, each of them ??annulating two terms in the right-hand side of (4.16). It makes possible to find expressions for $\tilde g_0^+$ , $\tilde g_0^-$ , $\tilde P$ subsequently:

$\displaystyle \tilde g_0^+=\frac{\tilde\nu(\tilde\nu+1)}{2},\quad \tilde g_0^-=\frac{\tilde\nu(\tilde\nu-1)}2,$

(4.18)

$\displaystyle \tilde P=1-\tilde\nu^2.$

(4.19)

It is easy to convince comparing (4.16) and (4.3) that $\tilde g_0^+$ , $\tilde g_0^-$ , $\tilde P$ concise with $\tilde G_\nu^+$ , $\tilde G_\nu^-$ , $\tilde G_\nu^0$ correspondingly. Expressions (4.19), (4.17) give us the zeroth sum rule for dissipative modes:

$\begin{displaymath} \sum\limits _{i=1}^m\tilde g_0^i=1-\tilde\nu^2. \end{displaymath}$

(4.20)

So we see, that the single weight coefficient of m time degenerated dissipative mode of matrix $\tilde\nu$ is equal to the sum of m weight coefficients of different dissipative modes of matrix $\tilde T_0$ . It is worth noticing that $\tilde P$ , as a weigth coefficient of a matrix $\tilde\nu$ , have the features of projection operator:

$\begin{displaymath} \tilde P^2=\tilde P, \end{displaymath}$

(4.21)

that follows from the property (2.16) of matrix $\tilde\nu$ . Let us consider (4.15) now. It is easy to simplify functions $\{\tilde\nu^{k},\tilde\varphi \}$ (4.11) using (2.16). For even and odd powers of k we have, correspondingly:

$\begin{displaymath} \{\tilde\nu^{2s},\tilde\varphi \}=\{\tilde\nu^{2},\tilde\va... ...){{\cdot}} \tilde\nu\{\tilde\nu,\tilde\varphi \}\tilde\nu , \end{displaymath}$

(4.22)

We can rewrite (4.15), splitting ``sound'' terms from ``dissipative'':

$\begin{displaymath} \sum\limits _{k\geq1}\frac{f^{(k)}(0)}{k!}\{\tilde\nu^{k-1}... ...ilde g_0^i+ D_*\sum\limits _{+,-}f'(\pm1)\tilde g_0^{\pm}. \end{displaymath}$

(4.23)

from which we receive at once, taking f(x)=1:

$\begin{displaymath} \sum\limits _{i=1}^{m+2}\tilde g_1^i=0\quad\mbox{or}\quad\s... ...ts _{i=1}^{m}\tilde g_1^i= -(\tilde g_1^++\tilde g_1^-). \end{displaymath}$

(4.24)

Putting a function f(x)=x^2p+1-x^2s+1, where p,s>1 in (4.23), we find:

$\begin{displaymath} \tilde\nu\tilde\varphi \tilde\nu+\tilde\nu^2\tilde\varphi \... ...nu+\tilde\nu\tilde\varphi \tilde\nu^2=2D_*{\cdot}\tilde\nu, \end{displaymath}$

(4.25)

which is some additional identity bonding matrices $\tilde\varphi$ and $\tilde\nu$ . Considering functions $f_1(x)=\frac{x^{2p}}{2p}-\frac{x^{2s}}{2s}$ and $f_2(x)=\frac{x^{2p+1}}{2p+1}-\frac{x^{2s+1}}{2s+1}$ subsequently, from the equation (4.23) we receive:

$\begin{displaymath} \tilde g_1^+-\tilde g_1^-=\{\tilde\nu^2,\tilde\varphi \}-... ...e g_1^-=\{\tilde\nu,\tilde\varphi \}-2D_*{\cdot}\tilde\nu, \end{displaymath}$

(4.26)

where identities (4.25) are taken into account. From (4.26) it follows, that:

$\begin{displaymath} \tilde g_1^\pm=\frac12\left(\{\tilde\nu,\tilde\varphi \}-... ...de\nu^2, \tilde\varphi \}\mp3D_*{\cdot}\tilde\nu^2\right). \end{displaymath}$

(4.27)

Taking f(x)=x after some simple algebra we receive:

$\begin{displaymath} \sum\limits _{i=1}^m D_i\tilde g_0^i=\tilde P\tilde\varphi \tilde P, \end{displaymath}$

(4.28)

from which the equality $(\tilde P\tilde\varphi \tilde P)^n=\sum\limits _{i=1}^m D_i^n\tilde g_0^i$ can be proved, using property (4.8), or for more general case of arbitrary analytical function F(x) for which F(0)=0 is satisfied, we can show following:

$\begin{displaymath} F(\tilde P\tilde\varphi \tilde P)=\sum\limits _{i=1}^m F(D_i)\tilde g_0^i. \end{displaymath}$

(4.29)

The equality (4.29) can be generalized for the case of $F(0)\neq 0$ as follows:

$\begin{displaymath} \tilde P{\cdot}F(\tilde P\tilde\varphi \tilde P){\cdot}\tilde P=\sum\limits _{i=1}^m F(D_i)\tilde g_0^i. \end{displaymath}$

(4.30)

The weight coefficients $\tilde g_o^k$ which we are interested in can be obtained by putting functions $F_k(x)=\prod\limits_{i=1(\neq k)}^m(D_i-x)$ into the expressions (4.30):

$\begin{displaymath} \tilde g_0^k=\prod\limits_{i=1(\neq k)}^m\tilde P\frac{D_i-\tilde\varphi }{D_i-D_k}\tilde P. \end{displaymath}$

(4.31)

We see, that consideration of (4.15) gave us possibility to find $\tilde g_0^{\pm}$ ; from the equation (4.15) the weight coefficients for dissipative modes $\tilde g_0^i$ , ${i=1,\dots,m}$ , and $\tilde g_1^{\pm}$ were received. So, we can expect the equation (4.15) to give us possibility to find $\tilde g_1^i$ , ${i=1,\dots,m}$ . Let us analyze (4.15). Analogically as (4.22), the functions $\{\tilde\nu^{k},\tilde\varphi ^2\}$ can be simplified:

$\displaystyle \{\tilde\nu^{2s},\tilde\varphi ^2\}$

=

$\displaystyle \{\tilde\nu^{2},\tilde\varphi ^2\}+(s-1) \left[2D_*\{\tilde\nu^{2... ...nu^2\{\tilde\nu\tilde\varphi ^2\}\tilde\nu\right]+ 2D_*^2(s-1)(s-2)\tilde\nu^2,$

(4.32)

$\displaystyle \{\tilde\nu^{2s-1},\tilde\varphi ^2\}$

=

$\displaystyle \{\tilde\nu,\tilde\varphi ^2\}+(s-1) \left[2D_*\{\tilde\nu,\tilde... ...lde\nu\{\tilde\nu\tilde\varphi ^2\}\tilde\nu\right]+ 2D_*^2(s-1)(s-2)\tilde\nu.$

(4.33)

Putting the functions $f_1(x)=\frac{x^{2m}}{2m}-\frac{x^{2p}}{2p}$ and $f_2(x)=\frac{x^{2m+1}}{2m+1}-\frac{x^{2p+1}}{2p+1}$ into the (4.22), after some algebra we obtain:

$\displaystyle \tilde g_2^++\tilde g_2^-$

=

$\displaystyle \{\tilde\nu^{2},\tilde\varphi ^2\}-2\left[2D_*\{\tilde\nu^{2}, \t... ...}+\tilde\nu^2\{\tilde\nu,\tilde\varphi ^2\}\tilde\nu\right]+12D_*^2\tilde\nu^2,$

(4.34)

$\displaystyle \tilde C\equiv\tilde g_2^+-\tilde g_2^-$

=

$\displaystyle \{\tilde\nu,\tilde\varphi ^2\}-\frac32 \left[2D_*\{\tilde\nu,\til... ...lde\nu\{\tilde\nu,\tilde\varphi ^2\}\tilde\nu\right] +\frac{15}2D_*^2\tilde\nu.$

(4.35)

To find $\tilde g_1^i$ , we will need the next equalities, which can be easily proved using the (4.25):

$\displaystyle \tilde g_0^+{\cdot}\tilde\varphi {\cdot}\tilde g_0^+=D_+{\cdot}\t... ...ad \tilde g_0^-{\cdot}\tilde\varphi {\cdot}\tilde g_0^-=D_-{\cdot}\tilde g_0^-,$

(4.36)

$\displaystyle \tilde g_0^j{\cdot}\tilde\varphi {\cdot}\tilde P=\hat P{\cdot}\tilde\varphi {\cdot}\tilde g_0^j= D_j{\cdot}\tilde g_0^j,$

(4.37)

where we distinguish diffusive coefficients for `+' and `-' sound modes (see (3.11)) which is necessary for further consideration. Putting f(x)=x into the (4.15), we will have:

$\begin{displaymath} \sum\limits _{i=1}^{m+2}(D_i\tilde g_1^i+\gamma_i\tilde g_0^i)=-\tilde C, \end{displaymath}$

(4.38)

where $\tilde C$ is defined in (4.35) Let us introduce operators $\widehat Q_i$ , which perform due to the rule:

$\begin{displaymath} \widehat Q_i{\cdot}\tilde A=\sum\limits _{j=1(\neq i)}^{m+2... ...}\tilde A{\cdot}\tilde g_0^i\right),\quad i=1,\dots,m+2. \end{displaymath}$

(4.39)

Applying operator $\widehat Q_i$ to both sides of equality (4.38), we see, that the second term in the left-hand side at the summation symbol will vanish due to (4.8). Then, using the equality (4.9), we obtain:

$\begin{displaymath} \widehat Q_i{\cdot}(-\tilde C)=\widehat Q_i{\cdot}\sum\limits _{j=1}^{m+2}D_j{\cdot}\tilde g_1^j=\tilde g_1^i. \end{displaymath}$

(4.40)

So, for the weight coefficients $\tilde g_1^i$ we have the following expressions:

$\begin{displaymath} \tilde g_1^i=\sum\limits _{j(\neq i)}^{m+2}\frac1{D_j-D_i}\... ...}\tilde C{\cdot}\tilde g_0^i\right),\quad i=1,\dots,m+2. \end{displaymath}$

(4.41)

Now it is obvious, that while finding matrices $\tilde g_1^\pm$ we should distinguish between coefficients D₊ and D_- to avoid peculiarities of type 0/0. Using the equalities (4.36), (4.37), it can be easily shown, that:

$\displaystyle \tilde g_0^+{\cdot}\tilde C\tilde g_0^-=\frac12(D_--D_+)\, \tilde... ...{\cdot}\tilde C\tilde g_0^+=\frac12(D_--D_+)\, \tilde g_0^-\tilde\varphi g_0^+,$

(4.42)

$\displaystyle \tilde g_0^+{\cdot}\tilde C\tilde g_0^j=(D_j-D_+)\, \tilde g_0^+ ... ... g_0^j{\cdot}\tilde C\tilde g_0^+=(D_j-D_+)\, \tilde g_0^j \tilde\varphi g_0^+,$

(4.43)

$\displaystyle \tilde g_0^-{\cdot}\tilde C\tilde g_0^j=(D_--D_j)\, \tilde g_0^- ... ... g_0^j{\cdot}\tilde C\tilde g_0^-=(D_--D_j)\, \tilde g_0^j \tilde\varphi g_0^-,$

(4.44)

$\displaystyle \tilde g_0^j{\cdot}\tilde C\tilde g_0^k=\tilde g_0^j\tilde\varphi \tilde\nu\tilde\varphi \tilde g_0^k,$

(4.45)

where $j,k=1,\dots,m$ . On the basis of these equalities and (4.41) we receive:

$\displaystyle \tilde g_1^\pm=\pm\tilde g_0^\pm\tilde\varphi (\tilde P+\frac12\tilde g_0^\mp)\pm (\tilde P+\frac12\tilde g_0^\mp)\tilde\varphi \tilde g_0^\pm,$

(4.46)

$\displaystyle \tilde g_1^j=(\tilde R_j{\cdot}\tilde\varphi -1)\tilde\nu\tilde\v... ...e g_0^j+\tilde g_0^j\tilde\varphi \tilde\nu(\tilde\varphi {\cdot}\tilde R_j-1),$

(4.47)

where

$\begin{displaymath} \tilde R_j=\sum\limits _{k=1(\neq j)}^{m}\frac{\tilde g_0^k}{D_k-D_j}, \quad j=1,\dots,m. \end{displaymath}$

(4.48)

Being developed, expression (4.46) coincide with (4.27) received above. So, we managed to obtain the zeroth and the first approximations for the weight coefficients of multicomponent mixture in analytical form. With accuracy to the first power of $\delta$ we can write for an arbitrary analytical function f(x):

$\begin{displaymath} f(\tilde\nu+\delta{\cdot}\tilde\varphi ) = \sum\limits _{+,... ...}{D}_i) \cdot(\tilde g_0^i + \delta {\cdot}\tilde g_1^i), \end{displaymath}$

(4.49)

where all variables in right-hand side of the (4.49), are given by equations (3.11), (3.16), (4.18), (4.31), (4.46) - (4.48). The expression (4.49) should be regarded as an interpolational, treating the multipliers like $f(\pm 1 +\delta{\cdot}{D}_*)$ as series which they yield. Taking into account the relation (2.13), the solutions for the time correlation functions (4.1) in the hydrodynamic limit can be written as:

$\begin{displaymath} \tilde F(k,z) = \left\{\sum\limits _{+,-}{\tilde g_0^\pm +\... ...\tilde g_1^i\over z+k^2v_s{D}_i} \right\}\cdot\tilde F_0. \end{displaymath}$

(4.50)

Simple fluid

Let us apply now results obtained above to the one-component fluid, being described by three parameters of abbreviated description $\{\hat n(k), \hat p(k), \hat h(k)\}$ . The sound velocity can be presented as:

$\begin{displaymath} v_s^2 = \left({\partial P\over\partial \rho} \right)_{NS} = {1\over\rho}{\gamma\over\kappa_т}, \end{displaymath}$

(5.1)

where $\rho=m/V$ is a mass density, and other thermodynamical functions are defined as follows:

$\displaystyle \kappa_т = -{1\over V}\left({\partial V \over \partial P} \right)... ...\quad \alpha_p = {1\over V}\left({\partial V \over \partial T} \right)_{P,N,M},$

(5.2)

$\displaystyle c_V=\frac TV\left({\partial S\over\partial T}\right)_{VN}, \quad c_P=\frac TV\left({\partial S\over\partial T}\right)_{PN},$

(5.3)

$\displaystyle \gamma = c_P/c_V,$

(5.4)

and are the isothermal compressibility, isobaric thermal expansion coefficients, specific heats in ensembles with constant volume, pressure and their ratio, respectively. Dimensionless matrices of frequencies $\tilde\nu$ and of memory functions $\tilde\varphi$ will read:

$\begin{displaymath} \tilde\nu=\frac1v_s\left( \begin{array}{ccc} 0&{n\over\rh... ...l\over\rho}&0\\ 0&0&{\lambda\over c_V} \end{array}\right), \end{displaymath}$

(5.5)

where n=N/V is a concentration, $\eta_l=L_{pp}={4\over3}\eta+\xi$ is a longitudinal viscosity, $\lambda=L_{hh}/T$ is a coefficient of thermoconductivity. The sound damping coefficient will read (see (3.11)):

$\begin{displaymath} \Gamma=v_s{\cdot}D_*= {1\over2} \left( {\eta_l\over\rho}+{\lambda (\gamma-1)\over c_P} \right). \end{displaymath}$

(5.6)

The damping coefficient of the heat mode can be written as

$\begin{displaymath} {\cal D}_h=v_s{\cdot}D_h= {\eta_l\over\rho} +{\lambda\over c_V} -2\Gamma ={\lambda\over c_P}. \end{displaymath}$

(5.7)

Fulfilling simple calculations with the help of equations (4.18), (4.31), (4.46) - (4.48) we can receive all the time correlation functions. These results are well-known in the literature, [32]. Expression for the dynamic structure factor, for example, is given by the generalized Landau-Plachek formula:

$\begin{displaymath} \mbox{Re}{S(k,\omega)\over S(k)}= {1\over2\gamma}\sum\limit... ...cdot}\lambda/c_P \over\omega^2 +(k^2{\cdot}\lambda/c_P)^2}, \end{displaymath}$

(5.8)

Mixture of magnetic and nonmagnetic particles in paramagnetic state

Let us consider now the mixture of magnetic and nonmagnetic particles described by parameters of abbreviated description (2.7). The frequency matrix $\tilde\nu$ for such a system reads [28]

$\begin{displaymath} \tilde\nu={1\over v_s}\left( \begin{array}{ccccc} 0&0&c_1... ... 0&0&{T\alpha_p\xi/(\rho\kappa_т)}&0&0 \end{array}\right), \end{displaymath}$

(6.1)

where c and $\alpha_p$ are defined by (5.2)-(5.3), $\xi= 1 +{\pi^2/(\chi{\cdot}\kappa_т)}$ ,

$\displaystyle \chi= {1\over V}\left(\partial \langle \hat S\rangle \over\partia... ...ght)_{NTV}, \quad \pi={1\over V}\left({\partial V\over\partial b}\right)_{PNT},$

(6.2)

$\displaystyle c_i= N_i/V, \qquad u_i= \left({\partial V\over\partial N_i}\right)_{TPbN_{\bar\imath}}$

(6.3)

(b-external magnetic field) are the magnetic susceptibility, coefficient of magnetostriction, partial concentration and partial molar volume per molecule of species `i', respectively. Matrix of static correlation functions for our system can be introduced as

$\begin{displaymath} \tilde F_0 =\left(\begin{array}{ccccc} S_{ n_1, n_1}&S_{ n... ... 0&0&0&F_{ s, s}&0\\ 0&0&0&0&F_{ h, h} \end{array}\right), \end{displaymath}$

(6.4)

and matrix of memory functions have the form opposite to that of $\tilde\nu$ , with elements which can be presented via the transport coefficients L_ik:

$\begin{displaymath} \varphi _{ij}= {TV\over v_s}\sum\limits _k L_{ik} (\tilde F^{-1}_0)_{kj}. \end{displaymath}$

(6.5)

For the paramagnetic case which we are interested in, we have

$\begin{displaymath} \nu_{p,s}= \nu_{s, p} =0, \quad \varphi _{ s,\hat Y}=\varphi _{\hat Y, s}=0, \end{displaymath}$

(6.6)

for all $\hat Y\neq s$ . Due to the (6.6) hydrodynamic matrix (2.12) decays into 2 blocks: submatrix $\tilde \nu_1 +\delta{\cdot}\tilde\varphi _1$ of dimension 4 x 4 and diagonal element $\delta{\cdot}\varphi _{ s s}$ , where $\tilde\nu_1$ and $\tilde\varphi _1$ are the matrices $\tilde\nu$ and $\tilde\varphi$ with deleted 4-th row and 4-th column, which correspond to the variable s. Let us introduce vectors $\vec\xi$ and $\vec\xi'$ :

$\begin{displaymath} \xi_i=\sum\limits _j\varphi _{i,j}\nu_{j,p}, \quad \xi_i'=\sum\limits _j\nu_{p,j}\varphi _{j,i}. \end{displaymath}$

(6.7)

Indices without dashes $(i,j,\dots)$ here and further take values from the set { n₁, n₂, h} and indices with dashes $(i',j',\dots)$ - from the set { p, s}. Indices, indicated by the capital latin letters $(I,J,\dots)$ take all values from the set { n₁,n₂,p,s,h}. From the expression (2.10) (see also [28]) we can prove the square of the sound velocity to be the inverse adiabatic compressibility defined in the ensemble with constant magnetization and number of particles of both species:

$\begin{displaymath} v_s^2 = -{V^2\over {\bar m}} \left({\partial P\over \partial... ...{NSM} = \left({\partial P\over\partial \rho} \right)_{NSM}, \end{displaymath}$

(6.8)

where ${\bar m}=N_1m_1+N_2m_2$ , V, $\rho$ , P are a mass, volume, mass density and pressure of the system, respectively. The sound damping coefficient (3.11) will read:

$\begin{displaymath} D_*=\frac12\left(\sum\limits _{i} \nu_{p,i}\xi_i + \varphi _{p,p}\right). \end{displaymath}$

(6.9)

where in $\nu_{p,i}$ the first index `p' stands for momentum variable, the second take values from the set {n₁,n₂,h}, as it was told above. Equation (3.15) for finding of the dissipative coefficients of diffusive modes can be easily factorized, and we derive:

$\displaystyle D_1=\varphi _{ s, s},$

(6.10)

$\displaystyle D_2=-\frac12b+\frac12\sqrt{b^2-4c},$

(6.11)

$\displaystyle D_3=-\frac12b-\frac12\sqrt{b^2-4c},$

(6.12)

where D₁, D₂, D₃ are the coefficients of spindiffusion, interdiffusion and termodiffusion, respectively, and

$\begin{displaymath} b=2D_* - \mathop{\mbox{\rm Sp}}\nolimits\,\tilde\varphi _1,\quad c=\det{(\tilde\nu_1+\tilde\varphi _1)}. \end{displaymath}$

(6.13)

In the zeroth approximation for the weight coefficients with respect to $\delta$ we will have from the expression (4.18)

$\displaystyle (\tilde g^\pm_0)_{i,j}=\frac12\nu_{i,p}\nu_{p,j},\quad (\tilde g^\pm_0)_{ s, J}=(\tilde g^\pm_0)_{J, s}=0,$

(6.14)

$\displaystyle (\tilde g^\pm_0)_{p, i}=\pm\frac12\nu_{p,i}, \quad (\tilde g^\pm_0)_{i,p}=\pm\frac12\nu_{i,p},\quad (\tilde g^\pm_0)_{p, p}=\frac12$

(6.15)

for the sound modes;

$\begin{displaymath} \tilde g^1_0=\mbox{diag}(0,0,0,1,0) \end{displaymath}$

(6.16)

for the spindiffusive mode; for the termo- and interdiffusive modes nonzero coefficients are:

$\displaystyle (\tilde g^2_0)_{i,j}=(D_3-D_2)^{-1}\left(D_3\delta_{ij} +\nu_{i,p... ...rphi _{p,p}-2D_*-D_3) +\nu_{i,p}\xi'_j +\xi_i\nu_{p,j} -\varphi _{i,j} \right),$

(6.17)

$\displaystyle (\tilde g^3_0)_{i,j}=(D_2-D_3)^{-1}\left(D_2\delta_{ij} +\nu_{i,p... ...rphi _{p,p}-2D_*-D_2) +\nu_{i,p}\xi'_j +\xi_i\nu_{p,j} -\varphi _{i,j} \right).$

(6.18)

After some simplifications of the expressions (4.46) - (4.48) nonzero linear approximations of the weight coefficients can be written as

$\displaystyle (\tilde g^\pm_1)_{i,j}=\pm \frac12\left(\nu_{i,p}\xi'_j +\xi_i\nu_{p,j} +\nu_{i,p},\nu_{p,j}(\varphi _{p,p}-3D_*)\right),$

(6.19)

$\displaystyle (\tilde g^\pm_1)_{p, j}=\frac12\left(\xi'_j +\nu_{p,j}(\varphi _{... ...tilde g^\pm_1)_{i,p}=\frac12\left(\xi_i +\nu_{i,p}(\varphi _{p,p}-2D_*)\right),$

(6.20)

$\displaystyle (\tilde g^\pm_1)_{p,p}=\pm\frac12(\varphi _{p,p}-2D_*),\quad (\tilde g^\pm_1)_{ s, J}=(\tilde g^\pm_1)_{I, s}=0$

(6.21)

for the sound modes

$\displaystyle (\tilde g^2_1)_{i,p}= (D_{3}-D_{2})^{-1} \left(D_{2}\left[\nu_{i,... ... _{p,p}+\varphi _{i,i}) +\xi_i -\varphi _{i,i}\nu_{i,p} \right]+\eta_i \right),$

(6.22)

$\displaystyle (\tilde g^2_1)_{p, j}= (D_{3}-D_{2})^{-1} \left(D_{2}\left[\nu_{p... ...+\xi'_j -\nu_{p,j}\varphi _{i,i} \right]+\eta'_i \right),\mbox{\hspace*{0.3cm}}$

(6.23)

$\displaystyle (\tilde g^3_1)_{i,p}= (D_{2}-D_{3})^{-1} \left(D_{3}\left[\nu_{i,... ... _{p,p}+\varphi _{i,i}) +\xi_i -\varphi _{i,i}\nu_{i,p} \right]+\eta_i \right),$

(6.24)

$\displaystyle (\tilde g^3_1)_{p, j}= (D_{2}-D_{3})^{-1} \left(D_{3}\left[\nu_{p... ...{p,p}+\varphi _{j,j}) +\xi'_j -\nu_{p,j}\varphi _{i,i} \right]+\eta'_i \right),$

(6.25)

for the termo- and interdiffusive modes. For the spindiffusive mode $\tilde g^1_1=\tilde0$ . Vectors $\vec\eta$ and $\vec\eta'$ in expressions (6.22-6.25) are defined as follows:

$\begin{displaymath} \eta_i=(\tilde\varphi ^{-1})_{i,j}\nu_{j,p}{\cdot}\frac{\de... ...}\frac{\det\tilde\varphi }{\varphi _{ s, s}\varphi _{p,p}}, \end{displaymath}$

(6.26)

here and further we imply summation when the indices are repeated. It is easy to see, that, for example:

$\displaystyle \eta_{ n_2}= -\varphi _{{ n_{{2}}, n_{{1}}}}\nu_{{ n_{{1}}},p}\va... ..._{{2}}, h}}-\nu_{{ n_{{2}}},p}\varphi _ {{ h, n_{{1}}}}\varphi _{{ n_{{1}}, h}}$

$\displaystyle +\nu_{{ n_{{2}}},p}\varphi _{{ n_{{1}}, n_{ {1}}}}\varphi _{{ h, ... ...{{1} }, h}}-\nu_{{ h},p}\varphi _{{ n_{{2}}, h}}\varphi _{{ n_{{1}}, n_{{1}}}}.$

Real parts of the correlation functions (4.50) read

$\displaystyle {\rm Re}\; \tilde F(k,\omega)_{i,j} = \sum\limits _{+,-} {k^2 v_s... ...(\tilde g_1^{\pm}\tilde F_0)_{i,j} \over (\omega\mp kv_s)^2 + (v_s D_*k^2)^2} +$

$\displaystyle +\sum\limits _{\mu=1}^3 {k^2 v_s D_\mu\cdot(\tilde g_0^\mu\tilde ... ...+ k \omega(\tilde g_1^\mu\tilde F_0)_{i,j} \over \omega^2 + (v_s D_\mu k^2)^2},$

(6.27)

To receive for example density-density partial dynamical structure factor $S_{ n_1, n_1}(k,\omega)$ we should calculate products $(\tilde g_{0,1}^\mu\tilde F_0)_{ n_1, n_1}$ , due to the equation (6.27). It can be easily shown that nonzero contributions have the following structure:

$\displaystyle (\tilde g_{0}^\pm\tilde F_0)_{ n_1, n_1}$

=

$\displaystyle \frac12\nu_{ n_1,p}\nu_{p, n_1}{\cdot}S_{ n_1, n_1} +\frac12 \nu_{ n_1,p}\nu_{p, n_2}{\cdot}S_{ n_2, n_1},$

(6.28)

$\displaystyle (\tilde g_{0}^2\tilde F_0)_{ n_1, n_1}$

=

$\displaystyle (D_3-D_2)^{-1}\big( D_3 +\nu_{ n_1,p}\nu_{p, n_1}(\varphi _{p,p}-2D_*-D_3)$

$\displaystyle \left.+\nu_{ n_1,p}\xi'_{ n_1} +\xi_{ n_1}\nu_{p, n_1} -\varphi _{ n_1, n_1} \right){\cdot}S_{ n_1, n_1}$

(6.29)

$\displaystyle +(D_3-D_2)^{-1}\big( D_3 +\nu_{ n_1,p}\nu_{p, n_2}(\varphi _{p,p}-2D_*-D_3)$

$\displaystyle \left.+\nu_{ n_1,p}\xi'_{ n_2}+\xi_{ n_1}\nu_{p, n_2} -\varphi _{ n_1, n_2} \right){\cdot}S_{ n_2, n_1},$

$\displaystyle (\tilde g_{1}^\pm\tilde F_0)_{n_1, n_1}$

=

$\displaystyle \pm \frac12\left(\nu_{ n_1,p}\xi'_{ n_1} +\xi_{ n_1}\nu_{p,n_1} +\nu_{ n_1,p}\nu_{p, n_1}(\varphi _{p,p}-3D_*)\right){\cdot} S_{ n_1, n_1}$

$\displaystyle \pm \frac12\left(\nu_{ n_1,p}\xi'_{ n_2} +\xi_{ n_1}\nu_{p, n_2} +\nu_{ n_1,p}\nu_{p, n_2}(\varphi _{p,p}-3D_*)\right){\cdot} S_{ n_2, n_1},$

(6.30)

$(\tilde g_{0}^3\tilde F_0)_{ n_1, n_1}$ can be obtained from the $(\tilde g_{0}^2\tilde F_0)_{n_1, n_1}$ by substitution $D_2\leftrightarrow D_3$ . There is no spin contribution to $S_{ n_1, n_1}(k,\omega)$ as it could be expected. Taking the expressions for the static structure factors

$\begin{displaymath} S_{ij}= TV \left({\partial c_i\over\partial \mu_j}\right)_{T,V,b,\mu_{\bar\jmath}}, \end{displaymath}$

(6.31)

frequency matrix and matrix of memory functions via the thermadynamical parameters [28], we can introduce the following abbreviations:

$\displaystyle \alpha=2\Gamma-\eta_l/\rho,$

(6.32)

$\displaystyle \gamma_k=L_{kl}{u_l\over\kappa_т} + L_{kh}{\alpha_p\over c_V\kappa_т}, \quad\gamma_h=L_{hl}{u_l\over\kappa_т} + L_{hh}{\alpha_p\over c_V\kappa_т},$

(6.33)

small Latin indices here adopt values {1,2} or variables $\{\hat n_1,\hat n_2\}$ ; subscript `h' corresponds to the heat mode. The ij-th partial dynamical structure factor can be written as follows:

$\displaystyle S_{ij}(\omega,k)$

=

$\displaystyle {TV\over2\rho v_s^2}\sum\limits _{\pm} {k^2{\Gamma}{\cdot}c_ic_j-... ..._ic_j-c_ic_j{\cdot}(\alpha+\Gamma) \big) \over(\omega\pm kv_s)^2+(k^2\Gamma)^2}$

(6.34)

+

$\displaystyle {k^2{\cal D}_2{\cdot}TV \over{\cal D}_3-{\cal D}_2}\cdot{ {\cal D... ...\alpha+{\cal D}_3)\big]/(\rho v_s^2)} -L_{ij} \over\omega^2 +(k^2{\cal D}_2)^2}$

+

$\displaystyle {k^2{\cal D}_3{\cdot}TV \over{\cal D}_2-{\cal D}_3}\cdot{ {\cal D... ...alpha+{\cal D}_2)\big]/(\rho v_s^2)} -L_{ij} \over\omega^2 +(k^2{\cal D}_3)^2},$

where $\rho=\bar m/V$ is a mass density, L_ij are the coefficients of (inter)diffusion between i-th and j-th component, L_ih are the coefficients of thermoparticle diffusion with i-th component, $\eta_l=L_{pp}={4\over3}\eta+\xi$ is a longitudinal viscosity, L_hh is a coefficient of thermodiffusion, ${\cal D}_i \equiv v_s D_i$ , $\Gamma\equiv v_s D_*$ . The first approximations of the dissipative modes (6.22)-(6.25) will be nonzero only for structure factors which involve momentum as one of their variables, as it can be easily proved from analysis of the first approximations of the weight coefficients (6.19)-(6.25). Thus, for example for Fourrier-components of two time correlation functions $F_{p,p}(k,\omega)$ and $F_{p, h}(k,\omega)$ , the second of them will involve nonzero linear contributions from the dissipative modes:

$\displaystyle F_{p,p}(k,\omega)/F_{p,p}(k)$

=

$\displaystyle \frac12\sum\limits _{+,-} {k^2 v_s D_* \pm k (\omega\mp kv_s)(\varphi _{p,p}-2D_*) \over (\omega\mp kv_s)^2 + (v_s D_*k^2)^2},$

(6.35)

$\displaystyle F_{p, h}(k,\omega)/F_{ h, h}(k)$

=

$\displaystyle \frac12\sum\limits _{+,-} {\pm k^2 v_s D_*\cdot \nu_{p, h} + k (\... ...\nu_{p, h}(\varphi _{p,p}-2D_*)\big) \over (\omega\mp kv_s)^2 + (v_s D_*k^2)^2}$

$\displaystyle + \sum\limits _{\mu=2}^3 {k \omega(\tilde g_1^\mu)_{p, h} \over \omega^2 + (v_s D_\mu k^2)^2},$

(6.36)

where $(\tilde g_1^\mu)_{p, h}$ are given by the equations (6.23), (6.25) for $\mu=2,3$ . Due to the separation of spin subsystem, the spin-spin dynamical structure factor will have the simple form:

$\begin{displaymath} F_{ s, s}(k,\omega)/F_{ s, s}={k^2 v_s D_1 \over \omega^2 + (v_s D_1 k^2)^2}. \end{displaymath}$

(6.37)

It is worth noticing that other correlation functions constructed on the variables {n₁,n₂,p,h} are similar to that for the binary mixture of simple liquids [30,31]. One can be easily convinced that all time correlation functions have the structure similar to one of three correlation functions given by expressions (6.28)-(6.37) and thus will not be written explicitly.

The dynamical structure factors for three-component mixture of nonmagnetic particles

Let us consider now simple three-component mixture, described by 5 parameters of abbreviated description: $\{\hat n_i, i=1..3; \hat p; \hat h\}$ , which are the partial densities of particle numbers, the momentum and the enthalpy, respectively. Here and further all Latin indices adopt values 1..3 or variables {n₁,n₂,n₃}, as we are interested in the dynamical structure factors, which actually correspond to the subsystem of densities from the parameters of abbreviated description. The sound damping coefficient ( $\Gamma\equiv v_s D_*$ ) and other damping coefficients ( ${\cal D}_i \equiv v_s D_i$ ) for this system can be derived from the expressions (3.11), (3.15), (3.16). The static correlation functions $S_{ij}\equiv S_{n_i,n_j}$ and the elements of frequency matrix, corresponding to the block of densities ( $\nu_{p,i}$ , $\nu_{i,p}$ , i=1..3) are generalizations of that for the previous case (see (6.1), (6.31)). Considering the first approximations of the matrices of weight coefficients (4.47), (4.48) we can conclude that they have the same structure as matrix $\tilde\nu$ and thus give no contribution to the dynamical structure factors. The dynamical structure factors for this system can be written as follows:

$\displaystyle S_{ij}(k,\omega)= {TV\over2\rho v_s^2} \sum\limits _\pm { \Gamma ... ...\mu=1}^3 {{\cal D}_\mu k^2{\cal G}^\mu_{ij}\over\omega^2+ ({\cal D}_\mu k^2)^2}$

(7.1)

where

$\begin{displaymath} \alpha={1\over\rho v_s^2}\left( \gamma_k{u_k\over\kappa_т}... ..._{N_{\bar l},V,T} \gamma_l +{\gamma_h^2\over T c_V} \right), \end{displaymath}$

(7.2)

$\gamma_k$ , $\gamma_h$ are given by the (6.33). Taking for example $\mu=1$ , the coefficients ${\cal G}^1_{ij}$ can be written as follows:

$\displaystyle {\cal G}^1_{ij}= {1\over({\cal D}_2-{\cal D}_1)({\cal D}_3-{\cal D}_1)}$

$\textstyle \bigg\{$

$\displaystyle {\cal D}_2{\cal D}_3 \left({\partial c_i\over\partial \mu_j}\righ... ...u_m\over\partial c_k}\right)_{N_{\bar k},V,T}L_{kj} +{L_{ih} L_{hj}\over T c_V}$

+

$\displaystyle { (c_i\gamma_j+\gamma_ic_j)(\alpha+{\cal D}_2+{\cal D}_3) +c_ic_j... ...l D}_3)] -c_i\delta_j-\delta_ic_j-\gamma_i\gamma_j \over \rho v_s^2 } \bigg\} ,$

(7.3)

where $\delta_j=L_{il} \left({\partial \mu_l/\partial c_k}\right)_{N_{\bar k},V,T} \gamma_k +{L_{ih}\gamma_h/ (Tc_N)}$ . The coefficients ${\cal G}^2_{ij}$ and ${\cal G}^3_{ij}$ can be derived from (7.3) by cyclic substitution of diffusion coefficients $\{{\cal D}_1$ , ${\cal D}_2$ , ${\cal D}_3\}$ . Summarizing obtained results, we see that while analysing hydrodynamic equations for the multicomponent mixtures, we have derived hydrodynamic excitation spectrum and time correlation functions. It was shown that hydrodynamic excitation spectrum includes two sound propagation modes and m diffusion modes, which are responsible for the interdiffusion, spin- and termodiffusion. Within the hydrodynamic approximation the exact analytical expressions for the time correlation functions were derived, the case of binary magnetic mixture in a paramagnetic state analyzed. This consideration is interesting from the experimental point of view because it gives mere dependance of dynamical structure factors on the thermodynamical functions.

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$\displaystyle \mbox{\rm i}\tilde\Omega(k)$	=	$\displaystyle (\mbox{\rm i}\hat L{\cdot}{\hat Y}(k),\hat Y(-k)) \ (\hat Y(k),\hat Y^+(-k))^{-1},$	(2.3)
$\displaystyle \tilde\Phi(k,z)$	=	$\displaystyle \left((1-{\cal P})\,\mbox{\rm i}\hat L{\cdot}\hat Y, \frac{1}{z+(... ...ox{\rm i}\hat L{\cdot}\hat Y^+\right) \left(\hat Y(k),\hat Y^+(-k)\right)^{-1},$	(2.4)

$\displaystyle \det$	$\textstyle \tilde B(\lambda,D)$	=0,	(3.3)
	$\textstyle \tilde B(\lambda,D)$	$\displaystyle \equiv\tilde\nu-\lambda+\delta{\cdot}(\tilde\varphi -D),$	(3.4)

$\displaystyle \det\tilde B(\lambda,D)$	$\textstyle \simeq$	$\displaystyle \det(\tilde\nu-\lambda)+ \delta\cdot\sum\limits _{i,j}(\varphi _{... ...\lambda,D)\over\partial (\delta{\cdot}\varphi _{ij})} \right\vert _{\delta=0} =$	(3.6)
	=	$\displaystyle \det(\tilde\nu-\lambda)+\delta\cdot\sum\limits _{i,j}(\varphi _{ij}-D\delta_{ij}) A_{ij}(\lambda) =0,$

		$\displaystyle z_i=0+\delta{\cdot}{D}_i+\delta^2{\cdot}\gamma_i,\quad\quad i=1,\dots,m,$	(3.19)
		$\displaystyle z_+=z_{m+1}=1+\delta{\cdot}{D}_*+\delta^2{\cdot} \gamma_+,$	(3.20)
		$\displaystyle z_-=z_{m+2}=-1+\delta{\cdot}{D}_*+\delta^2{\cdot}\gamma_- ,$	(3.21)

		$\displaystyle f(\tilde\nu)=f(1)\,\tilde G_\nu^++f(-1)\,\tilde G_\nu^-+f(0)\, \tilde G_\nu^0 ,$	(4.3)
		$\displaystyle f(\tilde T_\delta)=\sum\limits _{i=1}^{m+2}f(z_i)\,\tilde G_\delta^i.$	(4.4)

		$\displaystyle \tilde g_0^i{\cdot}\tilde g_0^j=\tilde g_0^i{\cdot}\delta_{ij},$	(4.8)
		$\displaystyle \tilde g_0^i{\cdot}\tilde g_1^j + \tilde g_1^i{\cdot}\tilde g_0^j = \tilde g_1^i{\cdot}\delta_{ij}.$	(4.9)
		$\displaystyle \dots\dots\dots$

		$\displaystyle \{\tilde\nu^s,\tilde\varphi \}\equiv\sum\limits _{k=0}^{s}\tilde\... ...phi +\tilde\nu^{s-1}\tilde\varphi \tilde\nu+\dots +\tilde\varphi \tilde\nu^{s},$	(4.11)
		$\displaystyle \{\tilde\nu^s,\tilde\varphi ^2\}\equiv\sum\limits _{k=0}^{s}\sum\... ...l=0}^{s-k}\tilde\nu^k \tilde\varphi \tilde\nu^l\tilde\varphi \tilde\nu^{s-k-l}.$	(4.12)

		$\displaystyle f(\tilde\nu)=\sum\limits _{i=1}^{m+2}f(\lambda_i){\cdot}\tilde g_0^i,$	(4.13)
		$\displaystyle \sum\limits _{k\geq1}\frac{f^{(k)}(0)}{k!}\{\tilde\nu^{k-1},\tild... ...eft[f(\lambda_i){\cdot}\tilde g_1^i+f'(\lambda_i)D_i{\cdot}\tilde g_0^i\right],$	(4.14)
		$\displaystyle \sum\limits _{k\geq2}\frac{f^{(k)}(0)}{k!}\{\tilde\nu^{k-2},\tild... ...bda_i)\gamma_i+\frac{f''(\lambda_i)}{2!}D_i^2\right){\cdot}\tilde g_0^i\right].$	(4.15)

		$\displaystyle \tilde g_0^+=\frac{\tilde\nu(\tilde\nu+1)}{2},\quad \tilde g_0^-=\frac{\tilde\nu(\tilde\nu-1)}2,$	(4.18)
		$\displaystyle \tilde P=1-\tilde\nu^2.$	(4.19)

$\displaystyle \{\tilde\nu^{2s},\tilde\varphi ^2\}$	=	$\displaystyle \{\tilde\nu^{2},\tilde\varphi ^2\}+(s-1) \left[2D_\{\tilde\nu^{2... ...nu^2\{\tilde\nu\tilde\varphi ^2\}\tilde\nu\right]+ 2D_^2(s-1)(s-2)\tilde\nu^2,$	(4.32)
$\displaystyle \{\tilde\nu^{2s-1},\tilde\varphi ^2\}$	=	$\displaystyle \{\tilde\nu,\tilde\varphi ^2\}+(s-1) \left[2D_\{\tilde\nu,\tilde... ...lde\nu\{\tilde\nu\tilde\varphi ^2\}\tilde\nu\right]+ 2D_^2(s-1)(s-2)\tilde\nu.$	(4.33)

$\displaystyle \tilde g_2^++\tilde g_2^-$	=	$\displaystyle \{\tilde\nu^{2},\tilde\varphi ^2\}-2\left[2D_\{\tilde\nu^{2}, \t... ...}+\tilde\nu^2\{\tilde\nu,\tilde\varphi ^2\}\tilde\nu\right]+12D_^2\tilde\nu^2,$	(4.34)
$\displaystyle \tilde C\equiv\tilde g_2^+-\tilde g_2^-$	=	$\displaystyle \{\tilde\nu,\tilde\varphi ^2\}-\frac32 \left[2D_\{\tilde\nu,\til... ...lde\nu\{\tilde\nu,\tilde\varphi ^2\}\tilde\nu\right] +\frac{15}2D_^2\tilde\nu.$	(4.35)

		$\displaystyle \tilde g_0^+{\cdot}\tilde\varphi {\cdot}\tilde g_0^+=D_+{\cdot}\t... ...ad \tilde g_0^-{\cdot}\tilde\varphi {\cdot}\tilde g_0^-=D_-{\cdot}\tilde g_0^-,$	(4.36)
		$\displaystyle \tilde g_0^j{\cdot}\tilde\varphi {\cdot}\tilde P=\hat P{\cdot}\tilde\varphi {\cdot}\tilde g_0^j= D_j{\cdot}\tilde g_0^j,$	(4.37)

		$\displaystyle \tilde g_0^+{\cdot}\tilde C\tilde g_0^-=\frac12(D_--D_+)\, \tilde... ...{\cdot}\tilde C\tilde g_0^+=\frac12(D_--D_+)\, \tilde g_0^-\tilde\varphi g_0^+,$	(4.42)
		$\displaystyle \tilde g_0^+{\cdot}\tilde C\tilde g_0^j=(D_j-D_+)\, \tilde g_0^+ ... ... g_0^j{\cdot}\tilde C\tilde g_0^+=(D_j-D_+)\, \tilde g_0^j \tilde\varphi g_0^+,$	(4.43)
		$\displaystyle \tilde g_0^-{\cdot}\tilde C\tilde g_0^j=(D_--D_j)\, \tilde g_0^- ... ... g_0^j{\cdot}\tilde C\tilde g_0^-=(D_--D_j)\, \tilde g_0^j \tilde\varphi g_0^-,$	(4.44)
		$\displaystyle \tilde g_0^j{\cdot}\tilde C\tilde g_0^k=\tilde g_0^j\tilde\varphi \tilde\nu\tilde\varphi \tilde g_0^k,$	(4.45)

$\displaystyle \tilde g_1^\pm=\pm\tilde g_0^\pm\tilde\varphi (\tilde P+\frac12\tilde g_0^\mp)\pm (\tilde P+\frac12\tilde g_0^\mp)\tilde\varphi \tilde g_0^\pm,$			(4.46)
$\displaystyle \tilde g_1^j=(\tilde R_j{\cdot}\tilde\varphi -1)\tilde\nu\tilde\v... ...e g_0^j+\tilde g_0^j\tilde\varphi \tilde\nu(\tilde\varphi {\cdot}\tilde R_j-1),$			(4.47)

		$\displaystyle \kappa_т = -{1\over V}\left({\partial V \over \partial P} \right)... ...\quad \alpha_p = {1\over V}\left({\partial V \over \partial T} \right)_{P,N,M},$	(5.2)
		$\displaystyle c_V=\frac TV\left({\partial S\over\partial T}\right)_{VN}, \quad c_P=\frac TV\left({\partial S\over\partial T}\right)_{PN},$	(5.3)
		$\displaystyle \gamma = c_P/c_V,$	(5.4)

		$\displaystyle \chi= {1\over V}\left(\partial \langle \hat S\rangle \over\partia... ...ght)_{NTV}, \quad \pi={1\over V}\left({\partial V\over\partial b}\right)_{PNT},$	(6.2)
		$\displaystyle c_i= N_i/V, \qquad u_i= \left({\partial V\over\partial N_i}\right)_{TPbN_{\bar\imath}}$	(6.3)

		$\displaystyle D_1=\varphi _{ s, s},$	(6.10)
		$\displaystyle D_2=-\frac12b+\frac12\sqrt{b^2-4c},$	(6.11)
		$\displaystyle D_3=-\frac12b-\frac12\sqrt{b^2-4c},$	(6.12)

$\displaystyle (\tilde g^\pm_0)_{i,j}=\frac12\nu_{i,p}\nu_{p,j},\quad (\tilde g^\pm_0)_{ s, J}=(\tilde g^\pm_0)_{J, s}=0,$			(6.14)
$\displaystyle (\tilde g^\pm_0)_{p, i}=\pm\frac12\nu_{p,i}, \quad (\tilde g^\pm_0)_{i,p}=\pm\frac12\nu_{i,p},\quad (\tilde g^\pm_0)_{p, p}=\frac12$			(6.15)

		$\displaystyle (\tilde g^2_0)_{i,j}=(D_3-D_2)^{-1}\left(D_3\delta_{ij} +\nu_{i,p... ...rphi _{p,p}-2D_*-D_3) +\nu_{i,p}\xi'_j +\xi_i\nu_{p,j} -\varphi _{i,j} \right),$	(6.17)
		$\displaystyle (\tilde g^3_0)_{i,j}=(D_2-D_3)^{-1}\left(D_2\delta_{ij} +\nu_{i,p... ...rphi _{p,p}-2D_*-D_2) +\nu_{i,p}\xi'_j +\xi_i\nu_{p,j} -\varphi _{i,j} \right).$	(6.18)

		$\displaystyle (\tilde g^\pm_1)_{i,j}=\pm \frac12\left(\nu_{i,p}\xi'_j +\xi_i\nu_{p,j} +\nu_{i,p},\nu_{p,j}(\varphi _{p,p}-3D_*)\right),$	(6.19)
		$\displaystyle (\tilde g^\pm_1)_{p, j}=\frac12\left(\xi'_j +\nu_{p,j}(\varphi _{... ...tilde g^\pm_1)_{i,p}=\frac12\left(\xi_i +\nu_{i,p}(\varphi _{p,p}-2D_*)\right),$	(6.20)
		$\displaystyle (\tilde g^\pm_1)_{p,p}=\pm\frac12(\varphi _{p,p}-2D_*),\quad (\tilde g^\pm_1)_{ s, J}=(\tilde g^\pm_1)_{I, s}=0$	(6.21)

		$\displaystyle (\tilde g^2_1)_{i,p}= (D_{3}-D_{2})^{-1} \left(D_{2}\left[\nu_{i,... ... _{p,p}+\varphi _{i,i}) +\xi_i -\varphi _{i,i}\nu_{i,p} \right]+\eta_i \right),$	(6.22)
		$\displaystyle (\tilde g^2_1)_{p, j}= (D_{3}-D_{2})^{-1} \left(D_{2}\left[\nu_{p... ...+\xi'_j -\nu_{p,j}\varphi _{i,i} \right]+\eta'_i \right),\mbox{\hspace*{0.3cm}}$	(6.23)
		$\displaystyle (\tilde g^3_1)_{i,p}= (D_{2}-D_{3})^{-1} \left(D_{3}\left[\nu_{i,... ... _{p,p}+\varphi _{i,i}) +\xi_i -\varphi _{i,i}\nu_{i,p} \right]+\eta_i \right),$	(6.24)
		$\displaystyle (\tilde g^3_1)_{p, j}= (D_{2}-D_{3})^{-1} \left(D_{3}\left[\nu_{p... ...{p,p}+\varphi _{j,j}) +\xi'_j -\nu_{p,j}\varphi _{i,i} \right]+\eta'_i \right),$	(6.25)

$\displaystyle \eta_{ n_2}= -\varphi _{{ n_{{2}}, n_{{1}}}}\nu_{{ n_{{1}}},p}\va... ..._{{2}}, h}}-\nu_{{ n_{{2}}},p}\varphi _ {{ h, n_{{1}}}}\varphi _{{ n_{{1}}, h}}$
$\displaystyle +\nu_{{ n_{{2}}},p}\varphi _{{ n_{{1}}, n_{ {1}}}}\varphi _{{ h, ... ...{{1} }, h}}-\nu_{{ h},p}\varphi _{{ n_{{2}}, h}}\varphi _{{ n_{{1}}, n_{{1}}}}.$

$\displaystyle {\rm Re}\; \tilde F(k,\omega)_{i,j} = \sum\limits _{+,-} {k^2 v_s... ...(\tilde g_1^{\pm}\tilde F_0)_{i,j} \over (\omega\mp kv_s)^2 + (v_s D_*k^2)^2} +$
$\displaystyle +\sum\limits _{\mu=1}^3 {k^2 v_s D_\mu\cdot(\tilde g_0^\mu\tilde ... ...+ k \omega(\tilde g_1^\mu\tilde F_0)_{i,j} \over \omega^2 + (v_s D_\mu k^2)^2},$			(6.27)

$\displaystyle (\tilde g_{0}^\pm\tilde F_0)_{ n_1, n_1}$	=	$\displaystyle \frac12\nu_{ n_1,p}\nu_{p, n_1}{\cdot}S_{ n_1, n_1} +\frac12 \nu_{ n_1,p}\nu_{p, n_2}{\cdot}S_{ n_2, n_1},$	(6.28)
$\displaystyle (\tilde g_{0}^2\tilde F_0)_{ n_1, n_1}$	=	$\displaystyle (D_3-D_2)^{-1}\big( D_3 +\nu_{ n_1,p}\nu_{p, n_1}(\varphi _{p,p}-2D_*-D_3)$
		$\displaystyle \left.+\nu_{ n_1,p}\xi'_{ n_1} +\xi_{ n_1}\nu_{p, n_1} -\varphi _{ n_1, n_1} \right){\cdot}S_{ n_1, n_1}$	(6.29)
		$\displaystyle +(D_3-D_2)^{-1}\big( D_3 +\nu_{ n_1,p}\nu_{p, n_2}(\varphi _{p,p}-2D_*-D_3)$
		$\displaystyle \left.+\nu_{ n_1,p}\xi'_{ n_2}+\xi_{ n_1}\nu_{p, n_2} -\varphi _{ n_1, n_2} \right){\cdot}S_{ n_2, n_1},$
$\displaystyle (\tilde g_{1}^\pm\tilde F_0)_{n_1, n_1}$	=	$\displaystyle \pm \frac12\left(\nu_{ n_1,p}\xi'_{ n_1} +\xi_{ n_1}\nu_{p,n_1} +\nu_{ n_1,p}\nu_{p, n_1}(\varphi _{p,p}-3D_*)\right){\cdot} S_{ n_1, n_1}$
		$\displaystyle \pm \frac12\left(\nu_{ n_1,p}\xi'_{ n_2} +\xi_{ n_1}\nu_{p, n_2} +\nu_{ n_1,p}\nu_{p, n_2}(\varphi _{p,p}-3D_*)\right){\cdot} S_{ n_2, n_1},$	(6.30)

		$\displaystyle \alpha=2\Gamma-\eta_l/\rho,$	(6.32)
		$\displaystyle \gamma_k=L_{kl}{u_l\over\kappa_т} + L_{kh}{\alpha_p\over c_V\kappa_т}, \quad\gamma_h=L_{hl}{u_l\over\kappa_т} + L_{hh}{\alpha_p\over c_V\kappa_т},$	(6.33)

$\displaystyle S_{ij}(\omega,k)$	=	$\displaystyle {TV\over2\rho v_s^2}\sum\limits _{\pm} {k^2{\Gamma}{\cdot}c_ic_j-... ..._ic_j-c_ic_j{\cdot}(\alpha+\Gamma) \big) \over(\omega\pm kv_s)^2+(k^2\Gamma)^2}$	(6.34)
	+	$\displaystyle {k^2{\cal D}_2{\cdot}TV \over{\cal D}_3-{\cal D}_2}\cdot{ {\cal D... ...\alpha+{\cal D}_3)\big]/(\rho v_s^2)} -L_{ij} \over\omega^2 +(k^2{\cal D}_2)^2}$
	+	$\displaystyle {k^2{\cal D}_3{\cdot}TV \over{\cal D}_2-{\cal D}_3}\cdot{ {\cal D... ...alpha+{\cal D}_2)\big]/(\rho v_s^2)} -L_{ij} \over\omega^2 +(k^2{\cal D}_3)^2},$

$\displaystyle F_{p,p}(k,\omega)/F_{p,p}(k)$	=	$\displaystyle \frac12\sum\limits _{+,-} {k^2 v_s D_* \pm k (\omega\mp kv_s)(\varphi _{p,p}-2D_) \over (\omega\mp kv_s)^2 + (v_s D_k^2)^2},$	(6.35)
$\displaystyle F_{p, h}(k,\omega)/F_{ h, h}(k)$	=	$\displaystyle \frac12\sum\limits _{+,-} {\pm k^2 v_s D_\cdot \nu_{p, h} + k (\... ...\nu_{p, h}(\varphi _{p,p}-2D_)\big) \over (\omega\mp kv_s)^2 + (v_s D_*k^2)^2}$
		$\displaystyle + \sum\limits _{\mu=2}^3 {k \omega(\tilde g_1^\mu)_{p, h} \over \omega^2 + (v_s D_\mu k^2)^2},$	(6.36)

$\displaystyle {\cal G}^1_{ij}= {1\over({\cal D}_2-{\cal D}_1)({\cal D}_3-{\cal D}_1)}$	$\textstyle \bigg\{$	$\displaystyle {\cal D}_2{\cal D}_3 \left({\partial c_i\over\partial \mu_j}\righ... ...u_m\over\partial c_k}\right)_{N_{\bar k},V,T}L_{kj} +{L_{ih} L_{hj}\over T c_V}$
	+	$\displaystyle { (c_i\gamma_j+\gamma_ic_j)(\alpha+{\cal D}_2+{\cal D}_3) +c_ic_j... ...l D}_3)] -c_i\delta_j-\delta_ic_j-\gamma_i\gamma_j \over \rho v_s^2 } \bigg\} ,$	(7.3)