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

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

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)

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

(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)

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

(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:

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 $\tilde T_\delta$ :

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

(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)

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)):

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

(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)

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

(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:

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:

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

(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

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

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:

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]):

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)

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

(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)

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:

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:

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)

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

(4.23)

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

(4.24)

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

(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)

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

(4.27)

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

(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)

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

(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)

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

(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)

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

(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 interpo

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

Introduction

Equations of Molecular Hydrodynamics

Hydrodynamic Excitation Spectrum

Time-Dependent Correlation Functions