On the statistical hydrodynamics for a binary mixture of magnetic and nonmagnetic particles

On the statistical hydrodynamics for a binary mixture of magnetic and nonmagnetic particles
Mryglod I.M., Rudavskii Yu.K., Tokarchuk M.V., Batsevych O.F.

Dynamic properties of a binary mixture of magnetic and nonmagnetic particles are considered with the help of the method of nonequilibrium statistical operator. The generalized hydrodynamic equations are derived and analyzed. On this basis hydrodynamic collective modes spectrum was calculated. The expressions for sound velocity and damping coefficients of collective hydrodynamic modes are found. We propose the consistent scheme for calculation of time correlation functions in the hydrodynamic limit and discuss the expressions found for them in paramagnetic case.

Introduction

Magnetic liquids, mixtures of magnetic and nonmagnetic particles in the external fields of mechanical or electromagnetic origin, have already taken their significant place in chemical, electronic and other modern technologies. That is why the investigations of the thermodynamical, structural and dynamical properties of liquid magnets are very actual for more deep understanding and forecasting of their behaviour [1,2,3].

The investigation of time-dependent correlation functions as well as generalized transport coefficients of a liquid mixture of magnetic and nonmagnetic particles give us possibility of deep insight into the processes in the systems with coupled classical and quantum peculiarities. One of the most interesting problems is investigation of the behaviour of hydrodynamic collective modes, which describe the properties of heat, sound, and mass fluctuations. From the experimental point of view the investigation of these phenomena as well as effects of ``speed'' sound, spin waves and their dependance on temperature, concentration are very valuable. The explicit information about the volume features of mixtures of magnetic and nonmagnetic particles, molecules is essential in studying of their interaction with active surfaces (metals, semiconductors), where the processes of metallic arrangement, adsorption, desorption, chemical transformation can take place. Another important aspect of this problem is the derivation of expressions for dynamic structure factors. It is known, that these functions can be extract from neutron scattering experiments. Such theoretical study should be based on the statistical approach, on the equations of generalized hydrodynamics, particularly. Similar approach was applied to one-component magnetic mixture [4,5,6,7,8]. The collective modes of Heisenberg ferrofluid were also considered in [8].

Statistical hydrodynamics for a mixture of magnetic and nonmagnetic particles in an external nonhomogeneous magnetic field ${\bf B}({\bf r};t)$ was studied in [9]. There was formulated the problem of derivation of generalized hydrodynamic equations for magnetic and nonmagnetic subsystems in the external nonhomogeneous magnetic field with the help of nonequilibrium statistical operator method for description both strong and weak nonequilibrium states. Magnetic and nonmagnetic subsystems were characterized by individual nonequilibrium thermodynamics parameters. As a result nonequilibrium thermodynamical relations and generalized equations of hydrodynamics were derived.

This paper is dedicated to the study of a binary magnetic mixture consisting of magnetic and nonmagnetic particles within the method of nonequilibrium statistical operator. From the physical point of view this case is more interesting then that [4,5,6,7,8] of the one-component mixtures, because for the great ratios of masses of different component particles it can be treated as a ferrocoloid mixture. On the base of a set consisting of five parameters of abbreviated description, for the weak nonequilibrium case the equations of generalized hydrodynamics are derived. For the microscopic quantities which make up a set of abbreviated description the expressions of currents were obtained and their contribution to the thermodynamical quantities and kinetic coefficients, determining dynamics of the system calculated. The calculation of hydrodynamic collective modes are carried out with the help of the perturbation theory, yielding two sound-propagation modes and three purely diffusive modes. The expression for sound velocity derived generalize that of a singlecomponent ferrofluid [8]. In the chapter 2 we give briefly the main relations of the method of nonequilibrium statistical operator [10], which will be used for finding the equations of generalized hydrodynamics for the mixture of magnetic and nonmagnetic particles. The definition of the Hamiltonian of the system will be given in the next chapter. The chapter 4 is dedicated to finding of the static correlation functions in the limit $k\to0$ . This gives us possibility to find the matrices of frequencies and memory functions in the Markovian approximation. In the sixth chapter the collective excitation spectrum in the hydrodynamic limit is derived and analyzed. In the chapter 7 the problem of calculation of time-correlation functions is considered. We propose here the scheme which allows to calculate the weight coefficients describing a partial contribution of each mode to the hydrodynamic time correlation functions. In some limiting cases (pure non-magnetic fluid, simple magnetic liquid, etc.) the obtained results are discussed in comparison with the results known in the literature.

Theoretical framework: Nonequilibrium statistical operator method

where classical part of ${\rm i}\hat L\,$ is determined as a Poisson brackets of the function $\rho$ with a classical part of the Hamiltonian of the system and the quantum one - as a commutator with its quantum part, $\rho$ is a function of phase variables $x^N=\left\{{\boldsymbol r},{\bf p}, \hat{\boldsymbol s}\right\}^N$ , where N is a total number of particles.

Following Zubarev's method of nonequilibrium statistical operator [10] we can rewrite equation (2.1) in the form:

$\begin{displaymath} \left(\frac{\partial}{\partial t}+{\rm i}\hat L\,\right)\rho(x^N)=-\varepsilon (\rho(x^N)-\rho_q(x^N)), \end{displaymath}$

(2.2)

where $\varepsilon \to0$ after the thermodynamical limit, $\rho_q$ is so-called quasi-equilibrium statistical operator. Nonzero right-hand side of the equation (2.2) imposes the boundary conditions, which destroy the time reversal symmetry of the Liouville equation and choose the retarded solutions which correspond to the ideas of abbreviated description of the nonequilibrium state of the system. To restrict our consideration to the set of slow physical quantities $\left\{\hat P_\alpha\right\}$ which are thought to determine nonequilibrium state (the set of conserved quantities). The Gibbs-like statistical operator $\rho_q$ can be found from the extremum problem of information entropy with fixed parameters of abbreviated description $\langle \hat P_\alpha \rangle ^t$ and normalizing condition $\mathop{\rm Sp}\nolimits \rho_q(t)=1$ :

where the parameters $\{F_\alpha(t)\}$ are defined from the self-consistency conditions:

$\begin{displaymath} \langle \hat P_\alpha \rangle ^t = \langle \hat P_\alpha \ra... ...\rm Sp}\nolimits \left( \hat P_\alpha \rho_q (x^N,t) \right). \end{displaymath}$

(2.4)

The index $\alpha=\{i,{\boldsymbol k}\}$ denotes a combination of discrete index i which numerates the variables and wave vector k so, that summation in (2.3) means: $\sum\limits _\alpha\ldots = \sum\limits _i\sum\limits _{{\boldsymbol k}}\ldots,$ $\langle \dots\rangle ^t=\mathop{\rm Sp}\nolimits (\dots\rho(t))$ . Taking into account projecting, the formal solution of equation (2.2) can be written [10]:

$\displaystyle \rho(t)=\rho_q(t)+\sum_\alpha \int\limits _{-\infty}^t dt'\, e^{-\varepsilon (t-t')} F_\alpha (t')T (t,t')$

$\displaystyle \times \int\limits _0^1 d\tau \rho^\tau_q (t') \hat I_\alpha (t') \rho^{1-\tau}_q (t'),$

(2.5)

are the generalized fluxes, $\dot{\hat P}_\alpha \equiv {\rm i}\hat L\, \hat P_\alpha.$

$\begin{displaymath} T(t,t') = exp_+\left\{- \int\limits _{t'}^t d\tau (1-{\mbox{$\cal P$}} (\tau)) {\rm i}\hat L\, (\tau) \right\} \end{displaymath}$

(2.7)

is the operator of time evolution with the Kawasaki-Ganton projection operator ${\mbox{$\cal P$}}_q(t)$ , which is connected with the generalized Mori projection operator ${\mbox{$\cal P$}}(t)$ by the relation:

$\begin{displaymath} {\mbox{$\cal P$}} (t) \ldots = \langle \ldots \rangle + \sum... ...\{\hat P_\alpha - \langle \hat P_\alpha \rangle ^t \right\}. \end{displaymath}$

(2.8)

which have the following properties: $\mbox{$\cal P$}(t) \hat P_\alpha = \hat P_\alpha$ , $\mbox{$\cal P$}(t) \mbox{$\cal P$}(t') = \mbox{$\cal P$}(t)$ , $\mbox{$\cal P$}(t)(1-\mbox{$\cal P$}(t')) = 0$ .

Statistical operator (2.5) determines generalized transport equations in the form [10]:

$\begin{displaymath} \frac{\partial}{\partial t}\langle \hat P_\alpha \rangle ^t ... ...ilon (t-t')} \phi_{\alpha\beta} (t,t') F_\beta (t') dt' \, , \end{displaymath}$

(2.9)

$\begin{displaymath} \phi_{\alpha\beta} (t,t') = \int\limits _0^1 d\tau \, \langl... ...o_q^\tau \; \hat{I}_\beta (t') \, \rho_q^{-\tau} \rangle _q^t \end{displaymath}$

(2.10)

are the generalized memory functions, or generalized transport kernels. Equations (2.9), (2.10) and (2.4) make up a closed system of nonlinear equations which describe both strong and weak nonequilibrium.

Now we will consider weak nonequilibrium, which allows us to linearize system (2.4), (2.9), (2.10). For the small deviations

$\begin{displaymath} \delta\hat P_\alpha(t)=\langle \hat P_\alpha\rangle ^t-\langle \hat P_\alpha\rangle _0 \end{displaymath}$

(2.11)

of averages $\langle \hat P_\alpha\rangle ^t=\mathop{\rm Sp}\nolimits \hat P_\alpha\rho(t)$ from the equilibrium values $\langle \hat P_\alpha\rangle _0=\mathop{\rm Sp}\nolimits \hat P_\alpha \rho_0(x^N)$ , where

$\begin{displaymath} \rho_0(x^N)=exp\left\{-\Phi_0-\sum_\alpha \hat P_\alpha F_\alpha^0\right\}, \end{displaymath}$

(2.12)

is the equilibrium statistical operator, deviations of the intensive quantities $\delta F_n(t)=F_n(t)-F_n^0$ can be easily found from the self-consistency conditions (2.4). In matrix form:

$\displaystyle \vert\vert(\Delta\hat P,\Delta\hat P^+)\vert\vert _{ij}=(\Delta\hat P_i,\Delta\hat P_j),$

(2.14)

$\displaystyle \Delta\hat P_i=\hat P_i-\langle \hat P_i\rangle ,$

$\begin{displaymath} (\hat A,\hat B) = \int\limits _0^1 \langle \hat A \rho_0^\tau \hat B \rho_0^{-\tau} \rangle \, d\tau, \end{displaymath}$

(2.15)

which transforms into simple average $\langle \hat A\hat B\rangle _0$ in classical case. Now we can rewrite (2.5) in the form

$\displaystyle \delta \rho (t) = \delta \rho_q (t) + \sum_\alpha$

$\displaystyle \int\limits _{-\infty}^t dt'\, e^{-\varepsilon (t-t')} \; \delta F_\alpha (t') \;T_0 (t-t')$

$\displaystyle \times \int\limits _0^1 d\tau \;\rho_0^\tau \;(1 - {\mbox{$\cal P$}}) \; \dot{\hat P}_\alpha \;\rho^{1-\tau}_0 ,$

(2.16)

$\begin{displaymath} {\mbox{$\cal P$}} \ldots = (\ldots \, , \Delta\hat P^+)(\Delta\hat P,\Delta\hat P^+)^{-1}\Delta\hat P. \end{displaymath}$

(2.17)

$\begin{displaymath} \frac{\partial}{\partial t}\langle \Delta\hat P\rangle ^t=\langle {\rm i}\hat L\,\hat P\rangle ^t, \end{displaymath}$

(2.18)

$\begin{displaymath} \left({\rm i}\omega-{\rm i}\Omega+\phi_\varepsilon (\omega)\right)\Delta\hat P(\omega)=0, \end{displaymath}$

(2.19)

$\begin{displaymath} {\rm i}\Omega = (\dot{\hat P} , \Delta\hat P^+ ) (\Delta\hat P, \Delta\hat P^+)^{-1}, \end{displaymath}$

(2.20)

$\displaystyle \phi_\varepsilon (\omega) = \left( (1-{\mbox{$\cal P$}}) \dot{\ha... ...mbox{$\cal P$}}) {\rm i}\hat L\, } (1-{\mbox{$\cal P$}}) \dot{\hat P}^+ \right)$

$\displaystyle \times ( \Delta\hat P , \Delta\hat P^+ )^{-1}$

(2.21)

are the matrices of frequencies and memory functions. The matrix equation for the Laplace-transforms $(\Delta\hat P, \Delta\hat P^+)^z$ of time correlation functions

$\begin{displaymath} (\Delta\hat P, \Delta\hat P^+)^z = \left(\Delta\hat P, { 1 \over z + {\rm i}\hat L\, } \Delta\hat P^+ \right), \end{displaymath}$

(2.22)

$\begin{displaymath} \left\{z -{\rm i}\Omega + \phi_\varepsilon (z) \right\}(\Delta\hat P, \Delta\hat P^+)^z = (\Delta\hat P, \Delta\hat P^+) . \end{displaymath}$

(2.23)

Retarded correlation Green functions can be expressed in terms of time correlation functions $(\Delta\hat P,\Delta\hat P^+)^t$ (which are connected with $(\Delta\hat P, \Delta\hat P^+)^z$ by Laplace transformation):

Hence, the poles of the retarded Green functions, which give the spectrum of collective modes are determined by matrix equation

Model Hamiltonian and hydrodynamic variables

Let us consider the system consisting of N₁ nonmagnetic and N₂ magnetic particles posed in external magnetic field. Hamiltonian of such a system can be written as in [12,13], taking into account the interaction with nonmagnetic subsystem:

Here and further subscripts ${}_1\,,\,\,{}_2$ or superscripts in parentheses ${}^{(1)}\,,\,\,{}^{(2)}$ indicate nonmagnetic and magnetic subsystem, respectively. Thus H₁ and $\hat H_2$ are the Hamiltonians of nonmagnetic and magnetic subsystems separately, H_int describes their interaction and $\hat H_{ex}$ is the energy of spin interaction with external magnetic field. The Hamiltonian H₁ of nonmagnetic subsystem can be taken in classical form

$\begin{displaymath} H_1 = \sum\limits _{j=1}^{N_1} { \mbox{${\mbox{\boldmath$p$}... ...imits _{j \neq l}^{N_1, N_1} V^{(11)} ({\boldsymbol r}_{jl}), \end{displaymath}$

(3.2)

where $\mbox{${\mbox{\boldmath$p$}}_{j}^{(1)}$}$ is the momentum of nonmagnetic particle, V⁽¹¹⁾ (r_jl) is the potential of interaction between two nonmagnetic particles j and l, which can be chosen for calculations in any convenient form; and m₁ is a mass of nonmagnetic particle. The term $\hat H_2$

$\begin{displaymath} \hat H_2 = H_{2L} - {1 \over 2} \sum\limits _{j \neq l}^{N_2... ...mbol r}_{jl}) (\hat{\boldsymbol s}_j, \hat{\boldsymbol s}_l) \end{displaymath}$

(3.3)

consists of classical, ``liquid'' part H_2L which has the same form as H₁ and quantum part, which describes spin subsystem and can be taken in Heisenberg-like form. Other terms in (3.1) could be written in the form

$\displaystyle H_{int} = \sum\limits _{i,j}^{N_1,N_2} V^{(12)} ({\boldsymbol r}_... ...N_2} \hat{\boldsymbol s}_i {\cdot} \hat{\boldsymbol B} ( {\boldsymbol r}_i, t),$

(3.4)

where V⁽¹²⁾ (r_ij) is the potential of interaction between i-th nonmagnetic and j-th magnetic particle, $\hat{\boldsymbol B} ( {\boldsymbol r}_i, t)$ - external magnetic field. Liouville operator which corresponds to Hamiltonian (3.1) can be written as follows:

$\begin{displaymath} {\rm i}\hat L\, = {\rm i}L_1 + {\rm i}\hat L\,_2 + {\rm i}L_{int} + {\rm i}\hat L\,_{ex} + {\rm i}\hat L\,_s, \end{displaymath}$

(3.5)

$\displaystyle {\rm i}L_1 = \sum\limits _i^{(1)} { {\mbox{\boldmath$p$}}_{i}^{(1... ...{i}^{(1)}} - {\partial \over \partial {\mbox{\boldmath$p$}}_{j}^{(1)}} \right),$

(3.6)

$\displaystyle {\rm i}\hat L\,_2 = {\rm i}L_{(1 {\to} 2)} + {1\over 2} \sum\limi... ...{i}^{(2)}} - {\partial \over \partial {\mbox{\boldmath$p$}}_{j}^{(2)}} \right),$

(3.7)

$\displaystyle {\rm i}L_{int} = - \sum\limits _{i , j}^{(1,2)} {\partial \over \... ...{i}^{(1)}} - {\partial \over \partial {\mbox{\boldmath$p$}}_{j}^{(2)}} \right),$

(3.8)

$\displaystyle {\rm i}\hat L\,_{ex} = \sum\limits _i^{(2)} {\partial \left(\hat{... ...ymbol r}_i} {\cdot} {\partial \over \partial {\mbox{\boldmath$p$}}_{i}^{(2)} },$

(3.9)

and ${\rm i}\hat L\,_s$ is a purely quantum part of Liouville operator, it is determined as a commutator

$\begin{displaymath} {\rm i}\hat L\,_s{\cdot}\hat A = {{\rm i}\over\hbar} \left[-... ...\cdot} \hat{\boldsymbol s}_i \right)\, , \,\, \hat A \right]. \end{displaymath}$

(3.10)

To study the dynamics near the equilibrium, we have to consider all the conserved quantities and most slow ones associated with them. For our model five parameters of abbreviated description $\{\hat P_i\}$ , i=1..5 can be chosen, namely: partial densities of particle number $\hat n_1({\boldsymbol r})$ , ${\hat n_2}({\boldsymbol r})$ , densities of momentum $\mbox{\boldmath$\hat p$}({\boldsymbol r})$ , magnetization $\hat{\boldsymbol m}({\boldsymbol r})$ and total energy $\hat{\varepsilon}({\boldsymbol r})$ . After the Fourier transformation ( $f({\boldsymbol k})=\int dr\, f({\boldsymbol r}) e^{{\rm i}{\boldsymbol k}{\boldsymbol r}}$ ), these values can be written as follows:

$\displaystyle \hat n_1({\boldsymbol k}) = \sum\limits _i^{(1)} e^{{\rm i}{\bold... ...symbol k}) = \sum\limits _i^{(2)} e^{{\rm i}{\boldsymbol k}{\boldsymbol r}_i} ,$

(3.11)

$\displaystyle \mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}) = \sum\limits _i... ...{\boldmath$p$}}_{i}^{(2)}$}^\alpha e^{{\rm i}{\boldsymbol k}{\boldsymbol r}_i},$

(3.12)

$\displaystyle \hat{\boldsymbol m}^\alpha({\boldsymbol k}) = \sum\limits _i^{(2)} \hat{\boldsymbol s}_i^\alpha e^{{\rm i}{\boldsymbol k}{\boldsymbol r}_i},$

(3.13)

$\displaystyle \hat{\varepsilon}({\boldsymbol k}) =\sum\limits _i^{(1)} \hat{\va... ...ol r}_i} %%+ \suml_i^{(2)} \lp \hat{\boldsymbol B} ({\boldsymbol r}_i,t) \, ,$

(3.14)

$\displaystyle \hat{\varepsilon}_i^{(1)} = { \mbox{${\mbox{\boldmath$p$}}_{i}^{(... ..._{ij} ) + {1\over 2} \sum\limits _{j} ^{(2)} V^{(12)} ( {\boldsymbol r}_{ij} ),$

(3.15)

$\displaystyle \hat{\varepsilon}_i^{(2)} = \hat{\varepsilon}_i^{(1 {\to} 2)} - {... ...(2,2)} J ({\boldsymbol r}_{ij}) (\hat{\boldsymbol s}_i, \hat{\boldsymbol s}_j).$

(3.16)

For our set of variables $\hat P_i$ , the quantum equations of motion have the following form (see Appendix)

$\displaystyle \dot{\hat P}_i({\boldsymbol k}) = {\rm i}{\boldsymbol k}^\alpha J_i^\alpha ({\boldsymbol k}) + R_i ({\boldsymbol k}),$

(3.17)

$\displaystyle \dot{\hat P}_l^\alpha ({\boldsymbol k}) = {\rm i}{\boldsymbol k}^\beta J_l^{\alpha \beta} ({\boldsymbol k}) + R_l^\alpha ({\boldsymbol k}),$

(3.18)

for the scalar $\{ \hat n_1({\boldsymbol k}),\, {\hat n_2}({\boldsymbol k}),\, \hat{\varepsilon}({\boldsymbol k}) \}$ and vector $\{ \mbox{\boldmath$\hat p$}({\boldsymbol k}) , \, \hat{\boldsymbol m}({\boldsymbol k}) \}$ variables respectively. Terms R_i (k), $R_l^\alpha ({\boldsymbol k})$ appeared due to nonhomogeneous external magnetic field. When we assume that $\hat{\boldsymbol B}({\boldsymbol r},t)$ is homogeneous, these terms disappear and variables $\{\hat P\}$ become conserved.

Static correlation functions

When we assume the external magnetic field is static, homogeneous and directed along the '0z' axis, $\hat{\boldsymbol B}({\boldsymbol r},t) = (0,0,b)$ , the '0z' component of magnetization density $\hat m ({\boldsymbol k}) \equiv \hat{\boldsymbol m}^z({\boldsymbol k})$ becomes conserved, see A.10. That is why for such conditions $\hat m$ can be chosen as a parameter of abbreviated description for the hydrodynamic region. So, for analysis of the generalized hydrodynamic equations (2.19) one needs to calculate the static correlation functions constructed on the variables

$\begin{displaymath} \hat P({\boldsymbol k})=\{\hat n_1({\boldsymbol k}),{\hat n... ...hat{m}({\boldsymbol k}),\hat{\varepsilon}({\boldsymbol k})\} \end{displaymath}$

(4.1)

Let us define static correlation function $(\hat a, \hat b)$ as an average of deviations:

$\begin{displaymath} (\hat a, \hat b) = \int\limits _0^1 d\tau \langle \Delta \hat a \rho_0^\tau \Delta \hat b \rho_0^{-\tau } \rangle \, , \end{displaymath}$

(4.2)

contrary to (2.15), where it was defined as a simple average. In order to give some thermodynamical interpretation of correlation functions we chose the equilibrium statistical operator (2.12) as a Gibbs distribution for the grand canonic ensemble $(\mu,\, V,\, T , \, {b} )$ , $\mu=\{\mu_1,\mu_2\}$ :

$\displaystyle \rho_0 = \exp{[\beta(\Omega-{\hat\omega})]} ,$

(4.3)

$\displaystyle {\hat\omega} \equiv \hat{\varepsilon} - \mu_1\hat n_1 - \mu_2{\hat n_2} -{b}\hat{m},$

(4.4)

$\displaystyle \Omega = \Omega(\mu,{b},T),$

(4.5)

where $\Omega$ is the thermodynamical potential, $\beta$ is the inverse temperature, $\hat n_1,\,...\,,\hat{\varepsilon}$ are the quantities (4.1), taken with k=0: $\hat n_1=\hat n_1({\boldsymbol k}=0),\,...\,,\hat{\varepsilon}=\hat{\varepsilon}({\boldsymbol k}=0)$ ; and b is an '0z' component of the external magnetic field.

For an arbitrary operator $\hat a$ and parameter $\gamma$ it is easy to prove the equation

$\begin{displaymath} {\partial \langle \hat a\rangle \over \partial \gamma} = - \... ... { \partial (\beta\hat\omega) \over \partial \gamma} \right), \end{displaymath}$

(4.6)

where the average $\langle \ldots\rangle _0$ is performed with the distribution (4.3). So, for example, if $\hat a=\hat n_1$ , $\gamma=\mu_1$ , we get:

Here and further the quantity, written by a capital letter denotes the average value of corresponding operator written by a small letter, for instance, $N_1=\langle \hat n_1\rangle _0$ . In the same manner with the help of (4.6) we can connect other correlation functions with thermodynamical quantities:

$\displaystyle (\hat{n}_i, \hat{n}_j)={1\over\beta} \left({\partial N_i \over \p... ... \left({\partial N_j \over \partial \mu_i} \right)_{T,b} , \quad i,j=1,2 \quad,$

(4.8)

$\displaystyle (\hat{n}_i,\hat m) = {1\over\beta} {\partial N_i \over \partial b} = {1\over\beta} \left({\partial M \over \partial \mu_i} \right)_{T,b},$

(4.9)

$\displaystyle (\hat m,\hat m)={1\over\beta}\left({\partial M \over \partial b} \right)_{T,\mu},$

(4.10)

$\displaystyle (\hat{\varepsilon},\hat m) = {1\over\beta}\left({\partial E\over\partial b}\right)_{T,\mu},$

(4.11)

As we see in (4.8)-(4.11), the set of $\hat P_i({\boldsymbol k}=0)$ is not orthogonal, in sense that nondiagonal elements of matrix $(\hat P,\hat P^+)$ do not vanish. But often it is more convenient to work with orthogonalized set of dynamic variables. Here we use the procedure of orthogonalization, which orthogonalize all variables one by one except of first two - $\hat n_1$ and ${\hat n_2}$ , so only the block (2 x 2) of matrix $(\hat P,\hat P^+)$ which includes correlation functions $(\hat{n}_i,\hat{n}_j)$ will be nondiagonal. At first we introduce the projection operator:

$\displaystyle {\mbox{$\cal P$}}_{\hat n} \ldots = \sum\limits _{i,j=1}^2(\ldots, \hat n_i) (\hat n,\hat n)^{-1}_{i,j} \cdot \hat n_j, <tex2html_comment_mark>262$

(4.12)

where $\left(\hat{n}, \hat{n}\right)^{-1}_{ij}$ means {ij}-element of matrix (2 x 2), which is inverse to the block $(\hat n,{\hat n}^+)$ of the matrix of static correlation functions $\big(\hat P,\hat P^+\big)$ . And consider `projected' magnetization defined by

It is obvious that $\hat s$ is orthogonal to $\hat n_1$ and ${\hat n_2}$ in the sense, that

$\begin{displaymath} {\mbox{$\cal P$}}_{\hat {s}} \ldots = (\ldots, \hat {s}) (\hat {s},\hat {s})^{-1} \hat {s}, \end{displaymath}$

(4.15)

$\begin{displaymath} \hat h = (1 - {\mbox{$\cal P$}}_{\hat n} - {\mbox{$\cal P$}... ... P$}}_{\hat n} - {\mbox{$\cal P$}} _{\hat {s}}) \hat\omega, \end{displaymath}$

(4.16)

The momentum operator $\mbox{\boldmath$\hat p$}({\boldsymbol k})$ is orthogonal to all variables intrinsically.

We must note, that projection with the help of operators (4.12), (4.15) means transition to another ensemble, for example, projection like in (4.13) means transition from $(\mu,V,T,b)$ to (N,V,T,b) ensemble. Really, magnetic susceptibility in (N,V,T)-ensemble is defined on `projected' variable $\hat {s}$ :

$\displaystyle \chi_{T,N} = \left({\partial M \over \partial b} \right)_{N,V,T} ... ...\partial \mu_1} \right)^{-1}\!\! \left({ \partial N_1 \over \partial b} \right)$

$\displaystyle \mbox{\hspace*{4mm}} - \left({ \partial M \over \partial \mu_2} \... ...\partial \mu_2} \right)^{-1}\!\! \left({ \partial N_2 \over \partial b}\right)=$

(4.18)

$\displaystyle =\beta\Big\{ (\hat m,\hat m) - (\hat m,\hat n_1)(\hat n_1,\hat n_... ...m) - (\hat m,{\hat n_2})({\hat n_2},{\hat n_2})^{-1} ({\hat n_2},\hat m) \Big\}$

$\displaystyle = \beta \Big( (1-{\mbox{$\cal P$}}_{\hat n} ) \hat m \, , \, \hat m \Big) = \beta(\hat {s},\hat {s}).$

Using (4.6) we can prove one more equality for entropy and arbitrary parameter $\gamma$

$\begin{displaymath} {\partial S\over\partial \gamma} = - \beta \left(\hat\omega, {\partial (\beta\hat\omega ) \over\partial \gamma}\right)\;, \end{displaymath}$

(4.19)

and for specific heat in $(\mu,V,T,b)$ -ensemble we will have: $C_{\mu,b} = T\left({\partial S/\partial T}\right)= \beta^2 (\hat\omega,\hat\omega)\;.$

$\begin{displaymath} C_{N,M} = \beta^2 \left((1- {\mbox{$\cal P$}}_{\hat n} - {... ...hat\omega\,, \; \hat\omega \right)= \beta^2 (\hat h,\hat h). \end{displaymath}$

(4.20)

For the operator $\mbox{\boldmath$\hat p$}({\boldsymbol k})$ we have the equalities

$\displaystyle \left(\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}),\mbox{\bol... ...ta}\over\beta} (m_1 N_1 + m_2 N_2) = {\delta_{\alpha\beta}\over\beta} {\bar m},$

(4.21)

$\displaystyle \left(\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}),\hat P_j(-... ... \hat P_j({\boldsymbol k})\neq\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}),$

(4.22)

where ${\bar m}=c_1m_1+c_2m_2$ is a mass of our mixture, c₁ and c₂ are the concentrations of nonmagnetic and magnetic particles, correspondingly.

Taking into account the relations (4.13), (4.14), (4.16), (4.17), (4.20) - (4.22), one sees, that the set of variables $\{\hat n_1,{\hat n_2},\mbox{\boldmath$\hat p$},\hat {s},\hat{\varepsilon}\}$ is orthogonalized in the sense discussed above. Generalizing obtained results, we can introduce new set of dynamic variables ( k-dependent)

$\begin{displaymath} \hat Y({\boldsymbol k}) = \{ \hat n_1({\boldsymbol k}), {\ha... ... k}), \hat {s}({\boldsymbol k}), \hat h({\boldsymbol k}) \}, \end{displaymath}$

(4.23)

which are mutually orthogonal. One exception is for variables $\hat n_1$ and ${\hat n_2}$ which are not mutually orthogonal. For $\hat {s}({\boldsymbol k})$ and $\hat h({\boldsymbol k})$ one has

$\displaystyle \hat {s}({\boldsymbol k})$

=

$\displaystyle (1- {\mbox{$\cal P$}}_{\hat n}({\boldsymbol k}) %%- {\mbox{$\cal P$}}_{{\hat n_2}}({\boldsymbol k}) ) \cdot\hat m({\boldsymbol k}),$

(4.24)

$\displaystyle \hat h({\boldsymbol k})$

=

$\displaystyle (1- {\mbox{$\cal P$}}_{\hat n}({\boldsymbol k}) -%% {\mbox{$\cal ... ...\cal P$}}_{\hat {s}}({\boldsymbol k})) \cdot \hat{\varepsilon}({\boldsymbol k})$

=

$\displaystyle (1- {\mbox{$\cal P$}}_{\hat n}({\boldsymbol k}) - %%{\mbox{$\cal {\mbox{$\cal P$}}_{\hat {s}}({\boldsymbol k})) \cdot \hat\omega({\boldsymbol k}),$

(4.25)

$\displaystyle {\mbox{$\cal P$}}_{\hat n} \ldots = \sum\limits _{i,j=1}^2 \big(\... ...l k}),\hat n(-{\boldsymbol k})\big)^{-1}_{i,j} \cdot \hat n_j({\boldsymbol k}),$

(4.26)

$\displaystyle {\mbox{$\cal P$}}_{\hat {s}} \ldots = (\ldots, \hat {s}(-{\boldsy... ...oldsymbol k}),\hat {s}(-{\boldsymbol k}))^{-1} \cdot \hat {s}({\boldsymbol k}).$

(4.27)

The correlation functions of these quantities can be considered as generalization of well-known thermodynamical derivatives (4.7) - (4.11), (4.18), (4.20), (4.21), for nonzero values of k. Hence,

$\displaystyle (\hat{n}_i({\boldsymbol k}), \hat{n}_j(-{\boldsymbol k})) = (N_i,N_j)^{1\over 2}\, S_{ij}({\boldsymbol k}), \quad i,j=1..2,$

(4.28)

$\displaystyle (\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}),\mbox{\boldmath$\hat p$}^\beta(-{\boldsymbol k})) = {\delta_{\alpha\beta}\over\beta} {\bar m},$

(4.29)

$\displaystyle (\hat {s}({\boldsymbol k}),\hat {s}(-{\boldsymbol k})) = {1\over\beta} \chi_{T,N} ({\boldsymbol k}),$

(4.30)

$\displaystyle (\hat h({\boldsymbol k}),\hat h(-{\boldsymbol k})) = {1\over\beta^2} C_{N,M}({\boldsymbol k}),$

(4.31)

where S_ij(k), defined in (4.28) are so-called partial structure factors, $\chi_{T,N} ({\boldsymbol k})$ and C_N,M(k) are the generalized susceptibility and specific heat for the mixture of magnetic and nonmagnetic particles, respectively.

Frequency matrix and matrix of memory functions

Taking into account the symmetry of the correlation functions under time inversion and spatial symmetry operations, or performing direct calculations, one can prove, that:

$\displaystyle \left(\dot{\hat Y}_i({\boldsymbol k}), {\hat Y}_j(-{\boldsymbol k... ...ox{\boldmath$\hat p$} \; \mbox{and} \; \hat Y_j\neq\mbox{\boldmath$\hat p$}) \;$

$\displaystyle \mbox{or} \; (\hat Y_i=\mbox{\boldmath$\hat p$} \; \mbox{and} \; \hat Y_j=\mbox{\boldmath$\hat p$}) .$

(5.1)

for calculation of frequency matrix we must find only correlation functions, which involve momentum:

$\begin{displaymath} \left(\dot{\mbox{\boldmath$\hat p$}}^\alpha({\boldsymbol k})... ...bol k})\right), \quad \hat Y_j\neq \mbox{\boldmath$\hat p$}. \end{displaymath}$

(5.2)

$\begin{displaymath} \langle J^{\alpha\beta}_p({\boldsymbol k}) \rangle _{{\bolds... ...^N) \over \partial {\boldsymbol r}_i} \right\rangle \right\}, \end{displaymath}$

(5.3)

Let us find now pressure of the system, which follows from the equilibrium treatment

where $\Omega$ is the thermodynamical potential (4.5). Imposing $\Omega$ to be V-dependent by substitution ${\boldsymbol r}={{\mbox{\boldmath$R$}}} V^{1\over 3}$ in potentials V⁽¹¹⁾, V⁽¹²⁾, V⁽²²⁾, J, we can get, after some simplifications

$\begin{displaymath} P = T {N_1+N_2\over V} - {1\over 3V} \left\langle \sum\limit... ...mbol r}^N) \over \partial {\boldsymbol r}_i} \right\rangle . \end{displaymath}$

(5.5)

$\begin{displaymath} \langle J^{\alpha\beta}_p({\boldsymbol k}) \rangle = \delta_{\alpha\beta}PV = - \delta_{\alpha\beta} \Omega . \end{displaymath}$

(5.6)

Results (5.6) show that in system with `isotropic' potential energy (which is a function of |r|, and does not depend on its direction) in homogeneous external field $\left({\partial \hat{\boldsymbol B} ({\boldsymbol r},t) / \partial {\boldsymbol r}} = 0 \right)$ , the pressure, defined from equilibrium treatment can be expressed through the average of the stress tensor. With the help of (4.6) now we get

$\displaystyle \left(\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}), \hat n_1(-{\boldsymbol k}) \right)_{{\boldsymbol k}\to 0}$

=

$\displaystyle {\rm i}{\boldsymbol k}^\beta \left( J^{\alpha\beta}_p, \hat n_1 \right)$

=

$\displaystyle {{\rm i}{\boldsymbol k}^\beta \over \beta} {\partial \langle J^{\... ... \Omega \over \partial \mu_1} = {{\rm i}{\boldsymbol k}^\alpha \over \beta}N_1.$

(5.7)

Let us introduce now scalar longitudinal momentum $\hat p({\boldsymbol k})$ , as a component of vector momentum $\mbox{\boldmath$\hat p$}({\boldsymbol k})$ , oriented along the vector k . From the equation (5.7) we conclude, that in isotropic system only longitudinal components make contribution to the matrix of memory functions, correlation functions of transverse components are equal to zero. In the same way as (5.7) was obtained we can obtain other correlation functions with the longitudinal current $\hat p{\boldsymbol k}$ in the limit $k\to0$ , (k=|k|)

$\displaystyle (\dot{\hat p}, \hat{n}_i) = {{\rm i}k\over \beta} N_i,$

(5.8)

$\displaystyle (\dot{\hat p}, \hat m) = {{\rm i}k\over \beta} V \left({ \partial... ...\rm i}k\over \beta^2} V \left({ \partial P\over \partial T } \right)_{\mu,V,b},$

(5.9)

$\begin{displaymath} (\dot{\hat p}, \hat {s}) = {{\rm i}k\over \beta} V \left({ \... ...ta^2} V \left({ \partial P\over \partial T } \right)_{N,V,M}. \end{displaymath}$

(5.10)

$\begin{displaymath} \left({ \partial P\over \partial b}\right)_{N,V,T} = {{\pi}\... ...ver V}\left({\partial V \over \partial P} \right)_{T,N,M} }, \end{displaymath}$

(5.12)

$\displaystyle {\pi} = {1\over V}\left({\partial V \over \partial b} \right)_{P,... ...quad \kappa_T = -{1\over V}\left({\partial V \over \partial P} \right)_{T,N,b},$

(5.13)

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

(5.14)

are the coefficients of magnetostriction, isothermal compressibility and isobaric thermal expansion for the mixture of magnetic and nonmagnetic particles, respectively. The value $\left({\partial V / \partial P} \right)$ in equation (5.12) is defined at constant magnetization M. After some algebra with the help of Gibbs-Duhem and Maxwell relations, we get

$\displaystyle \left({\partial V \over \partial P} \right)_{T,N,M} = - V \kappa_T + {V^2{\pi}^2 \over \chi + {{\pi}^2 V \over\kappa_T}},$

$\begin{displaymath} \left({ \partial P \over \partial T} \right)_{V,N,M} = {\alp... ...er\kappa_T} \left(1 + {V{\pi}^2\over\chi\,\kappa_T} \right). \end{displaymath}$

(5.15)

Now making generalization for ${\boldsymbol k}\neq 0$ we can write some elements of matrix ${\rm i}\Omega$ as follows

$\displaystyle {\rm i}\Omega_{p,s}({\boldsymbol k}) = {\rm i}k V{{\pi}({\boldsym... ...) = {\rm i}k V{{\pi}({\boldsymbol k})\over\kappa_T({\boldsymbol k}) {\bar m} },$

(5.16)

$\displaystyle {\rm i}\Omega_{p,h}({\boldsymbol k}) = {\rm i}k V{\alpha_p({\bold... ...({\boldsymbol k}) \over\chi({\boldsymbol k})\kappa_T({\boldsymbol k})} \right),$

(5.17)

$\displaystyle {\rm i}\Omega_{h,p}({\boldsymbol k}) = {\rm i}k {V\over \beta} {\... ...^2({\boldsymbol k})\over\chi({\boldsymbol k})\kappa_T({\boldsymbol k})}\right).$

(5.18)

Elements, involving densities of particle number, as follows from (5.8) and (2.20), read:

$\displaystyle \rho(t)=\rho_q(t)+\sum_\alpha \int\limits _{-\infty}^t dt'\, e^{-\varepsilon (t-t')} F_\alpha (t')T (t,t')$
$\displaystyle \times \int\limits _0^1 d\tau \rho^\tau_q (t') \hat I_\alpha (t') \rho^{1-\tau}_q (t'),$			(2.5)

		$\displaystyle \vert\vert(\Delta\hat P,\Delta\hat P^+)\vert\vert _{ij}=(\Delta\hat P_i,\Delta\hat P_j),$	(2.14)
		$\displaystyle \Delta\hat P_i=\hat P_i-\langle \hat P_i\rangle ,$

$\displaystyle \delta \rho (t) = \delta \rho_q (t) + \sum_\alpha$		$\displaystyle \int\limits _{-\infty}^t dt'\, e^{-\varepsilon (t-t')} \; \delta F_\alpha (t') \;T_0 (t-t')$
		$\displaystyle \times \int\limits _0^1 d\tau \;\rho_0^\tau \;(1 - {\mbox{$\cal P$}}) \; \dot{\hat P}_\alpha \;\rho^{1-\tau}_0 ,$	(2.16)

$\displaystyle \phi_\varepsilon (\omega) = \left( (1-{\mbox{$\cal P$}}) \dot{\ha... ...mbox{$\cal P$}}) {\rm i}\hat L\, } (1-{\mbox{$\cal P$}}) \dot{\hat P}^+ \right)$
$\displaystyle \times ( \Delta\hat P , \Delta\hat P^+ )^{-1}$			(2.21)

		$\displaystyle {\rm i}L_1 = \sum\limits _i^{(1)} { {\mbox{\boldmath$p$}}_{i}^{(1... ...{i}^{(1)}} - {\partial \over \partial {\mbox{\boldmath$p$}}_{j}^{(1)}} \right),$	(3.6)
		$\displaystyle {\rm i}\hat L\,_2 = {\rm i}L_{(1 {\to} 2)} + {1\over 2} \sum\limi... ...{i}^{(2)}} - {\partial \over \partial {\mbox{\boldmath$p$}}_{j}^{(2)}} \right),$	(3.7)
		$\displaystyle {\rm i}L_{int} = - \sum\limits _{i , j}^{(1,2)} {\partial \over \... ...{i}^{(1)}} - {\partial \over \partial {\mbox{\boldmath$p$}}_{j}^{(2)}} \right),$	(3.8)
		$\displaystyle {\rm i}\hat L\,_{ex} = \sum\limits _i^{(2)} {\partial \left(\hat{... ...ymbol r}_i} {\cdot} {\partial \over \partial {\mbox{\boldmath$p$}}_{i}^{(2)} },$	(3.9)

		$\displaystyle \hat n_1({\boldsymbol k}) = \sum\limits _i^{(1)} e^{{\rm i}{\bold... ...symbol k}) = \sum\limits _i^{(2)} e^{{\rm i}{\boldsymbol k}{\boldsymbol r}_i} ,$	(3.11)
		$\displaystyle \mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}) = \sum\limits _i... ...{\boldmath$p$}}_{i}^{(2)}$}^\alpha e^{{\rm i}{\boldsymbol k}{\boldsymbol r}_i},$	(3.12)
		$\displaystyle \hat{\boldsymbol m}^\alpha({\boldsymbol k}) = \sum\limits _i^{(2)} \hat{\boldsymbol s}_i^\alpha e^{{\rm i}{\boldsymbol k}{\boldsymbol r}_i},$	(3.13)
		$\displaystyle \hat{\varepsilon}({\boldsymbol k}) =\sum\limits _i^{(1)} \hat{\va... ...ol r}_i} %%+ \suml_i^{(2)} \lp \hat{\boldsymbol B} ({\boldsymbol r}_i,t) \, ,$	(3.14)

		$\displaystyle \hat{\varepsilon}_i^{(1)} = { \mbox{${\mbox{\boldmath$p$}}_{i}^{(... ..._{ij} ) + {1\over 2} \sum\limits _{j} ^{(2)} V^{(12)} ( {\boldsymbol r}_{ij} ),$	(3.15)
		$\displaystyle \hat{\varepsilon}_i^{(2)} = \hat{\varepsilon}_i^{(1 {\to} 2)} - {... ...(2,2)} J ({\boldsymbol r}_{ij}) (\hat{\boldsymbol s}_i, \hat{\boldsymbol s}_j).$	(3.16)

		$\displaystyle \dot{\hat P}_i({\boldsymbol k}) = {\rm i}{\boldsymbol k}^\alpha J_i^\alpha ({\boldsymbol k}) + R_i ({\boldsymbol k}),$	(3.17)
		$\displaystyle \dot{\hat P}_l^\alpha ({\boldsymbol k}) = {\rm i}{\boldsymbol k}^\beta J_l^{\alpha \beta} ({\boldsymbol k}) + R_l^\alpha ({\boldsymbol k}),$	(3.18)

		$\displaystyle \rho_0 = \exp{[\beta(\Omega-{\hat\omega})]} ,$	(4.3)
		$\displaystyle {\hat\omega} \equiv \hat{\varepsilon} - \mu_1\hat n_1 - \mu_2{\hat n_2} -{b}\hat{m},$	(4.4)
		$\displaystyle \Omega = \Omega(\mu,{b},T),$	(4.5)

		$\displaystyle (\hat{n}_i, \hat{n}_j)={1\over\beta} \left({\partial N_i \over \p... ... \left({\partial N_j \over \partial \mu_i} \right)_{T,b} , \quad i,j=1,2 \quad,$	(4.8)
		$\displaystyle (\hat{n}_i,\hat m) = {1\over\beta} {\partial N_i \over \partial b} = {1\over\beta} \left({\partial M \over \partial \mu_i} \right)_{T,b},$	(4.9)
		$\displaystyle (\hat m,\hat m)={1\over\beta}\left({\partial M \over \partial b} \right)_{T,\mu},$	(4.10)
		$\displaystyle (\hat{\varepsilon},\hat m) = {1\over\beta}\left({\partial E\over\partial b}\right)_{T,\mu},$	(4.11)

		$\displaystyle \chi_{T,N} = \left({\partial M \over \partial b} \right)_{N,V,T} ... ...\partial \mu_1} \right)^{-1}\!\! \left({ \partial N_1 \over \partial b} \right)$
		$\displaystyle \mbox{\hspace*{4mm}} - \left({ \partial M \over \partial \mu_2} \... ...\partial \mu_2} \right)^{-1}\!\! \left({ \partial N_2 \over \partial b}\right)=$	(4.18)
		$\displaystyle =\beta\Big\{ (\hat m,\hat m) - (\hat m,\hat n_1)(\hat n_1,\hat n_... ...m) - (\hat m,{\hat n_2})({\hat n_2},{\hat n_2})^{-1} ({\hat n_2},\hat m) \Big\}$
		$\displaystyle = \beta \Big( (1-{\mbox{$\cal P$}}_{\hat n} ) \hat m \, , \, \hat m \Big) = \beta(\hat {s},\hat {s}).$

		$\displaystyle \left(\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}),\mbox{\bol... ...ta}\over\beta} (m_1 N_1 + m_2 N_2) = {\delta_{\alpha\beta}\over\beta} {\bar m},$	(4.21)
		$\displaystyle \left(\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}),\hat P_j(-... ... \hat P_j({\boldsymbol k})\neq\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}),$	(4.22)

$\displaystyle \hat {s}({\boldsymbol k})$	=	$\displaystyle (1- {\mbox{$\cal P$}}_{\hat n}({\boldsymbol k}) %%- {\mbox{$\cal P$}}_{{\hat n_2}}({\boldsymbol k}) ) \cdot\hat m({\boldsymbol k}),$	(4.24)
$\displaystyle \hat h({\boldsymbol k})$	=	$\displaystyle (1- {\mbox{$\cal P$}}_{\hat n}({\boldsymbol k}) -%% {\mbox{$\cal ... ...\cal P$}}_{\hat {s}}({\boldsymbol k})) \cdot \hat{\varepsilon}({\boldsymbol k})$
	=	$\displaystyle (1- {\mbox{$\cal P$}}_{\hat n}({\boldsymbol k}) - %%{\mbox{$\cal {\mbox{$\cal P$}}_{\hat {s}}({\boldsymbol k})) \cdot \hat\omega({\boldsymbol k}),$	(4.25)

		$\displaystyle {\mbox{$\cal P$}}_{\hat n} \ldots = \sum\limits _{i,j=1}^2 \big(\... ...l k}),\hat n(-{\boldsymbol k})\big)^{-1}_{i,j} \cdot \hat n_j({\boldsymbol k}),$	(4.26)
		$\displaystyle {\mbox{$\cal P$}}_{\hat {s}} \ldots = (\ldots, \hat {s}(-{\boldsy... ...oldsymbol k}),\hat {s}(-{\boldsymbol k}))^{-1} \cdot \hat {s}({\boldsymbol k}).$	(4.27)

		$\displaystyle (\hat{n}_i({\boldsymbol k}), \hat{n}_j(-{\boldsymbol k})) = (N_i,N_j)^{1\over 2}\, S_{ij}({\boldsymbol k}), \quad i,j=1..2,$	(4.28)
		$\displaystyle (\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}),\mbox{\boldmath$\hat p$}^\beta(-{\boldsymbol k})) = {\delta_{\alpha\beta}\over\beta} {\bar m},$	(4.29)
		$\displaystyle (\hat {s}({\boldsymbol k}),\hat {s}(-{\boldsymbol k})) = {1\over\beta} \chi_{T,N} ({\boldsymbol k}),$	(4.30)
		$\displaystyle (\hat h({\boldsymbol k}),\hat h(-{\boldsymbol k})) = {1\over\beta^2} C_{N,M}({\boldsymbol k}),$	(4.31)

$\displaystyle \left(\dot{\hat Y}_i({\boldsymbol k}), {\hat Y}_j(-{\boldsymbol k... ...ox{\boldmath$\hat p$} \; \mbox{and} \; \hat Y_j\neq\mbox{\boldmath$\hat p$}) \;$
$\displaystyle \mbox{or} \; (\hat Y_i=\mbox{\boldmath$\hat p$} \; \mbox{and} \; \hat Y_j=\mbox{\boldmath$\hat p$}) .$			(5.1)

$\displaystyle \left(\mbox{\boldmath$\hat p$}^\alpha({\boldsymbol k}), \hat n_1(-{\boldsymbol k}) \right)_{{\boldsymbol k}\to 0}$	=	$\displaystyle {\rm i}{\boldsymbol k}^\beta \left( J^{\alpha\beta}_p, \hat n_1 \right)$
	=	$\displaystyle {{\rm i}{\boldsymbol k}^\beta \over \beta} {\partial \langle J^{\... ... \Omega \over \partial \mu_1} = {{\rm i}{\boldsymbol k}^\alpha \over \beta}N_1.$	(5.7)

		$\displaystyle (\dot{\hat p}, \hat{n}_i) = {{\rm i}k\over \beta} N_i,$	(5.8)
		$\displaystyle (\dot{\hat p}, \hat m) = {{\rm i}k\over \beta} V \left({ \partial... ...\rm i}k\over \beta^2} V \left({ \partial P\over \partial T } \right)_{\mu,V,b},$	(5.9)

		$\displaystyle {\pi} = {1\over V}\left({\partial V \over \partial b} \right)_{P,... ...quad \kappa_T = -{1\over V}\left({\partial V \over \partial P} \right)_{T,N,b},$	(5.13)
		$\displaystyle \alpha_p = {1\over V}\left({\partial V \over \partial T} \right)_{P,N,M}$	(5.14)

		$\displaystyle {\rm i}\Omega_{p,s}({\boldsymbol k}) = {\rm i}k V{{\pi}({\boldsym... ...) = {\rm i}k V{{\pi}({\boldsymbol k})\over\kappa_T({\boldsymbol k}) {\bar m} },$	(5.16)
		$\displaystyle {\rm i}\Omega_{p,h}({\boldsymbol k}) = {\rm i}k V{\alpha_p({\bold... ...({\boldsymbol k}) \over\chi({\boldsymbol k})\kappa_T({\boldsymbol k})} \right),$	(5.17)
		$\displaystyle {\rm i}\Omega_{h,p}({\boldsymbol k}) = {\rm i}k {V\over \beta} {\... ...^2({\boldsymbol k})\over\chi({\boldsymbol k})\kappa_T({\boldsymbol k})}\right).$	(5.18)

Abstract:

Introduction

Theoretical framework: Nonequilibrium statistical operator method

Model Hamiltonian and hydrodynamic variables

Static correlation functions

Frequency matrix and matrix of memory functions