Generalized Hydrodynamics Of Multi-Component Fluids

Starting from the rigorous microscopic approach and taking into account all the conserved quantities of multi-component, chemically nonreactive uids, the basic equations of generalized hydrodynamics are derived. The explicit expressions for the transport coeecients as well as the damping coeecients and dispersions of collective hydrody-namic modes are found. It is shown that these results are in agreement with those obtained by phenomenological theories.


Introduction
During the last decade an essential progress in understanding of dynamical properties of uids has been achieved in connection with the method of the generalized collective modes.The concept of generalized collective modes proposed initially in Ref. 1,2] for the study of time correlation functions of dense one-component uids, represents now a modern and powerful method, which allows to obtain the self-consistent description of dynamical properties within wide range of wave-numbers and frequencies starting from the hydrodynamic values and up to the Gaussian-like range.In this framework, time correlation functions could be written as a weighted sum of partial terms, each of them being associated with the relevant generalized collective mode could be expressed via the corresponding eigenvector and eigenvalue of so-called generalized operator of evolution.Some of the generalized collective modes correspond in the hydrodynamic limit to the well-known hydrodynamic excitations.The other ones are called the kinetic modes and have higher eigenvalues with nite damping coe cients in the hydrodynamic limit.It is important to note that this approach is based on the extended set of dynamic variables containing, in addition to the conserved variables, their higher-order time derivatives.
In 3] the parameter-free generalized collective modes approach, based on Markovian approximation for the higher order memory functions, has been suggested.In this paper, the ve-and seven-variable descriptions of longitudinal uctuations for a pure Lennard-Jones liquid has been developed.The extension of the formalism to a nine-variable description has been presented in 4,5].The calculations of time correlation functions for a Lennard-Jones uid 6] showed that in a wide range of k and ! a reasonable agreement of the theory with MD data can be already observed within lower-order mode approximations.
Very promising results have been aslo obtained for binary mixtures.The theoretical and experimental investigations showed that the dynamic structure factors S(k; !) of liquid water 7], liquid Li 4 Pb 8] and of dense gas mixtures He-Ne 9] as well as He- Ar 10,11] display a behavior, which can be explained only in terms of new generalized collective modes, namely, so-called fast sound.
Until recently, the theoretical description of the dynamics for multicomponent uids was based to a great extent on phenomenological approaches 12,13] and on the methods of kinetic theory (see, e.g., 14,15]) for low density gas mixtures.The main goal of this paper is: to derive the basic equations of generalized hydrodynamics for multi-component, chemically nonreactive uids; to study these equations in the hydrodynamic limit of small wave-numbers k and frequencies !; to analyse the spectrum of the hydrodynamic modes.Considering these problems one may create the basis for the next study of multi-component uids within the generalized mode approach.
The structure of the paper is following: section 2 outlines the main initial relations which are used in the paper; in section 3 we present the basic equations of the generalized hydrodynamics for a general mixture; the results for static correlation functions in the hydrodynamic limit are derived in section 4 by means of thermodynamic uctuation theory; in sections 5 and 6 we present the calculations for the elements of hydrodynamic frequency and memory functions matrixes; the study of the hydrodynamic collective modes for a mixture is performed in Section 7; we end with a discussion in section 8.

Initial relations
We consider for a multi-component uid an open region of volume V containing N 1 ; N 2 ; : : : ; N particles of species 1; 2; : : : ; .Hence, N = P =1 N is a total number of particles.
Let us introduce the following notations n = N V (1) is the number density of particles in the -th species, is the total number density of particles, For mixtures the partial structure factors are de ned as follows 24,25] Z dr r 2 (g (r) ? 1) e irk = + 4 (n n ) 1=2   1 Z 0 dr r 2 (g (r) ? 1) sin kr kr (7) where g (r) is the partial (two-particle) radial distribution function which is given by g On occasion the Faber-Ziman partial structure factors are also used 24,25] a (k) = 1 + n Z dr r 2 (g (r) ? 1) e irk = 1 + 4 n 1 Z 0 dr r 2 (g (r) ? 1) sin kr kr : It immediately follows from (7) and (9) that S (k) = + (c c ) 1=2 (a (r) ? 1) : (10) Note that in the limit k ! 1 one has S (k) ! ; and a (k) !1: The total static structure factor could be de ned as The microscopic set of the conserved (hydrodynamic) variables may be introduced in the form P h k = f Nk ; Ĵk ; Êk g; (12) where Nk = f Nk; g (13) is a column-vector with components Nk; , Nk; = N X i=1 expfikR i g; (14) is the number density of particles in the -th species; is the density of the total current, Ĵk = f Ĵx k ; Ĵy k ; Ĵz k g and m v i expfikR i g; (16) are the current densities of particles in the -th species; e i expfikR i g; (17) is the total energy density, where and V (jr ?r 0 j) is a potential of interparticle interactions, and the symbol ( 0 ) in (18) indicates that i 6 = j when = .
For all the dynamic variables introduced above the microscopic equations of motion could be written in the form P h k;l ?ikI h k;l = 0 (19) where P h k;l iL N P h k;l and iL N is a Liouville operator.The equation ( 19) is in fact the conservation law in the local form.The quantities I h k;l are the (microscopic) hydrodynamic uxes.
Turning our attention to the conserved dynamic variables, we should in principle consider a ( + 3 + 1)-component set containing -component of the number density Nk; of particles in each species, the three components of the total current density Ĵk , and the total energy density Êk .However, from the microscopic equations of motion (19) it follows that the number densities Nk; and the energy density Êk are coupled only with the longitudinal component of Ĵk , directed along k.This is because of space isotropy of the system.Thus, we have the same situation as in the case of simple liquids.As a result, one may split the set of the hydrodynamic variables into two separate subsets.The rst set P h;l k includes the variables f Nk ; Ĵl k ; Êk g and may be called as the set of longitudinal hydrodynamic variables, and the second one P h;t k comprises two components of the total current perpendicular to k (transverse currents).It is evident that the nal equations for two transverse components should be the same because of isotropy of the system.Therefore, one may suppose that the vector k is directed along 0Z axis, so that we have nally the ( + 2)-component longitudinal set P h;l k f Nk ; Ĵz k ; Êk g and the two component transverse set P h;t k f Ĵx k ; Ĵy k g of dynamic variables describing multi-component uids.

General framework
It has been shown 16,17] that the macroscopic equations of motion for an arbitrary set of the dynamic variables Pk = f Pk;1 ; Pk;2 ; : : : ; Pk;l g; (20) where Pk is a column-vector, could be written in the matrix form as follows fi!I l l ?i l l + 'l l (!)gh P i != 0 (21) where I l l is the l l unit matrix and Pk;i = Pk;i ?h Pk;i i, i l l = ( P ; P + ) ( P ; P + ) ?
The matrix equation for the Laplace transform Fl l (k; z) of the equilibrium time correlation functions F l l (k; t), where F l l (k; t) = h Pk e ?iLNtP + ?k i; (26) has the similar structure to (21), namely, f zI l l ?i l l + 'l l (z)g Fl l (k; z) = F l l (k; 0): (28) The matrix of memory functions could be also written in the another form 'l l (z) = ( P ; P + ) z ??(P ; P + ) z 1 ( P ; P + ) z ( P ; P + ) z 1 ( P ; P + ) ; (29) which may be useful for some applications.
It is worth noting that the time correlation functions F l l (k; t) are closely related to the classical retarded correlation Green functions G (r)   AB (t ?t 0 ) = ?i(t ?t 0 )hA(t)B(t 0 )i; (30) where (t) = 1 or 0 according to whether t > 0 or t > 0, so that as it follows from (28) the spectrum of collective modes to be found from the equation Det j zI l l ?i l l + 'l l (z) j= 0 (31) gives the poles of the Green functions constructed on the set of dynamic variables f Pk;i g.It should be also stressed that the matrix equation for the equilibrium time correlation functions (28) is in fact the exact equation until the explicit expressions for the frequency and memory function matrixes are used.This statement can be proved using the expressions for the frequency matrix (22) and the matrix of memory functions (29).
Note that as it follows from ( 22), (23), and (24) it is convenient for concrete calculations to use the set orthogonalized dynamic variables possessing the properties h Pk;i P?k;j i = ij h Pk;j P?k;j i; (32) so that one has ( Pk ; P + k ) ?1 ji = ji ( Pk;i ; P?k;i ) ?1 : The linearized transport equations (21), the equations for the equilibrium time correlation functions (28), and the equation for collective mode spectrum (31) form a general basis for the study of the dynamic behavior of uids in the memory function formalism.
We note that in the expressions given above the sum over wave-vector k has to be performed together with a summation of the index i of the dynamic variables.

Frequency matrix and matrix of memory functions for a multi-component uid
Let us apply the results presented above for the set of dynamic variables which contains the conserved variables of a multi-component uid P h k .By analogy with a case of simple uid, for longitudinal components we will start from the set of dynamic variables P l k = f Nk ; Ĵk ; Ĥk g; is Mori-like projection operator.We note that in contrary to the case of simple uids the hydrodynamic set (33) is not orthogonal in sense of (32) because of ( Nk; ; N?k; ) 6 = 0 for 6 = .
Using the properties of correlation functions under time inversion and spatial symmetry operations, one can show that in general the structure of the ( + 2) ( + 2) hydrodynamic frequency matrix is following where the notation 0 denotes the matrix, all the elements of which are zero.
For arbitrary k all the elements of the memory functions matrix are not equal to zero, so that Using the equations ( 21), ( 28) and (31) for the hydrodynamic set of dynamic variables P l k we obtain the basis equations of generalized hydrodynamics for longitudinal components of a multi-component uids.The generalized transport coe cients of the system can be introduced in usual way 17] via the elements of matrix 'H (k; z).
For the transverse uctuations with one variable hydrodynamic set P t k = Ĵt k ; (38) one has the generalized transport equation in the form fi! + 't jj (k; !+ i")gh Ĵt k i != 0; (39) which is quite similar as it is for a simple uid.The generalized shear viscosity can be de ned via the memory function 't jj (k; z).

Static correlation functions and thermodynamic theory of uctuations
The space and time response of the system near the equilibrium state may be calculated using: (i) the thermodynamic uctuation theory to provide the initial values for the static correlation functions as well as (ii) the linearized hydrodynamic equations to determine the modes by which the system returns to equilibrium.Let us consider the expressions for the static correlation functions F(k) = ( P h k ; P h ?k ) in the limit k !0 and discuss the thermodynamic relations for these functions which follow from the theory of thermodynamic uctuations.We start with the equilibrium Gibbs distribution ^ for the grand canonical ensemble (V; T; f g)-ensemble] ;N = exp where = (V; T; f g) = ?PV is the thermodynamic potential depending on temperature k b T = 1= , chemical potentials of the -th species.V is xed and P indicates the pressure.The index \ " in (40) labels the value of E N .The potential is de ned as usual by the condition that ^ is normalized to unity.
Finally, we get or in the more general form The equation ( 58) is in fact the de nition of the generalized speci c heat at constant volume in the grand canonical ensemble.Note that the orthogonal variables Ĵk and Nk; are orthogonal in sense of the equation (20) to Ĥk .
For the calculations of the frequency matrix we need also to know the static correlation function ( J k ; ĥ?k ) in the limit k !0. Taking into account the relation J k = ik zz (k); where zz (k) is the zz-component of stress tensor, one can nd from (42) where we used the equality h zz i = PV = ?: Generalized hydrodynamics of multi-component uids 125 As a result one gets ( zz ; Ĥ) = ( zz ; Ê) ?X ; @h Êi @ !T;V; @ @N T;V;N hN i = T " S ?X =1 N @S @N T;V;N # = TV @P @T V;N ; (61) or ( J ; Ĥ) = ik TV @P @T V;N : (62) In order to prove the equality S ?X =1 N @S @N T;V;N = V @P @T V;N one may consider the free energy F, dF = ?SdT ?PdV + X =1 dN : Introducing the normalized quantity f = F=V , N = V , s = S=N, one nds f = ?P + X =1 ; and df = ?sdT?PdV + X =1 d : Hence, we have @f @T = ?s= ?@P @T + X =1 @ @T : By making use now Maxwell relation @ @T = ?@s @ T; ; we get nally @P @T = s ?X =1 @s @ T; = 1 V " S ?X =1 N @S @N T;N # : 127 Thus, the hydrodynamic frequency matrix can be written in quite general form as follows i where c and v are column-vectors with the components c and v (k), respectively.In the hydrodynamic limit from (72) one gets Using the result (73) we can study the ideal hydrodynamics (without dissipations) of a many-component uid and calculate the dispersion of sound waves for the system.
To nd the dispersion of sound waves we have to solve the equation detjzI ( +2) ( +2) ?i H j = 0: Taking into account the structure of the matrix i H , the last equation can be written in the form z " z 2 ?i jh i hj ?X is the ratio of the speci c heats at constant pressure and constant volume.
Note that the generalized speci c heat at constant pressure c p (k) and the generalized ratio of the speci c heats (k) can be de ned, using the expressions for c v (k), t (k) and p (k) given above.Thereafter, from the equation We note also that the elements of the hydrodynamic frequency matrix for the transverse components of current are equal to zero as it should be.

Hydrodynamic matrix of memory functions
The memory functions are de ned on the basis of generalized uxes where the explicit expression for I k;h follows from the de nition and Q z (k) iL N Êk =ik is the ux of energy.
Thereafter the Laplace transforms of the hydrodynamic memory functions can be written via the generalized kinetic coe cients L lg (k; z) as follows Generalized hydrodynamics of multi-component uids 129 where l; l 0 ; g = f ; j; h; g and = 1; 2; : : : ; .We have introduced the generalized kinetic coe cients Llg (k; z) as the Laplace transforms of timedependent functions L lg (k; t) de ned on the set of the generalized uxes I d k;l , L lg (k; t) = V I d k;l ; expf?(1 ?P H )iL N tgI d ?k;l : Let us consider now in more detail the hydrodynamic limit k !0. In such a case one gets for the uxes where The property (87), as it will be seen below, is very important from the view-point of the properties of transport coe cients describing di usivity.
Let us consider the leading terms in wave vector k of the memory functions (83) when k is small.In such a case the Markovian approximation is valid.Hence, we may neglect the elements ' jh , ' 1  jn , and ' hj , ' 1 nj in (37) because they contribute to the collective mode spectrum in the next order of k (the static correlation functions (iL N Ĵ; iL N ĥ) and (iL N Ĵ; iL N N ) are equal to zero as follows from the symmetry reasons).Then all other terms in the matrix (37) are proportional to k 2 .
A further simpli cation can be reached by observing that the equality (89) It is seen in (89) that the kinetic coe cients L ll 0 are given via Green-Kubo like formulas.
The symmetry properties of the transport coe cients immediately follow from the symmetry properties of the generalized I d l uxes under time reversal transformation, L ll 0 = L l 0 l for any non-zero L ll 0 and l; l 0 are from the set f ; j; hg where = 1; 2; : : : ; .The kinetic coe cients L ll 0 can be considered as the well-known trans- port coe cients, namely: a) L jj = l = 4   3 + is the longitudinal viscosity, where and are the bulk and shear viscosities, respectively; b) L hh de nes the thermal conductivity (L hh = T ) 1 ; c) L nn D are the di usion coe cients; d) L nh K describes thermal di usion in the -th species.
For the transverse components of current we found L (t)  jj = V 1 Z 0 dt zx ; e ?iLt zx = ; (90) where is the shear viscosity and zx is a nondiagonal element of the stress tensor.
The di usion D and thermal di usion K coe cients possess the additional properties which immediately follow from (87), Thereafter , we have all the needed quantities for the subsequent calculations of collective mode spectrum in the hydrodynamic limit.where ; are from the set 1; 2; : : : ; .The equation ( 94) is of ( + 2)th order with respect to z.Looking for the solutions in the form z = z 0 k + z 1 k 2 + ; (95) the coe cients z 0 and z 1 may be calculated.
After some mathematical manipulations it can be shown that the equation (94) has in general: two complex-conjugated solutions in the form z s = c s k ?D s k 2 ; (96) and purely real solutions describing a heat mode with z h = ??k 2 (97) as well as ? 1 di usion modes with z r d = ?D r k 2 ; r = f1; 2; : : : ; ?1g; (98) where ?and D r are the damping coe cients of the heat and di usion modes, respectively.One can show that the sound velocity c s is given by (78) as it was expected.Using the structure of (94) we found also the expression for the sound damping coe cient D s in the form where D and K are the matrix and the vector-column with the elements D and K , respectively.The result (99) agrees with the expression found previously by phenomenological theories 13,20] To calculate the di usion damping coe cients D r , where r = f1; : : : ; ?
1g, one has to consider each from the cases with xed value of separately.We will study below as an example a binary mixture of simple uids with = 2.For transverse components one has the purely real mode z t = ?k 2 ; (100) similar as it is for a simple uid.(103) and m 1 @ 1 @n 1 V;T;n2 ? 1 m 2 @ 1 @n 2 V;T;n1 = 1 @ 1 @x 1 V;T;M ; (105) Thereafter, for a binary mixture the equation (94) can be rewritten in the form det 2 (113) and z x = @ x @x V;T;M ; x = 1 m 1 ? 2 m 2 : (114) c) a hydrodynamic di usion mode with z d = ?Dk 2 ; D = 1 2 a 1 ?q a 2 1 ?4a 0 ; (115) and the coe cients a 1 and a 0 are de ned by equations ( 112)-(113).
It can be shown that the obtained results (110), ( 111) and (115) agree with the results known previously from phenomenological treatment 23,22,13].However, the damping coe cients are written in forms which differ from most commonly quoted because of using di erent de nitions of phenomenological coe cients (see also 21]).

Conclusion
We have applied the scheme of the generalized hydrodynamics to a multicomponent, nonreacting uid for deriving the basic equations of generalized hydrodynamics.This makes possible to use the generalized mode approach developed previously for a simple uid for the subsequent study of timecorrelation functions, generalized collective mode spectrum and generalized transport coe cients of a mixture.
We have demonstrated one to one correspondence between the results for hydrodynamic collective modes obtained from the equations of generalized hydrodynamics and those found by phenomenological theories.It is worth to note also that the expressions for the generalized k-dependent thermodynamic quantities of a mixture have been derived rigorously for the rst time herein.

2 '
are now in position to present the matrix of memory functions in the hydrodynamic limit.This matrix is frequency independent and only contains terms of order k 7.2.Hydrodynamic modes for a binary mixtureTaking into account the relations (91) and (91), the normalized di usion d and thermal di usion k coe cients can be de ned as follows Generalized hydrodynamics of multi-component uids 133 where the sound velocity c s is given by the expression (78) and the damping coe cient D s may be written as follows 2D s = l + ( ? 1)