fluctuations and stochastic noise in systems with hyperbolic mass transport

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Physica A 366 (2006) 149–158

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Fluctuations and stochastic noise in systemswith hyperbolic mass transport

David Joua, Peter Galenkob,�

aDepartament de Fısica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Catalonia, SpainbInstitute for Space Simulation, German Aerospace Center, Cologne 51147, Germany

Received 29 July 2005; received in revised form 11 October 2005

Available online 17 November 2005

Abstract

A binary system in which the diffusion flux has a nonvanishing relaxation time is considered. We study the spectra of

fluctuations of the solute density and the solute diffusion flux. The role of the diffusion flux is analyzed in two descriptions.

First, the shortest observable time interval is shorter than the flux relaxation time, and the diffusion flux behaves as

fluctuating independent variable. Second, the shortest observable time interval is longer than the flux relaxation time, and

the diffusion flux behaves as a Markovian hydrodynamic noise.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Diffusion; Fluctuations; Stochastic noise

1. Introduction

Memory effects in generalized transport equations play a relevant role at high frequency or high speed ofperturbations. The influence of the non-vanishing relaxation time tD of the diffusion flux on the propagationof fast crystallization fronts has been studied [1,2] in consistency with extended thermodynamics [3]. Thememory effects play an important role in the propagation of phase interfaces during fast phase transitions [4].

Fluctuations for both slow variables (i.e., temperature, solute density, etc.) and fast variables (e.g., diffusionflux and heat flux) have been considered frequently. The fluctuations of the heat flux and the viscous pressurewere stressed for the first time by Landau and Lifshitz [5], who derived the expressions for their correlation. InRefs. [6,7], the role of rapid fluctuations of the heat flux as a stochastic noise has been considered within theextended thermodynamic formalism. In these works, a unified description of slow and fast heat fluctuationshas been made [8] for equilibrium and nonequilibrium steady states. The same idea about separation of slowand fast variables to study stochastic noise in a system of particles with inertia has been realized within asupersymmetric path-integral representation [9].

In the present paper, we consider the system of equations for the solute density (concentration) and the fluxapplied to the phenomenon of density fluctuations in an equilibrium state. For frequencies lower compared to

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ysa.2005.10.027

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ess: [email protected] (P. Galenko).

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ARTICLE IN PRESSD. Jou, P. Galenko / Physica A 366 (2006) 149–158150

t�1D , the usual diffusional description (with tD ¼ 0) is satisfactory, but for frequencies comparable or higherthan t�1D , the finite value of the relaxation time cannot be ignored. These frequencies are experimentallyaccessible, for instance, by means of neutron scattering experiments.

Besides density fluctuations, we explore the fluctuations of the diffusion flux and investigate their role in twodifferent kinds of descriptions. In the first of them, the shortest measurable time interval tmin is shorter than tD,and the diffusion flux behaves as an independent fluctuating variable. In the second description, tmin is largerthan tD, and the fluctuating part of the flux behaves as a stochastic noise in the evolution equation for thedensity.

The paper is organized as follows. In Section 2, the features of the hyperbolic transport are described.In Section 3, an explicit form of the fluctuation spectra for solute density and solute diffusion fluxare derived. The fluctuations of the density and of diffusion flux in the two time domains tminotD andtmin4tD are specially emphasized in Section 4. Finally, in Section 5 we present a summary of ourconclusions.

2. Hyperbolic transport

The system under study is described by the particle balance equation:

qn

qt¼ �r � J, (1)

where n is the particle density of a solute in a binary system, J the diffusion flux, and t the time. The diffusionflux is assumed to be described by a relaxational Maxwell–Cattaneo equation [1–4]

tD

qJqtþ J ¼ �Drn, (2)

where tD and D are the relaxation time and diffusion constant, respectively. The relaxation term is negligiblefor steady states or low-frequency perturbations. It becomes dominant at high frequencies or fast speed ofpropagation.

2.1. Density profiles

Combining Eqs. (1) and (2), one gets the following equation of a hyperbolic type:

tD

q2nqt2þ

qn

qt¼ Dr2n. (3)

Eq. (3) predicts the propagation of the density profile with a sharp front moving with a finite speed V D insidethe undisturbed system. To show this feature of hyperbolic transport, we find an analytical solution of Eq. (3)for the semi-infinite (one-dimensional) space by choosing the initial and boundary conditions in the formnðt; 0Þ ¼ nf , nð0;xÞ ¼ nðt;x!1Þ ¼ n0, qnð0;xÞ=qt ¼ 0 (where x is a spatial coordinate). Under theseconditions, the solution is described by the following expressions:

�
behind the diffusive front, 0pxotVD,
nðt;xÞ ¼ n0 þ ðnf � n0Þ expð�x=laÞ þ ðnf � n0Þðx=laÞ

Z t

t¼x=VD

f ðt;xÞdt,

f ðt; xÞ ¼expð�t=2tDÞ

ðt2 � x2=V2DÞ

1=2I1ðt2 � x2=V 2

DÞ1=2

2tD

" #, ð4Þ

�
at the diffusive front, x ¼ tV D,
nðt;xÞ ¼ n0 þ ðnf � n0Þ expð�x=laÞ � n0 þ ðnf � n0Þ expð�t=2tDÞ, (5)

ARTICLE IN PRESS

Fig. 1. Profiles of density n at different moments t1ot2ot3 as predicted by solution (4)–(6). Every profile moves with the sharp

discontinuity front which has the diffusion speed VD. The x-coordinate of this discontinuity front is given by tVD, and the amplitude of

the front is decreasing in time as expð�t=2tDÞ. The density profile at t ¼ t3\10tD is matched to the one described by a partial differential

equation of a parabolic type (i.e., of the form of Eq. (3) with tD ¼ 0).

D. Jou, P. Galenko / Physica A 366 (2006) 149–158 151

�
ahead of the diffusive front, tV Doxo1,
nðt;xÞ ¼ n0. (6)

Here, VD ¼ ðD=tDÞ1=2 and la ¼ 2ðDtDÞ

1=2 are the diffusive speed and the attenuation distance in the high-frequency limit [10], respectively, and I1 is a modified Bessel function of the first order.

The concentration profiles described by Eqs. (4)–(6) are shown in Fig. 1. In contrast with the concentrationprofiles described by the parabolic differential equation (Fick’s diffusion) the concentration profiles in thehyperbolic case have a sharp diffusive front which moves with the speed VD (Fig. 1). This diffusive frontseparates the spatial regions where diffusion occurs [n4n0 at xoV Dt, Eq. (4)] and where diffusion is absent[n ¼ n0 at x4VDt, Eq. (6)]. Therefore, the position of the diffusive front may be examined as a depth, tV D, ofdensity penetration into a binary system. As it is shown in Fig. 1, the amplitude of the diffusive front atx ¼ V Dt decreases with increasing time and spatial coordinate, according to Eq. (5).

2.2. Fluctuations and stochastic noise

Even though solution (4)–(6) describes a smooth profile of density (with sharp diffusion front) fluctuationsalways exist in a thermodynamic system. Indeed, techniques of light scattering or neutron scattering allow toexplore details of the dynamics of density perturbations in the system, and it has fostered progress innonequilibrium statistical mechanics [11]. Here, we study features of the system described by Eqs. (1) and (2)related to n and J fluctuations in equilibrium and to stochastic noise.

The equilibrium second moments of fluctuations of n and J are obtained from the Einstein’s equation forthe probability of fluctuations [5,12], namely

Pr� expd2s

2kBT

� �, (7)

where entropy sðn;JÞ is based on the independent thermodynamic variables n and J. It is known (see, e.g., Ref.[12, Chapter 15]) that Einstein’s (7), considered as an approximate Gaussian distribution function, predicts thesecond moments correctly, but it does not predict third and higher moments accurately. However, since we areonly interested in the second moments, we restrict ourselves to the use of the simple Einstein formula (7).

To obtain the second differential d2s of entropy in Einstein’s (7), one needs to choose the form of the Gibbsequation for entropy. The generalized Gibbs equation which incorporates slow and fast thermodynamicvariables is written as [3]

ds ¼1

Tdu�

mT

dn�tD

TDJ � dJ, (8)


where u is a density of internal energy, D is related to the usual diffusion coefficient D through D ¼ Dðqm=qnÞ,and m ¼ m1 � m2 is the relative chemical potential of the solute with respect to the one of the solvent.

We focus our attention on the fluctuations of n and J and assume du negligible for the sake of simplicity.Then, from Eq. (8), we get the second differential of the entropy as

d2s ¼ �1

T

qmqnðdnÞ2 �

tD

TDðdJÞ2. (9)

With definition (7) and taking the second variation of s from Eq. (9), the probability of fluctuations isdescribed by

Prðdn; dJÞ� exp �v

2kBT

qmqn

� �ðdnÞ2 �

vtD

2kBTDðdJÞ2

� �, (10)

where v is a small volume in which the fluctuations dn and dJ occur. The second moments of fluctuations aregiven by

hðdnÞ2i ¼kBT

vðqm=qnÞT; hðdJÞ2i ¼

kBTD

vtD

¼kBTD

vtDðqm=qnÞT. (11)

In what follows we discuss two important points:

(i)
the power spectra of the fluctuations of n and J; (ii) the description of the stochastic noise in system (1) and (2).
3. Power spectra of density and flux fluctuations

Let us define the correlation functions for the fluctuations of n and J in the following usual form

Cnðr; r0; t; t0Þ � hdnðr; tÞdnðr0; t0Þi,

CJðr; r0; t; t0Þ � hdJðr; tÞdJðr0; t0Þi, ð12Þ

where r is the position vector of a point in the system. Since we consider equilibrium (homogeneous, time-invariant state), one has

Cnðr; r0; t; t0Þ ¼ Cnðr� r0; t� t0Þ,

CJðr; r0; t; t0Þ ¼ CJðr� r0; t� t0Þ, ð13Þ

i.e., the correlation functions depend only on relative distances r� r0 and on the difference in time t� t0. Weare interested in the Fourier transforms of the quantities in Eq. (13), namely

Snðo; kÞ ¼Z

eioteikrCnðr; tÞdrdt,

SJðo; kÞ ¼Z

eioteikrCJ ðr; tÞdrdt. ð14Þ

These expressions represent fluctuation spectra and have special theoretical and practical interest, as they maybe measured by means of light scattering or neutron scattering techniques [11].

To obtain an explicit form of the fluctuation spectra we first write Fourier transform (in space) and Laplacetransform (in time) of Eqs. (1) and (2). Using the standard procedure described in Refs. [3,11] we arrive at

SdnkðSÞ þ ikdJkðSÞ ¼ dnkð0Þ,

tDSdJkðSÞ þ dJkðSÞ þ ikDdnkðSÞ ¼ dJkð0Þ, ð15Þ

where dnkðSÞ and dJkðSÞ are the Fourier–Laplace components of dn and dJ, respectively. Then, we have

dnkðSÞ

dJkðSÞ

!¼

1

Sð1þ tDSÞ þDk2

1þ tDS �ikD

�ik S

� � dnkð0Þ

dJkð0Þ

!. (16)

ARTICLE IN PRESSD. Jou, P. Galenko / Physica A 366 (2006) 149–158 153

In equilibrium (where k! 0), the crossed second moments hdnkð0ÞdJkð0Þi vanish because they have oppositetime-reversal parity. Then, from Eq. (16) we have

hdnkðSÞdnkð0Þi ¼1þ tDS

Sð1þ tDSÞ þDk2hjdnkð0Þj

2i,

hdJkðSÞdJkð0Þi ¼S

Sð1þ tDSÞ þDk2hjdJkð0Þj

2i. (17)

To obtain the time Fourier transform, one may write

Snðo; kÞ ¼ 2Re½hdnkðS ¼ ioÞdnkð0Þi�,

SJ ðo; kÞ ¼ 2Re½hdJkðS ¼ ioÞdJkð0Þi�. (18)

Finally, we obtain

Snðo; kÞ ¼2Dk2

t2Do4 þ ð1� 2DtDk2Þo2 þ ðDk2

Þ2hjdnkð0Þj

2i,

SJ ðo; kÞ ¼2o2

t2Do4 þ ð1� 2DtDk2Þo2 þ ðDk2

Þ2hjdJkð0Þj

2i. (19)

The corresponding expressions for hjdnkð0Þj2i and hjdJkð0Þj

2i in equilibrium, obtained from Eq. (11), aredescribed by

hjdnkð0Þj2i ¼

kBT

vðqm=qnÞT; hjdJkð0Þj

2i ¼2kBT

vtDðqm=qnÞT. (20)

Note, that in Eq. (19) the function Snðo; kÞ has a maximum at a frequency om given by

om ¼2Dk2tD � 1

2t2D

� �1=2

. (21)

The fact that the maximum is at oma0 indicates propagation of density waves with the speed k=om, incontrast with the situation when the maximum is at om ¼ 0, which means purely diffusive transport. It is clearfrom Eq. (21) that to observe such a maximum, i.e., the propagation of density wave, it is needed thatk4kc � ð2DtDÞ

�1=2. Thus, for kokc transport is diffusive, and for k4kc the density waves may propagate.This analysis is analogous to the analysis of the transverse velocity correlation function in generalizedthermodynamics for the Maxwell visco-elastic model [11], which is consistent with the formalism of EIT [3].

4. Stochastic noise in hyperbolic transport

In the former section we have analyzed the fluctuations of density and of diffusion flux, with specialemphasis to the former one. Here, in contrast, we focus our attention on a more conceptual question related tothe fluctuations of the flux J or, more concretely, to their conceptual interpretation. We shall see that,depending on the value of the relaxation time tD and of the observational time scale, such fluctuations can beinterpreted in two different ways. To discuss these ideas we must recall some results concerning hydrodynamicstochastic noise.

4.1. Hydrodynamic stochastic noise

Stochastic noise is an idealized representation of the dynamical effects of the fast variables excluded fromthe macroscopic hydrodynamic description. We first deal with a multiple abstract representation and,afterwards, with our specific system.


Let us consider a system described by one variable a whose evolution is given by

_a ¼ �Laþ Z. (22)

Here Z is the stochastic noise which reflects the effect of the variables much faster than a, and L is a coefficientrelated to friction forces. An equation of this form is known as a Langevin equation. In the original analysis ofLangevin, a was identified as the speed of a Brownian particle in a viscous fluid, and L was the inverse of thedecay time set by a viscous force. Namely, it was assumed that L � 6pZ0r=m with Z0 the viscosity of the fluid,and m and r the mass and radius of the Brownian particle, respectively. The stochastic noise Z was attributedto the very rapidly changing forces due to collisions with the microscopic particles of the solvent. Thus, in thisdescription, the only slow variable refers to the velocity of the Brownian particle.

The stochastic noise is assumed to be Gaussian and white, i.e., it is delta-correlated in time. It isdescribed by

hZðtÞi ¼ 0; hZðtÞZðt0Þi ¼ Gdðt� t0Þ, (23)

where h. . .i describes the average over the realizations of the noise. The constant G may be obtained byconsistency with the condition that ha2ð0Þi must be given by the theory of fluctuations at equilibrium. Then, itfollows that

G ¼ 2Lha2ð0Þieq. (24)

If, instead of one variable, a represents a set of several independent variables, Eq. (24) is written as [14]

C ¼ L � haaieq þ haaieq � LT, (25)

where LT stands for the transport matrix. Relations (24) and (25) are called ‘‘fluctuation–dissipationrelations’’, as they relate the second moments of the noise, G, to the friction coefficient L.

4.2. Hyperbolic transport with noise

We discuss now the stochastic noise in a system

t€aþ _a ¼ �Laþ Z, (26)

which may generally represent the hyperbolic transport (3) with noise. Let us compare the form of noise Zfrom Eq. (26) with the noise features introduced in system (22). Taking into account both independentvariables a and _a � b, Eq. (26) may be written as

_a ¼ bþ eZa; _b ¼ �L

ta�

1

tbþ eZb. (27)

The set of equations (27) represents the second-order (26) as two first-order evolution equations, in a way thatthe system becomes Markovian. In Eq. (27), Za and Zb are the respective stochastic noise of both equations,whose second-order moments have to be obtained. Applying Eqs. (25)–(27), we get

heZðtÞeZðt0Þi ¼ 0 �1L

t1

t

24 35 ha2ieq habieq

hbaieq hb2ieq

" #þha2ieq habieq

hbaieq hb2ieq

" # 0L

t

�11

t

26643775. (28)

In Eq. (28) we have habieq ¼ hbaieq ¼ 0 because a and _a have opposite time-reversal symmetry. Thus it isfound

heZ2ai ¼ 0; heZ2bi ¼ 2

thb2ieq,

heZaeZbi ¼ heZbeZai ¼ L

tha2ieq � hb

2ieq. (29)


To obtain the second moment of equilibrium fluctuations of a and b we assume, as before, that a and b areindependent variables. Then, including both of these variables into the entropy, one can write

Sða;bÞ ¼ Seq �12

Aa2 � 12

Bb2, (30)

where only second-order terms have been considered. According to Eq. (7), the probability of fluctuations isdescribed by

Prða;bÞ� exp �A

2kB

a2 �B

2kB

b2� �

. (31)

Note that in Eq. (31) we have identified the fluctuations da and db with a and b, respectively, because theirequilibrium average values are zero for both of them due to the form of the evolution equations (27). Then, inequilibrium, we have

ha2ieq ¼kB

A; hb2ieq ¼

kB

B. (32)

To evaluate the ratio A=B, we obtain the entropy production corresponding to Eq. (30). This yields

dS

dt¼ �Aa_a� Bb _b ¼ �Aa_a� B_a€a ¼ �_a½Aaþ B€a�X0. (33)

From this, and by following the usual methods of nonequilibrium thermodynamics [13], it follows the linearrelation between thermodynamic flux �_a and its conjugated force Aaþ B€a. This is

Aaþ B€a ¼ �m_a. (34)

Comparison of Eq. (34) with Eq. (26) yields B ¼ t, m ¼ 1, and A ¼ L. Thus, it follows that the secondmoments of a and b are related by

ha2ieq

hb2ieq

¼B

A¼

tL. (35)

Introducing this relation into Eq. (29) it is found

heZ2ai ¼ 0; heZaeZbi ¼ heZbeZai ¼ 0,

heZ2bi ¼ 2

thb2ieq ¼

2L

t2ha2ieq. (36)

Since heZ2ai ¼ 0, the first equation in Eq. (27) may be introduced into the second one. Therefore, we get

t€aþ _a ¼ �Laþ teZb, (37)

with

hZbð0ÞZbðtÞi ¼ hteZbð0ÞteZbðtÞi ¼ 2Lha2ieq. (38)

Thus, the expression for the noise keeps the same form as in the case with t ¼ 0. This is in agreement with theideas of fluctuation–dissipation, which relate the fluctuations to the dissipative part of the equation [the termin L in Eq. (37)].

The transition from noisy hyperbolic transport described by Eq. (26) to Langevin (22) might be analyzed byconsidering the generalized entropy (30) in the following form:

Sða;bÞ ¼ Seq �A

2a2 �

At2L

b2. (39)

With t! 0, the last term in b2 disappears together with the term in €a in Eq. (26). In this case, the dynamics ofa is described by a simple relaxation with a temporal constant given by L�1. One interesting situation may befound when t5L�151. In this case, a decays slowly and b decays fastly. Assume, for instance, that L � 1 s�1

and t�10�3 s. The typical relaxation of a will be of the order of 1 s and b will decay in a millisecond scale. In


this case, eZb describes the effect of all the variables whose relaxation time is much less than 10�3 s, in such away that they may be assumed to decay instantaneously in comparison with b.

4.3. Stochastic noise or independent variable

Now, consider a special system with two independent variables a and b in such a way that the relaxationrate of b is characterized by its high but finite value. In this case, one may write db � bþ La, i.e.,b ¼ �Laþ db, where db being the independent part of b. This part of b is orthogonal to the slow subspacegenerated by a. Then, we may write

_a ¼ �Laþ db; d _b ¼ �1

tdb (40)

and

hdbð0ÞdbðtÞi ¼LkB

Ate�jtj=t ¼

L

tha2ie�jtj=t. (41)

The following three cases may be outlined in considering Eqs. (40) and (41).

(i)
If t is sufficiently short, db acts as a ‘‘noise’’ in the equation for a [the first equation of system (40)]. (ii) If t is not completely negligible as compared to L�1, db acts as a colored noise. (iii) In the limit t! 0, one has
hdbð0ÞdbðtÞi ! 2Lha2ieqdðtÞ (42)

in agreement with Eq. (24).

Thus, it is seen that the transition of Eqs. (39) and (40) from small t to vanishing t is conceptuallyinteresting. It is illustrative of how the variable b (i.e., _a) goes from an independent variable with its owndynamics to a purely Markovian stochastic noise. In physical terms, the frontier between small t andvanishing t is settled by the time scale one is able to measure. E.g., if L ¼ 1 s�1 and t ¼ 10�8 s, the system willhave two independent variables, a and _a, for an observer which is able to measure picoseconds ð10�12 sÞ. But itwill have only one independent variable, a, plus a stochastic noise, in a form of _a, for an observer which is onlyable to measure milliseconds ð10�3 sÞ. The latter observer will be able to work in the ‘‘adiabatic’’approximation with very slow a in comparison with b.

The situation described by Eqs. (1) and (2) is interesting from this perspective. For negligible values of thediffusion relaxation time tD, one should use the Landau–Lifshitz formalism for the fluctuating hydrodynamicsand write

qn

qt¼ Dr2nþ Zn, (43)

where the stochastic noise Zn is interpreted as

Zn ¼ �r � dJ (44)

with dJ being a fluctuating part of the diffusion flux, i.e.,

J ¼ �Drnþ dJ. (45)

According to the Landau–Lifshitz approach [5], one has

hdJð0ÞdJðtÞi ¼ 2kBDTdðtÞ. (46)

Assume, in contrast, that tD is small but still measurable. In this case, we get

qn

qt¼ �r � J; tD

qJqtþ J ¼ �Drnþ ZJ , (47)

with n and J being independent variables of the entropy given by the generalized Gibbs (8). The secondmoments of the fluctuation of n and J are given by Eq. (11). The noise ZJ would correspond to values relaxing


in time scales much shorter than tD, in such a way that it may be considered as Markovian noise:

hZJð0ÞZJðtÞi ¼ 2kBDTdðtÞ. (48)

In the limit of vanishing tD, the fast part of J becomes a stochastic noise, and we get Zn in Eq. (43) described byEq. (46).

Still another form to discuss the interpretation of noise in the context of Eq. (26) is to write Eq. (27) withoutany added noise, i.e., in the following form:

_a ¼ b; _b ¼ �L

tD

a�btD

. (49)

This set of equations may be integrated to give

_aðtÞ ¼ �Z t

t0

Mðt� t0Þaðt0Þdt0 þ bðtÞ, (50)

with the memory function

Mðt� t0Þ ¼L

tD

exp �t� t0

tD

� �, (51)

and the exponential relaxation

bðtÞ ¼ _aðt0Þ exp �t� t0

tD

� �. (52)

In this case, the ‘‘noise’’ is due to the uncertainty in the value of _aðtÞ at the initial time t0.Mori’s expression for the fluctuation–dissipation theorem states that

hbðtÞbðt0Þi ¼Mðt� t0Þha2ieq. (53)

Indeed, if we use result (35), we obtain that

h_a2ðt0Þieq � hb2ðt0Þieq ¼ ha

2ieq

L

t. (54)

Combination of Eq. (54) with Eqs. (51) and (52) gives Eq. (53).Again, when tD becomes negligible, Mðt� t0Þ given in Eq. (51) becomes

Mðt� t0Þ ¼ Ldðt� t0Þ, (55)

and Eq. (50) becomes

_aðtÞ ¼ �LaðtÞ þ Za, (56)

with

hZ2ai ¼ 2Lha2i. (57)

Note that in limit (55) we consider t4t0, whereas in limit (57) a factor 2 appears because one considersjt� t0j40 rather than t� t040, i.e., one considers both t� t040 and t0 � t40.

5. Conclusions

The power spectra of a solute number density and a solute diffusion flux have been reviewed. The latter hasa nonvanishing relaxation time leading to a hyperbolic transport equation for the evolution of the density.

Several interpretations of the stochastic noise related to fast variables eliminated from the description havebeen examined. Our aim was not to review the well-known results of adiabatic elimination of slow variables,but, rather, to show up the role of variables which are ‘‘relatively’’ fast, i.e., which do not have a strictlyvanishing relaxation time.

The descriptions provided by two different observers, able to measure a minimum time interval t1 and t2,respectively, were compared. For one of the observers, t1otD: for him, the fast variable is not a stochastic


noise, but a dynamical variable with its own evolution equation and its own contribution to the entropy of thesystem. For the other observer, for which t24tD, the fast variable may be simply interpreted as a stochasticnoise appearing in the equation for the slow variable, but it has neither an evolution equation of its ownneither it contributes to the entropy of the system. Thus, the first observer should use an extendedthermodynamic approach, whereas the second observer would describe the same system by using fluctuatinghydrodynamics.

We considered the fluctuations of the solute density and the solute diffusion flux at the equilibrium steady-state. Another interesting question may appear considering evolution of strongly nonequilibrium systems.This concerns the correlation of fluctuations near nonequilibrium steady states. One may expect that thepurely statistical aspect of the problem of nonequilibrium fluctuations may be described by the correspondingnonequilibrium extension of the Einstein relation (7), but, of course, the dynamical aspects may be different.

The suggested analysis has a general meaning. In particular, it can be applied to the fluctuations in systemsunder rapid phase transitions [4] for both slow and fast thermodynamic variables.

Acknowledgements

D. J. acknowledges financial support from the Direccion General de Investigacion of the Spanish Ministryof Science and Technology BFM 2003-06033 and the Direccio General de Recerca of the Generalitat ofCatalonia under Grant 2001 SGR-00186. P. G. acknowledges financial support from the German ResearchFoundation (DFG—Deutsche Forschungsgemeinschaft) under the Project no. HE 1601/13.

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fluctuations and stochastic noise in systems with hyperbolic mass transport

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