A Gaussian theory for fluctuations in simple liquids

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by  Matthias Krüger

A Gaussian theory for fluctuations in simple liquids
Matthias Kru¨ger1 and David S. Dean2
14th Institute for Theoretical Physics, Universita¨t Stuttgart,
Germany and Max Planck Institute for Intelligent Systems, 70569 Stuttgart, Germany
2Univ. Bordeaux and CNRS, Laboratoire Ondes et Matire d’Aquitaine (LOMA), UMR 5798, F-33400 Talence, France
(Dated: January 27, 2017)
AssuminganeffectivequadraticHamiltonian,wederiveanapproximate,linearstochasticequation
ofmotionforthedensity-fluctuationsinliquids,composedofoverdampedBrownianparticles. From
this approach, time dependent two point correlation functions (such as the intermediate scattering
7
function) are derived. We show that this correlation function is exact at short times, for any
1
interactionand,inparticular,forarbitraryexternalpotentialssothatitappliestoconfinedsystems.
0
Furthermore, we discuss the relation of this approach to previous ones, such as dynamical density
2
functionaltheoryaswellastheformallyexacttreatment. Thisapproach,inspiredbythewellknown

n Landau-GinzburgHamiltonians,andthecorresponding“ModelB”equationofmotion,maybeseen
a as its microscopic version, containing information about thedetails on the particle level.
J

6 PACSnumbers: 05.40.-a,61.20.Gy,61.20.Lc,82.70.Dd
2

] I. INTRODUCTION beenusedtoderivethevanHovefunctionfromit[34–36].
t
f On the experimental side, the intermediate scattering
o
The statics and dynamics of simple liquids is of great function is an important quantity characterizing the dy-
s
. importance both in fundamental research[1–3], but also namicsofliquids,e.g.asregardstheglasstransition[37],
t
a in industry, technology and biology. The statics have andcanalsobemeasuredinconfinement[38]. Moregen-
m been investigated for many years and are well under- erally,thedynamicsoffluidsinconfinementhavereceived
- stood, forinstance viathe frameworkof classicaldensity alotofrecentattention[39–41],amongotherreasonsdue
d functional theory [1, 2, 4]. to improved experimental precision on small scales [42–
n
Studies of the dynamical properties of fluids, such as 45], and in microfluidic devices [46, 47] or blood flow in
o
c the viscosity,have a long history [5], and the field is still capillaries [48, 49].
[ veryactive [6, 7]. Linearresponse theory [8, 9], connects Previous approaches that discuss the stochastic dy-

transportcoefficientstotimedependentcorrelationfunc- namics of particle densities, including noise, have been
1
tions measured in thermal equilibrium, the time depen- presented in Refs. [50] and [51], see also Ref. [52]. We
v
9 dentcorrelationfunctions[1,10,11]studiedherearethus will discuss their relation with the approach developed
4 of particular importance.. here.
6 Timedependentcorrelationfunctionscanbecomputed In this manuscript, we propose a description of fluc-
7 fromvariousfundamentalequations,suchastheLiouville tuations of fluids near equilibrium by use of a Gaus-
0
[1] or Fokker Planck equations [10, 11]. Dilute systems sian field theory, corresponding to an effective quadratic
.
1 havebeenexaminedusingexactdynamicalformulations, Hamiltonian for the density fluctuation field and a cor-
0 for instance via the Boltzmann equation [12, 13] or us- responding Langevin equation. The Hamiltonian is con-
7
ing the Fokker Planck equation [10, 11, 14]. In dense structed to yield the correct static equilibrium averages.
1
systems, approximatedynamicalformulations have been The corresponding Langevin equation is constructed to
:
v used: Here, Mode Coupling Theory (MCT) [3] is useful yield the dynamics of overdampedparticles. Within this
Xi forthe computationoftime correlationfunctionsinbulk theory, we derive a closed, approximate expression for
systems, and has recently also been applied in confine- the time depend equilibrium correlation function, which
r
a ment [15–18]. Classical Density Functional Theory finds agrees with the exact result (found from the Smolu-
static equilibrium quantities [1, 2, 4], while Dynamical chowski equation) for short times. It is thus expected
Density Functional Theory (DDFT) [19–21] is powerful to describe well the dynamics at not too high densities,
for describing out of equilibrium situations. In addition and might especially provide insight into dynamics in
totheevolutionintimedependentpotentials,DDFThas confined systems. We also demonstrate the connection
also been used to study driven suspensions with spher- between the derived dynamics and the dynamics follow-
ical obstacles [22, 23] or with constrictions [24], driven ingfromdynamicaldensityfunctionaltheory,aswellthe
liquid crystals [25], suspensions under shear [26–30] and exact stochastic equation for the density operator.
for microswimmers [31]. Such research directions have The manuscriptis structuredasfollows. InSection II,
also benefitted from formal improvements within power we lay out the theoretical framework, starting with the
functional theory [32, 33]. system considered in Section IIA. We define the phys-
Despite these many applications, DDFT provides no ical observables of interest in Sec. IIB. The quadratic
immediate accesstothe timedependentequilibriumcor- effective Hamiltonian is introduced in Section IIC, and
relation functions, however the test particle trick has the stochastic equation of motion is introduced in Sec-
2
Brownian particles is denoted by D. Each particle thus
Symbol Meaning
obeys the stochastic differential equation
ρ(x) Density operator: ρ(x)=Piδ(x−xi) dx
hρ(x)i Mean density in equilibrium. i =DβF +√2Dξ (1)
dt i i
φ(x,t) Fluctuationofdensitynearaboutitsequi-
librium value, φ(x,t)=ρ(x)−hρ(x)i. where ξ is white noise in the Ito Stochastic Calculus
i
hφ(x,t)φ(x′,t′)i Time dependent correlations of density with ξi,µ(t)ξj,ν(t′) =δijδµνδ(t t′), and Fi is the force
h i −
fluctuations in equilibrium, the quantity actingonparticlei,duetothepotentialΦ. (Throughout,
of interest of this work. i and j label particles, while Greek indices label spacial
hρ(x,t)ineq Mean density in nonequilibrium state. components). In other words, in the absence of Φ, each
particle performs isotropic Brownian motion.
δρ(x,t) Averagedifferencefromequilibriumvalue
in a perturbed system, δρ(x,t) =
hρ(x,t)ineq−hρ(x)i.
B. Obervables – mean and fluctuating
TABLEI.Observablesstudiedinthismanuscript. Thelower
We summarize the arising observables in Table I. The
tworows, i.e., thedensityin nonequilibriumstates, aregiven
density operator, ρ(x) = δ(x x ) [1] is the start-
for comparison to dynamical density functional theory in i − i
ing point for all considerations that follow. If averaged
Sec. IV. P
over the equilibrium distribution, one obtains the mean
equilibrium density
tion IIE. The time dependent correlation function from
thisframeworkiscomputedinSectionIII,andwedemon- ρ(x) = δ(x x ) . (2)
i
strateinSectionIIIB,thatitisexactforshorttimes. In h i * − +
i
SectionIV,weshowthatproposedfoundequationofmo- X
tion is in close correspondence with DDFT. Sec. V dis- Here, we have introduced the equilibrium average ... ,
h i
cussestheconnectiontotheexactequationofmotionfor which,fortheoverdampedsystemisexactlygivenby(we
the density field of Ref. [50]. We summarize in Section introduce the phase space abbreviation Γ xi )
≡{ }
VI.
dΓ...e−βΦ(Γ)
... = . (3)
h i dΓe−βΦ(Γ)
R
II. SYSTEM, EFFECTIVE HAMILTONIAN AND
As noted above, for systeRms with infinite particle num-
EQUATION OF MOTION
ber, the grandcanonicalaverageagreeswith the canoni-
cal one given here. We introduce density fluctuations,
A. System
φ(x)=ρ(x) ρ(x) . (4)
−h i
We aim to analyze time dependent correlation func-
Thisquantitywillbeimportantforthismanuscript. Such
tions in liquids. For this, we choose a well studied, and
fluctuations are e.g. characterized by their correlation
alsoexperimentallyrelevantmodelsystem,whichisover-
function, which relates two points in space and in time
damped spherical (Brownian) particles. (The question
t, (in the following we will sometimes suppress the argu-
how well this model system describes aspects of molecu-
ments of C)
lar liquids also attracted recent interest [53].)
Regardingthe ensemble, usingBrowniandynamicsdi- C(x,x′,t t′)= φ(x,t)φ(x′,t′) . (5)
− h i
rectlyimpliesacanonicalorgrand-canonicaldescription,
Due to the fact that we restrict to equilibrium fluctua-
where the solvent acts as a bath at the given tempera-
tions, C is a function of t t′ [11], but depends on both
ture. We will generally have in mind systems for which
−
x and x′ in inhomogeneous systems. Computing C is
canonical and grand canonical descriptions are equiva-
themaingoalofthemanuscript. Iftransformedtorecip-
lent due to the large (infinite) particle number (such as
rocal Fourier (k-space), C˜ is the intermediate scattering
in the semi-infinite system bound by a planar surface).
function [10].
Extra care has thus to be taken for closed systems, such
For completeness, we also define the mean density in
as particles confined in a box of finite size (see Ref. [54]
an out of equilibrium situation, i.e., ρ(x,t) neq, and its
for an analysis of canonical systems in DFT).
h i
The Brownian particles with positions at x are sub- average deviation from equilibrium
i
ject to a potential Φ( xi ), including pairwise interac- δρ(x,t)= ρ(x,t) neq ρ(x) . (6)
{ }
tions (later denoted by V) as well as an external poten- h i −h i
tial (later denoted U). The thermal energy scale is de- Note the difference to Eq. (4), which is for a stochastic
noted by k T β−1, with Boltzmann constant k and fluctuation in an equilibrium system while Eq. (6) is an
B B
≡
the (solvent imposed) temperature T. The bare diffusiv- average deviation from the equilibrium average density
ity (the diffusivity in the absence of interactions) of the for a perturbed system.
3
C. Effective Hamiltonian that the direct correlation function is a rather feature-
less function, and typically zero if x y is larger than
| − |
Aiming at the correlation function C, we start by as- the interactionrangeofthe particles[1]. Incontrast,the
suring that the equal time value of C is found correctly. correlationfunction φ(x)φ(y) can extend to larger dis-
h i
We thus introduce the following effective Hamiltonian, tances, which, mathematically, is a consequence of tak-
which is a functional of the fluctuating field φ, ingtheoperatorinverse. Physically,itiswellknownthat
correlationsmayreachfurtherthaninterparticleinterac-
1 1 tions.
βH = dxdyφ(x) φ(y), (7)
2 φ(x)φ(y) Wefinishthissubsectionbyintroducingtheshorthand
Z h i
notation for the inverse of the static density correlation
where
1
1 φ(x)φ(y) −1, (8) ∆(x,y)≡kBT ρ(x) δ(x−y)−c(2)(x,y). (13)
φ(x)φ(y) ≡h i (cid:18)h i (cid:19)
h i
is to be understood in the sense of inverse operators. In ∆(x,y)playstheroleofaneffectiveinteractionpotential
the field theory description, the equilibrium average in between densities (we multiplied by kBT to obtain units
Eq. (3) is computed via the following functional integral of energy). In terms of it, the Hamiltonian is finally,
[55, 56]
1
H = dxdyφ(x)∆(x,y)φ(y). (14)
φ...e−βH 2
... = D . (9) Z
h i φe−βH
R D
φ denotes the measure ofRfunctional integration,which E. Equation of motion
D
ismosteasilyimplementedbydiscretizingspaceorwork-
ing with discrete Fourier transforms. As mentioned be-
While static equilibrium averages are determined via
fore, we point out that the Hamiltonian in Eq. (7) with
theHamiltonian(withEq.(9)),thereissomefreedom,or
Eq. (9) by construction correctly finds the static aver-
saying it differently, some lack of information, regarding
ages of φ, up to quadratic order. Its average, φ =0, as
h i thedynamics. TheHamiltonianinEq.(7)describesonly
required from Eq. (4), and its variance is indeed, using
a subset of degrees of freedom of the system. These de-
Eq. (7),
grees of freedom might however not capture all relevant
features of the dynamics [55, 58]. The tools of classical
φφ(x)φ(y)e−βH
φ(x)φ(y) = D . (10) mechanics are thus not applicable to deduce equations
h i φe−βH
R D ofmotion fromEq. (7). One possibility to overcomethis
problemistoresorttoLangevinequations[55,58],which
See e.g. Ref.[57]for usefulideRntities regardingGaussian
are based on deterministic (given by the explicit degrees
functional integrals.
offreedom)aswellasstochasticforces(duetointegrated
degrees of freedom) [59]. The former may be written in
terms of the driving force δH, which gives the force due
D. Static correlations – Theory of liquids δφ
todeviationsofH fromitsminimumvalue. Forourcase,
The variance of φ in Eq. (10) is a well studied object
δH φ(x)
inthetheoryofliquids,andcanbe expressedintermsof β = dyc(2)(x,y)φ(y) (15)
δφ ρ(x) −
the so calleddirect pair correctionfunction c(2) [1], (this h i Z
equation can be seen as one way of defining c2) β∆φ. (16)
≡
1 1 In the second line, we used the short hand notation of
= δ(x y) c(2)(x,y). (11)
φ(x)φ(y) ρ(x) − − Eq. (13), and also a short hand notation for operator
h i h i
products, so that the second line contains an integration
Usingthis,wecanmaketheHamiltonianinEq.(7)more over the joint coordinate. Clearly, δH = 0, vanish-
explicit, δφ
ing in equilibrium. This force transDformEs into changes
1 φ(x)2 1 in φ with application of the operator R = R(x,y), so
βH = 2 dx ρ(x) − 2 dxdyφ(x)c(2)(x,y)φ(y). that RδδHφ involves an integral over the joint coordinate.
Z h i Z (12) R involves among other things a mobility coefficient. R
having no time dependence, we have already restricted
This shows the nature of the Hamiltonian: It has a lo- to a time localdescriptionforsimplicity. Time non-local
cal term, corresponding to the local compressibility of dynamics can be realized in this framework as well. Via
an ideal gas, and a nonlocal term, which is given by the operator R, one can incorporate severaltypes of dy-
the direct correlation function. It is worth mentioning namics, such as dynamics conserving the density, or not
4
conserving it [55, 58, 60]. We aim at dynamics of over- III. TIME DEPENDENT CORRELATION
dampedBrownianparticlesgivenbyEq.(1),forwhich– FUNCTION
justified a posteriori – the proper choice for R is,
In this section, we finally compute andanalyze the re-
D
R= ρ(x) δ(x y). (17) sulting approximative form of the time dependent equi-
k T∇·h i∇ −
B libriumcorrelationfunction, as definedinEq.(5),as fol-
Note that, because R is written as a divergence, local lowing from Eq. (19).
conservationof density is given. Indeed, the chosenR in
Eq. (17) is a version of the famous “Model B” [55, 58].
We thus write the following equation of motion, A. General result
∂φ δH
=R + 2D ρ η(x,t), (18) We start by writing the Langevin equation, Eq. (19)
∂t δφ ∇· h i
in a shorter form, using Eq. (13),
D p δH
= ρ(x) + 2D ρ η(x,t). (19)
∂φ D δH
kBT∇·h i∇δφ ∇· h i = ρ(x) + 2D ρ η,
p ∂t kBT∇·h i∇δφ ∇· h i
The included stochastic force is fixed through the choice
p
of the operator R. The field η(x,t) is spatio-temporal =R∆φ+ 2D ρ η, (23)
∇· h i
vectorial white noise field with η =0, and whose com-
h i Eq.(23)canthenbeeasilpysolvedforthecorrelationfunc-
ponents have the correlationfunction
tion, from its general solution, (see, e.g., Ref. [60]),
η (x,t)η (y,t′) =δ δ(t t′)δ(x y). (20)
µ ν µν t
h i − −
φ(t)=φ(t )+ dse(t−s)R∆ 2D ρ η(s). (24)
The form of the last term in Eq. (18) and the variance 0 ∇· h i
Zt0
in Eq. (20) ensures the validity of the fluctuation dissi- p
pation theorem [60], and makes sure that Eq. (18) finds Theaverageofφ(t)φ(t′)overthenoisecontainsthensev-
the correct variance for φ. The explicit form of Eq. (18) eralterms,includingtermsdependingontheinitialvalue
reads att0. Aiming atthe equilibriumcorrelationfunction, we
let t and t′ formally go to infinity, and obtain the steady
equilibrium part, which depends only on t t′ (recall
∂φ −
=D φ φ log ρ ρ dyc(2)(x,y)φ(y) that, as before, ... denotes anaveragein equilibrium),
∂t ∇· ∇ − ∇ h i−h i∇ h i
+ (cid:20) 2D ρ η(x,t) Z (21(cid:21)) C = φ(x,t)φ(x′,t′) = kBTe|t−t′|∆R. (25)
∇· h i h i ∆
Examiningtphisequationforthecaseofanidealgasinthe
The correlation function is generally not an exponential
absence of an external potential, for which c(2) = 0 and
in time, because R and ∆ are operators. Eq. (25) is our
ρ spatiallyconstant,weobtain,asexpected,adiffusion
second main result.
h i
equation with a conservative noise term,
We can now show that Eq. (25) is exact for non-
∂φ interaction particles. To see this it is best to work in
=D 2φ+ 2D ρ η. (22) Fourier space where the density operator ρ takes the
∂t ∇ ∇· h i
form, for N particles,
p
While the above equation contains the correct diffusion
termforthe idealgas,ityields byconstructionGaussian N
density fluctuations (because it is implied that the noise ρ˜(k)= exp( ik xi) (26)
− ·
correlation in Eq. (20) is Gaussian, so that higher order i=1
X
correlations of η can be factorized). As a sidenote, we where x obeys Eq. (1) with F=0. The ensemble aver-
i
remark that, interestingly, even for an ideal gas, the dis-
ageinthisfreegasisoverthetrajectoriesoftheBrownian
tribution of φ is nontrivialand is in fact Poissonian[61]. motions ξ . The average of ρ˜(k) is given by
i
This difference in underlying statistics only showsup for
higher point correlation functions, so that the present ρ˜(k) =(2π)dδ(k) ρ , (27)
h i h i
theory is exact for the two point correlations in the case
withhereN/V = ρ the uniformbulk density. Asimple
of ideal gas. This will be demonstrated at the end of
h i
computation shows that the two point correlation func-
subsection IIIA.
tion of the fluctuations φ at different times is given for
TogetherwithEq.(15),Eq.(19)givesaclosedformfor
large N by
thedynamicsofthesystem,whichisourfirstmainresult.
Thisdynamicsischosentoyieldexactequilibriumcorre- φ˜(k,t)φ˜(k′,0) =(2π)dδ(k+k′) ρ exp( Dk2t). (28)
lation functions but also gives the exacttime-correlation h i h i −
function for non-interacting Brownian particles. In the TransformingEq.(25)(derivedfromEq.(22))toFourier
following sections, we will investigate the properties of space, the agreement to the independently obtained
the dynamics proposed here in more detail. Eq. (28) can easily be verified.
5
B. Comparing to exact solution for short times Comparison with Eq. (29) reveals that Eq. (25) agrees
withtheexactsolutionoftheSmoluchowskiequationfor
For small values of time t t′, we expand Eq. (25), short times, i.e., including the term linear in time. This
− linear term has been discussed in terms of the ”initial
φ(x,t)φ(x′,t′) = kBT (1+ t t′ ∆R+...). (29) decay rate” [63], or in terms of a wavevector dependend
h i ∆ | − | diffusivity [14, 64, 65]. The current formulation agrees
with these.
The dots representhigher orderterms in t t′. We shall
−
nowcomparethisresulttotheexactoneforEq.(1). (Re-
call that we assume that canonical and grand canonical IV. RELAXATION TO EQUILIBRIUM –
systems are equivalent). For this, we use the Smolu- AGREEMENT WITH DDFT
chowski equation corresponding to the set of stochas-
tic equations, Eq. (1). The Smoluchowski equation is a
While in the previous section, we compared the
partial differential equation for the distribution Ψ(Γ,t),
stochasticequationproposedhere(Eq.(19))totheexact
which is a function of phase space Γ [11],
Smoluchowskiequation,inthissection,weaimtodemon-
strate another equivalence: Near equilibrium, the relax-
∂
Ψ=ΩΨ. (30) ationdynamicsofEq.(19)agreesexactlywiththecorre-
∂t
sponding result of DDFT. This will be seen by studying
Ω = D ∂ [∂ βF ] is the Smoluchowski operator. the relaxation of a system which is initially out of equi-
i i· i− i
F is, as in Eq. (1), the force acting on particle i. The librium.
i
P
equilibrium time correlation function for density is then
written [62],
A. Mean relaxation to equilibrium from Eq. (19)
φ(x,t)φ(x′,t′) = dΓφ(x)e|t−t′|Ωφ(x′)Ψ (Γ). (31)
e
h i Letus assume,the systemis inan initialsituation out
Z
of equilibrium, so that the mean density deviates from
Here,Ψ istheequilibriumdistribution. Forshorttimes,
e the equilibrium one, and we define as in Tab. I,
Eq. (31) is expanded,
ρ(x) neq = ρ(x) +δρ(x). (36)
φ(x,t)φ(x′,t′) = φ(x)φ(x′) + h i h i
h i h i
If δρ(x) is small, we can use Eq. (19) to compute the
t t′ dΓφ(x)Ωφ(x′)Ψe(Γ)+... (32) relaxation of δρ to zero (δρ(x) must be small because
| − |
Z Eq. (19) is linear). Therefore, we replace φ in Eq. (19)
Ψ being the Boltzmann distribution, one has ∂ Ψ = by δρ, and remove the noise term, as it vanishes when
e i e
βF Ψ [11, 62]. Using this we can rewrite the second takingthemeanoftheequation. Weobtainthefollowing
i e
term in Eq. (32) by use of partial integrations (Einstein equation which is linear in δρ,
summation convention is implied),
∂δρ
=D δρ δρ log ρ
dΓφ(x)Ωφ(x′)Ψe(Γ)= D dΓ(∂iφ(x))(∂iφ(x′))Ψe. ∂t ∇·(cid:20)∇ − ∇ h i
−
Z Z (33) ρ dyc(2)(x,y)δρ(y) . (37)
−h i∇
Z (cid:21)
We now employ the definition of φ(x) = δ(x x )
i − i − In the next subsection, we will compute the analogous
ρ(x) , noticing that the mean density vanishes when
equation from DDFT, and demonstrate the agreement.
h i P
pluggedintoEq.(33): Itdoesnotdependonphasespace
and ∂ yields zero. With ∂ δ(x x )= ∂ δ(x x ), we
i i i x i
− − −
get B. DDFT expanded near equilibrium
D (∂ φ(x))(∂ φ(x′))
i i
− h i Quoting the equation of motion of dynamical density
= D δ(x x )δ(x′ x ) ←−′ functional theory for Brownian particles in an external
i i
− ∇h − − i∇
potential U [20], one has
i
X
= D δ(x x′) ρ (x′)←−′ =kBTR(x,x′). (34) ∂δρ
− ∇ − h i ∇ =D ( ρ +δρ)+β( ρ +δρ) U
Xi ∂t ∇· ∇ h i h i ∇
(cid:20)
Where we have identified the operator R from Eq. (17). δ ex
+β( ρ +δρ) F . (38)
We thus have the exact solution for short times, h i ∇ δρ
(cid:21)
φ(t)φ(t′) = kBT (1+ t t′ ∆R+...). (35) Fex is the so called excess free energy functional. This
h i ∆ | − | is a well known and well studied equation, which is an
6
approximative solution of the Smoluchowski equation, for the density operator ρ (see Table I) [50],
Eq.(30). Ithasbeensuccessfullyusedinmanyscenarios
∂ δβ
to describe the dynamics of interacting Brownian parti-
ρ(x)=D ρ E + 2Dρη(x,t). (45)
cles [20]. We now expand this equation for small values ∂t ∇· ∇δρ(x) ∇·
of δρ, as in Eq. (37). We first note that several terms p
Here, the noise η is distributed as in Eq. (20), and is
cancel,asthetimederivativemustvanishinequilibrium.
E
the energy functional
Specifically(notethateventheterminthesquarebrack-
ets vanishes),
=k T dxρ(x)ln(ρ(x))
B
δ ex E Z
0= ρ +β ρ U +β ρ F . (39) 1
∇·"∇h i h i∇ h i∇ δρ (cid:12)ρ=hρi# + 2 dxdyρ(x)V(x−y)ρ(y)
(cid:12) Z
(cid:12)
Furthermore, for small δρ, we expand the(cid:12) last term in + dxρ(x)U(x). (46)
Eq. (38) in a functional Taylor series,
Z
δ ex δ ex δ ex We have as before split the potential into an interaction
F = F + dy F δρ(y)
δρ(x) δρ(x) δρ(x)δρ(y) part V, and an external part U. Note the difference of
(cid:12)ρ=hρi Z (cid:12)ρ=hρi
(cid:12) (cid:12) Eq. (46) compared to the free energy functional of DFT
(cid:12) (cid:12) + (δρ2).
(cid:12) (cid:12) O [1,50,51]. AttemptingtolinearizeEq.(45)inthefluctu-
(40)
ationsφ, the firstnaturalchoiceis to replace the density
operator appearing in the noise term by its equilibrium
It is now important to note that the involved Taylor co-
average, i.e., √2Dρη(x,t) 2D ρ η(x,t). Interest-
efficientequals,bydefinition, thedirectcorrelationfunc- ≈ h i
tion c(2) [1], ingly, in order to keep detailedpbalance, this choice im-
plies that also the density operator on the right hand
δ ex side of Eq. (45) must be replaced by its mean, i.e., the
β F =c(2)(x,y). (41)
− δρ(x)δρ(y) resulting approximate, consistent linear equation reads
(cid:12)ρ=hρi
(cid:12)
(cid:12) ∂ δβ ′
Another useful relation is t(cid:12)he formal exact result for the φ(x)=D ρ E + 2D ρ η(x,t). (47)
equilibrium mean density, which is given by [1], ∂t ∇·h i∇δφ(x) ∇· h i
p
βδ ex Furthermore, we note Eq. (47) yields the correct result
ρ =zexp βU F , (42) for the variance of φ in equilibrium, φ(x)φ(y) , if the
h i "− − δρ(x) (cid:12)ρ=hρi# functional ′ coincides with H of Eq. (h7), ′ =Hi.
(cid:12) E E
(cid:12) It is thus interesting to note that after pre-averaging
with the (in the following irrelevant)(cid:12) normalization z.
thenoise,Eq.(19)appearstobe theonlyconsistent,lin-
With this equation, one can write
ear equation for φ. It has been recently shown that lin-
δ ex earizingthe interactionterminEq.(45)aboutthe mean
k T log ρ = U F . (43)
B ∇ h i −∇ −∇ δρ(x) bulk density, while using the mean bulk density in the
(cid:12)ρ=hρi noise term, leads to an analytically soluble theory in the
(cid:12)
We finally obtain for the expansion of Eq(cid:12)(cid:12). (38) linear in bulkwhichrecoverstherandomphaseapproximationfor
δρ, theequaltime correlationfunctions[66–70],notablythis
means that Debye-Hu¨ckel theory is obtained for Brown-
∂δρ ianelectrolytes. Theapproachhasbeenappliedtoavari-
=D δρ δρ log ρ
∂t ∇· ∇ − ∇ h i ety ofdrivenandoutofequilibrium systems. In particu-
(cid:20)
laritiscapableofreproducingthefullOnsagertheoryof
ρ dyc(2)(x,y)δρ(y) (44) electrolyte conductivity, both the Ohmic linear response
−h i∇
Z (cid:21) regime and the first Wien effect regime where the con-
which is identical to Eq. (37). We have thus shown that ductivityisenhancedbytheelectricfield[70]. While the
the new equation, Eq. (19) is in agreement with DDFT randomphaseapproximationisvalidonlyforweakinter-
forsmalldeviationsfromequilibrium. Thisdemonstrates actions or high temperatures, the approach here should
aconnectiontotheframeworkofRef.[51],without,how- allow the study of systems with form instance hard core
ever, an obvious direct equivalence. interactions, relevant for ionic liquids, both in the bulk
and under confinement.
V. COMPARISON TO THE EXACT
STOCHASTIC EQUATION VI. SUMMARY
StartingfromthesetofstochasticequationsinEq.(1), Wehavederivedaneffectivefieldtheoryforsimpleliq-
thefollowingexactstochasticequationofmotionisfound uids, which allows computation of dynamical correlation
7
functions of the density. The result for the dynamical be used to investigate out of equilibrium Casimir forces
correlation function is approximate, but exact for small in model B as in Ref. [71], however including effects of
times. The described dynamics also agrees exactly at all finite particle size.
times with dynamical density functional theory. Future
work will apply this theory to study the intermediate
scattering function in confinement (where recent exper-
ACKNOWLEDGMENTS
imental findings exist [38]). It will also be used to find
the local viscosity near surfaces and compare to previ-
ous theoreticalapproachesfor bulk [28] andconfinement We thank C. Rohwer for helpful discussions. This re-
[29, 30]. For this, an expression for the stress tensor in searchwassupportedbytheDFGgrantNo. KR3844/2-
this theory must be derived. Then this theory may also 1 and the ANR project FISICS.
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