A Derivation of the force correlator, Eq. (34)

# Hydrodynamic fluctuation-induced forces in confined fluids

## Abstract

We study thermal, fluctuation-induced hydrodynamic interaction forces in a classical, compressible, viscous fluid confined between two rigid, planar walls with no-slip boundary conditions. We calculate hydrodynamic fluctuations using the linearized, stochastic Navier-Stokes formalism of Landau and Lifshitz. The mean fluctuation-induced force acting on the fluid boundaries vanishes in this system, so we evaluate the two-point, time-dependent force correlations. The equal-time correlation function of the forces acting on a single wall gives the force variance, which we show to be finite and independent of the plate separation at large inter-plate distances. The equal-time, cross-plate force correlation, on the other hand, decays with the inverse inter-plate distance and is independent of the fluid viscosity at large distances; it turns out to be negative over the whole range of plate separations, indicating that the two bounding plates are subjected to counter-phase correlations. We show that the time-dependent force correlations exhibit damped temporal oscillations for small plate separations and a more irregular oscillatory behavior at large separations. The long-range hydrodynamic correlations reported here represent a “secondary Casimir effect”, because the mean fluctuation-induced force, which represents the primary Casimir effect, is absent.

###### pacs:
47.35.-i,05.40.-a,05.20.Jj

## I Introduction

The Casimir effect (1) is the most important example of a slew of phenomena usually referred to as fluctuation-induced interactions, their phenomenology extending from cosmology on the one side to nanoscience on the other (3); (7); (2); (4); (6); (8); (9); (5). The general idea tying these diverse phenomena together is that the confining surfaces constrain the quantum and thermal field fluctuations, inducing long-range interactions between these boundaries (3); (4). For electromagnetic fields, these confinement effects lead to the Casimir-van der Waals interactions that can be derived within the specific framework of QED, and quantum field theory more generally (6). Inspired by the close analogy between thermal fluctuations in fluids and quantum fluctuations in electromagnetism, Fisher and de Gennes predicted the existence of long-range fluctuation forces in other types of critical condensed matter systems (10) and the terms “Casimir” or “Casimir-like effect” now denote a range of other non-electromagnetic fluctuation-induced forces (3); (9).

Beyond detailed measurements of the Casimir-van der Waals interactions (6), attention has been directed toward Casimir-like forces engendered by density fluctuations in the vicinity of the vapor-liquid critical point (8); (11); (12); in binary liquid mixtures near the critical demixing point (13); (14); and in thin polymer (16); (15); (17) and liquid crystalline films (18); (19). Most recently, several studies have examined fluctuation-induced interactions for the Casimir-Lifshitz force out of thermal equilibrium (20); (21), for the temporal relaxation of the thermal Casimir or van der Waals force (22), and for nonequilibrium steady states in fluids (23); (24); (25), where fluctuations are anomalously large and long-range.

It is instructive to recall that the original 1955 derivation of the electromagnetic Casimir-van der Waals interactions by Lifshitz (26) was not fundamentally rooted in QED but rather in stochastic electrodynamics, first formulated by Rytov (27). In stochastic electrodynamics, Maxwell’s equations are augmented by fluctuating displacement current sources (28). This leads to two coupled electrodynamic Langevin-type equations, for each of the fundamental electrodynamic fields, that are then solved with standard boundary conditions. The interaction force is obtained by averaging the Maxwell stress tensor and taking into account the statistical properties of the fluctuating sources (29). This paradigmatic Lifshitz-route to fluctuation-induced interactions later became disfavored as other formal approaches gained strength (6), but appears to be reborn in recent endeavors regarding non-equilibrium fluctuation-induced interactions (23); (24). In fact, in the Dean-Gopinathan method there exists a mapping of the non-equilibrium problem characterized by dissipative dynamics onto a corresponding static (Lifshitz) partition function provided by the Laplace transform of the time-dependent force and the static partition function (30); (31).

Based on the success of stochastic electrodynamics, Landau and Lifshitz proposed by analogy the stochastic dissipative hydrodynamic equations (32), augmenting the linearized Navier-Stokes equations with fluctuating heat flow vector and fluctuating stress tensor (25); (33). This leads to three coupled hydrodynamic equations involving the fundamental hydrodynamic fields of mass density, velocity and local temperature, which can now be solved in different contexts. In the absence of thermal conductivity, this system further reduces to a Langevin-type equation for the velocity field, involving the stress tensor fluctuations, and a continuity equation for the mass density field. Since the fundamental hydrodynamic equations are non-linear, the derivation of fluctuating Landau-Lifshitz hydrodynamics already involves heavy linearity Ansätze and the possible generalization to a full non-linear fluctuating hydrodynamics is not clear (34); (35).

Although fluctuating electrodynamics is based on linear Maxwell’s equations with stresses quadratic in the field and fluctuating hydrodynamics stems from non-linear Navier-Stokes equations with stresses linear in velocities, the general similarity between these approaches might nevertheless lead one to assume that, in confined geometries, there should exist Casimir-like hydrodynamic fluctuation forces. But this notion is at odds with the standard decomposition of the classical partition function into momentum and configurational parts. This decomposition has far-reaching consequences, which were clearly understood as far back as van der Waals’ thesis (36). While there is an analogy between the description of fluctuations in these two areas of physics, caution should be exercised when trying to translate results from one field directly into the other. We will show that there does exist a type of Casimir effect in the hydrodynamic context, but that this effect has fundamentally different properties from the conventional Casimir effect.

The first step in bringing together the Casimir force in electrodynamics and its putative counterpart in hydrodynamics was made by Jones (37). Inspired by the obvious analogy between electrodynamics and hydrodynamics, Jones investigated the possible existence of a long-ranged, fluctuation-induced, effective force generated by confining boundaries in a fluid. He showed that in linearized hydrodynamics the net (mean) stochastic force vanishes, which led him to introduce a next-to-leading order formalism. The status of this formalism, however, is not entirely clear, because there are linearity assumptions rooted deep within fluctuational hydrodynamics (33); (25). Within the context of this next-to-leading order formalism, Jones demonstrated that long-range forces could exist in a semi-infinite fluid or around an immersed spherical body, and would be strongest in incompressible fluids, with much weaker forces in compressible fluids. This result is at odds with the momentum decomposition of the classical partition function and should be considered an artifact of the next-to-leading order analysis of the stochastic equations governing the hydrodynamic field evolution.

Chan and White (38), therefore, reconsidered the whole calculation. They concentrated on the planar geometry of two hard walls immersed in a fluid and argued that hydrodynamic fluctuations could give rise to a repulsive force in incompressible fluids, but that this force would vanish for classical compressible fluids. Since an incompressibility Ansatz does not translate directly into the interaction potential in the classical partition function (39), this fictional case could lead to a fluctuation-induced interaction that would not be contrary to the argument based on the momentum decomposition of the classical partition function. The repulsive fluctuation-induced force would also in itself not be that hard to envision since the existence of a repulsive force in the context of van der Waals interactions is well-established and was originally proposed in Ref. (40). The vanishing of the fluctuation-induced force for classical compressible fluids is based on a rough argument of analytic continuation of the viscosities into the infinite frequency domain (38). While this latter argument is appropriate in electrodynamics, because an infinite frequency corresponds to the vacuum, it is not reasonable in hydrodynamics, where the whole basis of the continuum hydrodynamic theory breaks down before any such limit could be enforced (33).

Therefore, both approaches to the problem of hydrodynamic Casimir-like interactions have strong limitations and subsequent developments failed to conclusively prove either point of view (41).

In this paper, we revisit the question of the existence of long-range, fluctuation-induced forces in classical fluids. We work strictly within the framework of linearized stochastic hydrodynamics and rather than considering the net force, which is zero trivially, we study the force correlators. In other words, we focus on the question: In what way do boundary conditions and statistical properties of the fluctuating hydrodynamic stresses affect the statistical properties (correlators) of the random forces acting on the bounding surfaces?

We formulate a general approach to this problem by considering a fluid of arbitrary compressibility, bounded between two plane-parallel, hard walls with no-slip boundary conditions. Thermal fluctuations lead to spatio-temporal variations in the pressure and velocity fields that can be calculated using the linearized, stochastic Navier-Stokes formalism of Landau and Lifshitz (32). Within this approach, we derive analytical expressions for the time-dependent correlators (for both the same-plate and the cross-plate correlators) of the fluctuation-induced forces acting on the walls. In particular, we express the variance of these forces in terms of frequency integrals that have simple plate-separation dependence in the small and large plate-separation limits.

Our results do not depend upon the next-to-leading order formalism of Jones (37), nor do they depend on the unrealistic validity of analytic continuation of the viscosities in the whole frequency domain (38). We show that, while the mean force vanishes, the variance of the fluctuation-induced normal force is finite and depends on the separation between the bounding surfaces. We call this the secondary Casimir effect, because the primary Casimir effect refers to the average value of the fluctuation-induced force (which is zero here) and not strictly its variance. Both quantities have been investigated in other Casimir-like situations (42); (22) and in disordered charged systems (43); (44); (45). The equal-time, cross-plate force correlation exhibits long-range behavior that is independent of the fluid viscosity and decays proportional to the inverse plate separation. Finally, we find that the time-dependent correlators exhibit damped oscillatory behavior for small plate separations that becomes irregular at large distances.

In Sec. II, we outline the stochastic formalism of Landau and Lifshitz and the strategy of our calculation of hydrodynamic fluctuation-induced forces in the general case of compressible fluids. Sections III and IV present the main steps of our calculation. We show results for the equal-time force correlators and the two-point, time-dependent correlators in Sections V and VI, respectively. We conclude our discussions in Sec.  VII.

## Ii Formalism

We consider the hydrodynamic fluctuations in a Newtonian fluid at rest and in the absence of heat transfer. These fluctuations are described by the stochastic Landau-Lifshitz equations (32)

 η∇2v+(η3+ζ)∇(∇⋅v)−∇p −ρ(∂v∂t+v⋅∇v)=−∇⋅S, (1) ∂ρ∂t+∇⋅(ρv)=0, (2)

where , and are the velocity, pressure and density fields and and are the shear and bulk viscosity coefficients, respectively (46). The randomly fluctuating microscopic degrees of freedom are driven by the random stress tensor , which is assumed to have a Gaussian distribution with zero mean and the two-point correlator

 ⟨Skl(r;t)Smn(r′;t′)⟩=2kBTδ(r−r′)δ(t−t′) ×[η(δkmδln+δknδlm)−(2η3−ζ)δklδmn]. (3)

Here the subindices () denote the Cartesian components , is Boltzmann’s constant and denote an equilibrium ensemble average at temperature . We do not consider any possible relaxation effects, which would formally correspond to frequency-dependent viscosities, but these effects can be easily incorporated (32). Denoting the frequency Fourier transform by a tilde, i.e.,

 ˜f(ω)=∫dteiωtf(t), (4)

we have and

 ⟨˜Skl(r;ω)˜S∗mn(r′;ω′)⟩=4πkBTδ(r−r′)δ(ω−ω′) ×[η(δkmδln+δknδlm)−(2η3−ζ)δklδmn], (5)

which hold independent of the boundary conditions imposed on the fluid system.

Before proceeding further, we should note that this form of fluctuating hydrodynamics is analogous to the Rytov fluctuating electrodynamics (26), where the basic equations for the electric and magnetic fields are

 ∇×E(r,t)= −∂B(r,t)∂t, (6) ∇×H(r,t)= ∂D(r,t)∂t+∂K(r,t)∂t, (7)

supplemented by and and appropriate boundary conditions. In this case, the fluctuating random polarization, , has Gaussian properties with , and

 ⟨˜Ki(r;ω)˜K∗j(r′;ω′)⟩=kBTεI(ω)ωδijδ(r−r′)δ(ω−ω′), (8)

where we have assumed a dispersive dielectric response function . We can immediately see the similarity between Eqs. (1)-(5) and Eqs. (6)-(8).

Thus, the stochastic approach to hydrodynamics is very close to Lifshitz’s original analysis of the electromagnetic problem (26), provided one fully takes into account the basic differences between the Maxwell equations and the Navier-Stokes equations (38): The former are linear in the fields with stresses quadratic in the fields, while the latter are non-linear in the fields with stresses linear in the fields. This difference leads to some important distinctions and precludes directly applying results from electrodynamics to the hydrodynamic domain.

### ii.1 Linearized stochastic hydrodynamics

For vanishing random stress tensor, the equilibrium solution of Eqs. (1) and (2) is , and , corresponding to a fluid at rest at constant temperature, , with uniform pressure, , and density, . The random stress tensor, , is of order and, consequently, macroscopically small. Thus, the corresponding fluctuations in the velocity, pressure and density fields are also macroscopically small. Therefore we introduce a linearized treatment of the Landau-Lifshitz equations, by setting , and , where the superscript denotes a term of order .

We assume local equilibrium, which enables us to relate the density and pressure as

 p(1)=c20ρ(1),withc20=(∂p∂ρ)0. (9)

Here is the adiabatic speed of sound, so that equals the inverse adiabatic compressibility (Newton-Laplace equation). Eqs. (1) and (2) can be linearized as

 η∇2v(1)+(η3+ζ)∇(∇⋅v(1)) Extra open brace or missing close brace (10) ∂ρ(1)∂t+ρ0∇⋅v(1)=0. (11)

or, in the frequency domain and using Eq. (9) (47),

 η∇2˜v(1)+(η3+ζ)∇(∇⋅˜v(1)) −c20∇˜ρ(1)+iωρ0˜v(1)=−∇⋅˜S, (12) ∇⋅˜v(1)−iωρ0˜ρ(1)=0. (13)

We now introduce transverse and longitudinal components of the velocity fluctuations , which we denote and , respectively. We have dropped the superscript for notational simplicity, i.e., , with

 ∇⋅vT=0and∇×vL=0. (14)

The random force density vector can be decomposed into transverse and longitudinal components as well, using , where

 ∇⋅ΣT=0and∇×ΣL=0. (15)

These random force density vector components have zero mean and zero cross correlations. Their self-correlations follow from Eq. (3) as

 ⟨˜ΣLi(r;ω)˜ΣLj(r′;ω′)⟩= 4πkBT(4η3+ζ)∇i∇′j ×δ(r−r′)δ(ω+ω′), (16) ⟨˜ΣTi(r;ω)˜ΣTj(r′;ω′)⟩= 4πkBTη(∇k∇′kδij−∇i∇′j) ×δ(r−r′)δ(ω+ω′). (17)

The stochastic Landau-Lifshitz equations can thus be written as

 η∇2˜vL+(η3+ζ)∇(∇⋅˜vL) −c20∇˜ρ(1)+iωρ0˜vL=˜ΣL, (18) η∇2˜vT+iωρ0˜vT=˜ΣT, (19) ∇⋅˜vL−iωρ0˜ρ(1)=0. (20)

We may simplify Eqs. (18)-(20) by using the vector identity

 ∇j∇j˜vLi=∇j∇j˜vLi+∇j(∇i˜vLj−∇j˜vLi)=∇i∇j˜vLj (21)

for the curl-free longitudinal component and by substituting Eq. (20) into Eq. (18) to obtain

 [4η3+ζ+iρ0c20ω]∇2˜vL+iωρ0˜vL=˜ΣL, (22) η∇2˜vT+iωρ0˜vT=˜ΣT. (23)

We have now decoupled the transverse and longitudinal components of the velocity fluctuations. Eqs. (22) and (23) are nothing but the Langevin equations for each component of the velocity field in the frequency domain. In fact, Eq. (22) is a scalar equation for the longitudinal component of the velocity fluctuation (33).

The density field fluctuations can be obtained from the longitudinal component of the velocity field fluctuations,

 ˜ρ(1)(r;ω)=−iρ0ω∇⋅˜vL(r;ω). (24)

## Iii Mean interaction force

To obtain the net effective interaction force between the fluid’s confining boundaries, we integrate the fluctuating hydrodynamic stress tensor, , over the bounding surfaces, , i.e.,

 ⟨Fi(t)⟩=∫Γ⟨σij(r;t)⟩dAj, (25)

where the fluctuating hydrodynamic stress tensor, which is (32)

 σij=η[∇ivj+∇jvi]−[(2η3−ζ)∇kvk+p]δij+Sij, (26)

can be written up to first order in the field fluctuations as , with

 σ(1)ij= η[∇iv(1)j+∇jv(1)i] −[(2η3−ζ)∇kv(1)k+c20ρ(1)]δij+Sij. (27)

The stress tensor is linear in the fluid fluctuations, which are themselves linear in the random stress tensor and, thus, their ensemble averages vanish and . As a result, at first order in field fluctuations, the net fluctuation-induced force acting on the fluid boundaries must vanish, irrespective of the geometry of the fluid system, i.e.,

 ⟨F(1)i(t)⟩=0. (28)

We note that the mean force at leading order stems from the equilibrium pressure and is simply . We exclude this contribution in the rest of our discussion and focus on the statistical properties of the force at first order in the field fluctuations.

In what follows, we limit our discussion to the plane-parallel geometry of two rigid walls of arbitrarily large surface area, . We assume that the walls are located along the axis at and at a separation distance of and that the fluid velocity satisfies no-slip boundary conditions on the walls.

## Iv Two-point, time-dependent correlations of the force

Although, as we have already noted, the mean inter-plate force due to hydrodynamic fluctuations in the fluid layer must vanish, its variance or correlation functions need not and do not. In this Section, we study the two-point, time-dependent correlators, including the variance, of the forces that act on the boundaries in the two-wall geometry. In this plane-parallel geometry, we are primarily concerned with the force perpendicular to the plane boundaries, in which case the two-point, time-dependent force correlator is given by

 C (z,z′;t,t′)=⟨F(1)z(z;t)F(1)z(z′;t′)⟩ =∬A⟨σzz(r;t)σzz(r′;t′)⟩dxdydx′dy′, (29)

where the integrals run over the surface areas of the two walls that are located at and . Throughout this paper, we use an uppercase to denote correlation functions of the normal forces acting on the fluid boundaries and a lowercase to refer to correlation functions of the fluctuating hydrodynamic fields. We express the former quantity in terms of the latter ones (see Appendix A). In the present case, the correlators of the velocity and density fluctuations are given by

 cTTij(r,r′;t,t′)= ⟨vTi(r;t)vTj(r′;t′)⟩, (30) cLLij(r,r′;t,t′)= ⟨vLi(r;t)vLj(r′;t′)⟩, (31) cρρ(r,r′;t,t′)= ⟨ρ(1)(r;t)ρ(1)(r′;t′)⟩. (32)

The cross-correlation function of the transverse and longitudinal components of the velocity vanishes by construction. Furthermore, the transverse velocity and density fluctuations are independent fields, with vanishing cross-correlation function. Therefore, the only other correlation function we need is the density-velocity cross-correlator,

 Missing or unrecognized delimiter for \big (33)

Not all of these correlators contribute to the time-dependent correlator of the forces between the two hard boundaries. In Appendices B and D, we show that the contributions to the normal force correlator generated by the correlation function of the transverse velocity field and by the correlation function between the velocity and density fields vanish for our geometry. Therefore, applying the formulae of the previous Section, we can write the time-dependent force correlator as the sum of three terms (see Appendix A for details),

 C(z,z′;t,t′)=2∑i=0Pi(z,z′;t,t′). (34)

Defining the dimensionless parameter

 χ=4/3+ζ/η, (35)

we can write the first term as

 P0(z,z′;t,t′)≡2kBTηχAδ(z−z′)δ(t−t′). (36)

This contribution stems directly from the integration of the random stress correlator, , over the bounding surfaces; this term vanishes unless and , in which case it reduces to an irrelevant constant that will be dropped in the rest of our analysis. The two other terms are

We note that, in the above, we have used Eq. (24), which relates the density fluctuations to the fluctuations of the longitudinal components of the velocity.

With this expression in hand, we can see that we need to determine the correlation functions of the density fields and the longitudinal component of the velocity fields. We proceed via the following steps (32):

1. Obtain the Green functions of Eq. (22);

2. Express the fluctuating fields and their correlation functions in terms of the Green functions above;

3. Integrate the resulting expressions over the boundaries of the fluid according to Eqs. (37) and (38).

### iv.1 Green functions

In the present model with no-slip walls, the velocity and, therefore, the corresponding Green function should vanish at the boundaries. Translational invariance in the two (transverse) directions perpendicular to the -axis prompts us to search for Green functions of the form

 ˜G(r,r′′;ω)=1(2π)2∫d2keik⋅(s−s′′)˜G(z,z′′;k;ω), (39)

where , with , and . The longitudinal Green function corresponding to Eq. (22) is a solution of the following equation:

 [∇2z−m2]˜GL(z,z′′;k;ω)=iλ2ωρ0δ(z−z′′), (40)

where and we have defined the longitudinal decay constant as

 λ2=−iω2ρ0(4η/3+ζ)ω+iρ0c20. (41)

The solution of Eq. (40) is well known (48); (49); (50), and with no-slip boundary conditions at and , the Green function is obtained as

 ˜GL(z,z′′;k;ω)= gL1e−mz+gL2em(z−L) −iλ22mωρ0e−m|z−z′′|, (42)

where

 gL1= iλ22mωρ0csch(mL)sinh(m(L−z′′)), (43) gL2= iλ22mωρ0csch(mL)sinh(mz′′), (44)

are constants of integration that satisfy the no-slip boundary conditions.

### iv.2 Characteristic scales and dimensionless parameters

We simplify the following analysis by introducing dimensionless parameters that characterize the fluid and the plane-parallel geometry of our system. There are two length scales that can be used for this purpose: The macroscopic plate separation, , and the microscopic scale at which the continuum hydrodynamic description breaks down, which we denote . There are two characteristic vorticity frequencies associated with each of these length scales (48),

 ω0=ηL2ρ0andω∞=ηa2ρ0. (45)

The inverse frequencies, and , correspond to the time that vorticity requires to diffuse a certain distance, in this case or , respectively. We also define the dimensionless parameter , which is given by

 γ=c20L2ω20=(Lρ0c0η)2. (46)

This parameter is the squared ratio of the vorticity time scale and the typical compression time scale in which a propagating sound wave travels a distance (48).

To facilitate our later discussions, we introduce the dimensionless ratios

 u=ωω0γandu∞=ω∞ω0γ, (47)

and define the function

 fm(u)=u2−m1+χ2u2. (48)

We can now express the real and imaginary parts of the longitudinal decay constant, , as

 ℓ+= λRL=ω0γLc0|u|√2√[1−√f2(u)]√f2(u), (49) ℓ−= λIL=−ω0γLc0u√2√[1+√f2(u)]√f2(u). (50)

The vorticity frequency scale marks the boundary between the low-frequency propagative regime, for which (or ) and sound waves propagate with speed , and the high-frequency diffusive regime, for which (or ) and viscosity effects damp compression perturbations (48). Furthermore, the dimensionless ratio can be expressed in terms of a new length scale :

 u∞=δ2a2whereδ=ηρ0c0=c0ω0γ. (51)

This length scale characterizes the boundary between the propagative and diffusive regimes at . We can also define a characteristic time scale,

 t0=δ/c0, (52)

associated with this boundary. Finally, then, we can write and as

 ℓ+= Lδ|u|√2√[1−√f2(u)]√f2(u), (53) ℓ−= −Lδu√2√[1+√f2(u)]√f2(u). (54)

For any reasonable choice of realistic parameters for a fluid far from the critical point, we have , i.e., we work in the propagative regime. In this case, the plate separation of a realistic experiment satisfies . For liquids close to the critical point, or polymers in solution, however, the crossover frequency can be much lower and, therefore, we can have . In this case, the system is in the diffusive regime and the crossover length scale, , may be macroscopic.

### iv.3 Correlation functions

Now that we have explicit expressions for the Green function solutions in hand, we turn to the correlation functions and , which enter in Eqs. (34)-(38), and express these correlation functions in terms of the corresponding Green functions. Here, we simply sketch the derivation for , as an example, and leave the details of the corresponding calculation of to Appendix C.

The longitudinal velocity fluctuations are given in terms of the longitudinal Green function as

 vLi(r;t)=∫dt′′∫d3r′′GL(r,r′′;t−t′′)ΣLi(r′′;t′′). (55)

We require the correlation function

 ⟨vLi(r;t) Extra open brace or missing close brace ×GL(r,r′′;t−t′′)GL(r′,r′′′;t′−t′′′) ×⟨ΣLi(r′′;t′′)ΣLj(r′′′;t′′′)⟩. (56)

Recalling the stochastic properties of the random stress tensor, Eq. (16), we obtain

 cLLij (r,r′;t,t′)=2kBTηχ∫dt′′∫d3r′′ ×∇′′iGL(r,r′′;t−t′′)∇′′jGL(r′,r′′;t′−t′′). (57)

We now introduce a Fourier representation of the Green functions

 cLLij (r,r′;t,t′)=2kBTηχ∫dt′′∫d3r′′ ×∫dω2πe−iω(t−t′′)∇′′i˜GL(r,r′′;ω) ×∫dω′2πe−iω′(t′−t′′)∇′′j˜GL(r′,r′′;ω′). (58)

The integral over generates a Dirac delta function for the frequencies, , and therefore one of the frequency integrals is trivial:

 cLLij (r,r′;t,t′)=2kBTηχ∫dω′2πeiω′(t−t′) ×∫d3r′′∇′′i˜GL(r,r′′;−ω′)∇′′j˜GL(r′,r′′;ω′). (59)

In principle, we could substitute our explicit expression for the Green function, Eq. (42), into this correlation function and attempt to directly calculate the integrals at this stage. We will see, however, that this is not the most straightforward approach: Spatial integrations over the fluid boundary will simplify our task considerably. We also take advantage of the fact that we only require the components of the velocity fields perpendicular to the plane boundaries. Therefore, we set in our expression for the correlation function, , and use the translational-invariant structure of the Green function, Eq. (39), to write

 cLLzz (r,r′;t,t′)=2kBTηχ∫dω′2πeiω′(t−t′) ×∫d3r′′∫d2k(2π)2eik⋅(s−s′′)∇′′z˜GL(z,z′′;k;−ω′) ×∫d2k′(2π)2eik′⋅(s′−s′′)∇′′z˜GL(z′,z′′;k′;ω′). (60)

The double integral over generates a wavenumber Dirac delta function, , that enables us to carry out one of the wavenumber integrals immediately and, thus, obtain

 cLLzz (r,r′;t,t′)=2kBTηχ∫dω′2πeiω′(t−t′) ×∫dz′′∫d2k(2π)2eik⋅(s−s′)∇′′z˜GL(z,z′′;k;−ω′) ×∇′′z˜GL(z′,z′′;−k;ω′). (61)

Analogous arguments apply to the density correlation function, which is (see Appendix C)

 cρρ(r,r′;t,t′)=kBTπρ20ηχ∫dω′ω′2eiω′(t−t′)∫dz′′ ×∫d2k(2π)2eik⋅(s−s′)(∇z∇′′z+k2)˜GL(z,z′′;k;−ω′) ×(∇′z∇′′z+k2)˜GL(z′,z′′;−k;ω′). (62)

### iv.4 Spatial integration over surface boundaries

Our final step is to integrate the correlation functions, Eqs. (61) and (62), over the boundaries of the fluid according to Eqs. (34)-(38). These integrals give our final result for the time-dependent correlators of the force acting on the fluid boundaries.

The double integrals over and in Eqs. (34)-(38) lead to a Dirac delta function over the transverse wavenumbers, . Thus, we can write these equations in terms of the Green function as

 P1(z,z′;t,t′)=kBTπη3χ3A∫dω′cos[ω′(t−t′)] ×∫dz′′∇z∇′′z˜GL(z,z′′;0;−ω′)∇′z∇′′z˜GL(z′,z′′;0;ω′), (63) P2(z,z′;t,t′)=kBTπρ20ηχc40A∫dω′ω′2cos[ω′(t−t′)] ×∫dz′′∇z∇′′z˜GL(z,z′′;0;−ω′)∇′z∇′′z˜GL(z′,z′′;0;ω′). (64)

These frequency integrals run over the frequency range and the spatial integral is over . In writing the above relations, we have used the fact that the integrands involved in calculating and (see Eqs. (61) and (62)) have odd imaginary parts, which thus vanish, leading to the factor from the real part of the exponential factor . We also note that , which follows from the reality of . Therefore, as expected, the final correlators are purely real.

Carrying out the derivatives and the remaining spatial integral is fairly straightforward. The results are

 P1(z,z′;t,t′)= kBTπρ0c20ALχ3[W0(z,z′;τ) +LV0(z,z′;τ)δ(z−z′)], (65) P2(z,z′;t,t′)= Missing or unrecognized delimiter for \Big +LV2(z,z′;τ)δ(z−z′)]. (66)

The relevant frequency integrals are given by (see Appendix E)

 Wm(0,0;τ)=2∫u∞0dufm(u)cos[uτ] ×1cosh[2ℓ+]−cos[2ℓ−][(ℓ22ℓ−−2ℓ−)sin[2ℓ−] +(ℓ22ℓ+−2ℓ+)sinh[2ℓ+]], (67) Wm(0,L;τ)=2∫u∞0dufm(u)cos[uτ] ×1cosh[2ℓ+]−cos[2ℓ−][(ℓ2ℓ−−4ℓ−)cosh[ℓ+]sin[ℓ−] +(ℓ2ℓ+−4ℓ+)cos[ℓ−]sinh[ℓ+]], (68)

and

 Vm(0,0;τ)=2∫u∞0dufm(u)cos[uτ]. (69)

In these equations , where we have defined and in Eqs. (53) and (54) and the function in Eq. (48). The dimensionless time parameter is

 τ=(t−t′)/t0, (70)

with the characteristic microscopic timescale defined in Eq. (52). We have used the symmetry of the integrand to integrate over the positive real axis up to the dimensionless microscopic cutoff, , of Eqs. (47) and (51). To simplify these expressions further, we note that

 V2(0,0;τ)+χ2V0(0,0;τ)=2τsin[u∞τ]. (71)

Putting together all of these results, from Eqs. (34) and (63)-(71), we find

 C(0,0;t,t′)= Missing or unrecognized delimiter for \bigg Missing or unrecognized delimiter for \bigg (72) C(0,L;t,t′)= kBTπρ0c20ALχ ×[χ2W0(0,L;τ