A Langevin description for driven granular gases

Math. Model. Nat. Phenom. Vol. 6, No. 4, 2011, pp. 87-117 DOI: 10.1051/mmnp/20116405 A Langevin Description for Driven Granular Gases P. Maynar and M. I. Garc´ ıa de Soria ∗ F´ ısica Te´ orica, Universidad de Sevilla, Apartado de Correos 1065, E-41080, Sevilla, Spain Abstract. The study of the fluctuations in the steady state of a heated granular system is reviewed. A Boltzmann-Langevin description can be built requiring consistency with the equations for the one- and two-particle correlation functions. From the Boltzmann-Langevin equation, Langevin equations for the total energy and the transverse velocity field are derived. The existence of a fluctuation-dissipation relation for the transverse velocity field is also studied. Key words: kinetic theory, Boltzmann equation, granular gases, hydrodynamics AMS subject classification: 76P05, 76T25, 82B40, 82C40, 82D05 1. Introduction A granular system is characterized by a continuous loss of energy in collisions. The simplest model that we can consider to describe this kind of systems is an assembly of smooth hard spheres (or disks in two dimensions) that collide inelastically, in such a way that part of the kinetic energy is dissipated in the encounter. The collision is characterized by a constant coefficient of normal restitution [16, 21]. The rich phenomenology that has been found for this kind of systems makes them of particular interest from both a fundamental and an applied point of view. In the dilute limit, it has been shown numerically that for a wide class of initial conditions an isolated granular gas reaches a homogeneous state, the so-called homogeneous cooling state (HCS), in which all the time dependence of the one-particle distribution function is embedded in the granular temperature which is defined as the second velocity moment of the distribution [17, 18]. This state, whose properties have been extensively studied in the literature, plays, for inelastic gases, a role similar to that of the equilibrium state for molecular gases. Nevertheless, from an experimental point of view, ∗Corresponding author. E-mail: [email protected] 87 Article published by EDP Sciences and available at http://www.mmnp-journal.org or http://dx.doi.org/10.1051/mmnp/20116405 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases it is not obvious to probe such a regime [19, 25]. However, it is possible to obtain a non-equilibrium stationary state by supplying energy to the system in such a way to compensate the dissipation due to the inelasticity of collisions. We can think of several different procedures to thermalise the system. An inhomogeneous stationary state can be reached if the energy is injected by a moving boundary such as a vibrating piston [8]. To obtain a spatially homogeneous stationary state, we can consider a mechanism coupling the velocity of each particle to a white noise, the so-called stochastic thermostat [10, 11, 12, 15, 22, 23, 24, 27, 30, 31, 32, 33]. For this kind of forcing, which is relevant for some two-dimensional experimental configurations with a rough vibrating piston [26], the system reaches a homogeneous stationary state after a transient regime. The Boltzmann equation for the steady state can be written and the distribution function evaluated approximately for thermal velocities. For large values of the modulus of the velocity, an exponential behaviour is found. An expression for the stationary temperature can be obtained as a function of the strength of the noise and of the inelasticity of collisions [30]. The aim of this paper is to review some recent results about the fluctuations for a system subject to the stochastic thermostat [14, 20] generalizing the theory to the study of dynamical properties. The study of the fluctuations of granular gases has been addressed for a system in the homogeneous cooling state [7]. The starting point for this study was the equations for the correlation functions obtained from an extension of the BBGKY hierarchy for granular gases. Later, following the ideas of Zwanzig [2], a Langevin equation for the distribution function was introduced, in which the properties of the noise of the Boltzmann-Langevin equation were identified requiring consistency with the equations for the correlations [6]. A similar method will be applied to the study of heated granular gases, identifying the Boltzmann-Langevin equation for the fluctuating distribution function close to the stationary state. Once the fundamental properties of the noise of the equations are calculated, we focus on the derivation of Langevin equations for the total energy and the transverse velocity. From these equations, the one-time and two-time correlation functions are evaluated. We pay special attention to the study of the transverse velocity remarking the differences with the elastic case (where we have a fluctuation-dissipation relation) and with the undriven inelastic case. We note that in previous studies fluctuating equations for the hydrodynamic fields were introduced phenomenologically, expecting to provide a valid description, at least, for small values of the inelasticity [31]. The Boltzmann-Langevin description, as we will show, is valid for all the range of values of the inelasticity, and allows us to address the problem of finding a fluctuating hydrodynamic description without the limitation of small inelasticity. The remainder of the paper is organised as follows. In Sec. 2. we recall some general results for a granular gas heated by a stochastic thermostat. Sec. 3. is devoted to the non-fluctuating hydrodynamic equations. In Sec. 4., the Boltzmann-Langevin description is presented and it is finally applied to the derivation of the Langevin equations for the energy and the transverse velocity field in Sec. 5. and 6., respectively. 88 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases 2. Kinetic theory description of heated granular gases The system considered is a dilute gas of Nsmooth inelastic hard particles of mass mand diameter σ. The position and velocity of the ith particle at time twill be denoted by Ri(t)and Vi(t), respectively. The effect of a collision between two particles iand jis to instantaneously modify their velocities according to the collision rule V0 i=Vi−1 + α 2(ˆ σ·Vij)ˆ σ, V0 j=Vj+1 + α 2(ˆ σ·Vij)ˆ σ,(2.1) where Vij ≡Vi−Vjis the relative velocity, ˆ σis the unit vector pointing from the center of particle jto the center of particle iat contact, and αis the coefficient of normal restitution. It is defined in the interval 0< α ≤1and it will be considered here as a constant, independent of the relative velocity. Between collisions, the system is heated uniformly by adding a random velocity to the velocity of each particle at equal times. The driving is implemented in such a way that the time between random kicks is small compared to the mean free time. Then, between collisions, the velocities of the particles undergo a large number of kicks due to the thermostat. In addition, we will assume that the “jump moments” of the velocities of the particles verify Bij,βγ ≡lim ∆t→0h∆Vi,β∆Vj,γinoise ∆t=ξ2 0δijδβγ +ξ2 0 N(δij −1)δβγ,(2.2) i, j = 1, . . . , N and β, γ = 1, . . . , d, where we have introduced ∆Vi,β ≡Vi,β(t+ ∆t)−Vi,β(t),Vi,β(t)being the βcomponent of the velocity of particle iat time t. We have also introduced the strength of the noise, ξ0, and h. . . inoise, which denotes average over different realizations of the noise. The non-diagonal terms (corresponding to i6=jand β=γ) are necessary in order to conserve the total momentum [14]. 2.1. Kinetic Equations Given a trajectory of the system, microscopic densities in phase space, Fs(x1, . . . , xs, t), at time t are defined by F1(x1, t) = N X i=1 δ[x1−Xi(t)],(2.3) F2(x1, x2, t) = N X i=1 N X j6=i δ[x1−Xi(t)]δ[x2−Xj(t)],(2.4) etc. Here Xi(t)≡ {Ri(t),Vi(t)}, while the xi≡ {ri,vi}are field variable referring to the one-particle phase space. The average of F1(x1, t)and F2(x1, x2, t)over the initial probability distribution of the system ρ(Γ,0), where Γ≡ {X1(t), . . . , XN(t)}, are the usual one-particle and two-particle distribution functions f1(x1, t) = hF1(x1, t)i, f2(x1, x2, t) = hF2(x1, x2, t)i,(2.5) 89 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases where we have introduced the notation hGi ≡ ZdΓG(Γ)ρ(Γ,0).(2.6) In the dilute limit, assuming molecular chaos, i.e. that no correlations exist between colliding particles, and that the sizes of the jumps due to the thermostat are small compared to the scale on which the distribution varies, the equation for the one-particle distribution function, f1(x1, t), is the Boltzmann-Fokker-Planck equation [30] ·∂ ∂t +L(0)(x1)¸f1(x1, t) = J[f1|f1] + ξ2 0 2 ∂2 ∂v2 1 f1(x1, t),(2.7) where we have introduced the free streaming operator L(0)(x1) = v1·∂ ∂r1 .(2.8) The collisional term can be expressed as J[f1|f1] = Zdx2δ(r12)¯ T0(v1,v2)f1(x1, t)f1(x2, t),(2.9) in terms of the binary collision operator ¯ T0(v1,v2) = σd−1Zdˆ σΘ(ˆ σ·v12)(ˆ σ·v12)[α−2b−1 ˆ σ−1],(2.10) where we have neglected the separation between the centers of colliding particles. The operator b−1 ˆ σchanges the velocities to its right into the pre-collisional velocities v∗ 1=v1−1 + α 2α(ˆ σ·v12)ˆ σ,(2.11) v∗ 2=v2+1 + α 2α(ˆ σ·v12)ˆ σ.(2.12) The last term in Eq. (2.7), which does not appear in the free cooling case, is the typical diffusive Fokker-Planck contribution from the heating. The two-particle distribution function is defined in such a way, see Eq. (2.4) and Eq. (2.5), that the quantity f2(x1, x2, t)dx1dx2is just the average number of pairs of particles in which one lies inside the differential volume dx1centered in x1and likewise, with dx2and x2for the second particle. This definition is easily generalized to higher n-particle distribution functions, fn(x1, . . . , xn). The evolution equation for f2, in the same limit in which the Boltzmann equation is valid, is [7] ·∂ ∂t +L(0)(x1) + L(0)(x2)¸f2(x1, x2, t) = δ(r12)σd−1¯ T0(v1,v2)f2(x1, x2, t) +σd−1Zdx3£δ(r13)¯ T0(v1,v3) + δ(r23)¯ T0(v2,v3)¤f3(x1, x2, x3, t) + FTH,(2.13) 90 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases where we have introduced FTH that accounts for the external driving. The evolution equation (2.13) contains essentially three parts: the free streaming in the left-hand side, the two terms in the right hand side corresponding to collisions, and the last term, FTH due to the thermostat. The collisional contribution is split into one part corresponding to collisions of particles with velocities v1and v2, and the other which involves collisions of particles with velocities v1or v2with a third particle with arbitrary velocity, v3. The collisional contribution is identical to the one that appears in the absence of forcing [7]. We concentrate now on the new term, FTH. Assuming that the sizes of the jumps due to the thermostat are small compared to the scale in which the distribution f2varies, we can expand FTH in the spirit of the Fokker-Planck description [29] FTH ≃1 2 d X β,γ=1 2 X i,j=1 Bij,βγ ∂ ∂vi,β ∂ ∂vj,γ f2(x1, x2, t) =1 2ξ2 0·∂2 ∂v2 1 +∂2 ∂v2 2−2 N ∂ ∂v1·∂ ∂v2¸f2(x1, x2, t),(2.14) where we have taken into account equation (2.2), and we have explicitly assumed that the jump moments Bij,βγ do not depend on the magnitude of the velocities of the particles. Two-time reduced distribution functions can also be defined in terms of the microscopic densities as fr,s(x1, . . . , xr, t;x0 1, . . . , x0 s, t0)≡ hFr(x1, . . . , xr, t)Fs(x0 1, . . . , x0 s, t0)i,(2.15) where it will be assumed that t>t0>0, for concreteness. Again, in the low-density limit and taking into account the influence of the thermostat, the lowest order distribution, f1,1, is seen to obey the equation [7], ·∂ ∂t +L(0)(x1)¸f1,1(x1, t;x0 1, t0) = σd−1Zdx2δ(r12)¯ T0(v1,v2)f2,1(x1, x2, t;x0 1, t0) +ξ2 0 2 ∂2 ∂v2 1 f1,1(x1, t;x0 1, t0),(2.16) which has to be solved with the initial condition f1,1(x1, t0;x0 1, t0) = δ(x1−x0 1)f1(x1, t0) + f2(x1, x0 1, t0).(2.17) Let us introduce the two-particle and three-particle correlation functions, g2and g3, through the usual cluster expansion f2(x1, x2, t) = f1(x1, t)f1(x2, t) + g2(x1, x2, t),(2.18) and f3(x1, x2, x3, t) = f1(x1, t)f1(x2, t)f1(x3, t) + g2(x1, x2, t)f1(x3, t) +g2(x1, x3, t)f1(x2, t) + g2(x2, x3, t)f1(x1, t) + g3(x1, x2, x3, t).(2.19) 91 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases Similarly, for the two-time correlation functions, we introduce the two-time correlation functions h1,1and h2,1through f1,1(x1, t;x0 1, t0) = f1(x1, t)f1(x0 1, t0) + h1,1(x1, t;x0 1, t0),(2.20) f2,1(x1, x2, t;x0 1, t0) = f1(x1, t)f1(x2, t)f1(x0 1, t0) + g2(x1, x2, t)f1(x0 1, t0) +h1,1(x1, t;x0 1, t0)f1(x2, t) + h1,1(x2, t;x0 1, t0)f1(x1, t) +h2,1(x1, x2, t;x0 1, t0).(2.21) The equation for the correlation function g2(x1, x2, t)can be obtained following the same lines as in reference [7]. Taking into account Eq. (2.18) and neglecting the three-body correlations, g3, in Eq. (2.13), we obtain ·∂ ∂t +L(0)(x1) + L(0)(x2)¸g2(x1, x2, t) = δ(r12)σd−1¯ T0(v1,v2)f1(x1, t)f1(x2, t) + [K(x1, t) + K(x2, t)] g2(x1, x2, t)−ξ2 0 N ∂ ∂v1·∂ ∂v2 f1(x1, t)f1(x2, t),(2.22) where we have introduced the linear operator K(xi, t)defined as K(xi, t) = σd−1Zdx3δ(ri3)¯ T0(vi,v3)(1 + Pi3)f1(x3, t) + ξ2 0 2µ∂ ∂vi¶2 ,(2.23) and where the permutation operator Pab interchanges the labels of particles aand bin the quantities on which it acts. In the same way, the equation for h1,1, neglecting the contribution coming from h2,1in Eq. (2.16), is ·∂ ∂t +L(0)(x1)−K(x1, t)¸h1,1(x1, t;x0 1, t0) = 0,(2.24) which have to be solved with the initial condition h1,1(x1, t;x0 1, t0) = δ(x1−x0 1)f1(x1, t0) + g2(x1, x0 1, t0).(2.25) To sum up, under the same hypothesis made to derive the Boltzmann equation, we obtain closed equations for the one-time and two-time correlation functions, Eqs. (2.22) and (2.24). These equations represent a starting point for the study of the fluctuations and correlations in the heated system. 2.2. The stationary state It has been shown numerically that, for a wide class of initial conditions, the system reaches a homogeneous stationary state [31] in which the energy input from the thermostat is compensated by the energy lost in collisions. In this case the Boltzmann-Fokker-Planck equation reads ξ2 0 2 ∂2 ∂v2 1 fH(v1) + J[fH|fH] = 0.(2.26) 92 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases For the sake of simplicity, let us introduce the following dimensionless distribution fH(v) = nH vd H χH(c),(2.27) with χH(c)an isotropic function of the modulus of c,vH= (2TH m)1/2,c=v vHand THthe granular temperature defined as d 2nHTH=Zdv1 2mv2fH(v).(2.28) By introducing equation (2.27) in equation (2.26), we obtain a closed equation for χH Zdc2¯ T(c1,c2)χH(c1)χH(c2) + ξ2 2 ∂ ∂c2 1 χH(c1) = 0,(2.29) where ¯ T(c1,c2) = Zdˆ σΘ(ˆ σ·c12)(ˆ σ·c12)[α−2b−1 ˆ σ−1],(2.30) is the dimensionless binary collision operator and ξ2=ξ2 0` v3 H is the dimensionless strength of the noise with `= (nHσd−1)−1proportional to the mean free path. An approximate expression for the scaled one-particle distribution function χH(c)was derived to second order in Sonine polynomials [30] χH(c) = e−c2 πd/2£1 + a2(α)S2 d/2−1(c2)¤,(2.31) where S2 d/2−1(c2) = 1 2c4−1 2(d+ 2)c2+1 8d(d+ 2).(2.32) The coefficient a2(α)is related to the forth velocity moment of χH(c) d d+ 2 hc4iH hc2i2 H = 1 + a2(α),(2.33) and an approximate expression is [30] a2=16(1 −α)(1 −2α2) 73 + 56d−24αd −105α+ 30(1 −α)α2.(2.34) In the stationary state, the expression for the temperature in the first Sonine approximation is given by TH=m·dξ2 0√π (1 −α2)ΩdnHσd−1¸2/3 (1 + O(a2)),(2.35) where Ωd= 2πd/2/Γ(d/2) is the d-dimensional solid angle. 93 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases In the case of the correlation functions for this homogeneous state, gH 2and hH 1,1, it is convenient to introduce the dimensionless counterparts, gHand hH, as gH 2(x1, x2) = nH `dv2d H gH(l12,c1,c2), hH 1,1(x1, t;x2, t0) = nH `dv2d H hH(l12,c1,c2, s −s0),(2.36) where we have introduced the dimensionless length scale l=r/`,l12 =l1−l2and the dimensionless time-scale s=vH `t. The reduced distribution, gH, fulfills ·Λ(c1) + Λ(c2)−c12 ·∂ ∂l12 ¸gH(l12,c1,c2) = −δ(l12)¯ T(c1,c2)χH(c1)χH(c2) +ξ2nH`d N ∂ ∂c1·∂ ∂c2 χH(c1)χH(c2), (2.37) where Λ(ci)is the linearized Boltzmann-Fokker-Planck operator Λ(ci)h(ci) = Zdc3¯ T(ci,c3)(1 + Pi3)χH(c3)h(ci) + ξ2 2 ∂2 ∂c2 i h(ci).(2.38) Equation (2.37) describes the one-time correlation between fluctuations in the stationary state. As can be seen, the correlation function is determined by the properties of the linearized Boltzmann- Fokker-Planck operator, Λ, and by the one-particle distribution function, χH. The evolution equation for the two-time correlation function is ·∂ ∂s −Λ(c1) + c1·∂ ∂l1¸hH(l12,c1,c2, s −s0) = 0,(2.39) which has to be solved with the initial condition hH(l12,c1,c2,0) ≡hH(l12,c1,c2) = χH(c1)δ(l12)δ(c12) + gH(l12,c1,c2).(2.40) Taking into account equation (2.37) and that the term c12 ·∂ ∂l12 χH(c1)δ(l12)δ(c12)vanishes identically, the equation for this quantity is ·Λ(c1) + Λ(c2)−c12 ·∂ ∂l12 ¸hH(l12,c1,c2) =−δ(l12)Γ(c1,c2) + ξ2nH`d N ∂ ∂c1·∂ ∂c2 χH(c1)χH(c2),(2.41) where Γ(c1,c2) = ¯ T(c1,c2)χH(c1)χH(c2)−[Λ(c1) + Λ(c2)]χH(c1)δ(c12).(2.42) 94 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases 3. Hydrodynamics Let us summarize briefly how the system can be analyzed from a hydrodynamic point of view. The hydrodynamic fields can be defined as usually done in kinetic theory n(r, t) = Zdvf1(x, t),(3.1) n(r, t)u(r, t) = Zdvvf1(x, t),(3.2) d 2n(r, t)T(r, t) = Zdvm 2V(r, t)2f1(x, t),(3.3) where the function V(r, t)≡v−u(r, t)is the peculiar velocity. The complete non-linear equations for these fields for a granular system heated by the stochastic thermostat are [15, 31] ∂ ∂tn=−∇·(nu),(3.4) ∂ ∂tu=−u·∇u−1 mn∇jPij,(3.5) ∂ ∂tT=−u·∇T−2 dn(∇·q+Pij∇jui)−ζT +mξ2 0,(3.6) where Pij is the pressure tensor, qis the heat flux and ζis the cooling rate, which is also a functional of the distribution function ζ=(1 −α2)mπd−1 2σd−1 4dΓ¡d+3 2¢nkBTZdv1Zdv2|v1−v2|3f1(r,v1, t)f1(r,v2, t).(3.7) 3.1. Evolution of homogeneous perturbations Now, we will focus on the study of a linear perturbation of the hydrodynamic fields around the homogeneous stationary state. We are interested, primarily, in the information related to the hydrodynamic eigenvalues that we can obtain from them. Considering a homogeneous state, the previous equations reduce to ∂ ∂tn= 0,∂ ∂tu=0,∂ ∂tT=−ζT +mξ2 0.(3.8) In the long time limit, the system is expected to approach a steady state with a constant temperature given by the equation ζH(fH)TH=mξ2 0.(3.9) Substituting the explicit form of the scaled one-particle distribution function in the first Sonine approximation, Eq. (2.31), in the equation above, we obtain the temperature given in (2.35). 95 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases The dimensionless field δε(s)is defined by δε(s) = δE(t) hE(t)iH =1 ˜ VZdc2c2 dδ˜ F(k=0,c, s),(5.2) where hE(t)iHis the total mean energy. We can identify δε(s)with the scalar product δε(s) = 2 ˜ Vh¯ ξ3|δ˜ F(k=0,c, s)i.(5.3) If we consider now that ¯ ξ3is approximately an eigenfunction of Λ(c)with eigenvalue λ3=−3 2ζ0, using the Boltzmann-Langevin equation we can write ·d ds +3 2ζ0¸δε(s) = Rε(s),(5.4) with Rε(s) = 2 ˜ Vh¯ ξ3|˜ R(k=0,c, s)i.(5.5) The next step is to find the properties of the noise term, Rε(s), making use of the properties of the noise of the Boltzmann-Langevin equation. From Eq. (4.20) it is clear that hRε(s)δε(s0)iH= 0, s > s0.(5.6) The correlation function of the noise is hRε(s1)Rε(s2)iH=4 ˜ V2Zdc1Zdc2χH(c1)−1χH(c2)−1¯ ξ3(c1)¯ ξ3(c2)H(l12,c1,c2)δ(s1−s2) =4 Nδ(s1−s2)£h¯ ξ3(c1)¯ ξ3(c2)|¯ T(c1,c2)χH(c1)χH(c2)i − h¯ ξ3(c1)¯ ξ3(c2)|(Λ(c1) + Λ(c2))χH(c1)δ(c12)i¤=Aεδ(s1−s2),(5.7) where we have made use of Eq. (4.22) and Eq. (2.42). The calculation is now straightforward although lengthy. Taking into account the results obtained in [14], we have, in first order in Sonine polynomials that h¯ ξ3(c1)¯ ξ3(c2)|¯ T(c1,c2)χH(c1)χH(c2)i= 3ζ0a33,(5.8) with a33 =−15 + 7d+ 14d2−3(−9 + d(9 + 2d))α+ 30(1 + d)α2−6(9 + d)α3) 9d(−19 + 2d(−7+3α) + 3α(9 + 2(−1 + α)α)) .(5.9) The other term can be easily evaluated in our approximation h¯ ξ3(c1)¯ ξ3(c2)|(Λ(c1) + Λ(c2))χH(c1)δ(c12)i≃−3ζ0Zdcµc2 d−1 6¶2 χH(c) =−3ζ0µ−5 36 +d+ 2 4d(1 + a2)¶.(5.10) 102 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases Finally, we can identify Aεto be Aε=12 Nζ0µa33 −5 36 +d+ 2 4d(1 + a2)¶.(5.11) With the properties we have about the noise Rε(s), it is easy to calculate the correlation function of δε. The solution of Eq. (5.4) can be formally written as δε(s) = e−3 2ζ0sδε(0) + Zs 0 ds0e−3 2ζ0(s−s0)Rε(s0)sÀ1 →Zs 0 ds0e−3 2ζ0(s−s0)Rε(s0).(5.12) So, we can evaluate hδε(s)2iH=Zs 0 ds1Zs 0 ds2e−3 2ζ0[(s−s1)+(s−s2)]hRε(s1)Rε(s2)iH=Zs 0 ds1e−3ζ0(s−s1)Aε sÀ1 →Aε 3ζ0 . (5.13) In Fig. 1, we show the comparison between the theoretical result (solid line) and the DSMC method results (circles and squares) for a two-dimensional system, as a function of the coefficient of normal restitution. As can be seen, the agreement is satisfactory for all the range of values of the inelasticity. In the limit, α→1−, we find the non-trivial result hδε2iH→d 3, while in the free cooling state, this quantity vanishes [7]. We emphasize that the elastic limit is singular: the behaviour for elastic systems with α= 1 is not approached by taking the quasi-elastic limit α→1−. We can also evaluate the two-time correlation function for the fluctuations of the total energy. If we multiply in Eq. (5.4) by δε(s0), for s0< s, and taking into account that the noise is not correlated with the distribution function for a time smaller, we obtain that hδε(s)δε(s0)iH=hδε(s0)2iHe−3 2ζ0(s−s0).(5.14) Then, the decay of the two-time correlation function is the same as we have found for a homogeneous perturbation of the total energy. As expected, the amplitude of the fluctuations coincides with the one obtained in [14] from the equations for the correlation functions. 6. The fluctuating transverse velocity The objective in this section is to derive a fluctuating equation for the transverse velocity field, w⊥(k, s). The reason to consider this field is that its equation is decoupled from the equations for the other hydrodynamic fields [31], and it can be derived exactly in the hydrodynamic limit with the knowledge we have about the spectrum of the linearized Boltzmann-Fokker-Planck operator, i.e. we do not need the approximation corresponding to the third eigenfunction. Mathematically, the transverse velocity is defined in the following way: Let us consider the d-dimensional vector kwhich belongs to the space <d. This space can be spanned by the base 103 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases 0 0.2 0.4 0.6 0.8 1 α 0.6 0.8 1 1.2 <(δ ε) >H 2 N=100 N=1000 Figure 1: Second moment of the fluctuations of the total energy as a function of the restitution coefficient α. The solid line is the theoretical prediction and symbols are the two-dimensional Monte Carlo simulation results of Ref. [32]. nˆ koSnˆ ki ⊥od−1 i=1 where ˆ kis a unitary vector parallel to kand nˆ ki ⊥od−1 i=1 are (d−1) unitary vectors orthogonal to k. The transverse velocity is defined as w⊥i(k, s) = Zdc(c·ˆ ki ⊥)δ˜ F(k,c, s), i = 1, . . . , d −1.(6.1) For the subsequent analysis, it is convenient to introduce the following projectors P(i)f(c)≡ h¯ ξ2⊥i(c)|f(c)iξ2⊥i(c),(6.2) Q(i)f(c)≡(1 −P(i))f(c).(6.3) Then, if we apply P(i)to δ˜ Fwe obtain P(i)δ˜ F(k,c, s) = w⊥i(k, s)ξ2⊥i(c),(6.4) and the transverse velocity is the component of δ˜ F(k,c, s)in the subspace generated by ξ2⊥i(c). In order to obtain an equation for w⊥, we apply the projectors Pand Q(for simplicity in the notation we skip the super-index) to the Langevin equation (4.17) ·∂ ∂s −PΛ(k,c)¸Pδ ˜ F(k,c, s) = P˜ R(k,c, s)−Pi(k·c)Qδ ˜ F(k,c, s),(6.5) 104 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases ·∂ ∂s −QΛ(k,c)¸Qδ ˜ F(k,c, s) = Q˜ R(k,c, s)−Qi(k·c)Pδ ˜ F(k,c, s).(6.6) Now, let us solve equation (6.6) formally Qδ ˜ F(k,c, s) = eQΛ(k,c)sQδ ˜ F(k,c,0) +Zs 0 ds0eQΛ(k,c)(s−s0)[Q˜ R(k,c, s0)−Qi(k·c)Pδ ˜ F(k,c, s0)].(6.7) Restricting ourselves to initial conditions with Qδ ˜ F(k,c, s) = 0, and by substituting Eq. (6.7) in Eq. (6.5), we obtain a closed equation for Pδ ˜ F(k,c, s) ·∂ ∂s −PΛ(k,c)¸Pδ ˜ F(k,c, s) + P(k·c)Zs 0 ds0eQΛ(k,c)(s−s0)Q(k·c)Pδ ˜ F(k,c, s0) =P˜ R(k,c, s) −Pi(k·c)Zs 0 ds0eQΛ(k,c)(s−s0)Q˜ R(k,c, s0).(6.8) In Appendix B it is shown that, in the hydrodynamic limit, i.e. to second order in kand in the long time limit, equation (6.8) reduces to the following equation for the transverse velocity ·∂ ∂s + ˜ηk2¸w⊥(k, s) = Rw(k, s).(6.9) The coefficient ˜ηis the shear viscosity given by ˜η=Z∞ 0 ds ZdccxcyeΛ(c)scxξ2,y(c) = −ZdccxcyΛ(c)−1cxξ2,y(c),(6.10) which agrees with the expression obtained in [15] by the Chapman-Enskog method, and Rw(k, s) is a noise term which can be decomposed as Rw(k, s) = S(k, s) + Π(k, s).(6.11) The term S(k, s)comes from the thermostat (which does not conserve momentum locally), and the second term, Π(k, s), is the fluctuating part of the pressure tensor. Their microscopic expressions in terms of the noise of the Boltzmann-Langevin equation are S(k, s) = Zdc(ˆ k⊥·c)˜ R(k,c, s),(6.12) and Π(k, s) = −ik Zs 0 ds0Zdc(ˆ k·c)(ˆ k⊥·c)eQΛ(k,c)(s−s0)Q˜ R(k,c, s0).(6.13) Of course, due to the fact that h˜ R(k,c, s)iH= 0, the mean value of the noise vanishes hRw(k, s)iH= 0.(6.14) 105 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases The autocorrelation function of the noise is evaluated in Appendix C. Due to symmetry considerations, there are only correlations between the kand −kcomponents, which yields hRw(k, s1)Rw(−k, s2)iH=˜ V2 N£ξ2δ(s1−s2) + k2Cxy(s2−s1)¤ +hS(k, s1)Π(−k, s2)iH, s1< s2.(6.15) The first term is the expected zeroth order term which comes from the heating. The Dirac delta function is an exact consequence of the fact that the external noise (the heating) is white. The function Cxy(s2−s1)reads Cxy(s2−s1) = Zdc1Zdc2c1xc1yc2xc2yeΛ(c2)(s2−s1)φH(c1,c2).(6.16) Here, we have introduced the function φH(c1,c2)as the space integral of the correlation function hH(l,c1,c2) φH(c1,c2) = ZdlhH(l,c1,c2) = χH(c1)δ(c12) + χ2(c1,c2),(6.17) where χ2(c1,c2) = RdlgH(l,c1,c2)and gHis the dimensionless correlation function defined in (2.36). The equation for φHcan easily be obtained by integration over space variable in Eq. (2.41), which gives [Λ(c1) + Λ(c2)]φH(c1,c2) = −Γ(c1,c2) + ξ2∂ ∂c1·∂ ∂c2 χH(c1)χH(c2).(6.18) As discussed in Appendix C, Cxy can be physically interpreted as the autocorrelation function of the global quantity PN i=1 Vx(t)Vy(t). In the elastic case, this correlation function is related to the shear viscosity but it is not the case for granular systems [1]. The formula for the correlation hS(k, s1)Π(−k, s2)iHis given in Appendix C, and it does not seem to admit a simple physical interpretation. As the two correlation functions, Cxy(s)and hS(k, s1)Π(−k, s2)iH, decay with the kinetic modes and, in the k→0limit, the fluctuating velocity is expected to be frozen (its time evolution is given by the null eigenvalue, Eq. (6.9)), we can consider that they are proportional to a Dirac delta function in time, and we have ˜ V2 Nk2Cxy(s2−s1) + hS(k, s1)Π(−k, s2)iH →2"˜ V2 Nk2Z∞ 0 dsCxy(s) + Z∞ 0 dshS(k,0)Π(−k, s)iH#δ(s1−s2). (6.19) Note that this is in contrast with the free cooling case where the noise can not be considered to be white [6]. In this case, the equation for the transverse velocity (rescaled with the thermal velocity) contains a term of order zero in kwhich is proportional to the cooling rate. Then, even in the 106 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases k→0limit, the velocity is not frozen on the kinetic scale. In Appendix D the second integral of equation (6.19) is evaluated obtaining Z∞ 0 dshS(k,0)Π(−k, s)iH→˜ V2 Nk2Zdc1Zdc2c1xc2xc2yΛ(c2)−1c2yφH(c1,c2),(6.20) which is valid in the hydrodynamic limit. If we substitute φH(c1,c2)by its expression in terms of the one- and two-particle distribution function, Eq. (6.17), we obtain (see Appendix E) that the one-particle contribution vanishes and the correlation function can be written in terms of the two-particle velocity correlation function, χ2(c1,c2) hRw(k, s1)Rw(−k, s2)iH =˜ V2 Nδ(s1−s2)½ξ2+ 2k2·−Zdc1Zdc2c1xc1yc2xc2yΛ(c2)−1χ2(c1,c2) +Zdc1Zdc2c1xc2xc2yΛ(c2)−1c2yχ2(c1,c2)¸¾.(6.21) As can be seen, the k2term has no relation, a priori, with the shear viscosity, Eq. (6.10), i.e. there is a priori no fluctuation-dissipation relation as that assumed in [31]. However, we now show that, under additional hypotheses (that in principle are not restricted to the elastic limit), the aforementioned term reduces to the shear viscosity. Let us assume that the most important contribution of the two-particle velocity correlation function, χ2(c1,c2), is the hydrodynamic part, i.e. we assume χ2(c1,c2)≃ d+2 X β=1 d+2 X β0=1 aβ,β0ξβ(c1)ξβ0(c2).(6.22) This assumption was already made in [14] where the coefficients aβ,β0were evaluated to calculate the total energy fluctuations. We emphasize that it led to an excellent agreement between analytical predictions and numerical data (Monte Carlo) for energy fluctuations, for all the values of the inelasticity [14]. In this approximation, the first integral in (6.21) vanishes because cxcyχH(c)is orthogonal to the hydrodynamic modes. The second term is Zdc1Zdc2c1xc2xc2yΛ(c2)−1c2y d+2 X β=1 d+2 X β0=1 aβ,β0ξβ(c1)ξβ0(c2) =a2x,2xZdc1c1xξ2x(c1)Zdc2c2xc2yΛ(c2)−1c2yξ2x(c2),(6.23) where, for symmetry considerations, the only term that remains is the one associated with β= β0= 2. If we use now that a2i2i=−1/2(see the reference [14]), and the formula for the shear viscosity, Eq. (6.10), we finally have hRw(k, s1)Rw(−k, s2) = ˜ V2 Nδ(s1−s2)(ξ2+ ˜ηk2).(6.24) 107 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases Then, in the hydrodynamic limit and assuming that the two-particle velocity correlation function, χ2(c1,c2), has only components in the hydrodynamic subspace, the correlation function of the noise reduces to the one introduced phenomenologically in [31], not only in the elastic limit, but for arbitrary inelasticity. Although we have no direct proof of the accuracy of approximation (6.22), we note that it is backed up by numerical data, see e.g. [14]. As an application of Eq. (6.9), let us calculate the two-time correlation for the transverse velocity field. Proceeding as in Sec. 5., the two-time correlation function can be calculated, obtaining hw⊥(k, s)w⊥(−k, s0)iH=˜ V2 N ξ2+ ˜ηk2 2˜ηk2e−˜ηk2(s−s0).(6.25) As seen, the correlation function decays as a linear macroscopic perturbation, with an amplitude which is a function of the heating and the shear viscosity. 7. Conclusions In this work, we have presented a theory of fluctuations for a dilute gas heated by a stochastic thermostat in the stationary state. The starting point has been the Boltzmann-Langevin equation: a stochastic equation for the fluctuating distribution function. This equation has the same form as the linearized Boltzmann equation but adding a noise term. The properties of the noise are determined enforcing consistency with the equations for the correlation functions, for both the one-time and the two-time correlation functions. We consider that the noise is white as it is the case for molecular fluids. From this equation it would be possible to construct a set of fluctuating hydrodynamic equations, and from them to evaluate the one-time and two-time correlation functions of the fluctuating fields. We have focused on the study of the simplest two fluctuating quantities: the total energy and the transverse velocity, whose equations are decoupled. The results obtained for the energy fluctuations match up with the ones calculated earlier by means of the equation for the correlation function [14]. We have also deduced the decay of the correlations for the fluctuations, finding that they decay in the same way as a linear perturbation. These results have been successfully tested against the numerical results obtained by the Direct simulation Monte Carlo method for all the range of values of the coefficient of normal restitution α[14, 32]. For the transverse velocity field, some important differences have been found in comparison with the free cooling case [5, 6], where it was shown that the relevant Langevin noise is not white and that there is no a fluctuation-dissipation relation of the second kind (the amplitude of the noise is not related to the shear viscosity). Firstly, the noise has two parts, one coming from the thermostat and the other one coming from the fluctuating pressure tensor. Moreover, the noise can be considered to be white as the dynamics of the velocity is as slow as desired in the hydrodynamic (low k) limit. With regard to the validity of the fluctuation-dissipation relation in the heated case in the hydrodynamic limit, the amplitude of the noise is priori not related to the shear viscosity. However, considering that the two-particle velocity correlation function has only hydrodynamic modes –which seems a reasonable assumption– the coupling between the two noises somehow restores fluctuation-dissipation and we obtain the expression assumed in [31], with the actual inelastic shear viscosity. In principle, this rather surprising result is not limited to small inelasticity. 108 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases Acknowledgements We thank Emmanuel Trizac for useful discussions and for a careful reading of the manuscript. We also thank Paolo Visco for providing us with the Monte Carlo data of Fig. 1. This research was supported by the Ministerio de Educaci´ on y Ciencia (Spain) through Grant No. FIS2008-01339 (partially financed by FEDER funds). A Eigenvalue problem for Λ We consider here the eigenvalue problem for the homogeneous linear Boltzmann-Fokker-Planck operator Λ, defined in (2.38) Λ(c)ξβ(c) = λβξβ(c).(A1) We are interested in the eigenfunctions and eigenvalues associated with linear hydrodynamics and, to perform the analysis, similar techniques as in [3, 7, 13] will be required. Consider first the function ψ1(c) = χH(c).(A2) When the linearized operator Λacts on χH, we have Λ(c1)χH(c1) = Zdc2¯ T(c2,c3)(1 + P12)χH(c2)χH(c1) + ξ2 2µ∂ ∂c1¶2 χH(c1).(A3) Taking into account the equation for χH, Ec. (2.29), we obtain the following relation Λ(c1)ψ1(c1) = −ξ2 2µ∂ ∂c1¶2 χH(c1).(A4) Now, let us considerer the function ψ2(c) = −∂ ∂cχH(c).(A5) Taking derivate in the equation obeyed by χH(c−w)with respect to w, and subsequently evaluating the result for w= 0, we obtain Λ(c1)ψ2(c1) = 0.(A6) Finally, we will consider the function ψ3(c) = c·∂ ∂cχH(c).(A7) From the equation obeyed by ψ3(λc1), we can take derivate with respect to λ, and evaluate the result for λ= 1. We arrive at an equation for ψ3(c1), Λ(c1)ψ3(c1) = (d+ 3)ξ2 2µ∂ ∂c1¶2 χH(c1).(A8) 109 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases From equations (A4), (A6) and (A8), we can identify two eigenfunctions of Λ. Making use of (A4) and (A8), it appears that Λ(c)µ1 3 ∂ ∂c·[cχH(c)] + χH(c)¶= 0.(A9) Hence, from Eqs. (A6) and (A9) we can conclude that the null eigenvalue is (d+ 1)-fold degenerate with the eigenfunctions ξ1(c) = 1 3 ∂ ∂c·[cχH(c)] + χH(c),ξ2=−∂ ∂cχH(c).(A10) B Derivation of the transverse velocity field equation In this Appendix we derive the equation for the transverse velocity field, w⊥(k, s), in the hydrodynamic limit. The starting point is equation (6.8) ·∂ ∂s −PΛ(k,c)¸Pδ ˜ F(k,c, s) + P(k·c)Zs 0 ds0eQΛ(k,c)(s−s0)Q(k·c)Pδ ˜ F(k,c, s0) =P˜ R(k,c, s) −Pi(k·c)Zs 0 ds0eQΛ(k,c)(s−s0)Q˜ R(k,c, s0).(B1) Let us first consider the term PΛ(k,c)Pδ ˜ F(k,c, s). As Λ(c)ξ2⊥(c) = 0 and Rdc(ˆ k⊥·c)(ˆ k· c)ξ2⊥(c) = 0, we easily have PΛ(k,c)Pδ ˜ F(k,c, s) = 0.(B2) Let us evaluate the last term of the left-hand side of Eq. (B1). To second order in k, we have P(k·c)Zs 0 ds0eQΛ(k,c)(s−s0)Q(k·c)Pδ ˜ F(k,c, s0) ≃k2ξ2⊥(c)Zs 0 ds0w⊥(k, s0)Zdc(ˆ k·c)(ˆ k⊥·c)eΛ(c)(s−s0)ˆ k·cξ2⊥(c) =k2ξ2⊥(c)Zs 0 ds0w⊥(k, s0)Gxy(s−s0)(B3) where we have introduced Gxy(s)≡Zdc(ˆ k⊥·c)(ˆ k·c)eΛ(c)s(ˆ k·c)ξ2⊥(c) =ZdccxcyeΛ(c)scxξ2y(c),(B4) and use has been made of the fact that the operator Λ(c)is isotropic. In the hydrodynamic limit, the velocity evolves in a scale much slower that the scale in which the function Gxy(s)decays. We then have Zs 0 ds0w⊥(k, s0)Gxy(s−s0)→˜ηw⊥(k, s),(B5) 110 P. Maynar and M.I. Garc´ ıa de Soria A Langevin description for driven granular gases where ˜ηis the dimensionless shear viscosity ˜η=Z∞ 0 dsGxy(s).(B6) The noise terms are the last two terms of Eq. (B1) P˜ R(k,c, s) = ξ2⊥(c)Zdc(ˆ k⊥·c)˜ R(k,c, s) =ξ2⊥(c)S(k, s),(B7) and P(ik·c)Zs 0 ds0eQΛ(k,c)(s−s0)Q˜ R(k,c, s0) =ξ2⊥(c)ik Zdc(ˆ k⊥·c)(ˆ k·c)Zs 0 ds0eQΛ(k,c)(s−s0)Q˜ R(k,c, s0) =−ξ2⊥(c)Π(k, s),(B8) where we have used the definitions of S(k, s)and the fluctuating pressure tensor, Π(k, s), Eqs. (6.12) and (6.13). Finally, by multiplying Eq. (B1) by ˆ k⊥·cand further integrating over velocities, we obtain the equation of the transverse velocity of the main text. C Autocorrelation function of Rw(k, s) In this Appendix we evaluate the correlation function of the noise of the transverse velocity field, Rw(k, s). We consider k6=0,s1< s2with s1large. It is convenient to introduce the following notation for the transverse and parallel components of the vector c ˆ k⊥·c=c⊥,ˆ k·c=ck.(C1) The autocorrelation function of Rw(k, s)reads, in terms of S(k, s)and Π(k, s), hRw(k, s1)Rw(−k, s2)iH=h(S(k, s1) + Π(k, s1))(S(−k, s2) + Π(−k, s2))iH =hS(k, s1)S(−k, s2)iH+hS(k, s1)Π(−k, s2)iH +hΠ(k, s1)S(−k, s2)iH+hΠ(k, s1)Π(−k, s2)iH. (C2) We now calculate each correlation function taking into account the microscopic expressions of S(k, s)and Π(k, s), Eqs. (6.12) and (6.13), and the correlation function of the noise of the Boltzmann-Langevin equation, Eqs. (4.21) and (4.22). 111