A Derivation of the fluctuating hydrodynamic equations

Fluctuating hydrodynamics for dilute granular gases

Abstract

Starting from the kinetic equations for the fluctuations and correlations of a dilute gas of inelastic hard spheres or disks, a Boltzmann-Langevin equation for the one-particle distribution function of the homogeneous cooling state is constructed. This equation is the linear Boltzmann equation with a fluctuating white noise term. Balance equations for the fluctuating hydrodynamic fields are derived. New fluctuating forces appear as compared with the elastic limit. The particular case of the transverse velocity field is investigated in detail. Its fluctuations can be described by means of a Langevin equation, but exhibiting two main differences with the Landau-Lifshitz theory: the noise is not white, and its second moment is not determined by the shear viscosity. This shows that the fluctuation-dissipation relations for molecular fluids do not straightforwardly carry over to inelastic gases. The theoretical predictions are shown to be in good agreement with molecular dynamics simulation results.

pacs:
45.70.Mg, 45.70.Qj, 47.20.Ky

I Introduction

The (modified) nonlinear Boltzmann equation for the one particle distribution function provides an accurate description of transport phenomena in a low density gas of inelastic hard spheres or disks (1); (2); (3); (4); (5). These particles are often used to model granular fluids (6), specially in the rapid flow regime (7). The Boltzmann equation does not provide any direct information about correlations and fluctuations in the gas, other that the particle velocity moments. Nevertheless, methods used in the derivation of the Boltzmann equation have been extended to obtain kinetic equations for the equal and different time correlations, in the same low density approximation. The general idea is that in order to obtain these equations the needed approximations are the same as those used to derive the Boltzmann equation itself.

One of the earliest and physically more transparent methods to study fluctuations is that of Langevin equations. Almost 40 years ago, in a seminal paper Bixon and Zwanzig (8) showed how a Boltzmann-Langevin equation could be constructed by generalizing the reasonings leading to the Boltzmann equation for molecular gases. The latter describes the behavior of the average value of the one-particle distribution function, while the former incorporates the effects of the fluctuations. As the authors indicated themselves in the paper, the derivation was based on physical intuition and analogy. A more systematic derivation of the same result, starting from first principles, was given in ref. (9).

A second approach to the study of correlations in dilute gases makes use of functional analysis. Its more general result is a kinetic equation for a generating functional at low density, from which all multi-point correlations can be obtained by functional differentiation (10). A closely related general scheme for the study of correlations is the hierarchical method (11); (12). The starting point are hierarchies of coupled equations for the time distribution functions describing the fluctuations and correlations. Then, the hierarchies are closed by using the same kind of approximations as needed to derive the kinetic equation, i.e. the Boltzmann equation in the case of dilute gases. This method has been recently extended to describe fluctuations and correlations of dilute inelastic gases in their simplest state, the homogeneous cooling state (HCS) (13). As an application, the fluctuations of the total energy were studied and a good agreement between theory and simulation results was found (13); (14).

One of the aims of this paper is to translate the above formalism in terms of kinetic equations for the correlation functions into a Langevin equation formulation, i.e. to extend the fluctuating Boltzmann equation to the case of inelastic hard spheres or disks. The relationship between kinetic equations and the fluctuating Boltzmann equation has been analyzed in detail in molecular gases (12); (15). One advantage of the Langevin formulation is that it is closer to the fluctuating hydrodynamic equations. Actually, the fluctuating Boltzmann equation for molecular systems has been shown (8); (16); (17) to lead to the same Langevin equations for the hydrodynamic fields as obtained by Landau and Lifshitz (18) using thermodynamic fluctuation theory. The noise terms in these equations are assumed to be white with second moments determined by the Navier-Stokes transport coefficients of the fluid. Their expressions are known as fluctuation-dissipation relations of the second kind (19).

The derivation of fluctuating hydrodynamic equations from the fluctuating Boltzmann equation for inelastic hard particles, will be also addressed here. Attention will be focussed on a particular state, the HCS, and on a specific hydrodynamic field, the transverse component of the velocity. The main conclusion will be that the fluctuation-dissipation relation for elastic gases can not be directly extrapolated to inelastic ones, but it needs to be significantly modified. The second moment of the noise is not determined by the Navier-Stokes shear viscosity. Moreover, the noise can not be assumed to be white. These theoretical predictions are in qualitative and quantitative agreement with molecular dynamics simulation results.

The consideration of the HCS does not imply by itself that the results obtained here are not relevant for other states more accesible experimentally. The HCS plays for inelastic gases a role similar to the equilibrium state for molecular gases. In the case of molecular systems, the expressions of the transport coefficients obtained by linearizing around equilibrium are the same as those appearing in the nonlinear Navier-Stokes equations as predicted by the Chapman-Enskog method and successfully used in many far from equilibrium problems (20). Also, the fluctuation-dissipation relations derived for near-local-equilibrium states in the original Landau and Lifshitz theory have proven to be accurate for many other hydrodynamic states (21). For dilute gases composed of inelastic hard particles, the equivalence between the transport coefficients obtained by linear perturbations of the HCS and by applying the Chapman-Enskog procedure has also been established (22). Something similar might be expected for the fluctuations and correlations.

In the system being considered here, the particles move freely and independently between consecutive collisions. More specifically, they are not coupled to any external energy source or thermal bath, contrary to the driven granular gas models. For these models, the linear response to an external perturbation (23) as well as the validity of the Einstein relation (24) have been investigated by numerical simulations, and some empirical models have been proposed. It is not evident a direct relation between the free model considered here and the above driven models.

The plan of the paper is as follows. In Sec. II, the kinetic equations for the one-time and two-time correlation functions of a dilute gas in the HCS derived in ref. (13) are shortly reviewed. These equations are translated into an equivalent Boltzmann-Langevin equation for the one particle distribution function in Sec. III. When written in the appropriate variables, this equation is the linear Boltzmann equation to which a fluctuating force term is added, similarly to what happens in molecular elastic gases. An expression for the second moment of the fluctuating force in terms of the collisional Boltzmann kernel is derived. In Sec. IV, the fluctuating hydrodynamic fields are defined, and balance equations for them are obtained from the Boltzmann-Langevin equation. They involve formal expressions for the fluctuating pressure tensor, the fluctuating heat flux, and the fluctuating cooling rate. In addition, an intrinsically inelastic fluctuating force shows up in the equation for the energy.

To get a closed description for the hydrodynamic fluctuations, expressions for the heat flux, the pressure tensor, and the cooling rate in terms of the fluctuating hydrodynamic fields are needed. This can be accomplished by means of the Chapman-Enskog procedure. Here only the case of the transverse component of the velocity field will be considered. As a consequence, only the expression for the non-diagonal elements of the pressure tensor is required. This is computed in Sec. V. The final result is a Langevin equation, that is the linear macroscopic equation for the transverse velocity field plus a fluctuating force term. Therefore, the structure is similar to what one could expect by extrapolating from the corresponding equation for molecular systems (25). Nevertheless, the noise term is not white and its second moment is not given by the usual fluctuation-dissipation relation. It is verified that the obtained theoretical predictions are in good agreement with molecular dynamics simulation results. Section VII contains some general comments and conclusions. Finally, the appendixes provide some details of the calculations needed to derive the results presented in the bulk of the paper.

Ii Kinetic equations for the homogeneous cooling state

The system considered is a dilute gas of smooth inelastic hard spheres () or disks () of mass and diameter . The position and velocity of the ith particle at time will be denote by and , respectively. The effect of a collision between particles and is to instantaneously modify their velocities according to the rule

(1)

where is the relative velocity, is the unit vector pointing from the center of particle to the center of particle at contact, and is the coefficient of normal restitution. It is defined in the interval and it will considered here as constant, independent of the velocities of the particles involved in the collision. A more realistic modeling of granular gases would require to consider a velocity dependent restitution coefficient (5).

Given a trajectory of the system, one-point and two-point microscopic densities in phase space at time are defined by

(2)

and

(3)

respectively. Here , while the are field variables referring to the one-particle phase space ( space). The density obeys the equation (11); (13)

(4)

with

(5)

where is the solid angle element for , , is the Heaviside step function, and is an operator replacing all the functions of and to its right by the same functions of the precollisional values and given by

(6)

It is seen that Eq. (4) for involves the two particle density . Actually, it is the first equation of an infinity hierarchy (13).

The averages of and over the initial probability distribution of the system , , are the usual one-particle and two-particle distribution functions,

(7)

where the notation

(8)

has been employed. Two-time reduced distribution functions can also defined from the microscopic densities and the initial probability distribution. The simplest one is the two-particle two-time distribution function,

(9)

From the definitions in Eqs. (7) and (9) it follows that

(10)

It is convenient to introduce one-time and two-time correlation functions by

(11)

and

(12)

respectively. Equation (10) translates into

(13)

In the low density limit, a closed set of kinetic equations for , , and can be derived (13) by extending the methods developed for molecular gases (12). They can be used to analyze the average properties as well as correlations and fluctuations in arbitrary states of a dilute granular gas. Here attention will be restricted to a particular state of a freely evolving granular gas, the so-called homogeneous cooling state (HCS) (26). Macroscopically, it is characterized by a uniform number of particles density , a vanishing velocity field, and a uniform time-dependent temperature . It is further defined by the one-particle distribution function having the scaled form (1)

(14)

where

(15)

is a thermal velocity and is an isotropic function of the scaled velocity . The distribution and the granular temperature are specified by the pair of coupled equations

(16)
(17)

In the above expressions,

(18)

is the dimensionless cooling rate in the time scale defined by

(19)

with , and is the inelastic Boltzmann collision term. Its explicit form is

(20)
(21)

The variable defined in Eq. (19) is proportional to the accumulated number of collisions per particle. For thermal velocities, i.e values of of the order of unity, a good approximation to the solution of Eqs. (16) and (17) is provided by the first Sonine approximation, in which (1); (27)

(22)

with

(23)

and

(24)

In the same approximation

(25)

A numerically exact solution of Eqs. (30) and (17) has been recently reported in (28). The two-particle one-time correlation function of the HCS is assumed to have also a scaled form (13)

(26)

where the scaled length scale has been introduced. The dimensionless correlation does not depend on and obeys the equation

(27)

where is the linearized Boltzmann collision operator (29),

(28)

The operator interchanges the labels of particles and of the quantities to its right. For the two-particle two-time correlation function the scaling reads (13)

(29)

and the kinetic equation is

(30)

valid for . The initial condition for this equation is

(31)

An equation for this distribution follows from Eqs. (17) and (27),

(32)

with

(33)

Equations (27) and (30) describe the correlations between fluctuations in the HCS. They become closed once the solution to Eqs. (16) and (17) is known. In the next section, an alternative and consistent description to that provided by these kinetic equations will be developed.

Iii Fluctuating Boltzmann equation around the HCS

Equation (4) is an exact consequence of the dynamical equations governing the motion of the particles. The aim of this section is to approximate it in such a way that give a closed description of the effective dynamics of a dilute granular gas in the HCS. To do so, the spatial separation between the centers of colliding particles will be neglected in the operator , and will be approximated by an effective (Boltzmann) two-particle phase space density at the mesoscopic level . Moreover, the dimensionless time scale and length scale introduced in the previous section will be used. Then, Eq. (4) becomes

(34)

where dimensionless phase space densities have been defined by

(35)
(36)

Comparison of the ensemble average of Eq. (34) with Eq. (17) gives the conditions

(37)
(38)

The subindex H in the angular brackets indicates that the ensemble average is taken over the probability distribution for the HCS.

The deviation of the microscopic density from its average value is defined by

(39)

An evolution equation for this quantity follows by subtracting Eqs. (34) and (17),

(40)

The structure of this equations suggests to introduce a cluster decomposition for of the form

(41)

This equation defines the microscopic correlation density . Substitution of its ensemble average in Eq. (38) yields

(42)

Moreover, use of Eq. (41) into Eq. (40) allows to rewrite the equation in the equivalent form

(43)

where

(44)

and the operator was defined in Eq. (28). Equation (43) can be interpreted as a fluctuating Boltzmann-Langevin equation for the one-particle distribution function (8); (16); (17), with the “noise term” being . Of course, this does add any new physical insight by itself in the understanding of the starting equation (34). The relevance and usefulness of this representation will depend on the properties of the noise term. A first one follows directly from Eq. (42), that is equivalent to

(45)

i.e. the noise has zero average. In the following, other properties of will be derived by requiring consistency with the results derived in the previous section. Multiplication of Eq. (43) by with , followed by averaging gives

(46)

From the definition of it is easily verified that

(47)

where is defined in Eq. (29). Therefore, consistency of Eqs. (46) and (30) implies that

(48)

for . Since, by hypothesis, the parameters of the system are such that the HCS is stable, the long time solution of Eq. (43) is

(49)

where the linear operator

(50)

has been introduced. Using Eq. (49), it is obtained

(51)

This equation must be compared with Eq. (32), having in mind Eq. (47). The time independence of the right hand side of Eq. (32) prompts to introduce the hypothesis that the noise term is Markovian, and write

(52)

On introduction of this into Eq. (51) and comparison with Eq. (32), it follows that

(53)

with defined in Eq. (33).

The properties given by Eqs. (45), (48), and (52) guarantee that the description provided by the Langevin equation (43) leads to the same expressions for the two-particle, one-time and two-time correlation functions as the formulation in terms of reduced distributions functions reviewed in Sec. II.

Iv Fluctuating hydrodynamic fields and balance equations

The fluctuating number of particles density, , momentum density, , and energy density, , are defined in terms of the microscopic phase space density as

(54)
(55)
(56)

Dimensionless deviations from their averages values in the HCS are given by

(57)
(58)
(59)

The quantity is the dimensionless velocity field. Balance equations for these fluctuating fields follow by taking velocity moments in the Langevin-Boltzmann equation (43) and using the properties of the noise . Some details of the calculations are given in appendix A. The resulting equations read

(60)
(61)
(62)

In the above equations, and are the fluctuating pressure tensor and heat flux, respectively. Their definitions in terms of the fluctuating one-particle distribution function are

(63)
(64)

where is the unit tensor of dimension , and

(65)
(66)

The term represents the fluctuations of the cooling rate about its average value in the HCS. Its formal expressions is

(67)

Finally, is a fluctuating force term having the properties

(68)

and

(69)

with given by Eq. (135). This noise term is intrinsic to the inelasticity of collisions and has no analogue in normal fluids. Of course, in the elastic limit , becomes a Gaussian and the fluctuating force is seen to vanish in agreement with the well known results for hydrodynamic fluctuations in molecular fluids (18). The other main differences between Eq. (62) and the one for molecular gases is the presence of the two terms involving the cooling rate, , and its fluctuations, . The presence of these terms is directly associated with existence of the cooling term in the macroscopic equation for the average energy (30); (31); (3).

V Langevin equation for the velocity field

Equations (60)-(62) do not provide a closed description of the hydrodynamic fluctuations of a granular gas in the HCS until , , and are expressed in terms of the fluctuating hydrodynamic fields. This turns out to be not a simple task, and attention will be restricted in the following to the equation of the velocity field , Eq. (61).

Given two functions and , their scalar product is defined as

(70)

where is the complex conjugate of . Next, a projection operator is introduced by

(71)

Here, the functions are the eigenfunctions of the linear Boltzmann operator defined in Eq. (28), corresponding to the hydrodynamic part of its spectrum. Therefore, they are solutions of the equation

(72)

Their expressions are (29); (22)

(73)

The associated eigenvalues are found to be

(74)

the eigenvalue being -fold degenerated. Finally the functions are

(75)

The sets of functions and are seen to have the biorthogonality property

(76)

. This guarantees that as defined by Eq. (71) is really a projector operator, i.e. it verifies . It projects any function of onto the subspace spanned by the hydrodynamic eigenfunctions of .

In the following, it will be more convenient to work in the Fourier representation. The Fourier transform of is

(77)

By means of , can be decomposed into its hydrodynamic and non-hydrodynamic parts,

(78)

where . The Fourier representation of the balance equation for the velocity fluctuations, Eq. (61), is