Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres

J. Stat. Mech. (2020) 034002 Kinetic model for a confined quasitwo-dimensional gas of inelastic hard spheres J Javier Brey, P Maynar and M I García de Soria Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080, Sevilla, Spain E-mail: [email protected] Received 1 October 2019 Accepted for publication 7 January 2020 Published 2 March 2020 Online at stacks.iop.org/JSTAT/2020/034002 https://doi.org/10.1088/1742-5468/ab7124 Abstract. The local balance equations for the density, momentum, and energy of a dilute gas of elastic or inelastic hard spheres, strongly confined between two parallel hard plates are obtained. The starting point is a Boltzmann-like kinetic equation, recently derived for this system. As a consequence of the confinement, the pressure tensor and the heat flux contain, in addition to the terms associated to the motion of the particles, collisional transfer contributions, similar to those that appear beyond the dilute limit. The complexity of these terms, and of the kinetic equation itself, compromise the potential of the equation to describe the rich phenomenology observed in this kind of systems. For this reason, a simpler model equation based on the Boltzmann equation is proposed. The model is formulated to keep the main properties of the underlying equation, and it is expected to provide relevant information in more general states than the original equation. As an illustration, the solution describing a macroscopic state with uniform temperature, but a density gradient perpendicular to the plates is considered. This is the equilibrium state for an elastic system, and the inhomogeneous cooling state for the case of inelastic hard spheres. The results are in good agreement with previous results obtained directly from the Boltzmann equation. Keywords: Boltzmann equation, kinetic theory of gases and liquids, transport properties J J Brey etal Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres Printed in the UK 034002 JSMTC6 © 2020 IOP Publishing Ltd and SISSA Medialab srl 2020 20 J. Stat. Mech. JSTAT 1742-5468 10.1088/1742-5468/ab7124 3 Journal of Statistical Mechanics: Theory and Experiment © 2020 IOP Publishing Ltd and SISSA Medialab srl ournal of Statistical Mechanics: J Theory and Experiment IOP 174 2 -5468/ 2 0 / 0 34002+19$ 33 . 00 Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 2 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 Contents 1. Introduction 2 2. Description of the system and balance equations 4 3. Kinetic model 9 4. The inhomogeneous cooling state 11 5. Summary and conclusions 13 Acknowledgments ................................................................................ 14 Appendix A. The macroscopic balance equations ............................... 14 Appendix B. The ICS distribution from the kinetic model equation ......................................................................................... 17 References 18 1. Introduction Fluids in extreme confinement between two parallel plates, have many peculiar properties as compared with the behavior of bulk systems far away from the boundaries. This includes both equilibrium [1–4] and non-equilibrium situations [5, 6] of molecular gases, and also granular gases, i.e. systems composed of macroscopic particles whose interactions are inelastic [7, 8]. For instance, in this latter case, it has been experimentally observed that when energy is continuously injected into the system to compensate the energy dissipation in collisions, the system reaches a spatially homogeneous steady state for a wide range of values of the density. Nevertheless, for certain densities, which depend on the values of the magnitudes characterizing the energy injection, the final state reached by the system exhibits two coexisting phases, one of them having a smaller number of particles density and a larger kinetic energy density than the other [9–12]. Although several eective models have been proposed to describe this phenomenology with more or less success [13–17], and one instability appearing in low density systems has been studied at the phenomenological level [18], a satisfactory explanation, based on a well established description of the dynamics of the system, of what is observed in the experiments is still lacking. One of the main reasons for this is that the form of the macroscopic hydrodynamic equations describing the evolution of the system is unknown. For strongly confined systems, the eect of the physical boundaries occurs not only through the boundary conditions to be imposed to the macroscopic evolution equations, but it is expected to modify also the structure of the transport equations themselves. The changes will be deeper if the dynamics observed by projecting on a plane parallel to the plates is considered, and the interest focusses in formulating macroscopic transport equations for that quasi-two-dimensional dynamics. Kinetic theory provides an intermediate level of description of a system of particles in which the details and basis for macroscopic balance equations have their origins. Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 3 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 To isolate the more important distinguished features of the extreme confinement, the simplest case of inelastic smooth hard spheres at low density is considered here. The bulk macroscopic hydrodynamic equations for a system composed of these particles have been extensively investigated, their structure has been established and explicit expressions for the coecients appearing in them has been obtained. The theoretical predictions following from them agree quite well with molecular dynamics simulations results [19–21]. The starting point for this research was the Boltzmann kinetic equation modified to account for inelastic two-particle collisions. From it, exact balance equations were derived for the mass, momentum, and energy. Then, to close those equations, a modified Chapman–Enskog method was employed to obtain a ‘normal solution’ of the kinetic equation as an expansion in spatial gradients of the macroscopic fields. The analysis is something more complex that in the case of elastic particles, due to the inelasticity of collisions that implies that there is no equilibrium state. Instead, a particular solution describing a homogeneous state in which the temperature decreases monotonically in time (the so-called homogeneous cooling state), was used as the reference state to carry out the expansion. In order to implement a similar program for strongly confined systems, the first issue is to formulate a kinetic equation describing the dynamic of the system. Recently, a Boltzmann-like kinetic equation has been derived for a system of hard spheres confined between two infinite parallel plates at rest, separated a distance smaller that two particles diameters. The equation can be applied, to both elastic [22, 23] and inelastic particles [24, 25]. The only dierence is in the collision rule used to determine the change of the velocities when two particles collide. For elastic systems, the kinetic equation has been proven to obey an H theorem, implying the tendency of any solution to a final equilibrium state. This state coincides with the one predicted by equilibrium statistical mechanics: the velocity distribution is Maxwellian with a uniform temperature and the density profile is inhomogeneous along the direction perpendicular to the plates. In the case of inelastic hard spheres, if no external energy is continuously injected into the system, trivially there is no equilibrium state. Instead there is again a particular state showing a uniform temperature that decreases monotonically in time, similarly to what happens in the homogeneous cooling state. On the other hand, as in the equilibrium state, the density is not uniform along the direction perpendicular to the plates. For this reasons, this state has been referred to as the inhomogeneous cooling state [25] In this work, balance equations for the density, momentum, and energy are derived from the Boltzmann equation for the confined system. This is done in the more general case of inelastic hard spheres, but the results for the elastic case are easily obtained by taking the appropriate limit. As a consequence of the extreme confinement, the pressure tensor and the heat flux have, in addition to the kinetic contributions associated to the motion of the particles, collisional transfer contributions that, in the present case, are due to the restriction on the allowed collisions imposed by the boundaries. An analysis of the structure of the collisional transfer contributions leads to a result that generalizes the ‘contact theorem’ of equilibrium statistical mechanics [26] to arbitrary non-equilibrium states and also to inelastic particles for extreme confinement. Actually, this is a particular case of a completely general property following from the dynamics itself of confined hard spheres [27]. Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 4 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 The complexity of the collision term makes quite dicult to apply the kinetic equation to realistic problems, beyond situations for which a linearization around the reference state (equilibrium or the ICS) can be done. For this reason, it is useful to formulate a kinetic model that permits analytical studies in fully nonlinear problems. A model kinetic equation is obtained by replacing the collision term with a much simpler form that preserves the most important properties. These include normalization and the local conservation laws for the number of particles, momentum, and energy. By preserving the equations, it is meant that, not only the structure of the equations is the same, but also that the expressions giving the pressure tensor and the heat flux, and in the inelastic case the energy sink term, are the same functionals of the distribution function as in the original description. The model proposed here is inspired in the well-known Bhatnagar–Gross–Kook (BGK) model for the usual Boltzmann equation [28, 29]. The BGK model has been extensively used and studied. Its limitations are well established, but also the means to correct them are known. As an illustration, the solution of the model kinetic equations describing the ICS is investigated and the results compared with the description provided by the original Boltzmann equation. The agreement can be considered as quite satisfactory. The remaining of the paper is organized as follows. In the next section, the Boltzmann equation for the confined system of hard spheres is reviewed. The local balance laws are derived from it by taking velocity moments. The collisional transfer contributions to the pressure tensor and the heat flux, due to the extreme confinement, are identified. The model kinetic equation is formulated in section 3, where its relation with the original Boltzmann equation is stressed. As an initial relevant test of the model, it is shown to have a solution describing the same macroscopic ICS as the original equation, namely uniform decreasing temperature and density gradient along the direction perpend icular to the plates. The comparisson of the cooling rates and the density profiles shows a good qualitative and quantitative agreement. The final section of the paper contains a short summary and some analysis of the possible applications of the model. Details of the calculations are provided in the appendices. 2. Description of the system and balance equations The Boltzmann equation, describing the time evolution of the one-particle distribution function, f (r,v,t ) for a dilute gas of inelastic hard spheres of mass m and diameter σ , confined between two large hard parallel plates separated a distance h smaller than twice the diameter of a particle, σ<h<2σ , has the form [22–24] ∂f ∂t +v ·∂f ∂r =J[r,v | f ], (1) with the binary collision term given by J [r,v|f]=σ  dv1  2π 0 dϕ  h−σ/2 σ/2 dz1|g·  σ| × Θ(g · σ)α−2b−1 σ− Θ( − g · σ)  f(r 1z ,v 1 ,t)f(r,v,t ). (2) Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 5 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 Here r is the position vector of components {x,y,z} , r1z≡{x,y,z1} , g≡v1−v , Θ is the Heaviside step function, and b−1 σ is an operator changing all the velocities v and v1 to its right into their pre-collisional values for a collision defined by the unit vector  σ , v ∗ ≡ b−1 σv=v+ 1+α 2α (g · σ)  σ , (3) v ∗ 1 ≡ b−1 σv1=v1 −1+α 2α (g · σ)  σ . (4) In the above expressions, and also in equation (2), α is the coecient of normal restitution, defining the inelasticity of collisions and defined in the interval 0<α⩽1 . The value α=1 corresponds to the limit of elastic collisions. The unit vector  σ is given by  σ≡{sin θsin ϕ, sin θcos ϕ, cos θ} (5) with ϕ being an azimuth angle and cos θ= z 1 −z σ , sin θ⩾ 0. (6) A scheme of the coordinates used to describe the collision of two particles is given in figure 1. The kinetic equation (1) holds for σ/2<z<h−σ/2 and it has to be solved with the appropriate boundary conditions [23]. These conditions guarantee that if the initial one-particle distribution function vanishes outside the system, this property is kept in time by the solution of the kinetic equation. For an arbitrary function φ(v) , the collision term has the useful property  dvφ(v)J[r,v|f]=σ  dv  dv1 h−σ/2 σ/2 dz1 2π 0 dφ|g·  σ|Θ(−g·  σ ) × f(r 1z ,v,t)f(r,v,t)(b σ− 1) φ(v), (7) where bσ is the inverse operator of b−1 σ , i.e. it changes all the velocities v and v1 to its right into the post-collisional values given by v  ≡ bσv=v+ 1+α 2 (g · σ)  σ , (8) v 1 ≡ bσv1=v1 −1+α 2 (g · σ)  σ . (9) Macroscopic density, n(r,t) , velocity, u(r,t) , and granular temperature, T(r,t) , fields are defined in the usual way, n (r,t) ≡ dvf(r,v,t ), (10) n(r,t)u(r,t) ≡ dvvf(r,v,t ), (11) Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 6 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 3 2 n(r,t)T(r,t) ≡m 2 dvV2(r,t)f(r,v,t ). (12) Here, V(r,t)≡v−u,(r,t) is the peculiar velocity of the particle relative to the local flow velocity. Notice that, as it is usual in the literature of granular gases, the Boltzmann constant does not appear in the definition of the granular temperature. This is done to emphasize the absence of any thermodynamic meaning in such definition. From equation (1), balance equations can be derived for the macroscopic fields by taking velocity moments. They have the form ∂n(r,t) ∂t +∂ ∂r· [n(r,t)u(r,t )] = 0, (13) mn (r,t)∂u(r,t) ∂t +mn(r,t)u(r,t) · ∂u(r,t) ∂r +∂ ∂r· P(r,t ) = 0, (14) 3 2n(r,t)∂T(r,t) ∂t +3 2n(r,t)u(r,t)·∂T(r,t) ∂r+P(r,t):∂u(r,t ) ∂r +∂ ∂r· Jq(r,t)= − 1 − α2  ω(r,t). (15) The pressure tensor P(r,t) , and the heat flux J q(r,t ) have both ‘kinetic’ and ‘collisional transfer’ contributions, P(r,t)=P (k) (r,t)+P (c) (r,t), (16) J q(r,t)=J(k) q (r,t)+J(c) q (r,t ). (17) The kinetic contributions are given by Figure 1. Coordinates used for the description of the collision of two hard spheres of diameters σ , confined in a quasi-two-dimensional system by means of two hard parallel plates separated a distance h, σ<h<2σ . Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 7 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 P (k)(r,t)=m  dvV(r,t)V(r,t)f(r,v,t ), (18) J (k) q(r,t)) = m 2 dvV2(r,t)V(r,t)f(r,v,t ). (19) The collisional transfer parts of the fluxes verify ∂ ∂z P(c) zi (r,t)= − m  dvviJ[r,v | f ], (20) ∂ ∂z J(c) q,z(r,t)= −m 2 dvv2J[r,v | f] −∂ ∂z  u(r,t) · P(c)(r,t) − (1 − α2)ω(r,t ). (21) Using these relations, the following explicit expressions can be identified P (c) ij (r,t)=δiz 1+α 4σ2m  dv  dv1  h−σ/2 σ/2 dz1  2π 0 dϕ  1 0 dλλ −1|g·  σλ|2  σλz  σλj ×Θ(−g· σλ)f(r1z,v1,t)f(rλz,v,t), (22) J (c) qi (r,t)=δiz 1+α 4σ2m  dv  dv1  h − σ/2 σ/2 dz1  2π 0 dϕ  1 0 dλλ −1|g·  σλ|2  σλzG·  σ λ ×Θ(−g· σλ)f(r1z,v1,t)f(rλz,v,t). (23) Here, G(r,t) ≡ V(r,t)+V1(r,t) 2 =v+v1 2− u(r,t ), (24) i.e. it is the center of mass velocity of the colliding particles relative to the macroscopic flow. Moreover rλz ≡{x,y,zλ}, (25) z λ ≡ z−(1 −λ)z1 λ, (26) and  σλ≡{sin θλsin ϕ, sin θλcos ϕ, cos θλ}, (27) where cos θλ= z 1 −z λσ , sin θλ=+  1 − cos2θλ . (28) Finally, the source term on the right hand side of the energy balance equation, equation (15), describes the kinetic energy dissipation in collisions due to inelasticity, and it can not be written as the divergence of a flux. Its expression is ω(r,t)=mσ 8  dv  dv1  h−σ/2 σ/2 dz1  2π 0 dϕ|g·  σ|3Θ(−g·  σ)f(r1z,v1,t)f(r,v,t ). (29) Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 8 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 Some details of the derivation of the balance laws and the expressions of the fluxes and the source term are given in appendix A. The collisional transfer contributions to the pressure tensor and the heat flux are due to the delocalization of the centres of the colliding pair of particles and, more specifically, to the dierent values of the two z coordinates. This is a consequence of the confinement of the system that restricts the possible values of the collision vector. The explicit form of this restriction for a particle, depends on the vertical coordinate of the particle and, for this reason, the vertical coordinates of the pair of colliding particle are relevant in describing the dynamics of the system even at low densities. When considering the expressions of the collisional transfer contributions to the pressure tensor and to the heat flow, equations (22) and (23), it must be kept in mind that, as already indicated, for physical initial conditions, the one-particle velocity distribution, f(r,v,t) vanishes for positions outside the system, i.e. for both z < σ/2 and z>h−σ/2 . As a consequence, the eective range of the λ -integration is restricted to values such that σ 2 <z − (1 − λ)z1 λ <h − σ 2. (30) This eliminates the apparent divergence of the integrands for λ=0 , and also guarantees that −1⩽cos θλ⩽1 . The previous comment is closely related with an important general property of a system of hard spheres or disks confined by means of hard walls. Consider z=σ/2 , so that z λ=σ/2 − (1 − λ)z1 λ. (31) For z1=σ/2 , it is zλ(z=σ/2, z1=σ/2) = σ/2 . Besides  ∂zλ ∂z1 z =−1−λ λ , (32) that is negative for 0<λ<1 . It follows that, as z1 increases from σ/2 , zλ decreases from the same vale. Let us stress that this holds for the particular value z=σ/2 . Consequently, zλ always takes values outside the system for z=σ/2 , and f(rλz,v,t)=0 , implying that P(c) ij (r,t )=0 (33) and J(c) q (r,t )=0 (34) for z=σ/2 . A similar argument shows that the collisional transfer contributions to the pressure tensor and the heat flux also vanish at the other plate, i.e. for z=h−σ/2 . The property that the collisional part of the pressure vanishes at a hard wall for a system of hard spheres or disks at equilibrium is known as the ‘contact theorem’ [26]. Recently, it has been extended to arbitrary non-equilibrium states and shown that, actually, all the components of the collisional transfer contribution to the pressure tensor vanish [27]. Here, it has been shown that the same property applies in the case of a strongly confined gas of inelastic hard spheres, and that the collisional transfer contribution to the heat flux also vanishes at the hard boundary. Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 9 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 3. Kinetic model The Boltzmann equation (1) is technically more involved than the usual Boltzmann equation for bulk systems, due to the space dependence of the collision term. For this reason, it is appealing to consider kinetic equations obtained as approximated representations of the original Boltzmann equation, that allow a more detailed analytical study. The basic idea of a kinetic model is to look for the maximum simplicity while retaining the most relevant physical and mathematical properties of the original equation. In the present case, these include the existence of the equilibrium state in the elastic limit and the exact form of the conservation laws. This will be accomplished by considering the collision term J as expanded in a complete set of velocity polynomials with a scalar product weighted with the local equilibrium distribution function. The contribution from the subspace leading to the balance equations for the macroscopic fields is kept exactly, while the rest is approximated by a single relaxation term. Similar models have been successfully formulated for the revised Enskog theory of a non-confined system of hard spheres, both for elastic [30, 31] and dissipative [32] collisions. To derive the kinetic model, it is useful to introduce a Hilbert space of real functions of the velocity v by means of the scalar product  g | h ≡ dvψl(v)g(v)h(v ), (35) where ψl(v) is the Maxwellian defined with the actual values of the local velocity and granular temperature at position r and time t, ψl(v) ≡ m 2πT(r,t)  3/2 exp − mV 2(r,t) 2T(r,t) . (36) This function is related with the usual local equilibrium distribution function fl(r,v,t) by fl(r,v,t)=n(r,t)ψl(v). (37) Notice that both ψl and f l depend on position and time through their functional dependence on the macroscopic fields, although it is not explicitly indicated for the former. In this sense, the Hilbert space is defined for given values of r and t, namely those corresponding to the position and time at which the distribution function will be evaluated. Consider next the set of functions { φβ}≡  1,  m T  1/2 V,  2 3  1/2  mV 2 2T−3 2 . (38) These functions are orthogonal with the scalar product definition in equation (35), φβ|φβ=δβ,β. (39) In the following the two notations {φβ}≡{φ1,φ2,φ5} (40) and Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 16 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 This integral is equal to λ−1 for σ/2<z λ<h−σ/2 and vanishes otherwise. Taking into account that f(r,v,t) also vanishes for z outside this interval, one easily gets equations (20) from (A.10), with the collisional transfer contribution to the pressure tensor given by equation (22). Finally, use of equations (20) into (A.1) directly leads to equation (14). To derive the balance equation for the energy, a similar procedure is followed. The Boltzmann equation is multiplied by mv2/2 and afterwards integrated over the velocity. Using the balance equation for the momentum, equation (14), it is found 3 2n(r,t)∂T(r,t) ∂t +3 2n(r,t)u(r,t)·∂T(r,t) ∂r+P(k)(r,t):∂u(r,t) ∂r +∂ ∂r· J(k) q(r,t) − ui(r,t)∂ ∂z P(c) zi (r,t)=  dvmv2 2 J[r,v | f ], (A.12) with the the kinetic part of the heat flux, J(k) q, given by equation (19). Here, the rule of implicit sum over repeated indexes is used. To compute the integral involving the collision term, the general property given in equation (7) is employed. From the collision rules it follows that v 2 − v2= − 1−α 2 4 (g · σ)2+(1+α)g · σG · σ , (A.13) where G is the center of mass velocity, G≡v+v 1 2. (A.14) In this way it is easily obtained that  dvmv 2 2 J[r,v | f]= − (1 − α2)ω(r,t)+I(r,t ). (A.15) The expression of the source term ω(r,t) is given in equation (29), while I (r,t)=−mσ 2(1 + α)  dv  dv1  h − σ/2 σ/2 dz1  2π 0 dϕ|g·  σ|2G·  σΘ(−g·  σ ) ×f(r1z,v1,t)f(r,v,t). (A.16) By using the same kind of method as for the pressure tensor this can be written as I (r,t)=−mσ2 4(1 + α)∂ ∂z  dv  dv1  h−σ/2 σ/2 dz1  2π 0 dϕ  1 0 dλλ −1|g·  σλ|2G·  σλ  σ λz ×Θ(−g· σλ)f(r1z,v1,t)f(rλ,v,t). (A.17) Now, the peculiar velocity of the center of mass G defined in equation (24) is introduced, and it is a simple task to get I (r,t)= −∂ ∂z  ui(r,t)P(c) zi (r,t) −∂ ∂z J(c) qz (r,t ). (A.18) The expression of the collisional transfer contribution to the heat flux, J (c ) qz is given in equation (23). To get equation (15) in the main text, it only remains to substitute the Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 17 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 above result into equation (A.15), and then use the derived expression for the collision term contribution into equation (A.12). Appendix B. The ICS distribution from the kinetic model equation In this appendix, the derivation of the results reported at the beginning of section 4 will be outlined. For the Maxwellian distribution in equation (51), it is P(k) (z,t)=n(z)T(t)I , (B.1) where I is the unit tensor in three dimensions. The kinetic part of the heat flux, J (k ) q , defined in equation (19), vanishes since the distribution function of the ICS is an even function of the velocity. From equations (20) and (A.2), ∂ ∂z P(c) zi (z,t)=1+α 2mσn(z)  dv  dv1  h−σ/2 σ/2 dz1  2π 0 dϕ|g· σ|2σ i ×Θ(−g· σ)n(z1)ψICS(v,t)ψICS(v1,t) =1+α 4mσn(z)dvdv1h−σ/2 σ/2 dz12π 0 dϕ|g·  σ|2  σ i × n(z 1 )ψ ICS (v,t)ψ ICS (v 1 ,t). (B.2) The velocity integrals are easily carried out to get ∂ ∂z P(c) zi (z,t)=1+α 2σn(z)T(t)  h−σ/2 σ/2 dz1  2π 0 dϕσin(z1) =δizπ(1 + α)n(z)T(t)  h−σ/2 σ/2 dz1(z1−z)n(z1 ). (B.3) Finally, for the collisional transfer contribution to the heat flux, use of the distribution (51) into equation (23) gives J (c) q,z(z,t)=mσ2 8(1 + α)n(z)  dv  dv1  h−σ/2 σ/2 dz1  2π 0 dϕ  1 0 dλ|g·  σλ| 2 ×G· σλσλzψICS(v,t)ψICS(v1,t), (B.4) and this expression vanishes because the integrand is and even function of the velocities. Taking into accout the above results, particularization of equation (50) for fICS(z,v,t) , given in equation (51), yields ∂fICS ∂t +vz ∂fICS ∂z =−ψICS T(t)  vz ∂P(c) z ∂z +  mv2 3T(t)−1  (1 −α2)ω  (B.5) or, equivalently, 3 2T(t)n(z)∂T(t) ∂t  mv2 3T(t)−1  −vz ∂n(z) ∂z =1 T(t)  vz ∂P(c) z ∂z +  mv2 3T(t)−1  (1 −α2)ω . (B.6) Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres 18 https://doi.org/10.1088/1742-5468/ab7124 J. Stat. Mech. (2020) 034002 This equation directly implies equations (53) and (54), just by equating coecients of the same power of the velocity. References [1] Thompson P A, Grest G S and Robbins M O 1992 Phase transitions and universal dynamics in confined films Phys. Rev. Lett. 68 3448 [2] Dietrich M and Haase A 1995 Scattering of x-rays and neutrons at interfaces Phys. 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