Mesoscopic description of the adiabatic piston: kinetic equations and H -theorem
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Noname manuscript No. (will be inserted by the editor)
Mesoscopic description of the adiabatic piston:kinetic equations and H -theorem Nagi Khalil
Received: date / Accepted: date
Abstract
The adiabatic piston problem is solved at the mesoscale using aKinetic Theory approach. The problem is to determine the evolution towardsequilibrium of two gases separated by a wall with only one degree of free-dom (the adiabatic piston). A closed system of equations for the distributionfunctions of the gases conditioned to a position of the piston and the distri-bution function of the piston is derived from the Liouville equation, under theassumption of a generalized molecular chaos. It is shown that the resultingkinetic description has the canonical equilibrium as a steady-state solution.Moreover, the Boltzmann entropy, which includes the motion of the piston,verifies the H -theorem. The results are generalized to any short-ranged repul-sive potentials among particles and include the ideal gas as a limiting case. Keywords
Adiabatic piston · Kinetic theory · Boltzmann equation ·H -theorem PACS · · · Mathematics Subject Classification (2010) · · · Nagi KhalilIFISC (CSIC-UIB), Instituto de F´ısica Interdisciplinar y Sistemas Complejos,Campus Universitat de les Illes Balears, E-07122, Palma de Mallorca, SpainTel.: +34971172008Fax: +34971173248E-mail: nagi@ifisc.uib-csic.es Nagi Khalil
The adiabatic piston problem is an old problem in Thermodynamics and Sta-tistical Mechanics. As described by Callen [1] many years ago, this is to predictthe equilibrium state of a cylinder containing two subsystems (usually twogases) separated by a movable adiabatic piston, i.e. a wall that isolates thetwo subsystems when kept fixed. The same author, and others [2], stated thatThermodynamics can only provide a partial solution to the problem, namelythe equality of pressures.Lieb [3] went one step further by pointing out that a Statistical Mechanicssolution to the problem is in contradiction with Thermodynamics. Namely, asdescribed by Landau and Lifshitz [4] and by Feynmann [5], the relaxation ofthe system, through the motion of the piston, is towards a global equilibrium,where all subsystems including the piston have the same temperature and thepressures of the gasses are equal. But this implies that the initially hotter sub-system decreases its entropy by doing work to the other subsystem, which maybe in contradiction with the Second Principle [6,7,8]. According to Lieb andYngvason [3,9,10] the adiabatic piston problem can be solved by reconsideringthe principle of maximizing the entropy in the absence of constrains.Most works on the adiabatic piston use simple models, such as non-interactingparticles. In this context, rigorous results can be found in the literature. If theinitial condition is of a fixed piston surrounded by two ideal gases in equilib-rium with the same pressures and temperatures, the dynamics of the pistonoccurs in several stages [11]: the initial condition becomes unstable after ashort time and the piston start oscillating [12,13,14], the amplitude of theoscillations attenuates exponentially, and a final thermalization leads to theequilibration of the whole system. For more general initial conditions, wherethe two gases may be in different states, the evolution is in two stages [7,15,16]: a first fast evolution towards mechanical equilibrium, where the two gaseshave the same pressure, is followed by a much slower relaxation towards ther-mal equilibrium, dominated by the energy flux through the piston. Moreover,if the effect of recollisions (when a particle collide with the piston more thanonce) can be removed under some limits, the motion of the piston convergesto an Ornstein-Uhlenbeck process [7,17,18]. See [7] for an attempt to includerecollisions.The last stage of the dynamics, under the condition of mechanical equilib-rium but with the temperatures of the gases still different, has attained muchattention. As a matter of fact, the existence of this stage is closely related tothe original controversy of the adiabatic piston problem. An interesting model,that keeps the system always in this last stage, considers two semiinfinite idealgases in each sides of the piston having the same pressure and temperaturedifference. Even the fact that the net force to the piston is zero, the systemreaches a steady state where the mean velocity of the piston is constant, differ-ent from zero, and towards the hotter gas [19,20,21,22,23,24]. When the massof the piston goes to infinity, its mean velocity vanishes [25,26,27]. See also [28]for a kinetic description of the model in close relation with Thermodynamics. esoscopic description of the adiabatic piston: kinetic equations and H -theorem 3 From a mesoscopic and macroscopic scales perspective, the major challengewhen dealing with the adiabatic piston is to derive closed equations for thepiston and the gases. This has been done rigorously [29,30] and approximately[31,32,33,34,35,36] for the case of non-interacting and similar models. Formore realistic ones, such as models with hard spheres or disks, most of theexisting results are numerical [37,38], or use rude approximations [39]. Asin the case of non-interacting particles, the numerical simulations show anexponential relaxation of the system towards thermal equilibrium [40,41].So far, the numerous theoretical results have not been compared againstexperiments. This is partly because the last stage of the evolution, actuallythe interesting one for the adiabatic piston problem, is too much slow to beobserved in macroscopic system. As far as we know, there is only one exper-imental work that can be relevant for the present discussion [42]. It providesexperimental measurements on the thermalization of a piston surrounded by afew hundreds of grains in a very dense configuration, close to the jamming tran-sition. What we learn from this experiment, and also from the theory done inthe dilute regime [43], is that the equilibration in the inelastic or granular caseis completely different from the classic or elastic one. Specifically, when thereis no energy source supplying the dissipation of the inelastic collisions, each ofthe gases reaches a state close to the homogeneous cooling state [44], and the“thermal” equilibration occurs when the cooling rates (or rates of energy loss)of both gases and the piston are the same. The new “equilibrium” criterion,as opposite of the temperatures being the same for the elastic case, explainsthe emergence of non-equilibrium phase transitions [45,46,47]. See also [43,48,49,50,50,51] for theoretical and numerical studies relevant for experimentalsituations.The works on the adiabatic piston in contact with granular matter suggestthe need to include at least two new ingredients in the study of the (elastic)adiabatic piston. Firstly, if we are interested in making some contact withreality, a systematic derivation of a theory for the piston in contact with a finitenumber of particles is needed. Secondly, the theory should include interactionsbetween particles. The aim of the present work is to propose a mesoscopicdescription of a realistic model of the adiabatic piston. The model considersa large but finite number of interacting particles, and reduces to the case ofnon-interacting particles when the diameters of the particles tend to zero.The organization of the work is as follow. The model is presented in Sec.2 where we also show, by computing some relevant quantities of the systemat thermal equilibrium, the importance of keeping the spatial correlations be-tween the gases and the piston. Section 3 contains the main results of thework, namely the derivation of a closed set of kinetic equations for the gases,conditioned to a position of the piston, and the piston. It is shown that theequations support the equilibrium solution of Sec. 2. As an application to theresults of Sec. 3, in Sec. 4 we demonstrate the H -theorem, which implies theevolution of the system towards thermal equilibrium from any initial condition.Finally, Sec. 5 includes a discussion together with some conclusions. Nagi Khalil d -dimensional cylinder of length L and section S divided intotwo parts by a moving wall of section S , zero width, and mass m p (the piston).The normal direction of the piston remains parallel to the axis of the cylinder,taken as the X -axis. The system contains N i d − dimensional hard spheres withmasses m i and diameter σ i , to the left of the piston for i = 1 and to the rightof the piston for i = 2. See figure 1 for a sketch of the system. Fig. 1
A representation of the system for d = 2. The microstate of the system is given by the set of positions and velocitiesas { r , . . . , r N , x p ; v , . . . , v N , v p } ≡ { R , x p ; V , v p } or the set of positions andmomenta Γ ≡ { r , . . . , r N , x p ; p , . . . , p N , v p } ≡ { R , x p ; P , p p } , where p i = m i v i , p p = m p v p , and N = N + N is the total number of particles. In thelatter expressions, the indexes i = 1 , . . . , N stand for the particles to the leftof the piston and i = N + 1 , . . . , N for the particles to the right.The state of the system changes because of the free motion of the parti-cles and the piston, as well as of instantaneous collisions among particles andbetween particles and the piston. The collisions conserve energy and linear mo-mentum. More specifically, for two particles colliding with velocities v k and v l , their postcollisional velocities, denoted by a prime, are v ′ k = v k − [( v k − v l ) · ˆ σ ] ˆ σ , (1) v ′ l = v l + [( v k − v l ) · ˆ σ ] ˆ σ , (2)where ˆ σ is a unit vector pointing from the center of particle k to the center ofparticle l when in contact. The collision rule for a particle with mass m i andvelocity v and the piston with velocity v p is v ′ x = v x − m p m i + m p ( v x − v p ) , (3) v ′⊥ = v ⊥ , (4) v ′ p = v p − m i m i + m p ( v p − v x ) , (5) esoscopic description of the adiabatic piston: kinetic equations and H -theorem 5 when v x > v p if the particle is on the left ( i = 1) and v x < v p if the particleis on the right ( i = 2). The symbol ⊥ represents the component of the ve-locity normal to the X -axis. An alternative characterization of the dynamics,more suitable when using the phase space representation, is provided by thehamiltonioan H = N X i =1 p i m + N X i = N +1 p i m + p p m p , ( R , x p ) ∈ V , + ∞ , ( R , x p ) / ∈ V , (6)where V is the volume of the phase space accessible to the system (no overlaps).The main focus in this work is on a mesoscopic description of the system.First, we consider the canonical equilibrium and the probability density ofthe phase space ρ ( R , x p ; P , p p ) and related probability densities. Later, a non-equilibrium description based on Kinetic Theory will be proposed and thedistribution functions for the gases and the piston, to be defined later, will beused instead. Many of the results to be obtained at equilibrium, although oftrivial derivation, will deserve as a reference and a guide for a more generalout-of-equilibrium description.2.2 Canonical equilibriumIf the system is big enough or if it is in contact with a thermal bath withtemperature T , i.e. if the walls of the container vibrates in equilibrium withtemperature T , then the probability density of the whole system is given bythe canonical distribution [52] ρ = e − β H h dN +1 Z , (7)where h is the Planck constant, β ≡ k B T with k B the Boltzmann constant, H is the hamiltonian of Eq. (6), and Z the partition function. The latteris difficult to compute in general, due to the exclusion effects. However, for N ( σ + σ ) ≪ L , that is if we neglect the volume of particles, an approximationto be taken along the work, the canonical distribution is well approximated as Z = h − ( dN +1) Z d R dx p d P dp p e − β H ≃ ( LS ) N Lλ dN λ dN λ p Z ds (cid:20)Z s dx (cid:21) N (cid:20)Z s dy (cid:21) N = ( LS ) N Lλ dN λ dN λ p N ! N !( N + 1)! . (8)where λ i ≡ q βh πm i is the thermal length associated to a mass m i , i = 1 , , p . Nagi Khalil
From the probability density we can obtain other relevant quantities, suchas the probability density of a particle to the left ρ , to the right ρ , and ofthe piston ρ p . For the first one, we have ρ ( r , p ) ≡ Z N Y i =2 d r i d p i dx p dp p ρ ( R , x p ; p , p p ) ≃ λ d e − β p m h d LS ( N + 1)! N ! N ! Z x/L ds (cid:20)Z s dx (cid:21) N − (cid:20)Z s dy (cid:21) N = ( N + 1) λ d N h d LS h − I (cid:16) xL , N , N + 1 (cid:17)i e − β p m , (9)where I ( x ; a, b ) is the regularized incomplete beta function. Proceeding anal-ogously, ρ ( r , p ) ≃ ( N + 1) λ d N h d LS I (cid:16) xL , N + 1 , N (cid:17) e − β p m , (10) ρ p ( x, p ) ≃ ( N + 1)! λ p N ! N ! hL (cid:16) xL (cid:17) N (cid:16) − xL (cid:17) N e − β p mp . (11)These expressions describe two non-homogeneous gases and a fluctuating pis-ton, as shown in Fig. 1 for two representative cases. Only for N , N → ∞ (regardless the mass of the piston) are the two gases spatially homogeneous,with the position of the piston fixed (although with a fluctuating velocity).Despite the previous results, the system is spatially homogeneous at equi-librium, namely the probability of finding any particle (including the piston)in a given position is uniform. This can be seen from the density of particles,defined as n i ( r , t ) ≡ N i Z d p ρ i ( r , p , t ) , i = 1 , , (12) n p ( x, t ) ≡ Z dp ρ p ( x, p, t ) , (13)for the particles on the left ( i = 1), the particles on the right ( i = 2), and thepiston. At thermal equilibrium, each of the quantities depends on x , howeverit is easily seen that the global density n is spatially uniform n ( x ) ≡ n ( x ) + n ( x ) + n p ( x ) = N + 1 LS . (14)Similar arguments allow us to show that the pressure is is N +1 LSβ along the X -axis and N +1 LSβ along any other normal direction. The calculation can be madeusing the partition function (then we should change the volume by fixing S inthe first case and fixing L in the second one) or by computing the net flux oflinear momentum across an imaginary ( d − esoscopic description of the adiabatic piston: kinetic equations and H -theorem 7 n /N n p n /N n/N N =5, N =10x/L n /N n p n /N n/N N =20, N =40x/L Fig. 2
Probability densities for particles and the piston at equilibrium for different numberof particles. ρ i ( r , p | x p ) for i = 1 ,
2, are spatially homogenous at equilibrium.Using the definition of the conditional probabilities, and neglecting volumeexclusion effects, we have ρ ( r , p | x p ) ≡ ρ ( r , x p , p ) n p ( x p )= Z x i ≤ x p N − Y i =1 d r i d p i Z x j ≥ x p N Y j = N +1 d r j d p j dp p ρ ( R , x p ; P , p p )= λ d h d Sx p e − β p m , x ≤ x p , (15)and ρ ( r , p | x p ) = λ d h d S ( L − x p ) e − β p m , x ≥ x p . (16)There are two remarkable aspects of the previous results. On the one hand,the conditional distributions characterize the equilibrium state as if the pistonwhere fixed. On the other hand, at thermal equilibrium the piston always“feels” homogeneous gases. The latter implies that we can also compute theequilibrium probability density of the piston as if it where in equilibrium underthe force F ( x p ) exerted by the gases. Namely, at equilibrium, F = Z pxm > ppmp d r ⊥ d p dp p (cid:12)(cid:12)(cid:12)(cid:12) p x m − p p m p (cid:12)(cid:12)(cid:12)(cid:12) ρ (( x p , r ⊥ ) , p | x p ) λ p h e − β p p mp ∆ − p − Z pxm < ppmp d r ⊥ d p dp p (cid:12)(cid:12)(cid:12)(cid:12) p p m p − p x m (cid:12)(cid:12)(cid:12)(cid:12) ρ (( x p , r ⊥ ) , p | x p ) λ p h e − β p mp ∆ + p, (17) Nagi Khalil where ∆ − p = 2 m m p m + m p (cid:18) p x m − p p m p (cid:19) ; ∆ + p = 2 m m p m + m p (cid:18) p p m p − p x m (cid:19) , (18)are the change of the linear momentum of the piston due to collisions with thegases. Operating, F = 2 m m p m + m p N λ d λ p h d +1 x p Z pxm > ppmp d p dp p (cid:18) p x m − p p m p (cid:19) e − β (cid:18) p m + p p mp (cid:19) − m m p m + m p N λ d λ p h d +1 ( L − x p ) Z pxm < ppmp d p dp p (cid:18) p p m p − p x m (cid:19) e − β (cid:18) p m + p p mp (cid:19) = 1 β (cid:20) N x p − N ( L − x p ) (cid:21) . (19)The associated “effective” potential U ( x ) verifies F ( x ) = − dU ( x ) dx and is, up toan additive constant, U ( x ) = − β ln (cid:2) x N ( L − x ) N (cid:3) . (20)Hence, the desired probability density of the piston is ρ p ( x, p ) = 1 Z p exp (cid:20) − β (cid:18) p m p + U ( x ) (cid:19)(cid:21) = x N ( L − x ) N Z p e − β p mp , (21)with Z p a normalization constant. The latter expression of ρ p coincides withEq. (11). As shown in the previous section, even at equilibrium, there are spatial cor-relations between the gases and the piston. These correlations should be keptin order to have an accurate description of a finite, out-of-equilibrium system.Moreover, a easier characterization of the thermal equilibrium of the gases ispossible using the conditional distributions. For these reasons, in this sectionwe derive equations for the conditional distribution functions f i ( r , v , t | x p ) for i = 1 , f p ( x p , v p , t ). esoscopic description of the adiabatic piston: kinetic equations and H -theorem 9 f i ( r , v , t ) are defined such as f i ( r , v , t ) d r d v is the mean number of particles on the left ( i = 1) or on the right ( i = 2) withposition between r and r + d r and velocity between v and v + d v at time t .They are related with the probability densities ρ i ( r , p , t ) as f i ( r , v , t ) = N i m di ρ i ( r , v /m i , t ) , i = 1 , . (22)The distribution function of the piston f p ( x p , v p , t ) is the probability densityof finding the piston with position x p and velocity v p at time t , now f p ( x p , v p , t ) = m p ρ p ( x p , v p /m p , t ) . (23)We define the conditional distribution of the gases as f i ( r , v , t | x p ) ≡ f i ( r , v , x p , t ) n p ( x p , t ) , i = 1 , , (24)where f i ( r , v , x p , t ) is the joint probability distribution for a particle of gas i with position r and velocity v and the piston with position x p , regardless itsvelocity, at time t . It is convenient to express the new probability density interms of the more general joint probability f ip ( r , v , x p , v p , t ) where the velocityof the piston is also specified, f i ( r , v , x p , t ) ≡ Z dv p f ip ( r , v , x p , v p , t ) . (25)The other quantity in Eq. (24) is the density of the piston n p ( x p , t ) alreadydefined in Eq. (13), now given in terms of f p as n p ( x p , t ) = Z dv p f p ( x p , v p , t ) . (26)Finally, it is convenient for forthcoming discussions to define the conditionalprobability distribution of a position and a velocity of the gases to a positionand velocity of the piston as f i ( r , v , t | x p , v p ) ≡ f ip ( r , v , x p , v p , t ) f p ( x p , v p , t ) , i = 1 , . (27)It is forth observing that f i ( r , v , t | x p ) = R f i ( r , v , t | x p , v p ) in general, due tovelocity correlations. An exception is when the system is at thermal equilib-rium. If we were to proceed as usual, we should write down a set of equations for thedistribution functions, of the gases and the piston, from the Liouville equation.This would result in a non-closed set of equations, since higher order distribu-tion function would appear. The difference in the present case with respect toother problems is that the conventional molecular chaos hypothesis does notwork, and new approximations are needed.Take first the kinetic equation for f p ( x p , v p , t ), resulting form integratingthe Liouville equation over all variables except the position and velocity of thesystem, ∂ t f p + v p ∂ x p f p = J p [ v p | f , f p ] + J p [ v p | f , f p ] . (28)It is nothing but a probabilistic balance equation, where the lhs accounts forthe free-of-collision motion of the piston and the rhs accounts for the collisionswith the gases. The latter includes the collision operators, J p = Z d r ⊥ Z v ∗ x >v ∗ p d v ( v ∗ x − v ∗ p ) f p (( x p , r ⊥ ) , v ∗ , x p , v ∗ p , t ) − Z d r ⊥ Z v x >v p d v ( v x − v p ) f p (( x p , r ⊥ ) , v , x p , v p , t ) , (29) J p = Z d r ⊥ Z v ∗ x The usual molecular chaos hypothesis for precollisions is trans-lated in this work to f i (( x p , r ⊥ ) , v , t | x p , v p ) ≃ f i (( x p , r ⊥ ) , v , t | x p ) , (33) esoscopic description of the adiabatic piston: kinetic equations and H -theorem 11 for precollisions. That is, we keep the correlations in the position but removeprecollisional velocity correlations.Observe that the functional dependence of the collision operators are evalu-ated when a particle is at the position of the piston, dissregarding the diameterof the particle. Since we are interested in a mesoscopic description where therelevant spatial variation of the distribution function occur over a mean freepath of the particles, the simplification is expected to be accurate for the gasesdilute enough. To complete the kinetic description, we need equations for the conditionalprobabilities f i ( r , v , t | x p ). For that purpose, and according to the definition ofEq. (24), we need equations for n p ( x p , t ) and f ip ( r , v , x p , v p , t ).Integrating Eq. (28), we get ∂ t n p ( x p , t ) = Z dv p ∂ t f p ( x p , v p , t ) = Z dv p (cid:0) − v p ∂ x p f p + J p + J p (cid:1) = − ∂ x p [ u p ( x p , t ) n p ( x p , t )] , (34)where u p ( x p , t ) = 1 n p ( x p , t ) Z dv p v p f p ( x p , v p , t ) (35)is the mean velocity of the piston. In Eq. (34) we have used the conservationof the number of particles in collisions, Z dv p J p = Z dv p J p = 0 . (36)The kinetic equation for the joint probability f p ( r , v , x p , v p , t ), resultingfrom the Liouville equation after a proper integration, is ∂ t f p = − ( v · ∂ r + v p ∂ x p ) f p + J p, ′ + J p,p ′ + J p,p , (37)where the collision operators take into account the different mechanisms thedistribution function f p can change through collisions. For the operator J p, ′ the collision is between two particles on the left of the piston, for J p,p ′ thecollision is between the piston and a second particle on the left, and J p,p accounts for collisions between the piston and a particle on the right. Thesecollision terms involve three-body distribution functions. There is still one kindof collision to consier, involving a particle with position and velocity given bythe argument of f p and the piston, when they are in contact ( x = x p ) andabout to collide ( v x ≥ v p ). In this case, the usual procedure is to imposea boundary condition to f p , which is a consequence of the conservation ofprobability:( v ′ p − v ′ x ) f p (( x p , r ⊥ ) , v ′ , x p , v ′ p , t ) = ( v x − v p ) f p (( x p , r ⊥ ) , v , x p , v p , t ) , (38) for v x ≥ v p . The primed variables are given by the collision rule in Eqs. (1)-(2). For the gas on the right, an analogous equation and boundary conditionresult.Before proceeding with the derivation of the equation for f i ( r , v , t | x p ), it isworth focusing on some consequences of the boundary condition in Eq. (38).If we integrate it over velocities for v x ≥ v p , after a change of variable andtaking into account that dv x dv p = dv ′ x dv ′ p , we get n p ( x p , t ) n i ( x p , t ) u p ( x p , t ) = n p ( x p , t ) n i ( x p , t ) u i,x ( x p , t ) , i = 1 , , (39)with the mean velocity of the gases u i ( r , t ) ( i = 1 , 2) defined as n i ( r , t ) u i ( r , t ) ≡ Z d v v f i ( r , v , t ) , i = 1 , . (40)For nonzero densities, the velocity of the gases and the piston in contact coin-cide.Integrating Eq. (37) over the velocity of the piston, we have ∂ t Z dv p f p = − v · ∂ r Z dv p f p − ∂ x p Z dv p v p f p + Z dv p J p, ′ , (41)where we have used the fact that the collision operators conserve the numberof particles. Using last equation and Eq. (34) with the temporal derivative ofEq. (24), after some manipulations, we arrive at ∂ t f ( r , v , t | x p ) = f ( r , v , t | x p ) ∂ x p [ u p ( x p , t ) n p ( x p , t )] n p ( x p , t ) − n p ( x p , t ) v · ∂ r Z dv p f p − n p ( x p , t ) ∂ x p Z dv p v p f p + 1 n p ( x p , t ) Z dv p J p, ′ , (42)which is not a closed equation yet. Assumptions 2 and 3. In order to close the equation we include additional ap-proximations that go beyond the standard molecular chaos hypothesis, namely Z dv p v p f p ( r , v , x p , v p , t ) = Z dv p v p f p ( r , v , t | x p , v p ) f p ( x p , v p , t ) ≃ f p ( r , v , t | x p ) Z dv p v p f p ( x p , v p , t )= n p ( x p , t ) u p ( x p , t ) f p ( r , v , t | x p ) , (43)and Z dv p J p, ′ ≃ J Z dv p f p ( r , v , t | x p ) = n p ( x p , t ) J , (44) esoscopic description of the adiabatic piston: kinetic equations and H -theorem 13 with J the collision operator of two particles on the left, which is a functionof v and a functional of f ( r , v , t | x p ). For hard-sphere collisions, the latterapproximation is Z dv p J p, ′ = Z dv p f p ( x p , v p , t ) Z d v d ˆ σ Θ [( v − v ) · ˆ σ ]( v − v ) · ˆ σ × ( b − − f ( r , v , t | x p , v p ) f ( r , v , t | x p , v p ) ≃ Z dv p f p ( x p , v p , t ) Z d v d ˆ σ Θ [( v − v ) · ˆ σ ]( v − v ) · ˆ σ × ( b − − f ( r , v , t | x p ) f ( r , v , t | x p )= n p ( x p , t ) J , (45)where b − is the restitution operator, which replaces the velocities of the col-liding particles by their precollisional values, and J = Z d v d ˆ σ Θ [( v − v ) · ˆ σ ]( v − v ) · ˆ σ × ( b − − f ( r , v , t | x p ) f ( r , v , t | x p ) . (46)Finally, we arrive at a closed equation for f ( r , v , t | x p ), ∂ t f ( r , v , t | x p ) ≃ − v · ∂ r f ( r , v , t | x p ) − u p ( x p , t ) ∂ x p f ( r , v , t | x p ) + J . (47)With similar approximation, we also obtain the kinetic equation for the gason the right of the piston, ∂ t f ( r , v , t | x p ) ≃ − v · ∂ r f ( r , v , t | x p ) − u p ( x p , t ) ∂ x p f ( r , v , t | x p ) + J , (48)with J = Z d v d ˆ σ Θ [( v − v ) · ˆ σ ]( v − v ) · ˆ σ × ( b − − f ( r , v , t | x p ) f ( r , v , t | x p ) . (49)For the boundary conditions, we have to differentiate from particles incontact with the walls of the cylinder, denoted by ∂V , and the piston. In thefirst case, the conservation of probability imposes f i ( r , v , t | x p ) = f i ( r , v ′ , t | x p ) , r ∈ ∂V, i = 1 , , (50)for v · ˆ n = − v ′ · ˆ n , where ˆ n is a unit vector normal to ∂V . Assumption 4. For the boundary conditions at the position of the piston, dif-ferent approximations are possible. We take the one resulting from Eq. (38), af-ter integration over the velocity of the piston and assuming f ( r , v , t | x p , v p ) ≃ f ( r , v , t | x p ) inside the integrals, Z v x >v p dv p ( v ′ p − v ′ x ) f (( x p , r ⊥ ) , v ′ , t | x p ) f p ( x p , v ′ p , t ) ≃ Z v x >v p ( v x − v p ) f (( x p , r ⊥ ) , v , t | x p ) f p ( x p , v p , t ) . (51)and Z v x 1, we could find a regime ofdilute gases where the postcollisional velocity correlations decay to zero on adistance much smaller than the mean free path of the particles, meaning thatthe boundary conditions become correct in the mesoscale. An exact analysisof this considerations needs a formulation with the functions f i ( r , v , t | v p , x p ),which is beyond the scope of the present work.Observe that the structure of the kinetic equations for the conditionaldistributions of the gases is different from the one expected from naive con-siderations, as terms proportional to the mean velocity of the piston appear.Although the presence of these terms can be justified by means of physical considerations, i.e they are important in order to compute the work done bythe piston correctly, they can compromise some key properties of the equa-tion. Namely, for a given physical initial condition, the kinetic equations forthe conditional distribution functions should provide positive and normalizedsolutions for all later times. Both aspects could be indirectly demonstratedfrom the equation of f ip and n p or directly. As an example, we show in thesequel that the kinetic equations (28)-(47)-(48) preserve the normalization ofdistribution functions.Consider first the equation of the density of the piston Eq. (34). Integratingover x p ∈ (0 , L ), using the divergence theorem, and the fact that the pistoncan not leave the system, n p (0 , t ) u p (0 , t ) = n p ( L, t ) u p ( L, t ) = 0, it turns outthat R dx p n p ( x p , t ) is a conserved quantity. Hence, the normalization of f p isensured, provided it is initially normalized.Consider now the gas on the left. After integration of Eq. (47) over v , weface several terms: Z d v ∂ t f ( r , v , t | x p ) = ∂ t n ( r , t | x p ) , (60)with n ( r , t | x p ) ≡ Z d v f ( r , v , t | x p ) (61)being the conditional density; Z d v v · ∂ r f ( r , v , t | x p ) = ∂ r · [ n ( r , t | x p ) u ( r , t | x p )] , (62)with the conditional velocity defined as u ( r , t | x p ) ≡ n ( r , t | x p ) Z d v v f ( r , v , t | x p ); (63) Z d v u p ( x p , t ) ∂ x p f ( r , v , t | x p ) = u p ( x p , t ) ∂ x p n ( r , t | x p ); (64)and Z d v J = 0 . (65)Hence, the balance equation for the conditional density is ∂ t n ( r , t | x p ) = − ∂ r · [ n ( r , t | x p ) u ( r , t | x p )] − u p ( x p , t ) ∂ x p n ( r , t | x p ) , (66)and a similar one for the conditional density of the gas on the right of thepiston n ( r , t | x p ). Integrating over x ∈ (0 , x p ) and all possible values of r ⊥ : ∂ t N ( t | x p ) = − Z d r ⊥ n ( r , t | x p ) u ,x ( r , t | x p ) (cid:12)(cid:12)(cid:12)(cid:12) x = x p − u p ( x p , t ) Z d r ⊥ Z x p dx∂ x p n ( r , t | x p ) , (67) esoscopic description of the adiabatic piston: kinetic equations and H -theorem 17 with N ( t | x p ) ≡ Z d r n ( r , t | x p ) , (68)We have used the condition u ( x ∈ ∂V, t | x p ) · ˆ n = 0 , (69)with ˆ n a unit vector normal to boundary of the cylinder ∂V , which is a directconsequence of the kinetic boundary condition (50). The quantity u ,x is the X component of u ( r , t | x p ). Now, Z x p dx∂ x p n ( r , t | x p ) = ∂ x p N ( t | x p ) − n ( x p , t | x p ) . (70)Hence, ∂ t N ( x p , t ) = − Z d r ⊥ n ( r , t | x p )[ u ,x ( r , t | x p ) − u p ( x p , t )] (cid:12)(cid:12)(cid:12)(cid:12) x = x p − u p ( x p , t ) ∂ x p N ( t | x p ) . (71)The first term on the rhs is zero, using the kinetic boundary condition at thepiston (51), Z v x >v p dv x dv p ( v x − v p ) f ( x p , v x , t | x p ) f p ( x p , v p , t )+ Z v x >v p dv x dv p ( v ′ x − v ′ p ) f ( x p , v ′ x , t | x p ) f p ( x p , v ′ p , t ) ≃ Z dv x dv p ( v x − v p ) f ( x p , v x , t | x p ) f p ( x p , v p , t ) ≃ ⇒ n ( x p , t | x p ) n p ( x p , t )( u ,x − u p ) ≃ ⇒ n ( x p , t | x p )( u ,x − u p ) ≃ , (73)if n p ( x p , t ) = 0. Hence, the balance equation for N results ∂ t N ( t | x p ) = − u p ( x p , t ) ∂ x p N ( t | x p ) . (74)From the nature of this equation we infer that if the conditional distributionfunction for the gas on the left is normalized to N for all allowed values of x p , that is N (0 | x p ) = N , then N ( t | x p ) = N for t ≥ N for all times. H -theorem In this section we demonstrate the H -theorem for the system of kinetic equa-tions derived along the previous section. It states that the H function, to bedefined below, mush approach a limit where the distribution functions are ofthermal equilibrium given by Eqs. (53)-(55). We first prove H is a decreas-ing function, then that its time derivative is zero at thermal equilibrium, andfinally that it is bounded from below by its value at equilibrium.4.1 Definition of H Following a recent work [53], take the H function as H ≡ Z dΓ ρ N ( Γ, t ) ln " ρ N ( Γ, t ) ρ N ρ N ρ p , (75)with ρ N the probability density of the whole system, Γ ≡ ( R , x p ; P , p p ), and ρ i constants that make the expression dimensionless. Removing velocity cor-relations, as usual, it is ρ N ( R , x p ; P , p p , t ) = ρ N ( R ; P , t | p p , x p ) ρ p ( x p , p p , t ) ≃ ρ N ( R ; P , t | x p ) ρ p ( x p , p p , t ) ≃ N Y i =1 ρ ( r i , p i , t | x p ) N Y j =1 ρ ( r j , p j , t | x p ) ρ p ( x p , p p , t ) , (76)and the H -function turns H ≃ N Z d r dx p d p dp p ρ ( r , p , t | x p ) ρ p ( x p , p p , t ) ln (cid:20) ρ ( r , p , t | x p ) ρ (cid:21) + N Z d r dx p d p dp p ρ ( r , p , t | x p ) ρ p ( x p , p p , t ) ln (cid:20) ρ ( r , p , t | x p ) ρ (cid:21) + Z dx p dp p ρ p ( x p , p p , t ) ln (cid:20) ρ p ( x p , p p , t ) ρ p (cid:21) . (77)In terms of the distribution functions, H ≃ Z d r dx p d v dv p f ( r , v , t | x p ) f p ( x p , v p , t ) ln (cid:20) f ( r , v , t | x p ) f (cid:21) + Z d r dx p d v dv p f ( r , v , t | x p ) f p ( x p , v p , t ) ln (cid:20) f ( r , v , t | x p ) f (cid:21) + Z dx p dv p f p ( x p , v p , t ) ln (cid:20) f p ( x p , v p , t ) f p (cid:21) , (78)with the new constants f i having the dimensions of their respective distribu-tion functions. esoscopic description of the adiabatic piston: kinetic equations and H -theorem 19 ddt H ≤ H as given by its last expression. Forthe sake of simplicity on the notation, we omit the obvious dependence of thedifferent functions.The time derivative of the first term of Eq. (78) is ddt Z d r dx p d v f n p ln (cid:18) f f (cid:19) = Z d r dx p d v (cid:20) (cid:18) f f (cid:19)(cid:21) n p ∂ t f + Z d r dx p d v f ln (cid:18) f f (cid:19) ∂ t n p , (79)which is, after using the equations for f and n p (47) and (34), Z d r dx p d v (cid:20) (cid:18) f f (cid:19)(cid:21) n p (cid:0) − v · ∂ r f − u p ∂ x p f + J (cid:1) − Z d r dx p d v f ln (cid:18) f f (cid:19) ∂ x p ( u p n p ) . (80)After some algebra, (cid:20) (cid:18) f f (cid:19)(cid:21) n p ( − v · ∂ r f ) = − v · ∂ r (cid:20) ln (cid:18) f f (cid:19) n p f (cid:21) , (81) (cid:20) (cid:18) f f (cid:19)(cid:21) n p (cid:0) − u p ∂ x p f (cid:1) = − ∂ x p (cid:20) ln (cid:18) f f (cid:19) n p u p f (cid:21) + f ln (cid:18) f f (cid:19) ∂ x p ( u p n p ) , (82)and their respective integrals include − Z d v Z L dx p Z d r ⊥ Z x p dx v · ∂ r (cid:20) ln (cid:18) f f (cid:19) n p f (cid:21) = − Z d v Z L dx p Z d r ⊥ v x (cid:20) ln (cid:18) f f (cid:19) n p f (cid:21) x = x p , (83) − Z d r ⊥ Z L dx p Z x p dx ∂ x p (cid:20) ln (cid:18) f f (cid:19) n p u p f (cid:21) = Z d r (cid:20) ln (cid:18) f f (cid:19) n p u p f (cid:21) x p = x . (84)After some manipulations, we finally arrive at ddt Z d r dx p d v f n p ln (cid:18) f f (cid:19) = Z d r d v ( u p − v x ) ln (cid:18) f f (cid:19) n p f (cid:12)(cid:12)(cid:12)(cid:12) x p = x + Z d r dx p d v ln (cid:18) f f (cid:19) n p J , (85) and similarly ddt Z d r dx p d v f n p ln (cid:18) f f (cid:19) = Z d r d v ( v x − u p ) ln (cid:18) f f (cid:19) n p f (cid:12)(cid:12)(cid:12)(cid:12) x p = x + Z d r dx p d v ln (cid:18) f f (cid:19) n p J . (86)It is still possible to rewrite the terms involving the piston on the last twoexpression in a form which will be useful soon. After some manipulations andthe use of the kinetic boundary conditions (51)-(52), we have Z d r d v ( u p − v x ) ln (cid:18) f f (cid:19) n p f (cid:12)(cid:12)(cid:12)(cid:12) x p = x = Z d r Z v x >v p d v dv p ( v x − v p ) f f p ln (cid:18) f ′ f (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x p = x (87)and Z d r d v ( u x − v p ) ln (cid:18) f f (cid:19) n p f (cid:12)(cid:12)(cid:12)(cid:12) x p = x = Z d r Z v x 0, the expression in Eq. (98) is positive. Asimilar reasoning shows that the second and third terms of Eq. (97) are alsopositive.The last three terms of Eq. (97) are − β (cid:20)Z d r dx p d v dv p ( f ,eq f p,eq − f f p ) m v (100)+ Z d r dx p d v dv p ( f ,eq f p,eq − f f p ) m v + Z dx p dv p ( f p,eq − f p ) m p v p (cid:21) , (101)where use has been made of the form of the equilibrium distribution functionand the fact that Z d r dx p d v dv p ( f i,eq f p,eq − f i f p ) = 0 , i = 1 , , (102) Z dx p dv p ( f p,eq − f p ) = 0 . (103) esoscopic description of the adiabatic piston: kinetic equations and H -theorem 23 Hence, Eq. (100) is − β E eq − E ) = 0 , (104)where E eq is the mean energy at thermal equilibrium, with is the same as themean energy E for any other state, since collisions conserve energy.This way, H ≥ H . In this work, we have proposed a mesoscopic description of a system made ofa cylinder including a finite number of hard spheres in d dimensions dividedby the adiabatic piston. We started by computing some equilibrium propertiesof the system as given by the canonical distribution. The analysis showed thatthe fluctuations of the position of the piston make the probability distributionof the gases to be spatially inhomogenous. However, a direct calculation alsoshowed that the conditional distributions of the gases to a position of the pistonare spatially homogenous, which provide a direct characterization of the globalequilibrium. Furthermore, it can be shown that the conditional distributionsof the gases to a position and a velocity of the piston depend on the positionbut not on the velocity, which reflect the presence of spatial correlation as wellas the absence of velocity correlations at thermal equilibrium.The derivation of the kinetic equations depend on few assumptions. As itis usual in Kinetic Theory, we have used the molecular chaos hypothesis byremoving velocity correlations when two particles or a particle and the pistonare about to collide. In addition, we have also remove some velocity correlationsbetween particles and the piston after a collision. As already discussed, this isexpected to be a good approximation if the piston suffers from many collisionsin a mean free time of the surrounding particles, when the gases are diluteenough. As a result of the approximations, we ended up with a closed systemof equations which solved the adiabatic piston problem, that is to say describesthe evolution of the system towards thermal equilibrium where the two gaseshave the same pressure and all temperatures, including that of the piston, areequal. As a main difference with respect to other theories one may find inthe literature, the one here is not restricted to the thermodynamic limit, norto small particle to piston masses ratio, and include short-ranged collisionsbetween particles.Many of the results of the present work are easily generalized along differentdirections. Even though the kinetic description is for a system of hard spheres,more general interactions among particle can be considered, by replacing thecollision operators J and J with the appropriate ones. Moreover, for mostthe of the collision operators conserving the number of particles, linear mo-mentum, and energy, the H -theorem still holds. More in general, we can alsoinclude dissipation, thermostats, vibrating walls, more pistons as in [55,56],and so on. An interesting aspect not addressed in this work is to precise the conditionsunder which the assumptions are expected to be correct. This would need aformulation including velocity correlations, which are seems to be present forsmall system even at thermal equilibrium [57]. This aspect is left for futureworks. Acknowledgements I dedicate this work to the memory of Mar´ıa Jos´e Ruiz Montero. References 1. H. Callen. Thermodynamics: Physical theories of equilibrium thermodynamics andirreversible thermodynamics (1963)2. A. Curzon, H.S. Leff, American Journal of Physics (4), 385 (1979)3. E.H. Lieb, Physica A: Statistical Mechanics and its Applications (1-4), 491 (1999)4. L.D. Landau, E.M. Lifshitz, Course of theoretical physics (Elsevier, 2013)5. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman lectures on physics, Vol. I: Thenew millennium edition: mainly mechanics, radiation, and heat , vol. 1 (Basic books,2011)6. P. Glansdorff, I. Prigogine, Thermodynamic theory of structure, stability and fluctua-tions , vol. 306 (Wiley-Interscience New York, 1971)7. N.I. Chernov, J. Lebowitz, Y.G. Sinai, Russian Mathematical Surveys (6), 1045 (2002)8. S.R. De Groot, P. Mazur, Non-equilibrium thermodynamics (Courier Corporation, 2013)9. E.H. Lieb, J. Yngvason, in Statistical Mechanics (Springer, 1998), pp. 353–36310. E.H. Lieb, J. Yngvason, Physics Reports (1), 1 (1999)11. N. Chernov, J. Lebowitz, Journal of Statistical Physics (3-4), 507 (2002)12. Y.G. Sinai, Theoretical and Mathematical Physics (1), 1351 (1999)13. A. Neishtadt, Y.G. Sinai, Journal of statistical physics (1-4), 815 (2004)14. P. Wright, Communications in Mathematical Physics (2), 553 (2007)15. C. Gruber, S. Pache, A. Lesne, Journal of statistical physics (5-6), 1177 (2003)16. E. Caglioti, N. Chernov, J. Lebowitz, Nonlinearity (3), 897 (2004)17. R. Holley, Probability Theory and Related Fields (3), 181 (1971)18. D. D¨urr, S. Goldstein, J. Lebowitz, Communications in Mathematical Physics (4),507 (1981)19. C. Gruber, J. Piasecki, Physica A: Statistical Mechanics and its Applications (3-4),412 (1999)20. J. Piasecki, C. Gruber, Physica A: Statistical Mechanics and its Applications (3-4),463 (1999)21. C. Gruber, L. Frachebourg, Physica A: Statistical Mechanics and its Applications (3-4), 392 (1999)22. J. Piasecki, Journal of Statistical Physics (5-6), 1145 (2001)23. N. Chernov, Mathematical Physics Electronic Journal [electronic only] , Paper No.2, 18 p. (2004). URL http://eudml.org/doc/124746 24. M. Itami, S.i. Sasa, Journal of Statistical Physics (1), 37 (2015)25. J. Piasecki, Journal of Physics: Condensed Matter (40), 9265 (2002)26. C. Gruber, S. Pache, Physica A: Statistical Mechanics and its Applications (1-4),345 (2002)27. C. Gruber, S. Pache, A. Lesne, Journal of Statistical Physics (3-4), 669 (2002)28. C. Gruber, S. Pache, A. Lesne, Journal of Statistical Physics (3-4), 739 (2004)29. J. Lebowitz, J. Piasecki, Y. Sinai, Scaling dynamics of a massive piston in an ideal gas (Springer, 2000)30. N. Chernov, J. Lebowitz, Y. Sinai, Journal of Statistical Physics (3-4), 529 (2002)31. B. Crosignani, P. Di Porto, M. Segev, American Journal of Physics (5), 610 (1996)32. C. Gruber, European journal of physics (4), 259 (1999)33. M. Cencini, L. Palatella, S. Pigolotti, A. Vulpiani, Physical Review E (5), 051103(2007)esoscopic description of the adiabatic piston: kinetic equations and H -theorem 2534. E.A. Gislason, American Journal of Physics (10), 995 (2010)35. M.M. Mansour, C. Van den Broeck, E. Kestemont, EPL (Europhysics Letters) (4),510 (2005)36. M.M. Mansour, A.L. Garcia, F. Baras, Physical Review E (1), 016121 (2006)37. P.I. Hurtado, S. Redner, Physical Review E (1), 016136 (2006)38. J. White, F. Roman, A. Gonzalez, S. Velasco, EPL (Europhysics Letters) (4), 479(2002)39. J.J. Brey, N. Khalil, Journal of Statistical Mechanics: Theory and Experiment (11),P11012 (2012)40. E. Kestemont, C. Van den Broeck, M.M. Mansour, EPL (Europhysics Letters) (2),143 (2000)41. M.E. Foulaadvand, M.M. Shafiee, EPL (Europhysics Letters) (3), 30002 (2013)42. F. Lechenault, K.E. Daniels, Soft Matter (13), 3074 (2010)43. J.J. Brey, N. Khalil, EPL (Europhysics Letters) (1), 14003 (2011)44. N. Khalil, Journal of Statistical Mechanics: Theory and Experiment (4), 043210(2018)45. R. Brito, M. Renne, C. Van den Broeck, EPL (Europhysics Letters) (1), 29 (2005)46. P.I. Hurtado, S. Redner, Physical Review E (1), 016137 (2006)47. J.J. Brey, N. Khalil, Physical Review E (5), 051301 (2010)48. J. Brey, M. Ruiz-Montero, EPL (Europhysics Letters) (6), 805 (2004)49. J.J. Brey, M. Ruiz-Montero, Journal of Statistical Mechanics: Theory and Experiment (09), L09002 (2008)50. J.J. Brey, M. Ruiz-Montero, Physical Review E (3), 031305 (2009)51. J.J. Brey, M. Ruiz-Montero, Physical Review E (2), 021304 (2010)52. R. Pathria, P.D. Beale, Statistical Mechanics (Elsevier, 2011)53. P. Maynar, M.G. de Soria, J.J. Brey, Journal of Statistical Physics (5), 999 (2018)54. J.A. McLennan, Introduction to nonequilibrium statistical mechanics (Prentice Hall,1989)55. R. Brito, Granular Matter (2), 133 (2012)56. L. Caprini, L. Cerino, A. Sarracino, A. Vulpiani, Entropy (7), 350 (2017)57. L. Cerino, G. Gradenigo, A. Sarracino, D. Villamaina, A. Vulpiani, Physical Review E89