Driven inelastic Maxwell gas in one dimension
V. V. Prasad, Sanjib Sabhapandit, Abhishek Dhar, Onuttom Narayan
DDriven inelastic Maxwell gas in one dimension
V. V. Prasad, Sanjib Sabhapandit, Abhishek Dhar, and Onuttom Narayan The Institute of Mathematical Sciences, Taramani, Chennai - 600113, India Raman Research Institute, Bangalore - 560080, India International centre for theoretical sciences, TIFR, Bangalore - 560012, India University of California, Santa Cruz, California 95064, USA (Dated: October 10, 2018)A lattice version of the driven inelastic Maxwell gas is studied in one dimension with periodic boundaryconditions. Each site i of the lattice is assigned with a scalar ‘velocity’, v i . Nearest neighbors on the latticeinteract, with a rate τ − c , according to an inelastic collision rule. External driving, occurring with a rate τ − w ,sustains a steady state in the system. A set of closed coupled equations for the evolution of the variance andthe two-point correlation is found. Steady state values of the variance, as well as spatial correlation functions,are calculated. It is shown exactly that the correlation function decays exponentially with distance, and thecorrelation length for a large system is determined. Furthermore, the spatio-temporal correlation C ( x , t ) = (cid:104) v i ( ) v i + x ( t ) (cid:105) can also be obtained. We find that there is an interior region − x ∗ < x < x ∗ , where C ( x , t ) hasa time-dependent form, whereas in the exterior region | x | > x ∗ , the correlation function remains the same asthe initial form. C ( x , t ) exhibits second order discontinuity at the transition points x = ± x ∗ and these transitionpoints move away from the x = PACS numbers: 45.70.-n, 47.70.Nd, 05.20.Dd
I. INTRODUCTION
It is well-known that for a system of interacting particlesin thermal equilibrium, the velocities of different particles arecompletely uncorrelated and the joint distribution of the ve-locities is given by the product of independent single particleMaxwell distributions. On the other hand, when a system isdriven out-of-equilibrium, for example through application ofa temperature gradient, non-zero correlations can build up be-tween the velocities of particles [1]. An important class ofnon-equilibrium systems is driven dissipative systems . An ex-ample of a dissipative system is granular gas, which, in theabsence of an external supply of energy, loses energy con-tinuously due to inelastic collisions. In the presence of ex-ternal driving, for example in vibrated granular systems, onecan obtain non-trivial steady states [2–8]. A signature of non-equilibrium in this system is that the single-particle velocitydistribution is no longer Maxwellian. It is thus interesting toask about the nature of correlations amongst the velocities inthis system. We investigate this question in a simple latticemodel of an inelastic gas in one dimension. We calculate theexact form of the spatial correlation function of velocity forthis model in its driven steady state.The presence of correlations in granular gases has beenobserved in unforced [9–12] as well as forced granulargases [13–20]. Different models studying unforced granu-lar gasses observed power-law behavior in the spatial cor-relation functions [9–11]. In an early numerical study of aone-dimensional granular gas, driven by uncorrelated whitenoise, Williams and Mackintosh [13] observed for the densitycorrelation function, a power-law behaviour when the inelas-ticity is large. An analytical study [15] of a similar systemof inelastic gas also found long-range correlations in densityand velocity in the large- N limit, for finite inelasticities. Hy-drodynamic analysis of inelastic hard-sphere systems drivenby white noise [16] proposed correlations with logarithmic and power-law ( / x ) form, respectively, for two and three di-mensions, which agreed with simulations in the near elasticregimes. In an experimental study of a granular gas on aninclined plane and driven by a vibrating wall at the bottom,Blair and Kudrolli [17] also observed a power-law decay inthe steady-state velocity correlations with the exponent rang-ing from 1 . et al. [18]found an exponential decay in the spatial correlation of thevelocities of the particles. The authors argued that the differ-ence between their results and the previous ones was due tothe different driving schemes used. In particular, the drivingin the analytical studies was modeled as diffusive driving , withthe rate of change of velocity due to driving equated to uncor-related white noise. However, the authors in [18] argue thatthe driving from the wall should also be treated as inelasticmomentum-nonconserving collisions, which suppresses long-range correlations. To account for the different dissipationmechanisms, Gradenigo et al . [19] considered driving witha phenomenological viscous term, in addition to the whitenoise. Assuming the separation of time-scales between thecollisions and driving, they obtained an exponential form forthe velocity correlations that agreed with the experimental ob-servations. In the present work, considering a specific modelof a dissipative gas, we try to understand the correlations in thecase in which one does not have a time-scale separation. Also,unlike the previous models in which the driving is done by anOrnstein-Uhlenbeck noise (driving with the viscous term), weconsider driving by wall-like collisions, that is motivated bythe experimental systems.The system in which we are interested is an inelastic gasliving on a one-dimensional lattice. In the model, a scalar ve-locity is ascribed to each lattice point. The velocities at eachpoint change as they interact, according to the rules of inelas-tic collisions. As in one-dimensional (1D) models of granular a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r gas with nearest-neighbor collisions, here the interactions areamong the nearest-neighbor points on the lattice. The modelhas been effective in describing the various qualitative fea-tures of cooling 1D granular gases, such as long-range corre-lations and the appearance of shocks in the system [10]. Themodel has also been of recent interest, in developing a hy-drodynamic description of granular fluids in cooling [20, 21]as well as boundary-driven steady states [22]. In the drivenmodel presented here, in addition to the inelastic collision be-tween nearest neighbors, each site has independent externaldriving.Considering any nearest-neighbor interaction occurringwith equal rates, we derive an exact set of coupled equationsfor the evolution of the variance of the single-particle distri-bution and the correlation functions for the system. Such aclosure has been observed before, for a system of Maxwellgas [23], where spatial correlations were ignored. The set ofequations allows one to characterize the steady-state proper-ties for a driven system. For instance, the coupled relationscan be used to find out whether the system goes to a steadystate or not for various values of the parameters in the drivensystem. One of our main results is the exact functional be-havior of the spatial correlation function of the velocity field,which shows an exponential decay at large distances. We alsoobtain the spatio-temporal correlation function, and we findthat it shows a second-order discontinuity.Similar models have been studied before [24–27] in thecontext of granular gases as well as in the broader context ofdriven dissipative systems. In these studies, each site has anenergy instead of a momentum variable associated with it. In-elastic collisions are represented in the model by changing theenergy of a randomly chosen particle to a fraction of the sumof its energy and that of any of its nearest neighbors. In ad-dition, there is dissipation and drive from a reservoir at eachsite or at the boundary. In the model considered here, onehas pairwise momentum-conserving and energy-dissipativeexchanges between neighboring particles, and it represents asomewhat more natural extension of the Maxwell model toincorporate spatial correlations [10, 20–22].The outline of the paper is as follows. First, in Sec. II weintroduce the model of Maxwell-like gas on a lattice with therules of interaction and driving. The time evolution of the ve-locity distribution involves a hierarchy of equations as seen inthe kinetic theory of granular gases. Later in Sec. III, an exactevolution of the variance and two-point correlation functionsis calculated for the system. This helps us to characterize thetime evolution of the system. In Sec. IV, we derive an ex-act formula for the steady-state variance and the equal-timecorrelation between the velocity variables at different sites.Using this, one obtains an asymptotic functional form for thecorrelation functions for a large system. We also show the ex-tension of the above model where a collision between a pairoccurs only when the left particle has a larger velocity thanthe right one, which mimics the real systems. Since this isdifficult to solve analytically, we use direct simulation resultsto compare it with the model without such a constraint. As forthe equal-time correlations, a set of equations for the spatio-temporal correlations are calculated in Sec. V. We summarize our results in Sec. VI. The details of some of the analysis aregiven in the Appendix. II. THE MODEL
We consider a one-dimensional lattice of N sites ( i = , , . . . , N ) with periodic boundary conditions ( N + i ≡ i ).Each lattice site i is associated with a real scalar variable v i ,which one calls the ‘velocity’. It should be kept in mind thatthis velocity does not correspond to any motion in the sys-tem. The system evolves in time t as follows: each nearest-neighbor pair ( i , i + ) interacts with each other with a rate τ − c according to the inelastic collision rule v i = ε v ∗ i + ( − ε ) v ∗ i + , v i + = ( − ε ) v ∗ i + ε v ∗ i + , (1)where, ( v ∗ i , v ∗ i + ) and ( v i , v i + ) respectively are the pre-collision and post-collision velocities of the two interactingparticles. Here ε = ( − r ) /
2, with r being the coefficient ofrestitution. For r = r < r ∈ ( , ) , one may consider the entire range r ∈ ( − , ) as awell-defined mathematical model of a dissipative gas.In addition to the binary inter-particle interaction, each par-ticle is driven with a rate τ − w according to v i = − r w v ∗ i + η , (2)where r w is the coefficient of restitution of the wall particlecollision with η taken to be Gaussian noise with variance σ and zero mean, acting up on each particle independently anduncorrelated in time. The above driving is motivated from thecollisions of the particle with a vibrating wall. The veloci-ties of the particle v ∗ i and the vibrating wall V ∗ w upon collisionchanges to new velocities v i and V w respectively which satisfya relation ( v i − V w ) = − r w ( v ∗ i − V ∗ w ) . Considering a massivewall so that V w ≈ V ∗ w , one can obtain Eq. (2) by substituting ( + r w ) V w by a random noise η . As explained before, fora Maxwell gas it is useful to extend the driving Eq. (2) fornegative values of r w such that r w ∈ [ − , ] .Note that r w = − V w → ∞ whilekeeping η = ( + r w ) V w finite] corresponds to the addition ofGaussian white noise [2, 13], which breaks the conservationof momentum of the system, unlike the inelastic interparticlecollisions. However, this causes an overall diffusion of thecenter of mass of the system and results in the energy of thesystem increasing linearly with time [23]. This was noted in[28], where the authors add additional terms in their drivingmechanism to ensure conservation of momentum.For − < r w ≤
1, the system reaches a non-trivial steadystate [23]. Note that 0 < r w ≤ III. EQUAL-TIME CORRELATIONS
Let us define the equal time correlations Σ i , j ( t ) = (cid:104) v i ( t ) v j ( t ) (cid:105) . To get the equation for the time evolution of Σ i , j ( t ) , we follow standard procedures [29] to use Eqs. (1,2)and average over all possible events occurring between times t and t + dt . In the limit dt → d Σ i , j dt = (cid:104) a ∆ − b (cid:105) Σ i , j , for | i − j | > d Σ i , i + dt = − [( + ε ) a + b ] Σ i , i + + a [ Σ i − , i + + Σ i , i + ]+ a ε [ Σ i , i + Σ i + , i + ] , d Σ i , i dt = [ − a ( + ε ) − b ( − r w )] Σ i , i + a ( − ε ) [ Σ i − , i − + Σ i + , i + ]+ ε a [ Σ i , i − + Σ i , i + ] + C , (3) where C = σ / τ w , a = ( − ε ) / τ c and b = ( + r w ) / τ w , (4)with b , a > b → r → α , L − → τ − c , Σ i + k , i → C k , and making thecorrection ( − α ) → ( − α ) / ∆ is the discrete two-dimensional Laplacian operatordefined by ∆ Σ i , j = Σ i + , j + Σ i − , j + Σ i , j + + Σ i , j − − Σ i , j .We note that Σ i , j = Σ j , i . We now consider translationally in-variant initial conditions such that Σ i , j ( t ) = Σ ( | i − j | , t ) . Wethen get ddt Z ( t ) = − A Z ( t ) + C (5)where Z ( t ) = [ Σ ( , t ) , Σ ( , t ) , .. Σ ( n , t )] T , n = N / ( N + ) / N even and odd, and the matrix A is an ( n + ) × ( n + ) tri-diagonal matrix of the form, A = [ ε a + b ( − r w )] − ε a − ε a [( + ε ) a + b ] − a − a ( a + b ) − a ... ... ... − a ( a + b ) − a − a ( a + b ) . (6) and the column vector C has ( n + ) -dimensions with the onlynon-zero element C = σ / τ w . The set of equations Eq. (3)can be derived alternatively from the BBGKY hierarchy forthe distributions, as explained in Appendix A).The evolution of Z ( t ) can be exactly calculated from Eq. (5)which is shown in Fig. 1 along with the numerical simulation.One can also consider a Maxwell gas with the rate which de-pends on the average kinetic energy of the system. However,the steady-state properties in both cases follow the same statis-tics. Further, one can extend the lattice model in the followingway. Instead of allowing the interaction (Eq. (1)) to occurwith a global rate, one can consider it to occur between thechosen nearest-neighboring pair only if their relative velocity( v i − v i + ), is positive. The condition, which is referred to askinematic constraint [10, 30], prevents collision if the veloci-ties correspond to a “receding” pair. We have not been able toobtain a closed set of equations for this system. One can ob-tain the evolution of the correlations from direct simulation,and this is plotted in Fig. 1. One finds that the behaviour ofthe system with the kinematic constraint is different from thatwithout the constraint. IV. STEADY STATE PROPERTIES
It suffices to know the eigenvalues of A to see whether thesystem goes to a steady state or not. Consider the special case t -3 -2 -1 Σ ( d , t ) Σ ( ) Σ ( ) Σ ( ) Σ ( ) FIG. 1. The figure shows the evolution of Σ ( x , t ) for x=0,1,2,3 for a10 particle system with r = / , r w = / σ = τ c = τ w =
1. Thetriangles depict the same system with the constraint that only thosepairs with positive relative velocity will collide. of r w = −
1, where the matrix has a simpler form with b = r w = − A vanishes, and so, no steady state exists (see Appendix B 1).On the other hand for r w (cid:54) = − -8 -6 -4 -2 0 2 4 6 8 v -10 -8 -6 -4 -2 P ( v ) SimulationGaussian
FIG. 2. The velocity distribution of a 50 particle system with r = / , r w = / σ = τ c = τ w =
1. The solid line shows theGaussian with variance calculated for the system. One can see thedeviation from Gaussian. with the left-hand side equated to zero. The elements of Z ss ,the steady-state correlation vector, Σ ss ( x ) = Σ ( x , t → ∞ ) , areobtained from, Z ss = A − C . (7)Here x ≡ | i − j | , denotes the separation between lattice points,which takes integer values. Only the first column of the matrix A − suffices to calculate all the elements as, Σ ss ( x ) = A − x σ / τ w . (8)Calculation of A − x is easy due to the tri-diagonal nature of A − . The explicit formula for x = A − = a n det A (cid:110) [ c − ( − ε )] (cid:104) ( s n − + s − ( n − ) (cid:105) − (cid:104) s [ n − ] + s − [ n − ] (cid:105)(cid:111) , (9)for x = , , .. n : A − x = ε a n det A (cid:104) s n − x + s − ( n − x ) (cid:105) , (10)where c ≡ ( + b / a ) and s ≡ ( c + (cid:112) c − ) . (11)As b and a takes positive values, c and s will always be greaterthan or equal to 1 (equal to 1 when r w = − A , denoted as det A has the formdet A = a n + (cid:110) K (cid:104) s n − + s − ( n − ) (cid:105) − K (cid:104) s n − + s − ( n − ) (cid:105)(cid:111) , (12)where K , K are functions of ( ε , c , r w ) given by: K = ε + ( c − )[ ε + ( − r w )( + ε )] + ( c − ) ( − r w ) , K = ε + ( − r w )( c − ) . (13) For a large system, one can calculate the asymptotic form ofthe correlation function Σ ss ( x ) . To do this, let us rearrangeEq. (10) to obtain A − x = ε a n s n det A (cid:104) ( s − x + s − ( n − x ) ) (cid:105) . (14)As s >
1, in the large n limit the Eq. (14) becomes, A − x = ε a n s n det A (cid:2) s − x (cid:3) . (15)Similarly, from Eq. (12), for large n , det A can be shown tohave the form,det A = a ( n + ) s n (cid:2) K cs − − K s − (cid:3) . (16)Thus in large n limit, Σ ss ( x ) has the following form: Σ ssd = B exp ( − x ln s ) , (17a) B = ε s ( − ε )( τ w τ c ) ( K − K ) . (17b)This shows that the system has a finite correlation length ξ = / ln s . In Fig. 3 we plot the asymptotic form (Eq. (17))along with the numerical (Eq. (8)) and simulation results. Byexpanding ln s near s =
1, one can see that the correlationlength ξ diverges as 1 / (cid:112) ( + r w ) when r w approaches − Σ ss ( x ) for a system with the constraint is obtained fromsimulation and is plotted in Fig. 3. The correlation in this caseis not the same as that of the model without the constraint. Asit is difficult to obtain Σ ss ( x ) for higher x values from simula-tions, the characteristics of the function are not clear. V. TWO-TIME CORRELATIONS
By proceeding as in the equal time case in Sec. III, it is easyto obtain the equations of motion for the time-dependent cor-relation functions defined by C i , j ( t ) = (cid:104) v i ( t ) v j ( ) (cid:105) , where theaverage is over the dynamics. The translation invariance of thesystem means that C i j ( t ) = C ( i − j , t ) . We get the followingequation for C ( x , t ) . dC ( x , t ) dt = (cid:104) a ∆ − b (cid:105) C ( x , t ) , (18)where ∆ C ( x , t ) = C ( x + , t ) − C ( x , t ) + C ( x − , t ) . Takingthe limit N → ∞ and defining the Fourier transform (cid:101) C ( q , t ) = ∑ x e iqx C ( x , t ) , d -8 Σ d ss analytical- asymptotic simulation N=50exact N=50simulations N=10exact N=10with constraint N=20 FIG. 3. Steady-state values of Σ ss ( x ) for the simulation of 10 and50-particle systems with r = / , r w = / σ = τ c = τ w =
1. The rate of collision is independent of the variance. The exactanalytical results, given by Eq. (8), are shown by the ‘ + ’ symbol for ( N = ) and ‘ × ’ for N =
10. The asymptotic expression Eq. (17)is represented by the solid green line. The triangles show simulationresults for the case in which particles collide only when their relativevelocity is positive. we get the following solution (cid:101) C ( q , t ) = exp [ − ( b + a ( − cos q )) t ] (cid:101) C ( q , t = ) , (19)where (cid:101) C ( q , t = ) = ∑ x e iqx C ( x , t = ) . (20)From Eq. (17) we have C ( x , t = ) = B exp ( −| x | / ξ ) , whichgives (cid:101) C ( q , t = ) = B s − s + − s cos q . (21)Therefore, the two-time correlation function can be obtainedas C ( x , t ) = π (cid:90) π − π (cid:101) C ( q , t ) e − iqx dq = B e − bt C ( x , t ) , (22)where C ( x , t ) is given by C ( x = (cid:96) at , t ) = ( s − ) π (cid:90) π − π exp (cid:0) − (cid:2) ( − cos q ) + iq (cid:96) (cid:3) at (cid:1) s + − s cos q dq . (23)It immediately follows from the above integral that C ( − x , t ) = C ( x , t ) . Therefore, in the following, we considerthe case x ≥
0. For large t , the above integral can be evaluatedby saddle point method, which suggests the form C ( x = (cid:96) at , t ) ∼ e − atI ( (cid:96) ) . (24) The saddle point is given by q ∗ = − i ln (cid:104) (cid:96) + (cid:112) + (cid:96) (cid:105) , (25)which lies on the negative imaginary q axis. However, beforeproceeding with the saddle-point calculation, we note that theintegrand has a simple pole on the negative imaginary q axisat q = − i ln s (there is also another one at + i ln s which donot interfere with the saddle point calculation). Now, for (cid:96) < ( s − ) / ( s ) the saddle point lies between the origin and q .Therefore, the contour of integration can be taken through thesaddle point without crossing the pole. On the other hand,for (cid:96) > ( s − ) / ( s ) , the pole lies between the origin and thesaddle point. Therefore, in this case the dominant contributionto the integral comes from the pole. Thus the function I ( (cid:96) ) isgiven by I ( (cid:96) ) = (cid:40) I ( (cid:96) ) for (cid:96) < (cid:96) ∗ I ( (cid:96) ) for (cid:96) > (cid:96) ∗ (26)where (cid:96) ∗ = ( s − ) / ( s ) , and I ( (cid:96) ) = ( − cos q ∗ ) + iq ∗ (cid:96) (27) = (cid:16) − (cid:112) + (cid:96) (cid:17) + (cid:96) ln (cid:104) (cid:96) + (cid:112) + (cid:96) (cid:105) , (28)and I ( (cid:96) ) = ( − cos q ) + iq (cid:96) (29) = − ( b / a ) + (cid:96) ln s , (30)where we have used the simplification ( s − ) / ( s ) = ( b / a ) .It is easy to check that I ( (cid:96) ) has a second order discontinuityat (cid:96) = (cid:96) ∗ , that is, I ( (cid:96) ∗ ) = I ( (cid:96) ∗ ) and I (cid:48) ( (cid:96) ∗ ) = I (cid:48) ( (cid:96) ∗ ) whereas I (cid:48)(cid:48) ( (cid:96) ∗ ) (cid:54) = I (cid:48)(cid:48) ( (cid:96) ∗ ) . It is interesting to note that, similar disconti-nuities of the rate function have been found recently in variousother contexts [31–34]. It follows from, Eqs. (22), (24), and(30), that for | x | > (cid:96) ∗ t , we have C ( x , t ) ∼ B e −| x | / ξ = C ( x , t = ) . (31)Therefore, while for | x | < (cid:96) ∗ t , the correlation function de-pends on time, for | x | > (cid:96) ∗ t , it still retains the initial form.Such dynamical transition has been found recently in a differ-ent context [34]. The physical reason is that in both of thesesystems, disturbances take a finite time to propagate from onepoint to another.Finally, following the method used in Ref. [32], we can alsowrite down a more complete asymptotic form of C ( x , t ) forlarge t as, C ( x = (cid:96) at , t ) ≈ e − atI ( (cid:96) ) √ π at ( s − )( + (cid:96) ) / (cid:16) s + − s √ + (cid:96) (cid:17) + sgn ( (cid:96) − (cid:96) ∗ ) (cid:113) (cid:2) I ( (cid:96) ) − I ( (cid:96) ) (cid:3) + e − atI ( (cid:96) ) (cid:20) θ ( (cid:96) − (cid:96) ∗ ) −
12 sgn ( (cid:96) − (cid:96) ∗ ) erfc (cid:113) at (cid:2) I ( (cid:96) ) − I ( (cid:96) ) (cid:3)(cid:21) , (32) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - � - � � � ��� - �� �� - � �� - � �� - � �� - � �� - � ℓ � � ( � ℓ � � � � ) FIG. 4. The points are obtained by numerically integratingEq. (23), whereas the solid line represents the analytical form givenby Eq. (32). The parameters used are τ c = τ w = r = r w = / t =
10. These correspond to a = b = / s = + √
3. Thevertical dashed lines plot the location of ± (cid:96) ∗ where (cid:96) ∗ = √ where I ( (cid:96) ) and I ( (cid:96) ) are given by Eqs. (28) and (30) respec-tively.Figure 4 compares the above result with the exact C ( x , t ) obtained by numerically integrating Eq. (23) and finds perfectagreement between the two.As a special case, we find for large t the form C ( , t ) ≈ B ( s + ) e − bt ( s − ) √ π at . (33)Thus there is an exponential decay as a function of time witha 1 / √ t prefactor. VI. CONCLUSION
In this work, we studied a simple model for driven inelasticgas in one dimension for which we find the equal-time spatial velocity correlation functions as well as two-time correlationfunctions in the steady state. The equal-time correlations de-cay exponentially in space. An interesting finding is that thereexists a velocity l ∗ a such that the decay of correlations doesnot propagate beyond a distance | x | = l ∗ at , which leads to sec-ond order dynamical transition in the spatio-temporal correla-tion function. Such transitions have never been discussed inthe context of granular physics, and therefore, this study opensup a new direction of research in granular physics. Hopefully,in future experiments, such transitions could be observed inreal granular systems.We also obtain the condition for the existence of a steadystate for the model. Experimental studies on granular gasesdriven by wall collisions, have found an exponential decay forthe spatial correlation functions of velocity [18, 19]. Simplebut exact models such as the one introduced here may facil-itate a better understanding of the observed features. It willbe interesting to study the nature of correlations in other mod-els of granular systems with different interactions and drivingmechanisms. ACKNOWLEDGMENTS
This research was supported in part by the InternationalCentre for Theoretical Sciences (ICTS) during a visit of V.V.P.and O.N. for participating in the program “Non-equilibriumstatistical physics” (Code:ICTS/Prog-NESP/2015/10).
Appendix A: BBGKY hierarchy
Here we show that the equations for the correlations Eq. (3)can also be derived by starting from the BBGKY hierarchy forthe distribution functions. Let P ( v i , t ) be the 1-point probabil-ity distribution function for the site i to have the velocity vari-able v i at time t . Similarly P ( v i , v i + x , t ) be the 2-site probabil-ity distribution function for the sites i , i + x to have velocities v i , v i + x at time t . Similarly defined is the 3-site probabilitydistribution function P ( v i − m , v i , v i + x ) ( { m , x } are integers lessthan N ). For the dynamics in Eqs. (1,2), one can immediatelywrite a set of evolution equation for the distributions as, ∂∂ t P ( v i , t ) = τ − c (cid:20) (cid:90) dv i + T ( v i , v i + ) P ( v i , v i + , t ) + T ( v i − , v i ) P ( v i − , v i , t ) (cid:21) + τ − w (cid:20) (cid:90) dv ∗ i P ( v ∗ i , t ) (cid:104) δ ( v i − [ − r w v ∗ i + η i ]) (cid:105) η i − P ( v i , t ) (cid:21) , (A1a) ∂∂ t P ( v i , v i + x , t ) = τ − c (cid:26) T ( v i , v i + x ) P ( v i , v i + x , t ) δ x , + (cid:90) dv i − T ( v i − , v i ) P ( v i − , v i , v i + x , t )+ (cid:20) (cid:90) dv i + T ( v i , v i + ) P ( v i , v i + , v i + x , t ) + (cid:90) dv i + x − T ( v i + x − , v i + x ) P ( v i , v i + x − , v i + x , t ) (cid:21) ( − δ x , )+ (cid:90) dv i + x + T ( v i + x , v i + x + ) P ( v i , v i + x , v i + x + , t ) (cid:27) + τ − w (cid:20) (cid:90) dv ∗ i P ( v ∗ i , v i + x , t ) (cid:104) δ ( v i − [ − r w v ∗ i + η i ]) (cid:105) η i + (cid:90) dv ∗ i + x P ( v i , v ∗ i + x , t ) (cid:104) δ (cid:0) v i + x − [ − r w v ∗ i + x + η i + x ] (cid:1) (cid:105) η i + x − P ( v i , v i + x , t ) (cid:21) . (A1b)and so on. Here, T ( v i , v j ) defined as, T ( v i , v j ) S ( v i , v j ) = r − S ( v ∗ i , v ∗ j ) − S ( v i , v j ) , and acts only on the two variables des-ignated by the arguments of the T operator. Also δ i , j is theKronecker delta function. The evolution of the distributionfunctions thus involves a hierarchy of equations. The solutionwould require a closure of this hierarchy. As for the Maxwellparticles [23], one may ask whether there exists such a closurein terms of the variance and two-point correlation functionsfor the one-dimensional lattice gas also.We calculate the evolution of the function Σ ( x , t ) , by multi-plying v i v i + x and integrating over v i and v i + x . This results inthe closed set of equations for Σ given in Eq. (3). Appendix B: Existence of steady states for various values of r w for the inelastic gas on a 1-D lattice1. Absence of steady state when r w = − Here, we show that the correlation vector Z ( t ) whichevolves according to Eq. (5), does not have a steady state when r w = −
1. To show this, we observe the properties of the eigen-values of the matrix A (Eq. (6)). We note that when r w = − b is equal to zero and the tri-diagonal matrix A has a simpler form (Eq. (B1)). We denote this matrix by A ( r w = − ) . A ( r w = − ) = a n + ε − ε − ε ( + ε ) − − − − − − . (B1)The determinant of the above ( n + ) -th order matrix denotedas det A ( r w = − ) , can be shown to satisfy the relation, when n > A ( r w = − ) = ε a n + (cid:2) det A (cid:48) n − − det A (cid:48) n − (cid:3) , (B2)where det A (cid:48) k is the determinant of A (cid:48) k , which is a matrix oforder k ∈ N , and has the form given below. A (cid:48) = − − − − − − − − (B3) One can find det A (cid:48) k , as follows. Let us denote det A (cid:48) k ≡ D (cid:48) k . Itcan be shown to satisfy the relation, D (cid:48) k − D (cid:48) k − + D (cid:48) k − = . (B4)Using the boundary conditions, D (cid:48) = D (cid:48) =
2, the solutionof Eq. (B4) can be easily obtained as, D (cid:48) k = det A (cid:48) k =
2. Substi-tuting this in Eq. (B2) we obtain the result, det A ( r w = − ) =
0. This shows that at least one of the eigenvalue is zero, whichimplies the lack of steady state for the system.
2. Presence of steady state when | r w | < Consider the matrix A (Eq. (6)) when r w (cid:54) = −
1. We canuse Gershgorin circle theorem [35] to predict the range ofthe eigenvalues of the matrix A . The theorem states that anyeigenvalue λ of the matrix A should satisfy the condition: | λ − A ii | ≤ ∑ j (cid:54) = i | A i j | , i = , , ... n (B5)From the first row of A , we find that: | λ − [ ε a + b ( − r w )] | ≤ ε a , (B6)which says, λ − b ( − r w ) ≥
0. Similarly for i >
1, usingEq. (B5) we obtain the result, λ − b ≥
0. Thus all the eigen-values are strictly greater than zero as b >
0. This proves thatwhen | r w | <
1, the system goes to a steady state.
3. Presence of steady state when r w = When r w =
1, Gershgorin circle theorem provides the in-equalities, λ ≥ A ( r w = ) and λ − b ≥ A ( r w = ) , to be satisfied by the eigen-values λ of A ( r w = ) . The above observations show that theeigenvalues of A ( r w = ) will satisfy the condition λ ≥ A ( r w = ) (cid:54) =
0. We show this in the following.As we are interested in the large system case, we considera system with n >
2. For the system, one can show as before,that det A ( r w = ) satisfies the equation,det A ( r w = ) = ε a n + (cid:104) ( c − ) det A (cid:48)(cid:48) n − − det A (cid:48)(cid:48) n − (cid:105) , (B7)where A (cid:48)(cid:48) k is a k × k matrix given by, A (cid:48)(cid:48) k = c − − c − − c − − c − − c . (B8) We define the determinant, det A (cid:48)(cid:48) k ≡ D (cid:48)(cid:48) k . From Eq. (B8), onecan show that D (cid:48)(cid:48) k satisfies the equation, D (cid:48)(cid:48) k − cD (cid:48)(cid:48) k − + D (cid:48)(cid:48) k − = , k = , .. (B9)with c = + b / a . The exact form of D (cid:48)(cid:48) k can be found bysolving the difference equation using the initial conditions D (cid:48)(cid:48) = c , D (cid:48)(cid:48) = c −
2. The general solution for Eq. (B9)has the form, D (cid:48)(cid:48) k = As k + Bs − k , (B10)with s = c + √ c −
1. Using the initial conditions, the exactform of D (cid:48)(cid:48) k is found as, D (cid:48)(cid:48) k = s k + s − k . (B11)Substituting det A (cid:48)(cid:48) k = ( s k + s − k ) in Eq. (B7), one gets:det A ( r w = ) = ε a ( n + ) (cid:16) ( + b / a ) (cid:104) s ( n − ) + s − ( n − ) (cid:105) − (cid:104) s ( n − ) + s − ( n − ) (cid:105)(cid:17) . (B12)One can rewrite the Eq. (B12) as,det A ( r w = ) = ε a ( n + ) × (cid:8) (cid:104) s ( n − ) − s ( n − ) + s − ( n − ) − s − ( n − ) (cid:105) + ba (cid:104) s ( n − ) + s − ( n − ) (cid:105) (cid:9) . (B13)Note that s >
1. The material within the first set of squarebrackets on the right-hand side of Eq. (B13) can be rewrittenas, (cid:20) s ( n − ) − s ( n − ) + s ( n − ) − s ( n − ) (cid:21) = (cid:0) s n − − (cid:1) s − s n − > s > n ≥
2. As the term in the second set of squarebrackets in Eq. (B13) is a positive definite quantity, the right-hand side of Eq. (B13) will be non-zero. So the determinantof A ( r w = ) is non-zero. [1] P. L. Garrido, J. L. Lebowitz, C. Maes, H. Spohn, Phys. Rev. A , 1954 (1990).[2] T. P. C. van Noije and M. H. Ernst, Granular Matter , 57(1998).[3] F. Rouyer and N. Menon Phys. Rev. Lett. , 3676 (2000).[4] E. Ben-Naim and P. L. Krapivsky, Phys. Rev. E , R5 (2000).[5] A. Santos and M. H. Ernst, Phys. Rev. E , 011305 (2003).[6] J. S. van Zon and F. C. MacKintosh Phys. Rev. Lett. , 038001(2004).[7] E. Ben-Naim and J. Machta, Phys. Rev. Lett. , 138001(2005).[8] V. V. Prasad, S. Sabhapandit, and A. Dhar, Europhys. Lett., ,54003 (2013) .[9] T. P. C. van Noije, M. H. Ernst, R. Brito and, J. A. G. Orza,Phys. Rev. Lett. , 411 (1997). [10] A. Baldassarri , U. Marini Bettolo Marconi, and A. Puglisi, Eu-rophys. Lett., ,14 (2002).[11] M. Shinde, D. Das, and R. Rajesh Phys. Rev. Lett. , 234505(2007).[12] J. Javier Brey and M. J. Ruiz-Montero, Phys. Rev. E 91, 012202(2015).[13] D. R. M. Williams and F. C. MacKintosh, Phys. Rev. E ,R9(R) (1996).[14] S. J. Moon, M. D. Shattuck, and J. B. Swift Phys. Rev. E ,031303 (2001).[15] M. R. Swift, M. Boamfa, S. J. Cornell, and A. Maritan, Phys.Rev. Lett. , 4410 (1998)[16] T. P. C. van Noije, M. H. Ernst, E. Trizac, and I. Pagonabarraga,Phys. Rev. E , 4326 (1999).[17] D. L. Blair and A. Kudrolli, Phys. Rev. E , 050301(R) (2001). [18] A. Prevost, D. A. Egolf, and J. S. Urbach, Phys. Rev. Lett. ,084301 (2002).[19] G. Gradenigo, A. Sarracino, D. Villamaina and A. Puglisi,Europhys. Lett. , 014704 (2012).[20] A. Lasanta, A. Manacorda, A. Prados and A. Puglisi, New J.Phys.
810 (2016).[23] V. V. Prasad, S. Sabhapandit, and A. Dhar, Phys. Rev. E ,062130 (2014).[24] D. Levanony and D. Levine, Phys. Rev. E , 055102R (2006).[25] A. Prados, A. Lasanta, and P. I. Hurtado, Phys. Rev. Lett. ,140601, (2011). [26] A. Prados, A. Lasanta, and P. I. Hurtado Phys. Rev. E ,031134 (2012).[27] P. I. Hurtado, A. Lasanta, and A. Prados, Phys. Rev. E ,022110 (2013).[28] P. Maynar, M. de Soria, and E. Trizac, Eur. Phys. J. Spec. Top. , 123 (2009).[29] V. Privman, Nonequilibrium Statistical Mechanics in One Di-mension (Cambridge University Press, 1997).[30] A. Baldassarri, U. Marini Bettolo Marconi, and A. Puglisi Phys.Rev. E , 051301 (2002).[31] S. Sabhapandit, Europhys. Lett. , 20005 (2011).[32] S. Sabhapandit, Phys. Rev. E , 021108 (2012).[33] A. Pal and S. Sabhapandit, Phys. Rev. E , 022138 (2013).[34] S. N. Majumdar, S. Sabhapandit, and G. Schehr, Phys. Rev. E , 052131 (2015); , 052126 (2015).[35] H. E. Bell, The American Mathematical Monthly72