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granular materials such as sand and powders have generated much interest of late . such an ensemble of particles with macroscopic size is challenging since it may behave as a solid , a liquid or a gas . size separation , pattern formation , avalanches , compaction and convection are just a few examples of the wide array of observed phenomena@xcite . flow underlies most of these phenomena and therefore , theoretical studies so far focus on formulating a hydrodynamical description appropriate to sand@xcite . these theories , stemming from the boltzmann equation , depend on the assumption of `` molecular chaos '' , _ i.e. _ , the assumption that no interparticle correlations exist . this assumption is far from obvious . as a dissipative dynamical system , a granular system has attractors in its phase space , which may cause correlations between particles . under certain conditions , these attractors lead to a singularity , inelastic collapse , which can not be explained by hydrodynamics . in one dimension , these attractors are so strong that hydrodynamics breaks down for the entire parameter space@xcite  in a confined geometry , all particles , save one , form a practically stationary clump against the elastic wall , while the remaining particle moves rapidly back and forth between the clump and the heated wall . such a state clearly violates partition of energy .    in this study , we investigate the corresponding situation in two - dimensions@xcite . we consider inelastic hard spheres in a box where one wall is kept at a fixed temperature and the other three are reflecting ( see fig . [ fig : syspict ] ) . energy input at the heated wall balances the dissipation due to inter - particle collisions and the system can achieve a steady state . unlike the one - dimensional situation , the density and the temperature profiles are smooth functions of the distance from the heated wall . in the steady - state , the momentum balance equations imply that the pressure is constant throughout the system . particles move faster close to the energy source and more slowly deeper inside the system due to energy loss . thus the density is greater farther away from the wall to maintain a constant pressure . density variations may be a consequence of the above mechanism or an effect of the intrinsic attractors in the system which build up correlations among particles . assuming that the particles are not coherent in the quasi - elastic limit , we derive a differential equation that describes the density variation throughout the system . heuristic expressions for the mean - free path , the equation of state , and the thermal conductivity are incorporated into the energy flux balance equation to obtain a closed equation describing the steady - state density profiles . the theory compares well with simulational results over a wide range of densities in the quasi - elastic limit . =    our theory is appropriate for a steady - state which is very close to an equilibrium state . we expect this to be applicable only in the nearly elastic limit . we argue that in the complementary situation of stronger inelasticity , hydrodynamics may still be relevant . to analyze the nature of the steady - state , we study velocity distributions throughout the system . we observe that the velocity distributions exhibit scaling . this observation is then used to obtain a qualitative description of the behavior of the system . granular materials are different from ordinary fluids or gases in that the diameter of a particle may be comparable to the collisional mean free path . to formulate a hydrodynamic theory , it is necessary to describe how the mean free path , the pressure , and the thermal diffusivity depend on the grain diameter , the number density , and the temperature . in this section , we obtain heuristic expressions for these quantities . let us denote the number density of grains by @xmath0 , the mean free path by @xmath1 , and the grain diameter by @xmath2 . without loss of generality , we set the particle mass and the boltzmann constant to unity : @xmath3 . the granular temperature can be defined as the average kinetic energy per particle : @xmath4 . this quantity is well defined for an equilibrium state in which the particles have a symmetric velocity distribution . in such a case , @xmath5 , can be used as an approximate value for the average grain speed . in our system , quantities such as temperature , density , etc . , are position dependent , and there is not a global equilibrium . however , in the quasi - elastic limit , we expect the system to be very close to local thermal equilibrium . let @xmath6 be the distance from the heated wall . in the steady state , all quantities vary only in this transverse direction . the energy balance equation can therefore be written @xmath7 where @xmath8 is the energy flux in the @xmath6 direction . the sink term @xmath9 accounts for the energy lost per unit area per unit time due to inelastic collisions . the energy flux is induced by a temperature gradient , @xmath10 , where the coefficient of thermal diffusivity , @xmath11 , is positive . consequently , we find @xmath12      since collisions between grains are inelastic , kinetic energy is continually transferred into heat . for simplicity we neglect rotation and thus , the degree of inelasticity can be parameterized by @xmath13 . when two particles collide , their tangential velocities are unchanged , while the relative normal velocity is decreased by a factor of @xmath14 , where the negative sign merely indicates that they move apart after a collision . using momentum conservation one can write the final velocities ( indicated by primes ) in terms of the initial velocities @xmath15 in the above equation , the subscript @xmath16 denotes the velocity component along the line connecting the centers of particles 1 and 2 . the energy lost in each collision is therefore @xmath17 in this study , we focus on the quasi - elastic limit , i.e.  @xmath18 . physically , this limit is relevant to hard particles such as glass or steel beads .    using the above expression for the energy dissipated in each collision , one can estimate the sink term , @xmath9 , the mean energy lost per unit area per unit time . consider a particle moving with speed @xmath19 . during each collision , it loses , on average , energy proportional to @xmath20 . in unit time , it collides roughly @xmath21 times . in unit area , there are @xmath0 particles , and consequently @xmath22      suppose there is a temperature gradient in the @xmath6-direction . as a result , there will be an energy flux along this axis . to calculate this flux , let us consider the number of particles crossing a line perpendicular to this direction in a time interval @xmath23 . we define `` crossing the line '' as having any part of the particle over the line during this time interval . to cross the line within @xmath23 , a particle with speed @xmath19 must have its rightmost point @xmath24 or closer to the line . thus , only grains in an area ( @xmath25 ) to the left of the line can pass the line from the left in @xmath23 . in fact , only one half of these particles are moving to the right , so the number of particles crossing the line per unit cross - sectional length in @xmath23 is @xmath26 . in a steady state , the number of crossing events from each side must balance . any energy flux is due to the fact that particles coming from the right are at a different temperature ( @xmath27 ) than those from the left ( @xmath28 ) and thus , @xmath29 . consider two grains on opposite sides of the line . their centers are approximately a distance @xmath30 apart , so the temperature difference is roughly @xmath31 . the coefficient of thermal conductivity is therefore @xmath32 . a natural choice for @xmath23 is the typical collision time @xmath33 . this choice is small enough to avoid multicounting and is sufficiently large to ensure that heat transfer does occur . our heuristic picture therefore estimates the thermal diffusivity by @xmath34 this is a rough approximation ; the actual prefactors depend on the velocity distribution of the grains . therefore , we generally assume @xmath35    the energy balance equation  ( [ eq : m2 ] ) takes the form @xmath36 = ( 1-r^2)\frac{\rho t^\frac{3}{2}}{\gamma l } \label{eq : m3}\ ] ] where @xmath37 is the ratio of prefactors in the expressions for @xmath9 and @xmath11 . in the following subsections , we discuss how to obtain the dimensionless coefficients @xmath38 and @xmath37 self - consistently . to proceed , it is necessary to relate @xmath0 and @xmath39 through the equation of state . additionally , the mean free path @xmath1 must be expressed in terms of @xmath0 . for the system to be in a steady state , the pressure must be constant throughout . the equation of state relates the pressure and the temperature to the density . for example , in the low density limit , the ideal gas law holds , @xmath40 . using @xmath41 and @xmath42 , one has @xmath43 on the other hand , in the high density limit , the mean free path , which is simply the interparticle spacing , is much less than the particle diameter ( @xmath44 ) . denoting the close - packing density by @xmath45 , one finds @xmath46 in this limit , the center of a grain is confined to an area of the order of @xmath47 , so the entropy per particle , @xmath48 , equals @xmath49 plus a function of temperature . from eq . ( [ eqn : denl ] ) , @xmath48 depends on density only through the term @xmath50 . using maxwell s relation , @xmath51 we obtain the pressure in the limit @xmath52 , @xmath53 we therefore propose the following interpolation formula for the pressure @xmath54 indeed , in the limits of high and low density this expression reduces to ( [ eqn : pressld ] ) and ( [ eqn : presshd ] ) , respectively . it is useful to compare this expression with the van der waal s equation of state which takes into account long range attraction and hard core repulsion @xcite . for an inelastic gas of hard spheres , there are no long range forces , and the van der waal s pressure for a two - dimensional gas reads @xmath55 for @xmath56 . in the low density limit the pressure given by eq . ( [ eqn : pressgen ] ) agrees with the van der waal s expression to second order in @xmath57 . furthermore , one can also compare ( [ eqn : pressgen ] ) to tonks series expansion  @xcite for the pressure of a two - dimensional gas of hard spheres , @xmath58/ [ 1 - 1.307(\rho/\rho_c)^3 + 0.307(\rho/\rho_c)^4]$ ] , valid for all densities . over the entire density range , the two expressions differ by less than 1.3% . in contrast , we found that the van der waal s formula is inadequate for describing the high density limit . hence , we use the interpolation formula ( [ eqn : pressgen ] ) for the pressure . the mean free path can be expressed in terms of the density and the diameter . in the low density limit one has @xmath59 while in the high density limit , eq .  ( [ eqn : denl ] ) gives @xmath60 again , we use these high and low density limits to interpolate a general expression for the mean free path . using the 2d close - packing value @xmath61 , we find @xmath62 where @xmath63 . ( [ eqn : pressgen ] ) and  ( [ eqn : lgen ] ) express the temperature and the mean free path in terms of the density . substituting these expressions into eq . ( [ eq : m3 ] ) yields a second order differential equation for @xmath0 . using for convenience the variable @xmath64 , we have @xmath65 =   \nonumber\\ & & \hspace{-6cm}\frac{32}{3d^2 } \frac{1-r^2}{\gamma } \frac{z - a}{(z+1)^{3/2 } } \sqrt{\frac{z-1}{z}}. \label{eqn : zgen}\end{aligned}\ ] ] this equation is complemented by the boundary conditions @xmath66 where @xmath67 is the width of the system and @xmath68 is the total number of particles . the latter condition merely reflects conservation of particles . the former condition is a consequence of the fact that the temperature gradient vanishes at the elastic wall .    in principle , @xmath37 and @xmath38 are two dimensionless factors which can be calculated exactly from the velocity distribution . however , this distribution is poorly understood . nevertheless , it is still possible to estimate these prefactors by comparing the theoretical predictions of eq . ( [ eqn : bcnorm ] ) with numerical simulations in the limit of high and low densities .    in both of these extreme cases the governing equation  ( [ eqn : zgen ] ) can be solved analytically . in the high density limit , @xmath69 , or equivalently , @xmath70 , eq . ( [ eqn : zgen ] ) reduces to @xmath71 . it is convenient to write this equation in terms of the temperature , @xmath72 . from  ( [ eqn : pressgen ] ) , @xmath73 when @xmath70 , and the temperature obeys @xmath74 with @xmath75 . solving this equation subject to the boundary condition @xmath76 at @xmath77 gives the temperature profile @xmath78 far from the elastic wall , @xmath79 , the temperature decays exponentially in agreement with haff s calculation @xcite @xmath80 eq .  ( [ eqn : lgen ] ) shows that in the high density limit , @xmath81 . both the temperature and the mean free path decay exponentially with the distance from the heated wall . the decay length @xmath82 is much larger than the mean free path . for the continuum description to be valid , @xmath82 must also be much larger than the diameter of a particle , @xmath2 , i.e. , @xmath83 since the prefactor @xmath37 is of order unity , we learn that the continuum theory is valid only in the quasi - elastic limit , @xmath84 . ( note that the restriction @xmath84 is stricter than @xmath18 . )    to test the theoretical predictions , we performed an event - driven simulation of the system@xcite . the heated wall is implemented in such a way that for @xmath85 , particles have a boltzmann velocity distribution with an average temperature equal to one . specifically , any particle that collides with the wall at @xmath86 is ejected with a positive @xmath87 drawn from the probability distribution @xmath88 and a @xmath89 drawn from @xmath90@xcite . however , the behavior of particles in the bulk of the system is independent of the details of the boundary condition ( see fig .  [ fig : vxscal ] below ) . this is not surprising , but rather a necessity for a thermodynamic theory to be valid . numerical simulations confirm the exponential decay in the quasi - elastic limit . furthermore , the decay length @xmath82 can be measured for various degrees of inelasticity @xmath91 ( see fig .  [ fig : denscal ] ) . we verified that indeed @xmath92 as suggested by eq . ( [ eqn : hdtemp ] ) . the value of the prefactor @xmath93 can be found from the simulations as well . =    we now turn to the low density limit . here , @xmath94 , and eq . ( [ eqn : zgen ] ) reduces to @xmath95 . as a result , @xmath96 where @xmath97 , and we have used the boundary condition of eq . ( [ eqn : bcnorm ] ) and the notation @xmath98 . simulations show that the density near the heated wall is significantly smaller than the density at the elastic wall . thus , for @xmath99 and @xmath100 , we find the following approximate solution @xmath101 the inverse of the density , and hence the temperature , decays linearly with @xmath6 in the low density regime near the heated wall . that the low density decay length @xmath102 is similar to the high density decay length @xmath103 , reflects the fact that the underlying differential equation is second order .    in the nearly elastic case ( @xmath18 ) , @xmath104 is close to @xmath105 , and eq . ( [ eqn : lddzdx ] ) can be integrated exactly @xmath106_{z_0}^{z(x)}.\ ] ] comparing this prediction with low density numerical simulations allows us to determine @xmath107 at various @xmath91 values ( see fig . [ fig : dilscal ] ) and verify that indeed @xmath108 . the constant of proportionality and the previously calculated @xmath37 yields @xmath109 . =    in the low density limit , the equation of state @xmath110 , implies that @xmath111 . for the state to be locally very close to equilibrium , the mean free path must be much less than the length scale over which the temperature is changing , i.e. , @xmath112 . this condition is @xmath113 when the temperature is of order unity . since @xmath38 and @xmath37 are also of order unity , the hydrodynamic description is again valid only in the quasi - elastic regime ( @xmath84 ) . our treatment so far has concentrated on either the high or the low density limit , where analytical expressions were possible . for systems that include both high and low density regions , eq . ( [ eqn : zgen ] ) can be solved numerically using the boundary and normalization conditions  ( [ eqn : bcnorm ] ) and the previously calculated values for @xmath37 and @xmath38 . by examining a number of simulations that include a range of densities we determined that the optimal values for the prefactors are @xmath114 and @xmath115 . while this value of @xmath37 is consistent with the value obtained in the high density calculation , this @xmath38 value is slightly lower than our prediction . a typical system with @xmath116 is shown in fig .  [ fig : samp ] , and it is seen that the predictions of hydrodynamic theory match the numerical data over wide density variations . for this simulation , the ratio @xmath117 is approximately @xmath118 , which is not of order unity , so the condition of eq . ( [ eq : valid2 ] ) is satisfied . the calculation in the previous section assumes that the steady state is very close to equilibrium , and that pressure and temperature can be used to describe the system . this requires that condition ( [ eq : valid1 ] ) is satisfied in the high density limit , or condition ( [ eq : valid2 ] ) is satisfied in the low density limit .    an important question is : when does the behavior of the system changes qualitatively ? one such transition occurs when @xmath91 becomes low enough for the system to undergo inelastic collapse . here strong correlations and large density variations develop , and applying hydrodynamics becomes impossible . even when @xmath91 is slightly higher than the critical value for inelastic collapse , the attractors mentioned in the introduction may still be strong enough to build correlations . where this breakdown occurs is determined by the degree of inelasticity , the density and the total number of particles@xcite . for each pair of values of density and total number of particles , there is a value of @xmath91 which divides two different kinds of behavior : loose sand and coherent sand . however , the theory developed in the previous section is strictly for the quasi - elastic limit , @xmath18 , so there may exist systems that , although elastic enough to avoid inelastic collapse , still have @xmath91 far enough from one that the hydrodynamics do not apply . since correlations between particles are built up through inelastic collisions , high density regions are more liable to inelastic collapse@xcite , while for low density regions , correlations are harder to establish . therefore , we will investigate the low density limit in order to observe the breakdown of the hydrodynamic description as the degree of inelasticity increases . when the temperature variation within a mean free path is significant , the system is unable to reach local equilibrium . therefore , particles carrying energy away from the heated wall can not share this energy with the slow particles returning from the higher density region near the elastic wall . this inefficient mixing leads to a temperature gap  the average energy of particles with @xmath119 is greater than that of the particles with @xmath120 ( see fig .  [ fig : tempdiff ] ) . furthermore , near the heated wall , the temperature drops by approximately @xmath121 over a mean free path , which suggests that such a system will be unable to reach local thermal equilibrium ( see fig . [ fig : mfp ] ) . =    =    the probability distribution functions for @xmath87 ( the velocity perpendicular to the heated wall ) provide an illuminating way to measure this deviation from equilibrium ( see fig .  [ fig : vxskew ] ) . note the asymmetry : the @xmath122 tail is longer than the @xmath120 tail . this is consistent with our understanding that the @xmath122 particles have more energy than the @xmath120 ones . a rough quantification of the deviation from equilibrium is provided by condition  ( [ eq : valid2 ] ) : when @xmath117 is of order unity , the theory breaks down . for @xmath123 ( the value used in the figures ) , this quantity is approximately @xmath124 . note that this condition involves @xmath91 only and is not sensitive to the local density @xmath0 . this suggests that the behavior of regions of the system with different densities should be similar . in fact , we observed that the velocity distribution function obeys scaling ( see fig . [ fig : vxscal ] ) , i.e. , @xmath125 the function @xmath126 is independent of boundary conditions ( see fig .  [ fig : vxscal ] ) . thus @xmath127 , where the constant of proportionality depends only on the shape of the function @xmath126 . note that this shape will depend on @xmath91 , the degree of inelasticity ; the more inelastic the particles are , the more skewed the velocity distribution is . the probability distribution for the @xmath128-component of the velocity behaves similarly : @xmath129 but here @xmath130 is a symmetric , nearly gaussian , function . additionally , the same velocity scale @xmath131 characterizes the transverse and the longitudinal velocity distributions . therefore , while at each position @xmath6 there is no longer a single hydrodynamic temperature , there is a well defined characteristic velocity scale , @xmath131 , so that the granular temperature , @xmath132 , is proportional to @xmath133 . this scaling suggests that , although the system may deviate significantly from equilibrium , it can still be treated using some of the tools of the previous sections . specifically , return to the equation for energy balance  ( [ eqn : enerbal ] ) : @xmath134 , where @xmath9 is the energy loss due to collisions per unit time per unit area , and @xmath8 is the energy flux , the heat transfer per unit cross - sectional length per unit time . as discussed previously , @xmath135 while the heat flux is approximately @xmath136 conservation of momentum flux suggests that @xmath137 is constant , and the energy balance equation gives @xmath138 which indicates that @xmath139 , and hence the temperature , depends linearly on @xmath6 . this prediction is consistent with our simulational results ( fig . [ fig : gandg2 ] ) . in this work , we examined the steady - state behavior of a weakly inelastic two - dimensional driven granular system . we found that a hydrodynamic formulation provides a satisfactory description of the near - equilibrium behavior of the system in the quasi - elastic limit . we extended haff s theory to the low density limit and found that the corresponding temperature profile varies linearly in space . for slightly higher inelasticities , the system is no longer close to equilibrium in the low density limit . however , the scaling behavior of the velocity distributions suggests that a hydrodynamic treatment can still be useful in describing the system .    in this non - equilibrium regime , we found that the particle velocity distributions were non - gaussian . indeed , deviations from normal distributions have been observed in theoretical @xcite and experimental studies@xcite . in addition to this variation in the velocity distributions , systems of this sort can produce highly inhomogeneous spatial distributions , as has been noted elsewhere in one @xcite and two @xcite dimensions . a recent experimental study examined the spatial distribution of hard particles in two dimensions in the presence of an energy input @xcite . their data is in qualitative agreement with our theoretical predictions , and inhomogeneous spatial distributions reminiscent of fig . ( 1 ) are observed . * acknowledgements : * we are indebted to leo kadanoff for stimulating our interest in this problem . we also thank bill young , tom witten , sergei esipov and thorsten pschel for useful discussions . this work was supported by the national science foundation under awards # dmr-9415604 and # dmr-9208527 and the mrsec program , award # dmr-9400379 . 99 h.  m.  jaeger , s.  r.  nagel , and r.  p.  behringer , phys .  today * 49 * , 32 ( 1996 ) . k.  haff , j.  fluid mech .  * 134 * , 401 ( 1983 ) . j.  t.  jenkins and m.  w.  richman , j.  fluid mech .  * 192 * , 313 ( 1988 ) . y.  du , h.  li , and l.  p.  kadanoff , phys . lett .  * 74 * , 1268 ( 1995 ) . e.  l.  grossman and b.  roman , phys . fluids * 8 * , 3218 ( 1996 ) . j.  maddox , nature * 374 * , 11 ( 1995 ) . s.  e.  esipov and t.  pschel , `` boltzmann equation and granular hydrodynamics , '' preprint . h.  hayakawa , s.  yue , and d.  c.  hong , phys . lett .  * 75 * , 2328 ( 1995 ) . l.  tonks , phys . 50 * , 955 ( 1936 ) . i.  goldhirsch and g.  zanetti , phys . , 1619 ( 1993 ) . s.  mcnamara and w.  r.  young , phys . e * 53 * , 5089 ( 1996 ) . y.  d.  lan and a.  d.  rosato , phys . fluids * 7 * , 1818 ( 1995 ) . y.  h.  taguchi and h.  takayasu , europhys . lett .   * 30 * , 499 ( 1995 ) . s.  warr , g.  t.  h.  jaques , and j.  m.  huntley , powder tech . * 81 * , 41 ( 1994 ) . s.  mcnamara and j .- l . barrat , `` the energy flux into a fluidized granular medium at a vibrating wall , '' preprint . a.  kudrolli , m.  wolpert , and j.  p.  gollub , `` cluster formation due to collisions in granular material , '' preprint .
we study steady - state properties of inelastic gases in two - dimensions in the presence of an energy source . 
 we generalize previous hydrodynamic treatments to situations where high and low density regions coexist . 
 the theoretical predictions compare well with numerical simulations in the nearly elastic limit . 
 it is also seen that the system can achieve a nonequilibrium steady - state with asymmetric velocity distributions , and we discuss the conditions under which such situations occur . 
 pacs numbers : 05.20.dd , 47.50.+d , 81.35.+k    2