Scaling and aging in the homogeneous cooling state of a granular fluid of hard particles
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Scaling and aging in the homogeneous cooling state of a granularfluid of hard particles
J.J. Brey, A. Prados, M.I. Garc´ıa de Soria, and P. Maynar
F´ısica Te´orica, Universidad de Sevilla,Apartado de Correos 1065, E-41080, Sevilla, Spain (Dated: November 4, 2018)
Abstract
The presence of the aging phenomenon in the homogeneous cooling state (HCS) of a granularfluid composed of inelastic hard spheres or disks is investigated. As a consequence of the scalingproperty of the N -particle distribution function, it is obtained that the decay of the normalized two-time correlation functions slows down as the time elapsed since the beginning of the measurementincreases. This result is confirmed by molecular dynamics simulations for the particular case ofthe total energy of the system. The agreement is also quantitative in the low density limit, forwhich an explicit analytical form of the time correlation function has been derived. Moreover,the reported results provide support for the existence of the HCS as a solution of the N-particleLiouville equation. PACS numbers: 45.70.-n,51.10.+y,05.20.Dd . INTRODUCTION In many far from equilibrium states, it has been observed that the relaxation (or response)rate decreases as the “age” of the process increases. Here “age” refers to the time elapsedsince the beginning of the considered experiment. Then, it is said that the system ages, orthat it exhibits an aging phenomenon. A revision of the concept of aging in spin glasses andin other systems can be found in ref. [1].One of the typical experiments for the analysis of aging in a given physical system is thestudy of the two-time correlation functions of some of its properties, since they are relatedwith the response of the system to a given perturbation. Let C AB ( t w , t ) denote the two-timecorrelation function of the magnitudes A and B of the system, the former being measured attime t w and the latter at time t ≥ t w . In a system at equilibrium, C AB ( t w , t ) only dependson the the time difference τ = t − t w , due to the time translational invariance. On the otherhand, in systems presenting aging, it depends on both τ and t w . It is important to stressthat the aging phenomenon is more than just the loss of the time translational invariance; itconsists in the relaxation of C AB ( t w , t w + τ ) slowing down as the “waiting time” t w increases.The simplest aging phenomenon occurs when the two-time correlation function dependsonly on the time ratio τ /t w , and it is sometimes called “full aging”. This behavior isexhibited by some simple models, such as mean field model of spin glasses [2, 3] and theone-dimensional Ising model at zero temperature [4, 5]. More complicated dependencieslike ln t/ ln t w have also been found [6] and, with more generality, behaviors of the form h ( t ) /h ( t w ), with different functions h [1, 7].Granular media are inherently non-equilibrium systems, due to the dissipative characterof the interactions between grains. There is a continuous loss of kinetic energy and the systemtends to a rest, unless energy is being continuously injected into the system, for instancethrough a vibrating wall. A kind of typical experiments carried out in dense granular systemsare those designed to investigate compaction [8]. Usually, the system is submitted to a seriesof separated pulses or taps of a given short duration. After each tap, the system is allowed torelax freely until it reaches a metastable configuration with all the particles at rest. In thisstate, there are many permanent contacts between particles. Next, the system is tappedagain and the process is repeated many times. Properties of interest like the volume orthe energy are measured at each rest configuration. In this way, the evolution of these2roperties as a function of time, measured in number of taps, is obtained. For large numberof taps, the system tends to a steady state with a density that is a monotonically decreasingfunction of the tapping intensity [8, 9]. The relaxation of the system towards the steadystate is very slow and clearly non-exponential. Furthermore, when the intensity of tappingis changed cyclically, hysteresis phenomena in the density are observed [9]. These behaviorsare similar to those found in structural glasses when submitted to cyclical variations of theirtemperature [10] and are a quite strong evidence of the presence of aging phenomena incompact granular media. Actually, aging has been observed both experimentally [11] andalso identified in simple models of compaction [12, 13].A completely different regime of granular systems are the so-called granular gases, inwhich there are not permanent contacts between particles, but they move freely and indepen-dently between collisions. The simplest possible state of a granular gas is the homogeneouscooling state (HCS), whose temperature decays monotonically in time. At a microscopiclevel, the HCS is assumed to be characterized by a phase space probability distribution inwhich all the time dependence occurs through the temperature. For the case of hard par-ticles, this implies a scaling property and the possibility of identifying some features of thetime dependence of many relevant properties of the system, without carrying out explicitcalculations. Among these properties are ensemble averages as well as time-correlation func-tions [14, 15]. The above peculiarities render the HCS a good candidate to investigate indetail and by means of analytical methods the possible existence of aging and, in the caseof a positive answer, its origin and properties. In spite of the above, it has not been untilvery recently that attention has been devoted to this particular aspect of the HCS [16]. Theaim of this paper is twofold. Firstly, to investigate in detail the possible existence of theaging phenomenon in the HCS of a granular fluid of inelastic hard particles, deepening intoits origin. This is done both analytically and by means of particle simulation methods. Thetheoretical analysis will be based on the existence of the HCS at the level of the N -particledistribution function and its scaling property, for arbitrary density and inelasticity. Then,the accuracy of the predictions following from the theory provides strong support for theexistence of the HCS in the context of a many body theory. This is precisely the second aimof the paper.The presentation here proceeds as follows. In Sec. II, some general properties of the N -particle distribution function defining the assumed HCS of a system of inelastic hard3pheres or disks are shortly reviewed. By means of an appropriate scaling of the dynamicsof the system, general features of the time dependence of the time-correlation functions canbe identified. This is discussed in Sec. III, and the results presented hold, in principle, for ageneral pair of dynamical variables and arbitrary density and inelasticity. It is shown thatthe system exhibits full aging as a direct consequence of the scaling property of the HCS.The particular case of the total energy of the system is considered in Sec. IV. The time self-correlation function of this property in the HCS is known in detail quite accurately in thelow density limit, then allowing to get very detailed information about its time behavior.Comparison of the theoretical predictions with molecular dynamics simulation results isalso presented. A quite good agreement is observed. Finally, Sec. V contains some generalcomments and final remarks. II. THE HOMOGENEOUS COOLING STATE OF A GRANULAR FLUID
Consider a system on N inelastic hard spheres ( d = 3) or disks ( d = 2) of mass m anddiameter σ . The position and velocity of particle r at time t will be denoted by q r ( t ) and v r ( t ), respectively. The dynamics of the system consists of free streaming, i.e. straight linemotion along the direction of the velocity until a pair of particles, r and s , is at contact, atwhich time their velocities v r , v s change instantaneously to v ′ r , v ′ s according to v ′ r = v r − α b σ · v rs ) b σ , (1) v ′ s = v s + 1 + α b σ · v rs ) b σ , (2)where v rs ≡ v r − v s is the relative velocity and b σ is a unit vector along q rs ≡ q r − q s atcontact. Finally, α is the coefficient of normal restitution, defined in the range 0 < α ≤ ρ (Γ , t ), with Γ denoting a point in the 2 N d dimensional phase space of thesystem, Γ ≡ { q , v , . . . , q N , v N } . The macroscopic variables of interest are the average ofmicroscopic observables A (Γ) at a given time t , defined in the two equivalent forms h A ( t ) i ≡ Z d Γ ρ (Γ) e tL A (Γ) = Z d Γ A (Γ) e − tL ρ (Γ) . (3)4n the above expressions, L is the generator of the dynamics for phase functions, while L isthe generator of the dynamics for distribution functions. Their expressions are L (Γ) ≡ N X r =1 v r · ∂∂ q r + 12 N X r =1 N X s = r T ( r, s ) , (4) L (Γ) ≡ N X r =1 v r · ∂∂ q r − N X r =1 N X s = r T ( r, s ) , (5)with the binary collisions operators T ( r, s ) and T ( r, s ) defined by T ( r, s ) ≡ σ d − Z d b σ Θ( − b σ · v rs ) | b σ · v rs | δ ( q rs − σ ) ( b rs − , (6) T ( r, s ) ≡ σ d − Z d b σ Θ( b σ · v rs ) | b σ · v rs | (cid:2) α − δ ( q rs − σ ) b − rs − δ ( q rs + σ ) (cid:3) . (7)In these expressions, d b σ is the solid angle element corresponding to b σ , σ ≡ σ b σ , q rs ≡ q r − q s ,and Θ( x ) is the Heaviside step function. Moreover, b rs is the substitution operator thatreplaces the velocities v r and v s to its right by their “postcollisional” values accordinglywith Eqs. (1) and (2). Thus for an arbitrary function F , b rs F ( v r , v s ) = F ( v ′ r , v ′ s ) . (8)Finally, the operator b − rs is the inverse of b rs , i.e. it changes the velocities v r , v s by their“precollisional” values, b − rs F ( v r , v s ) = F ( v ′′ r , v ′′ s ) , (9) v ′′ r = v r − α α ( b σ · v rs ) b σ , (10) v ′′ s = v r + 1 + α α ( b σ · v rs ) b σ . (11)In summary, the dynamics of the probability distribution function in phase space isgoverned by the Liouville equation (cid:18) ∂∂t + L (cid:19) ρ (Γ , t ) = 0 . (12)Due to the energy dissipation in collisions, there is no stationary solution to the above Liou-ville equation, except in the elastic limit α = 1. A granular temperature T is usually definedfrom the average of the energy density. For a homogeneous state of a system composed ofhard particles, it is given by T ( t ) = 2 N d h E ( t ) i , (13)5ith E being the total (kinetic) energy of the system. By using Eq. (3), it is found ∂T ( t ) ∂t = − ζ ( t ) T ( t ) , (14)where the “cooling rate” ζ ( t ) is identified as ζ ( t ) = − T ( t ) N d h LE ( t ) i ≥ . (15)Of course, there is a large class of time-dependent homogeneous states, depending on theinitial preparation. Here, it will be assumed that, after a few collisions per particle, there isa relaxation of the velocity distribution towards a “universal” form, characterized becauseits entire time dependence occurs through the cooling temperature. This special state iscalled the homogeneous cooling state (HCS) and, at the macroscopic level, it is definedby a uniform number density n h , a uniform but time-dependent temperature T h ( t ), and avanishing flow velocity. Because of the absence of any additional microscopic energy scalefor hard particles, its distribution function has the form ρ h (Γ , t ) = [ ℓv ( t )] − Nd ρ ∗ h (cid:18)(cid:26) q rs ℓ , v r v ( t ) ; r, s = 1 , . . . , N (cid:27)(cid:19) , (16)where v ( t ) ≡ (2 T h /m ) / is a thermal velocity and ℓ ≡ ( n h σ d − ) − a characteristic lengthproportional to the mean free path. The above special form of the N -particle distribu-tion function allows to determine the temperature (and time) dependence of many averageproperties without explicit calculations. This fact will be actually exploited in the following.The existence of the HCS solution to the Liouville equation has already been assumedseveral times in the literature [17, 18]. Although there is no direct proof of it, nor a con-structive solution of the Liouville equation for this state has been developed, moleculardynamics (MD) simulations have shown that some of its implications, e.g. the scaling lawfor the temperature mentioned below, are observed in detail. Additional support has beenprovided by means of a time scale change that transforms the assumed HCS distributioninto a time-independent distribution [19, 20]. MD simulations seem to confirm the existenceof the steady state, that is reached after a few collisions per particle. A more demandingevidence of the existence of the HCS with a distribution function having the scaling propertygiven in Eq. (16) is provided by the results to be reported in this paper.The temperature dependence of the cooling rate for the HCS, ζ h ( t ), can be determinedby dimensional analysis to be ζ h [ n h , T h ( t )] ∝ T h ( t ) / . Now, Eq. (14), particularized for the6CS of a system of hard spheres or disks, can be integrated, to obtain the time dependenceof the temperature T h ( t ) = T h ( t ′ ) (cid:20) ζ ∗ v ( t ′ )( t − t ′ )2 ℓ (cid:21) − , (17)with ζ ∗ ≡ ℓζ h ( t ) v ( t ) (18)being a dimensionless time-independent cooling rate. This algebraic decay of the tempera-ture of the HCS is known as the Haff law [21].For the analysis of the HCS, it is useful to introduce the dimensionless time scale s definedthrough s ( t ) = Z t dt ′ v ( t ′ ) ℓ . (19)Therefore, s is proportional to the accumulated average number of collisions per particle inthe time interval (0 , t ). In terms of this new time variable, the cooling law (17) becomes T ( s ) = T ( s ′ ) e − ( s − s ′ ) ζ ∗ . (20)The t and s time scales are related through s = 2 ζ ∗ ln (cid:20) ζ h (0)2 t (cid:21) , (21)as can be directly seen by comparison of Eqs. (17) and (20) or, equivalently, by directintegration of Eq. (19). III. TIME-CORRELATION FUNCTIONS IN THE HCS
As indicated in the previous section, the scaling property of the distribution function ofthe HCS implies that the time-dependence of many macroscopic properties of the systemcan be identified without carrying out explicit calculations. Let A (Γ) be a homogeneousfunction of degree a of the velocities of the particles. Then, it is A (Γ) ≡ A ( { q r , v r ; r = 1 , . . . , N } ) ≡ A ( { ℓ q ∗ r , v ( t ) v ∗ r ; r = 1 , . . . , N } )= v a ( t ) A ( { ℓ q ∗ r , v ∗ r ; r = 1 , . . . , N } ) , (22)where q ∗ r ≡ q r /ℓ and v ∗ r ≡ v r /v ( t ). Examples of this kind of properties are the center ofmass velocity or the total energy of the system. The average value of A in the HCS is h A ( t ) i h = Z d Γ A (Γ) ρ h (Γ , t ) = v a ( t ) h A i ∗ h , (23)7here h A i ∗ h ≡ Z d Γ ∗ ρ ∗ h (Γ ∗ ) A ( { ℓ q ∗ r , v ∗ r ; r = 1 , . . . , N } ) , (24)and Γ ∗ ≡ { q ∗ r , v ∗ r ; r = 1 , . . . , N } . Thus all the time dependence of h A ( t ) i h is in the factor v a ( t ).Suppose next that B (Γ) is also a homogeneous function of the velocities of degree b , B (Γ) = v b ( t ) B ( { ℓ q ∗ r , v ∗ r ; r = 1 , . . . , N } ) , (25)and consider the HCS time-correlation function for A and B defined as C AB ( t, t ′ ) ≡ h A ( t ) B ( t ′ ) i h − h A ( t ) i h h B ( t ′ ) i h , (26)for t ≥ t ′ ≥
0. By carrying out the transformation to dimensionless variables, it can beshown that [14] h A ( t ) B ( t ′ ) i h = v a ( t ) v b ( t ′ ) h A ( s − s ′ ) B i ∗ h , (27)with h A ( s ) B i ∗ h = Z d Γ ∗ ρ ∗ h (Γ ∗ ) A ( { ℓ q ∗ r , v ∗ r } , s ) B ( { ℓ q ∗ r , v ∗ r } ) . (28)Here A ( { ℓ q ∗ r , v ∗ r } , s ) = e s L ∗ A ( { ℓ q ∗ r , v ∗ r } ) , (29)where L ∗ is the new generator for the dynamics of the phase functions, L ∗ (Γ ∗ ) ≡ ζ ∗ N X r =1 v ∗ r · ∂∂ v ∗ r + L ∗ (Γ ∗ ) , (30) L ∗ (Γ ∗ ) = ℓv ( t ) L (Γ) = [ L (Γ)] { q r = q ∗ r , v r = v ∗ r } ,σ = σ ∗ . (31)The first term on the right hand side of Eq. (30), is due to the time-dependent scaling ofthe velocities with v ( t ). Use of Eqs. (23) and (27) into Eq. (26) yields C AB ( t, t ′ ) = v a ( t ) v b ( t ′ ) C ∗ AB ( s − s ′ ) , (32) C ∗ AB ( s ) = h A ( s ) B i ∗ h − h A i ∗ h h B i ∗ h . (33)To identify the aging phenomena clearer, it is convenient to normalize the correlation func-tion to unity for t = t ′ , by defining a relaxation function φ AB ( t, t ′ ) as φ AB ( t, t ′ ) ≡ C AB ( t, t ′ ) C AB ( t ′ , t ′ ) = (cid:20) v ( t ) v ( t ′ ) (cid:21) a φ ∗ AB ( s − s ′ ; ζ ∗ )= e − ( s − s ′ ) aζ ∗ φ ∗ AB ( s − s ′ ; ζ ∗ ) . (34)8pon writing the last equality above, use has been made of Eq. (20). The dimensionlessrelaxation function φ ∗ AB above is φ ∗ AB ( s ; ζ ∗ ) = h A ( s ) B i ∗ h − h A i ∗ h h B i ∗ h h AB i ∗ h − h A i ∗ h h B i ∗ h . (35)It follows from Eq. (34) that the two-time correlation function depends on time only throughthe difference s − s ′ . Consequently, there is no aging when time is measured in the dimen-sionless scale s . In ref. [16], the aging property of the velocity time-autocorrelation functionof a granular gas of inelastic hard particles was investigated by means of MD simulations.Time was measured by the average cumulated number of collisions per particles that, as saidabove, is proportional to the dimensionless time scale s used here. The simulations indicatethat the velocity autocorrelation function, C vv ( s, s ′ ) in the language used here, depends bothon s ′ and s − s ′ , and from this feature the authors conclude that the system exhibits aging. Ofcourse, the analysis developed here applies for the case of the velocity time-autocorrelationfunction, corresponding to a = b = 1 and, therefore, aging should not be expected accordingto the results derived above. This apparent discrepancy seems to occur because of a wronguse of the physical concept of aging in ref. [16]. As pointed out in the Introduction, for theexistence of aging, it is not enough the dependence of the time-correlation function on both s ′ and s − s ′ . In fact, the analysis developed here leads to C AB ( s, s ′ ) = C AB ( s ′ , s ′ ) φ AB ( s, s ′ ) = f ( s ′ ) e − aζ ∗ ( s − s ′ ) φ ∗ AB ( s − s ′ ; ζ ∗ ) . (36)Nevertheless, all the dependence on s ′ occurs in the prefactor f ( s ′ ) = C AB ( s ′ , s ′ ) and, there-fore, there is no real aging, since the decaying rate is always the same and only the initialvalue changes with s ′ for constant s − s ′ .On the other hand, the aging phenomenon shows up in the original time scale t in thelimits ζ h (0) t ′ ≫ ζ h (0) t ≫
1. In this regime, it follows from Eq. (21) that s − s ′ ∼ ζ ∗ ln tt ‘ , (37)and, therefore, Eq. (34) takes the form φ AB ( t, t ′ ) = (cid:18) t ′ t (cid:19) a φ ∗ AB (cid:20) ζ ∗ ln t ′ t ; ζ ∗ (cid:21) ≡ F (cid:18) tt ′ ; ζ ∗ (cid:19) , (38)valid for t, t ′ ≫ ζ h (0) − . This result is the mathematical expression of the aging phenomenon.The normalized time-correlation function depends on the initial and final times, t ′ and t , only9hrough their quotient t/t ′ , so the system exhibits full aging, as defined in the Introduction.The remaining parameter determining the long time behavior of the correlation functionis the dimensionless cooling rate ζ ∗ . It is worth to stress the generality of this result.No limitation on the degree of inelasticity or density has been introduced. In fact, bothmagnitudes are relevant in determining, through the value of ζ (0), the time region in whichthe aging behavior predicted by Eq. (36) is to be expected. The only hypothesis madehere is the existence of the HCS with a distribution function having the scaling form givenin Eq. (16). Of course, the system is assumed to stay in that state for all the relaxationtime t considered. This requires that the HCS, in addition to exist, be stable, at least ina determined region of parameters. The extensive measurements of the velocity correlationfunction by means of MD simulations reported in ref. [22] show that it is possible to covera wide range of density and inelasticity in which the observed homogenous state appearsto be stable. This is confirmed by the simulation results to be presented here in the nextsection. IV. ENERGY TIME CORRELATION FUNCTION IN THE HCS FOR A DILUTEGRANULAR GAS
Theoretical predictions for the explicit form of the function φ ∗ AB ( s, ζ ∗ ) are scarce in theliterature. An exception is the total energy autocorrelation function, C EE ( t, t ′ ), for a dilutegranular gas of hard particles. By projecting the Liouville equation onto the hydrodynamicmodes in the low density limit, an analytical expression for C EE was derived in ref. [15], C EE ( t, t ′ ) = N T ( t ) T ( t ′ ) e ( α ) e − ( s − s ′ ) ζ ∗ , (39)where e ( α ) is a given function of only the coefficient of restitution. Then, φ EE ( t, t ′ ) ≡ C EE ( t, t ′ ) C EE ( t ′ , t ′ ) = T ( t ) T ( t ′ ) e − ( s − s ′ ) ζ ∗ . (40)Since the energy E is a homogenous function of degree a = 2 of the velocities of the particles,comparison of the above expression with the general result given in Eq. (34) leads to theidentification φ ∗ EE ( s ; ζ ∗ ) = e − ζ ∗ s . (41)10herefore, using Eq. (38), it follows that the aging behavior of the total energy of a dilutegranular gas of hard particles is given by φ EE ( t, t ′ ) = e − ζ ∗ ( s − s ′ ) ∼ (cid:18) t ′ t (cid:19) , (42)for t, t ′ ≫ ζ h (0) − . An explicit expression for the cooling rate of a dilute granular gas hasbeen derived from the Boltzmann equation in the so-called first Sonine approximation [23].This expression has been shown to accurately agree with the numerical results obtained bymeans of the direct simulation method of the Boltzmann equation [20, 24] in the thermalvelocity region, i.e. for velocities of the order of v ( t ).Due to the simplicity of φ ∗ EE in the present case, the asymptotic form of the normalizedtime correlation function for the energy, φ EE ( t, t ′ ), does not depend on the value of ζ ∗ ,what implies that it is independent of the density n h and of the coefficient of restitution α .Consequently, if φ EE ( t, t ′ ) is plotted as a function of t/t ′ for times in the range t, t ′ ≫ ζ (0) − ,the curves corresponding to different densities and inelasticities should tend to collapse ona unique one. Of course, the density range to which this result applies is restricted by thevalidity of the Boltzmann description.To check the above theoretical predictions, we have performed MD simulations of a systemof inelastic hard disks ( d = 2). In Fig. 1, results obtained for a system of N = 10 particleswith coefficient of normal restitution α = 0 .
95 and density n h σ = 0 .
02 are reported. Forthese values of the parameters, the HCS is stable, since the critical length is larger than thesize of the system and, therefore, the velocity vortices and high density clusters characteristicof the clustering instability [25] cannot develop. Of course, in all the simulations it has beenverified that the system remains homogeneous. The reported results have been averagedover 1200 trajectories of the system. The several curves correspond to different values of ζ h (0) t ′ , as indicated in the figure. The value of ζ h (0) has been estimated by using the lowdensity expressions derived in [23].In agreement with the analysis presented here, it is observed that as t ′ increases the time-correlation function approaches a form that depends only on the value of the time ratio t/t ′ . More precisely, for ζ h (0) t ′ > ∼
24, all the plotted curves coincide within the statisticaluncertainties. Note that the latter increase as the value of t grows, for a given value of t ′ .Moreover, the asymptotic curve agrees with the theoretical prediction given by Eq. (42),whose graphical representation is the solid line in the figure. This good accuracy is consistent11 -4 -3 -2 -1 (t’/t) -4 -3 -2 -1 φ EE ×10 ζ h (0) t’ FIG. 1: Dimensionless time self-correlation function of the total energy, φ EE ( t, t ′ ), for a dilutegranular gas of inelastic hard disks in the HCS. The density is n h σ = 0 .
02 and the restitutioncoefficient α = 0 .
95. The different symbols correspond to different values of the initial time, asindicated. The solid line is the theoretical prediction describing the full aging phenomenon, Eq.(42). with the value of the density of the system, that is small enough as to expect a low densitydescription, at the level of the Boltzmann equation, to apply.In Fig. 2, a similar plot is given, but now two systems, one with n h σ = 0 . α = 0 . n h σ = 0 . α = 0 .
98, are considered. For the sake of clearness, onlysimulation data corresponding to large waiting times have been included. Again, a goodagreement with the behavior predicted by Eq. (42) is observed. The same behavior has beenobtained for other densities between the two above values. This confirms the independenceof φ EE ( t, t ′ ) from the density and the inelasticity at low density. In fact, the good agreementobserved for n h σ = 0 . n h σ = 0 . φ EE ( t, t ′ )has also been evaluated at a definitely non small density, namely n h σ = 0 .
2. For thisvalue, density corrections to the low density behavior are clearly identified in most of the12 -4 -3 -2 -1 (t’/t) -4 -3 -2 -1 φ EE ×10 ×10 ×10 ζ h (0) t’ FIG. 2: The same as in Fig. 1, but for different values of the (low) density and of the restitutioncoefficient. Filled symbols correspond to a system with n h σ = 0 .
02 and α = 0 .
85, while emptysymbols refer to a system with n h σ = 0 . α = 0 . equilibrium properties of a molecular gas. It must be mentioned that, in order to keepthe system well inside the stable region of parameters and with a number large enough ofparticles, the value of α must be rather close to unity. Moreover, they have been averagedover 1500 trajectories. The results shown in Fig. 3 have been obtained with α = 0 . N = 700. Once again, a tendency towards a behavior depending only on the ratio t/t ′ as t ′ increases is clearly identified, i.e. the system exhibits full aging. Besides, andrather surprisingly, the aging phenomenon seems to be accurately described over severaldecades by the law ( t ′ /t ) , that was obtained here in the context of very dilute granulargases (Boltzmann limit). V. DISCUSSION
The objective here has been to explore the existence of aging in the homogeneous coolingstate of a granular gas. This has been done by exploiting the assumed scaling property ofthe N -particle distribution of this state for a system of inelastic hard spheres or disks. In13 -4 -3 -2 -1 (t’/t) -4 -3 -2 -1 φ EE ×10 ζ h (0) t’ FIG. 3: The same as in Fig. 1, but for n h σ = 0 . α = 0 .
98. A clear tendency towards afunction depending only on t/t ′ , characteristic of full aging, is observed as t ′ increases. fact, the presence of aging and its specific form turn out to be directly associated to thescaling property of the distribution [14, 15]. From this perspective, the agreement betweenmolecular dynamics simulations and the theory discussed in this paper, is an almost directproof of the existence of the homogeneous cooling state also at the level of the full manybody pseudo-Liouville equation. This would extend the rather well established fact thatthe inelastic Boltzmann and Enskog equations have such a solution, consistently with someprevious results [19, 20].Granular media are inherently non-equilibrium systems due to the lack of energy conser-vation in the interactions between grains. They present a very rich phenomenology which,sometimes, is similar to that of normal, molecular systems. Moreover, it has been verified inthe last years that the methods of kinetic theory and non-equilibrium statistical mechanicsdeveloped for normal fluids, can be extended to granular fluids, yielding to results having ananalogous structure. There are also significant differences, but they are well understood asconsequences of the inelasticity. One characteristic feature of granular systems is that quiteoften they exhibit the phenomena in a much simpler context than molecular systems. Thisrefers to both, theoretical and experimental views. In this paper, the simplicity of the HCS14f a granular fluid has allowed the identification of aging and the derivation of its explicitform in the dilute limit, Eq. (42), for the correlation of the total energy of the system. Theanalytical expression, that corresponds to the so-called full aging, has been shown to bein perfect agreement with MD simulation results. In particular, the long time limit of thenormalized time-correlation function of the total energy is independent of the inelasticityand the density. Quite interestingly, the simulations indicate that this independence seemsto extend to densities beyond the dilute limit.To really appreciate the results presented here, it is important to differentiate betweenboth the existence of aging and the specific, particular, form of the law governing it. Equa-tion (38) implies the existence of full aging in the system, i.e. the normalized time-correlationfunction of the properties A and B is not a function of the time difference t − t ′ and dependson the time ratio t/t ′ . The only necessary condition to derive this equation is the existenceof the HCS itself, as discussed above. On the other hand, identification of the function F in Eq. (38) requires more detailed additional analysis, which has been carried out only inthe low density limit up to now. The above leads to Eq. (42) in the particular case ofthe properties A and B being both the total (kinetic) energy of the system. In this sense,the results presented in Fig. 3 strongly support the existence of the HCS and the scalingproperty of its N -particle distribution function at high densities, independently of whetheror not the convergence occurs towards the power law given by Eq. (42), as suggested by thesimulation results.In real granular gases, the restitution coefficient is not constant, but it depends on theimpact relative velocity. Then, the distribution function of the HCS does not scale in theform given by Eq. (16) and, therefore, the discussion in the present paper does not apply inprinciple, although it can provide an accurate approximation to the actual behavior of thesystem. To be more precise, consider a given model of granular gas with velocity-dependentcoefficient of normal restitution. Now, the problem being addressed has two energy scales.One is the total energy per particle or, equivalently, the cooling temperature T h ( t ). Theother energy scale, ǫ , is fixed by some property of the specific collision model. Define adimensionless parameter ǫ ∗ ≡ ǫmv ( T h ) . (43)For hard spheres, ǫ = 0 and so ǫ ∗ = 0. It is in this limit when the distribution function ofthe HCS has the scaling property (16) [26]. For ǫ ∗ >
0, the scaling does not hold exactly,15ut it can be an appropriate description as long as ǫ ∗ ≪
1, i.e. the interaction be sufficientlyhard and/or the kinetic energy of the particles be sufficiently large.
VI. ACKNOWLEDGEMENTS
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