Fluctuating hydrodynamics is used to describe the total energy fluctuations
of a freely evolving gas of inelastic hard spheres near the threshold of the
clustering instability. They are shown to be governed by vorticity fluctuations
only, that also lead to a renormalization of the average total energy. The
theory predicts a power-law divergent behavior of the scaled second moment of
the fluctuations, and a scaling property of their probability distribution,
both in agreement with simulations results. A more quantitative comparison
between theory and simulation for the critical amplitudes and the form of the
scaling function is also carried out

Brey, J. Javier

de Soria, M. I. Garcia

Dominguez, A.

Maynar, P.

English

arXiv

Fluctuating hydrodynamics is used to describe the total energy fluctuations of a freely evolving gas of inelastic hard spheres near the threshold of the clustering instability. They are shown to be governed only by vorticity fluctuations that also lead to a renormalization of the average total energy. The theory predicts a power-law divergent behavior of the scaled second moment of the fluctuations, and a scaling property of their probability distribution, both in agreement with simulations results. A more quantitative comparison between theory and simulation for the critical amplitudes and the form of the scaling function is also carried out.Ministerio de Educación y Ciencia FIS2005-01398FIS2005-0139

Domínguez Álvarez, Álvaro

García de Soria Lucena, María Isabel

Maynar Blanco, Pablo

Brey Abalo, José Javier

idUS. Depósito de Investigación Universidad de Sevilla

PRL 96, 158002 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 APRIL 2006Mesoscopic Theory of Critical Fluctuations in Isolated Granular GasesJ. Javier Brey,* A. Domı́nguez, M. I. Garcı́a de Soria, and P. MaynarFı́sica Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain(Received 24 November 2005; published 21 April 2006)0031-9007=Fluctuating hydrodynamics is used to describe the total energy fluctuations of a freely evolving gas ofinelastic hard spheres near the threshold of the clustering instability. They are shown to be governed onlyby vorticity fluctuations that also lead to a renormalization of the average total energy. The theory predictsa power-law divergent behavior of the scaled second moment of the fluctuations, and a scaling property oftheir probability distribution, both in agreement with simulations results. A more quantitative comparisonbetween theory and simulation for the critical amplitudes and the form of the scaling function is alsocarried out.DOI: 10.1103/PhysRevLett.96.158002 PACS numbers: 45.70.n, 05.20.Dd, 51.10.+yGranular gases are assemblies of macroscopic particlesevolving independently between inelastic collisions [1].The methods of nonequilibrium statistical mechanics, ki-netic theory, and hydrodynamics have been successfullyextended to describe the observed macroscopic behaviorand also, although in a much more limited form, thefluctuations around it [2]. The lack of energy conservationmakes these systems behave quite differently from mo-lecular fluids. A simple widely used model for them con-sists of smooth inelastic hard spheres (IHS’s), with mo-mentum conserving dynamics. Inelasticity is characterizedby means of a constant coefficient of normal restitution .Recently, molecular dynamics (MD) simulation resultshave been reported for the total energy fluctuations of atwo-dimensional freely evolving IHS gas, near the thresh-old of the clustering instability [3]. The dimensionlesssecond moment was found to exhibit a power-law diver-gent behavior with the distance to the instability. Also, thescaled cooling rate was found to tend to zero according to apower law, although in a weak way. Besides, the distribu-tion function for the energy fluctuations, when properlyscaled, turned out to be independent of the parametersdefining the system. This was associated with a scalingproperty of the distribution. Quite remarkably, the scalingfunction was very well fitted by the same expression asseveral equilibrium and nonequilibrium molecular systems[4,5]. The main goal of this Letter is to provide an expla-nation for the above results on the basis of fluctuatinghydrodynamics [6].Consider an isolated system of N IHS’s of mass m anddiameter . The total (kinetic) energy ~E of the system canbe expressed in the form [7]2 ~Et Zdrd~nr; t ~Tr; t m~nr; t~u2r; t; (1)where d is the dimension of the system, ~nr; t the numberdensity field, ~Tr; t the temperature field, and ~ur; t theflow field. The tildes indicate that all the quantities areunderstood as fluctuating variables. In the following, sys-tems in the homogeneous cooling state (HCS) will be06=96(15)=158002(4)$23.00 15800considered. At a macroscopic level, this state is character-ized by a constant uniform density nH, a vanishing flowfield uH  0, and a uniform time-dependent temperatureobeying the law [8] @tTHt  HTHTHt, where H /THt1=2 is the cooling rate. Moreover, we will restrictourselves to the region in which the amplitudes of thefluctuations of the fields around their HCS values remainsmall on the average [see below Eq. (12)]. Then retainingup to quadratic order in the deviations, Eq. (1) yields ~Et  ~Et  EHt12ZdrdnH ~Tr; t  d~nr; t ~Tr; tmnHj~ur; tj2: (2)Here, EHt  dNTHt=2, ~nr; t  ~nr; t  nH,~ur; t  ~ur; t, and  ~Tr; t  ~Tr; t  THt. It isnow convenient to introduce dimensionless position, l,and time, s, scales by l  r=l0 and ds  vHtdt=l0, re-spectively, where vH  2THt=m1=2 is the thermal ve-locity and l0  nHd11 is proportional to the meanfree path. Moreover, dimensionless fields are defined byl; s  ~nr; t=nH, !l; s  ~ur; t=vHt, andl; s   ~Tr; t=THt. Then, Eq. (2) takes the forms 0sV1V2Xkksks 2dj!ksj2; (3)where s   ~Es=EHs, V  Ld is the volume of thesystem in the new units, and the Fourier transforms of thefields have been introduced. It is assumed that after a timeof the order of the mean free time, the system reaches aregime in which all its energy is stored in the hydrody-namic modes. In this regime, the hydrodynamic fields areexpected to be described at a mesoscopic level by fluctuat-ing hydrodynamic equations. Here, they will be assumed tobe linear Langevin equations obtained by linearizing theNavier-Stokes equations for a granular gas around theHCS. Moreover, it is postulated that the noise terms aredefined by the same properties as for molecular, elastic2-1 © 2006 The American Physical SocietyPRL 96, 158002 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 APRIL 2006gases [9]. This is not expected to be true, except in thenearly elastic limit, i.e., when  is very close to unity.Consequently, the theory will be restricted in the followingto this limit. Thus, the transversal flow field or vorticityfield, !k?, obeys the following equation in the scaledvariables [6,9]:@s  	=2 	k2!k?s  k?s: (4)In this expression, 	  HTHtl0=vHt and 	 HTHt=mnHl0vHt, H being the shear viscosity.The noise term k?s is Gaussian, withhk?sk0?s0i V2Ns s0k;k0	k2l; (5)l being the unit tensor in the subspace perpendicular to k,and the angular brackets denoting average over the noiserealizations. A main advantage of using the scaled varia-bles is that the coefficients in Eq. (4) and the strength of thenoise are time independent, contrary to what happens in theoriginal variables. The equation shows that !k? grows intime for those values of k such that 	?k  	=2	k2 > 0. Although this does not imply by itself that theHCS is linearly unstable, due to the time-dependent scalingof the velocity introduced above, simulation results andnonlinear analytical analysis of the Navier-Stokes equa-tions have shown that this growth is the origin of theclustering instability [10,11]. The minimum value of kfor a system of linear extent L, measured in the l scale,is kmin  2=L. Then, for given values of the other pa-rameters, the system becomes unstable if L > Lc, withLc  22	=	1=2. For L< Lc, the HCS is stable andthe long time solution of Eq. (4) is! k?s Z s1ds0ess0	?kk?s0: (6)From this expression, it is easily obtainedh!k?s!k0?s0i  V2	k22N	?kess0	?kk;k0 l; (7)for s  s0  1. This shows that as L approaches Lc frombelow, the amplitudes of the fluctuations of the transversalcomponents of the velocity increase very fast due to con-tributions from values of k close to kc. For the same reasonthe decay of these fluctuations becomes very slow. This isnot the case for the fluctuations of the other hydrodynamicfields, whose Langevin equations are decoupled fromEq. (4) [6]. Therefore, it seems possible to consider a rangeof values of fL  Lc  L=Lc where the fluctuations of!k;?, although still small, dominate over the fluctuationsof density and temperature. But, although this is true forcomponents with k > 0, some care is needed when analyz-ing Eq. (3), since it involves 0s. The Langevin equationfor s is obtained from the linearization around the HCSof the macroscopic average equation for the total energy,15800@tEt  d2Zdrnr; tHn; TTr; t: (8)The result is@ss  	s 32V0s: (9)Here, the dependence of the cooling rate on the tempera-ture has been taken into account. Moreover, the noise termdiscussed in Ref. [12], associated with the localized char-acter of the energy dissipation, has been omitted. Althoughit can be expected to be negligible far from the instability inthe quasielastic limit, this may not be the case near theinstability. Equation (9) shows a coupling between thefluctuations of the volume averaged temperature and thoseof the total energy. Use of Eq. (3) into Eq. (9) and neglect-ing contributions from the density and longitudinal veloc-ity fluctuations gives@ss  	2s  !s;!s 6V2dXkj!k?sj2;(10)valid in the region fL 1. The long time limit of theaverage value of s is, therefore,hist  lims!1h !si  3d 1NdXk	k2	?k: (11)Since we are considering fL 1, the sum over k in theabove expression is dominated by the 2d modes with thelargest wavelength, for which 	?kmin ’ 	fL. Usingthis into Eq. (11), it follows that there is a renormalizationby fluctuations of the average total energy of the HCS,Et  h ~Eti, given byEt  EHt13d 1nHLdcfL1: (12)Consistency of the theory we are developing requires thatnHLdcfL1  1, a condition involving the inelasticityand the distance to the instability. Similarly, there is alsoa renormalization of the temperature of the HCS, Tt h ~Ttist, that can be evaluated directly from the long timelimit of the average of Eq. (9),Tt  THt1h0istV THt12d 1nHLdcfL1:(13)Alternatively, an effective temperature Teft can be de-fined as Teft  2Et=Nd. Of course, the form of therenormalized law for the temperature depends on the defi-nition used for the latter. In Ref., [3], what was actuallymeasured was 	ef  efTefl0=vHTef, with ef defined by@tTef  efTefTef . Then, it is found2-2PRL 96, 158002 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 APRIL 2006	ef  	"13d 1nHLdcfL1#1=2: (14)This result predicts that near the clustering instabilitythreshold, 	2ef 	2=	2AfL1 with A3d1=nHLdc , that is just the behavior observed in Ref. [3].Define fEs  ~EsEs=Es shist EHs=Es and  !s  !s  h !ist. Note that we areconsidering deviations from the renormalized average val-ues, i.e., including the fluctuations effects, and not from themacroscopic bare values. A standard calculation usingEq. (7) and exploiting the Gaussian character of the noise,gives that at the instability threshold and for s s01 it ish !s !s0ist 9d 1n2HL2dc dfL2ess0=sc ; (15)where sc  2	fL1 is a divergent ‘‘critical’’ relaxationtime. Now Eq. (10) can be easily solved with the resulthfEsfEs0ist  h !s !s0ist; (16)valid for s  s0  1. Thus below the instability, the scaledtotal energy fluctuations decay with the same rate as thefluctuations of the kinetic energy associated with the trans-versal modes of the velocity. For s  s0, Eq. (16) yields2E  hfE2ist  A2fL2; (17)with A2  9d 1=n2HL2dc d. Therefore, close to the in-stability point, the relative dispersion of the total energyfluctuations E presents a divergent behavior with a criti-cal exponent 1, and an amplitude A depending on nHand  (through the value of the critical length Lc). Again,this is the same behavior as reported in Ref. [3] from MDsimulations.To carry out a more detailed check of the theory pre-sented here, we have performed MD simulations of two-dimensional systems with different values of  and nH (seeTable I). In all cases, the dependence on fL of both thecooling rate and the dispersion of the total energy, i.e., theTABLE I. Comparison between the predicted and MD valuesfor the critical amplitudes of the cooling rate A and the totalenergy dispersion A. All the values of the amplitudes have beenmultiplied by 103.nH2  Atheory AMD Atheory AMD0.02 0.9 0.88 1.11 0.62 0.60.02 0.8 1.62 3.63 1.15 1.50.1 0.98 1.06 0.50 0.75 0.470.1 0.95 2.4 2.38 1.7 1.450.2 0.98 1.97 2.34 1.4 1.340.2 0.95 4.59 7.78 3.24 3.615800exponents in the power laws (14) and (17), was in agree-ment with the theoretical predictions. This was illustratedin Figs. 1 and 2 of Ref. [3] and no more details will begiven here. The comparison between the predicted criticalamplitudes and the MD results given in Table I can beconsidered as satisfactory, in the sense that the theorycorrectly predicts the order of magnitude of the amplitudes,especially taking into account the smallness of the quanti-ties being measured.Next, let us proceed to investigate the form of theprobability distribution of the energy fluctuations.Particularization of Eq. (4) for the modes with the smallestpossible value of k in the limit fL 1 gives@s  	fL!k?s  k?s; (18)where it is understood that jkj  kmin. Define a new timescale d  	fLds, and a new transversal velocity field by! 	k?  !k?=Ldc1=2E . Equation (18) becomes@  1!	k?  	k?; (19)withh	k?	k0?0i d1=26d 11=2k;k0 0l : (20)Equation (19) implies that the probability distribution for!	k? with jkj  kmin near the clustering instability dependsonly on the dimension d of the system. In fact, since thenoise term 	k? is Gaussian, it is trivial to write the longtime form of this distribution using Eq. (7) with s  s0,Pst!	k?  22!d1=2e!	2k?=22!; (21)with 2!  d1=2=12d 11=2. In the time scale , andkeeping only the dominant modes, Eq. (10) readsfL@y   12y6dXjkjkminj!	?kj2; (22)where y  =E and the sum is restricted to vectors kwithjkj  kmin. From the comparison of Eqs. (19) and (22) it isseen that, on the  scale and in the threshold of theinstability, y decays much faster than the dominant com-ponents of !	k?. Consequently, for large  the solution ofEq. (22) is y  6dPjkjkmin j!	?kj2, where the probabilitydistribution of the modes !	?k is given by Eq. (21). Sincethe latter does not depend on the parameters of the systemother than the dimensionality, the same property followsfor the probability distribution of both y and the variablefEE dd 11=2 6dXjkjkminj!	k?j2: (23)2-3-6-4-2 0-2  0  2  4ln [σE P(δ E)]δ E~~ / σEFIG. 1. Probability density function of the relative total energyfluctuations EPfE for a system of inelastic hard disks. Thebroken line is the theoretical prediction derived in this Letter andthe solid line Eq. (7).PRL 96, 158002 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 APRIL 2006This is equivalent to saying that the probability distributionfor fE verifies the scaling relationPfE  1EffEE; (24)where f is a scaling function. This is just the propertyassumed in Ref. [3] and verified by MD simulations. Sincethe probability distribution function for!	k? is known, it ispossible to numerically generate the probability distribu-tion function PfE. The result is shown in Fig. 1. Alsoplotted is the functionfE  Kexexa; xbs fE; a  =2;(25)with K  2:14, b  0:938, and s  0:374, that fits ex-tremely well the MD results for EPfE [3]. It is impor-tant to remark that fluctuations in a large number ofequilibrium and nonequilibrium systems exhibiting self-organized criticality as well as confined turbulent flowspresent the same kind of behavior [4,5]. Although theagreement between both plotted curves is not so bad forpositive values of fE, strong discrepancies are observedfor negative values. A major source for them is easilyidentified from Eq. (23), that for d  2 implies fE=E 2p, while smaller values are found in the MD simula-tions. Since Eq. (23) is a consequence of Eq. (9), it seems15800plausible that in order to elaborate a more accurate theorythe intrinsic noise associated with the cooling rate must betaken into account.In summary, we have developed a mesoscopic theory forthe fluctuations of the total energy of an isolated granulargas near the threshold of the clustering instability. Thetheory describes accurately the qualitative behavior ob-tained in MD simulations, namely, the divergent behaviorof the dimensionless second moment and the decrease ofthe apparent cooling rate. Also, it is consistent with theobserved scaling property of the probability distributionfunction of the fluctuations. On the other hand, it seemsclear that a more refined formulation is needed in order toget a more satisfactory quantitative agreement, especiallyfor the distribution function. Also, it should be interestingto check whether a similar behavior occurs in more real-istic models of granular gases in which  depends on therelative velocity and, although present, the clustering in-stability seems to be a transient phenomenon [13].This research was supported by the Ministerio deEducación y Ciencia (Spain) through Grant No.FIS2005-01398 (partially financed by FEDER funds).2-4*Electronic address: brey@us.es[1] H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Rev. Mod.Phys. 68, 1259 (1996).[2] I. Goldhirsch, Annu. Rev. Fluid Mech. 35, 267 (2003).[3] J. J. Brey, M. I. Garcı́a de Soria, P. Maynar, and M. J. Ruiz-Montero, Phys. Rev. Lett. 94, 098001 (2005).[4] S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton,Nature (London) 396, 552 (1998).[5] S. T. Bramwell, K. Christensen, J.-Y. Fortin, P. C. W.Holdsworth, H. J. Jensen, S. Lise, J. M. López,M. Nicodemi, J.-F. Pinton, and M. Sellitto, Phys. Rev.Lett. 84, 3744 (2000).[6] L. Landau and E. M. Lifshitz, Fluid Mechanics(Peragamon Press, New York, 1959).[7] R. Brito and M. H. Ernst, Europhys. Lett. 43, 497 (1998).[8] P. K. Haff, J. Fluid Mech. 134, 401 (1983).[9] T. P. C. van Noije, M. H. Ernst, R. Brito, and J. A. G. Orza,Phys. Rev. Lett. 79, 411 (1997).[10] I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619(1993); S. McNamara and W. R. Young, Phys. Rev. E 50,R28 (1994).[11] J. J. Brey, M. J. Ruiz-Montero, and D. Cubero, Phys.Rev. E 60, 3150 (1999).[12] J. J. Brey, M. I. Garcı́a de Soria, P. Maynar, and M. J. Ruiz-Montero, Phys. Rev. E 70, 011302 (2004).[13] N. Brilliantov, C. Salueña, T. Schwager, and T. Pöschel,Phys. Rev. Lett. 93, 134301 (2004).

Mesoscopic theory of critical fluctuations in isolated granular gases

Mesoscopic Theory of Critical Fluctuations in Isolated Granular Gases

Mesoscopic Theory of Critical Fluctuations in Isolated Granular Gases

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idUS. Depósito de Investigación Universidad de Sevilla