Energy Transport in a One-Dimensional Granular Gas
Italo Ivo Lima Dias Pinto, Alexandre Rosas, Katja Lindenberg
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Energy transport in a one-dimensional granular gas ´Italo’Ivo Lima Dias Pinto , Alexandre Rosas , and Katja Lindenberg Departamento de F´ısica, CCEN, Universidade Federal da Para´ıba,Caixa Postal 5008, 58059-900, Jo˜ao Pessoa, Brazil Department of Chemistry and Biochemistry, and Institute for Nonlinear Science,University of California San Diego, La Jolla, CA 92093-0340
We study heat conduction in one-dimensional granular gases. In particular, we consider twomechanisms of viscous dissipation during inter-grain collisions. In one, the dissipative force is pro-portional to the grain’s velocity and dissipates not only energy but also momentum. In the other,the dissipative force is proportional to the relative velocity of the grains and therefore conservesmomentum even while dissipating energy. This allows us to explore the role of momentum conser-vation in the heat conduction properties of this one-dimensional nonlinear system. We find normalthermal conduction whether or not momentum is conserved.
PACS numbers: 45.70.-n,05.20.Dd,05.70.Ln,83.10.Rs
I. INTRODUCTION
Energy transport in low-dimensional systems can bepathological in the sense that Fourier’s law for heatconduction might break down [1, 2, 3, 4]. In three-dimensional solids, the energy (heat) transported is gov-erned by Fourier’s law, which says that the heat flux J is proportional to the gradient of the temperature, J = − κ ∇ T, (1)where κ is the thermal conductivity. The thermal con-ductivity in Fourier’s law is independent of system sizeand of time. This equation assumes that a local equi-librium is established at each time, so that one can de-fine the local energy flux J ( x, t ) and temperature T ( x, t ),when the temperature gradient lies along the x direc-tion [4]. In one-dimensional systems one might expecta similar equation, with the gradient replaced by thederivative of T ( x, t ) with respect to x , and J simplybeing the energy flux per unit time. However, it hasbeen observed that in many one-dimensional models thethermal conductivity varies with the system size N as κ ∼ N α , with α lying between 0.32 and 0.44 [4, 5, 6]. Self-consistent mode coupling analysis for one-dimensionalnonlinear chains [7] predicts a universal value of α = 2 / α =1 /
3. Recently, Mai et al. [5, 6] revisited the problem andobtained the same α = 1 / all length scales even if the static order of thechain extends to very large N . They explain the dis-agreement with most numerical results as a consequenceof the numerical difficulties in reaching asymptotic be-havior, which requires extremely large N , but their con-clusions have in turn been disputed [6]. Whether thereis a single universality class or there are two, there is aconnection between heat conduction and diffusion in one-dimensional systems [8], that is, normal diffusion leads tonormal heat conductivity ( κ ∼ N ), whereas anomalous transport is associated with κ ∼ N α , with α > < d systems is momentum conserva-tion [4]. In fact, momentum conservation usually impliesthe divergence of the thermal conductivity in 1 d [1, 9],and yet it is not necessary for the occurrence of anoma-lous diffusion [10]. An exception to the momentum con-servation rule is a chain of particles interacting via thenearest-neighbor potential V ( q i − q i − ) = 1 − cos( q i − q i − ), the so-called rotator model [11]. This momentum-conserving model exhibits normal transport behavior tovery high numerical accuracy, and is said to occur be-cause the rotator model “cannot support a nonvanishingpressure, and thus infinite-wavelength phonons cannotcarry energy” [10]. In [4] this behavior is ascribed to theperiodicity of the potential and the associated indepen-dent jumps from valley to valley.In this contribution we discuss heat conduction in one-dimensional granular gases. Granular gases are dissi-pative systems, the associated energy loss usually beingmodeled by introducing a coefficient of restitution as aparameter in the description of granular collisions. In-stead, we follow a more dynamical approach and intro-duce energy dissipation explicitly via viscous terms in theequations of motion for the granules. There are a numberof different sources and descriptions of viscous effects inthe literature [12, 13, 14, 15, 16, 17, 18]. The differentways in which they appear in the dynamical descriptionssuits our particular focus of interest, which is the roleof momentum conservation in the heat transport pro-cess. In particular, one way to introduce viscosity con-serves momentum [19, 20, 21, 22], while the other doesnot [18, 23, 24]. Interestingly, we establish that thermalconduction exhibits normal behavior in both cases, thatis, that there is no divergence as the system size increases.In Sec. II we describe the two dissipative models, andin Sec. III we present simulation results to characterizethe transport of heat through the granular gas in bothcases. We conclude with a short summary in Sec.IV. II. DISSIPATIVE MODELS
We consider N identical granules constrained to moveon a line between two walls at different temperatures.The granules move freely except during collisions eitherwith the walls or with one another. The system is a one-dimensional “granular gas” because the distance betweenthe walls is much greater than the space occupied by thegranules. The walls act as heat baths, that is, whenever agranule collides with a wall at temperature T , its energyis absorbed by the wall and it acquires a new velocityaway from the wall according to the probability distribu-tion [25] P ( v ) = | v | T exp (cid:0) − v / k B T (cid:1) . (2)Interparticle collisions are governed by the power-law po-tential V ( δ k,k +1 ) = an | δ | nk,k +1 , δ ≤ ,V ( δ k,k +1 ) = 0 , δ > . (3)Here δ k,k +1 ≡ y k +1 − y k , a is a prefactor determinedby Young’s modulus and Poisson’s ratio, and the prin-cipal radius of curvature R of the surfaces at the pointof contact [26, 27]; and y k is the displacement of gran-ule k from its position at the beginning of the collision.The exponent n is 5 / n = 2 is entirely different from atwo-sided (repulsion and attraction) harmonic potential.The one-sidedness of the potential leads to analytic com-plexities even in the dissipationless case [24, 28, 29],and even greater complexities in the presence of dissi-pation [18, 19, 20, 24].In this paper we explore the low density limit, whichleads to enormous analytical and computational simplifi-cations. The low density feature of these approximationsis implemented via the assumption that the collisions arealways binary [30], that is, that only two granules at atime are members of any collision event, and that at anymoment of time there is at most one collision.Our approach starts with the equations of motion ofthe particles labeled by index k , and so we write y k as afunction of time τ . It is convenient to deal with scaledposition and time variables x k and t , related to the un-scaled variables y k and τ as follows, y k = (cid:18) mv a (cid:19) /n x k , τ = 1 v (cid:18) mv a (cid:19) /n t. (4)Here m is the mass of the granules, and the velocity v isan arbitrary choice in terms of which other velocities areexpressed. We also introduce a scaled friction coefficient γ = ˜ γmv (cid:18) mv a (cid:19) /n , (5) where ˜ γ is the friction coefficient for the unscaled vari-ables. In the low density limit we only need to con-sider the equations of motion for two colliding granules k = 1 ,
2. Furthermore, in this paper we only considercylindrical grains, which leads to considerable simplifica-tion while still capturing the important general featuresof the system that we seek to highlight. We stress thatthe one-sided granular potential (i.e., one with only re-pulsive interactions) even with n = 2 is entirely differentfrom a two-sided harmonic potential.Consider a viscous force that is proportional to the rel-ative velocity of the colliding granules. Such a force hasbeen considered not only theoretically [12, 19, 20] but ithas also been observed experimentally [21, 22]. While onemight be tempted to think of this force as arising becauseone grain rubs against the other, the actual mechanismis more complicated and involves the medium that sur-rounds the granules [21, 22]. The exact mechanism isstill a matter of conjecture. In any case, the appropriateequations of motion in this case are¨ x = [ − γ ( ˙ x − ˙ x ) − ( x − x )] θ ( x − x ) , ¨ x = [ γ ( ˙ x − ˙ x ) + ( x − x )] θ ( x − x ) , (6)where a dot denotes a derivative with respect to t . TheHeaviside function is defined as θ ( x ) = 1 for x > θ ( x ) = 0 for x <
0, and θ (0) = 1 /
2. It ensures thatthe two particles interact only when in contact, that is,only when the particles are loaded. The post-collisionalvelocities (called u below) can be written in terms of thevelocities of the two granules at the beginning of the col-lision (called v ) as u = 12 (cid:2) − e − γt (cid:3) v + 12 (cid:2) e − γt (cid:3) v ,u = 12 (cid:2) e − γt (cid:3) v + 12 (cid:2) − e − γt (cid:3) v , (7)where t = π/ ( p − γ ) is the duration of the collision.Since u + u = v + v , the momentum of the center ofmass is conserved.A model in which the center of mass velocity is notconserved is one governed by the equations of motion [18,24] ¨ x = [ − γ ˙ x − ( x − x )] θ ( x − x ) , ¨ x = [ − γ ˙ x + ( x − x )] θ ( x − x ) . (8)Here one might think of the viscosity arising from an in-teraction with the medium. However, such an interactionwould produce a damping term not only during a colli-sion but also while the granules are moving independentlybetween collisions. In the low density limit the granuleswould hardly collide before stopping entirely unless theviscosity is extremely low, and one can then not talk ofheat transport along the one-dimensional system. Thus,this model, while perhaps not realistic in any sense, is a“toy model” in which momentum is not conserved whilestill capable of supporting energy transport and thereforerelevant to our question. Indeed, we can again calculatethe post-collision velocities, u = 12 h e − γt − e − γt / i v + 12 h e − γt + e − γt / i v ,u = 12 h e − γt + e − γt / i v + 12 h e − γt − e − γt / i v , (9)where t = 2 π/ ( p − γ ) is again the duration of thecollision. It is easy to verify that in this case the momen-tum of the center of mass decays from v + v before acollision to u + u = [ v + v ] e − γt , (10)after the collision, so that the momentum is not con-served.Caution must be exercised in the choice of the dampingcoefficient γ . If it is too large, energy acquired from eitherwall is simply dissipated before it crosses the system, inwhich case the problem changes from one of energy trans-fer from one wall to the other through the granular gasto that of two walls pumping energy into the granular“sink.” To estimate the limit on the damping coefficientwe can imagine a sequence of collisions whereby a granulestarting from the hot wall with kinetic energy E collideswith the next granule, which in turn collides with thenext one, and so on, until the last granule, whose en-ergy is E N , collides with the cold wall. The change inthe kinetic energy due to each collision in the momentumconserving model is∆ E ≡
12 ( u + u ) −
12 ( v + v ) = − (cid:0) − e − γt (cid:1) ( v + v ) , (11)and in the momentum non-conserving model it is∆ E ≡
12 ( u + u ) −
12 ( v + v ) = − (cid:0) − e − γt (cid:1) ( v + v ) . (12)In both cases the kinetic energy before and after the col-lision are related by an expression of the form E after = E before (1 − bγt ) where b is a velocity-dependent dimen-sionless quantity of order unity, and where we have as-sumed that γt ≪
1. As an order of magnitude estimateit is sufficient to write E N ∼ E (1 − γt ) N . The averageenergy of a hot granule is k B T / k B T / γ . − (cid:18) T T (cid:19) /N . (13)We now wish to explore the following specific ques-tions. Is there a finite thermal conductivity for eitheror both of these models? If so, which model has ahigher/lower thermal conductivity? How does the ther-mal conductivity depend on temperature? On viscosity?On system size? These are the questions we address inthe next section. III. NUMERICAL SIMULATIONS -5 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 7 8 9 10 E ( x ) time (x 10 ) FIG. 1: (Color online) From top to bottom: energy injectionby the hot wall (red), energy absorption by the cold wall (ma-genta), energy injection by the cold wall (blue), and energyabsorption by the hot wall (green), for a momentum non-conserving system of N = 500 particles between walls main-tained at T l = 6 and T r = 3 and with γ = 0 . T = 1. The low density limit allows us to use an event drivenalgorithm in our simulations. As indicated earlier, whena granule collides with a wall at temperature T its en-ergy is absorbed by the wall and it acquires a new en-ergy as determined by the velocity distribution given inEq. (2). The initial temperature of the granular gas is ar-bitrarily set to T = 1, and we allow the system to arriveat a steady state before beginning our “measurements.”While ideally one would want the temperature gradientin the steady state to be strictly linear as assumed inthe linear response theory that leads to the Fourier law,the subtleties encountered in 1 d systems are well-knownand observed pretty much no matter how the thermal-ization is implemented [6, 25] and the resulting gradientsare only strictly linear away from the boundaries (non-linear behavior typically sets in near the boundaries). Inany case, we can address the question of whether or notthere are system size divergences in the transport of heat.That we do in fact achieve a steady state can be seen, forexample, in Fig. 1, where we show the energy absorbedand injected by each wall as a function of time for thedissipating momentum model. After a short transient,the rate of energy injection and absorption become con-stant, as they should in a steady state. We ran teststo ascertain that the initial temperature of the granu-lar gas is not important for this equilibration, that is,we find that the rates of energy injection and absorptionare independent of the initial velocity distribution of thegranular gas. The momentum-conserving model equili-brates equally well, also independently of initial condi-tion. Henceforth we set the initial temperature of thegranular gas to unity. -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 2 4 6 8 10 12 J ∆ T FIG. 2: (Color online) Rates of energy transmission as a func-tion of the temperature gradient. In this simulation the vis-cosity coefficient was set to γ = 0 . κ ( T ) T FIG. 3: (Color online) Thermal conductivity as a function ofthe temperature. In this simulation the viscosity coefficientwas set to γ = 0 . Next, we discuss the dependence of the energy flux onthe temperature gradient. The energy flux J = dE/dt was calculated as the slope of the transmitted energyas a function of time, that is [(energy injection by thehot wall - energy absorption by the hot wall) - (energyinjection by the cold wall - energy absorption by the cold κ ( γ ) γ (x 10 -5 ) FIG. 4: (Color online) Thermal conductivity as a function ofthe viscosity coefficient. In this simulation the temperatureof the cold wall is set to 3 and that of the hot wall to 6.The system is composed of 500 granules. The (red) plus signscorrespond to the momentum conserving system, the (blue)stars to the momentum dissipating system. J N FIG. 5: (Color online) Rates of energy transmission as a func-tion of the system size. In this simulation the viscosity co-efficient was set to γ = 0 . wall) - (energy dissipated during flow along the chain)]per unit time. As seen in Fig. 2, both models lead tobehavior fairly well described by Fourier’s law. Repeatingthis plot for different temperatures of the cold wall, weobtain the dependence of the thermal conductivity on thetemperature. As observed in Fig. 3, κ is an increasingfunction of T for both models, but it is larger for themomentum dissipating system. On the other hand, fora given temperature, the thermal conductivity decreases(almost linearly) with the viscosity (see Fig. 4). Onceagain, the dissipating system presents larger values of κ . J γ N FIG. 6: (Color online) Rates of energy transmission as a func-tion of scaled system size. The temperatures of the walls wereset to 3 and 6. The (red) plus signs correspond to the mo-mentum conserving system, the (blue) stars to the momentumdissipating system.
Finally, we have analyzed the dependence of the rateof energy transmission on system size, for a fixed temper-ature gradient and fixed viscosity. This is the crucial testof normal vs anomalous behavior. In Fig. 5 we can clearlysee that as the size of the system increases, the rate ofenergy transmission does not increase with system size.In fact, it decreases . Therefore, in our model the ther-mal conductivity does not diverge with increasing systemsize regardless of momentum conservation or dissipation.One may be tempted to think of this behavior entirelyas a consequence of the energy dissipation because forbigger systems the number of collisions necessary for theenergy to be transmitted from one end of the systemto the other increases. Hence more energy is dissipatedand less energy arrives at the cold wall. However, theintroduction of dissipation changes the dynamics moreprofoundly. This can be seen in Fig. 6, where we plot therates of energy transmission against γN , thus taking intoaccount the “simple” effect of energy dissipation [see dis-cussion preceding Eq.(13)]. The rate of energy transfer inthe momentum conserving system is essentially indepen-dent of system size in this scaled representation, but thatof the momentum dissipating system actually decreases even when scaled in this way. In any case, we note thevery small scale of variation of the ordinate in Fig. 6. IV. CONCLUSIONS
Heat transport in one-dimensional discrete systemscontinues to be a problem of theoretical interest, un-certainty, and even controversy [1, 2, 3, 4, 5, 6]. Veryrecently, experimental results in this arena have alsostarted to appear. In [31] the focus is the understandingof transport of heat along colliding granular beads in aliquid medium where the questions of interest involve thenature of the contact regions (“liquid bridges”) betweengranules. In [32] the issue is the breakdown of Fourier’slaw in nanotube thermal conductors in that the thermalconductivity diverges with length of the nanotube. Thepoint is that even after many years of study the condi-tions that lead to the validity or violation of Fourier’s laware not yet clear. While momentum conservation has of-ten been featured as a condition closely associated withthe system size divergence of the 1 d thermal conductiv-ity, we have examined two dissipative granular gases, oneof which involves momentum dissipation along with en-ergy dissipation, whereas in the other momentum is con-served. In both of these the thermal conductivity remainsfinite and in fact decreases as the number of granules inthe system increases, as it must in a system where eachcollision leads to energy dissipation. In the momentumdissipating model, it decreases more rapidly than can beaccounted for by the simply heat loss mechanism of thecollisions, indicating a more profound change in the dy-namics. This behavior points to the caution that mustbe exercised when associating momentum conservationwith anomalous behavior in one dimension. Acknowledgments
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