Energy transport in a one-dimensional granular gas

Energy transport in a one-dimensional granular gas Ítalo’Ivo Lima Dias Pinto1 , Alexandre Rosas1, and Katja Lindenberg2 1 arXiv:0902.2401v1 [cond-mat.stat-mech] 13 Feb 2009 Departamento de Fı́sica, CCEN, Universidade Federal da Paraı́ba, Caixa Postal 5008, 58059-900, João Pessoa, Brazil 2 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 two mechanisms of viscous dissipation during inter-grain collisions. In one, the dissipative force is proportional 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 conserves momentum even while dissipating energy. This allows us to explore the role of momentum conservation in the heat conduction properties of this one-dimensional nonlinear system. We find normal thermal 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 be pathological in the sense that Fourier’s law for heat conduction might break down [1, 2, 3, 4]. In threedimensional solids, the energy (heat) transported is governed 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 conductivity in Fourier’s law is independent of system size and of time. This equation assumes that a local equilibrium is established at each time, so that one can define the local energy flux J(x, t) and temperature T (x, t), when the temperature gradient lies along the x direction [4]. In one-dimensional systems one might expect a similar equation, with the gradient replaced by the derivative of T (x, t) with respect to x, and J simply being the energy flux per unit time. However, it has been observed that in many one-dimensional models the thermal conductivity varies with the system size N as κ ∼ N α , with α lying between 0.32 and 0.44 [4, 5, 6]. Selfconsistent mode coupling analysis for one-dimensional nonlinear chains [7] predicts a universal value of α = 2/5, while renormalization group analysis based on hydrodynamical models supports a universality class with α = 1/3. Recently, Mai et al. [5, 6] revisited the problem and obtained the same α = 1/3 behavior for one-dimensional nonlinear chains as obtained for the fluid-like systems. They argue that the dynamic equations are those of a fluid at all length scales even if the static order of the chain extends to very large N . They explain the disagreement with most numerical results as a consequence of the numerical difficulties in reaching asymptotic behavior, which requires extremely large N , but their conclusions have in turn been disputed [6]. Whether there is a single universality class or there are two, there is a connection between heat conduction and diffusion in onedimensional systems [8], that is, normal diffusion leads to normal heat conductivity (κ ∼ N 0 ), whereas anomalous transport is associated with κ ∼ N α , with α > 0 (< 0) for super(sub)diffusion. Thus, superdiffusion is associated with a divergent thermal conductivity. Another key concept in the understanding of the thermal conductivity in 1d systems is momentum conservation [4]. In fact, momentum conservation usually implies the divergence of the thermal conductivity in 1d [1, 9], and yet it is not necessary for the occurrence of anomalous diffusion [10]. An exception to the momentum conservation rule is a chain of particles interacting via the nearest-neighbor potential V (qi − qi−1 ) = 1 − cos(qi − qi−1 ), the so-called rotator model [11]. This momentumconserving model exhibits normal transport behavior to very high numerical accuracy, and is said to occur because the rotator model “cannot support a nonvanishing pressure, and thus infinite-wavelength phonons cannot carry energy” [10]. In [4] this behavior is ascribed to the periodicity of the potential and the associated independent jumps from valley to valley. In this contribution we discuss heat conduction in onedimensional granular gases. Granular gases are dissipative systems, the associated energy loss usually being modeled by introducing a coefficient of restitution as a parameter in the description of granular collisions. Instead, we follow a more dynamical approach and introduce energy dissipation explicitly via viscous terms in the equations of motion for the granules. There are a number of different sources and descriptions of viscous effects in the literature [12, 13, 14, 15, 16, 17, 18]. The different ways in which they appear in the dynamical descriptions suits our particular focus of interest, which is the role of momentum conservation in the heat transport process. In particular, one way to introduce viscosity conserves momentum [19, 20, 21, 22], while the other does not [18, 23, 24]. Interestingly, we establish that thermal conduction exhibits normal behavior in both cases, that is, that there is no divergence as the system size increases. In Sec. II we describe the two dissipative models, and in Sec. III we present simulation results to characterize the transport of heat through the granular gas in both cases. We conclude with a short summary in Sec.IV. 2 II. DISSIPATIVE MODELS We consider N identical granules constrained to move on a line between two walls at different temperatures. The granules move freely except during collisions either with the walls or with one another. The system is a onedimensional “granular gas” because the distance between the walls is much greater than the space occupied by the granules. The walls act as heat baths, that is, whenever a granule collides with a wall at temperature T , its energy is absorbed by the wall and it acquires a new velocity away from the wall according to the probability distribution [25] P (v) =
|v| exp −v 2 /2kB T . T (2) Interparticle collisions are governed by the power-law potential a |δ|n V (δk,k+1 ) = n δ ≤ 0, k,k+1 , (3) V (δk,k+1 ) = 0, δ > 0. Here δk,k+1 ≡ yk+1 − yk , a is a prefactor determined by Young’s modulus and Poisson’s ratio, and the principal radius of curvature R of the surfaces at the point of contact [26, 27]; and yk is the displacement of granule k from its position at the beginning of the collision. The exponent n is 5/2 for spheres (Hertz potential), it is 2 for cylinders, and in general depends on geometry. We stress that the one-sided (only repulsion) granular potential even with n = 2 is entirely different from a two-sided (repulsion and attraction) harmonic potential. The one-sidedness of the potential leads to analytic complexities even in the dissipationless case [24, 28, 29], and even greater complexities in the presence of dissipation [18, 19, 20, 24]. In this paper we explore the low density limit, which leads to enormous analytical and computational simplifications. The low density feature of these approximations is implemented via the assumption that the collisions are always binary [30], that is, that only two granules at a time are members of any collision event, and that at any moment of time there is at most one collision. Our approach starts with the equations of motion of the particles labeled by index k, and so we write yk as a function of time τ . It is convenient to deal with scaled position and time variables xk and t, related to the unscaled variables yk and τ as follows, 1/n 1/n   mv02 1 mv02 yk = xk , τ= t. (4) a v0 a Here m is the mass of the granules, and the velocity v0 is an arbitrary choice in terms of which other velocities are expressed. We also introduce a scaled friction coefficient 1/n  γ̃ mv02 γ= , (5) mv0 a where γ̃ is the friction coefficient for the unscaled variables. In the low density limit we only need to consider the equations of motion for two colliding granules k = 1, 2. Furthermore, in this paper we only consider cylindrical grains, which leads to considerable simplification while still capturing the important general features of the system that we seek to highlight. We stress that the one-sided granular potential (i.e., one with only repulsive interactions) even with n = 2 is entirely different from a two-sided harmonic potential. Consider a viscous force that is proportional to the relative velocity of the colliding granules. Such a force has been considered not only theoretically [12, 19, 20] but it has also been observed experimentally [21, 22]. While one might be tempted to think of this force as arising because one grain rubs against the other, the actual mechanism is more complicated and involves the medium that surrounds the granules [21, 22]. The exact mechanism is still a matter of conjecture. In any case, the appropriate equations of motion in this case are ẍ1 = [−γ(ẋ1 − ẋ2 ) − (x1 − x2 )] θ(x1 − x2 ), ẍ2 = [γ(ẋ1 − ẋ2 ) + (x1 − x2 )] θ(x1 − x2 ), (6) where a dot denotes a derivative with respect to t. The Heaviside function is defined as θ(x) = 1 for x > 0, θ(x) = 0 for x < 0, and θ(0) = 1/2. It ensures that the two particles interact only when in contact, that is, only when the particles are loaded. The post-collisional velocities (called u below) can be written in terms of the velocities of the two granules at the beginning of the collision (called v) as   1 1 1 − e−γt0 v1 + 1 + e−γt0 v2 , 2 2   1 1 −γt0 u2 = v1 + (7) 1+e 1 − e−γt0 v2 , 2 2 p where t0 = π/( 2 − γ 2 ) is the duration of the collision. Since u1 + u2 = v1 + v2 , the momentum of the center of mass is conserved. A model in which the center of mass velocity is not conserved is one governed by the equations of motion [18, 24] u1 = ẍ1 = [−γ ẋ1 − (x1 − x2 )] θ(x1 − x1 ), ẍ2 = [−γ ẋ2 + (x1 − x2 )] θ(x1 − x1 ). (8) Here one might think of the viscosity arising from an interaction with the medium. However, such an interaction would produce a damping term not only during a collision but also while the granules are moving independently between collisions. In the low density limit the granules would hardly collide before stopping entirely unless the viscosity is extremely low, and one can then not talk of heat 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 while still capable of supporting energy transport and therefore 3 u1 + u2 = [v1 + v2 ] e−γt0 , (10) after the collision, so that the momentum is not conserved. Caution must be exercised in the choice of the damping coefficient γ. If it is too large, energy acquired from either wall is simply dissipated before it crosses the system, in which case the problem changes from one of energy transfer from one wall to the other through the granular gas to that of two walls pumping energy into the granular “sink.” To estimate the limit on the damping coefficient we can imagine a sequence of collisions whereby a granule starting from the hot wall with kinetic energy E0 collides with the next granule, which in turn collides with the next one, and so on, until the last granule, whose energy is EN , collides with the cold wall. The change in the kinetic energy due to each collision in the momentum conserving model is
1 1 1 ∆E ≡ (u21 +u22 )− (v12 +v22 ) = − 1 − e−2γt0 (v1 +v2 )2 , 2 2 4 (11) and in the momentum non-conserving model it is
1 1 1 ∆E ≡ (u21 +u22 )− (v12 +v22 ) = − 1 − e−γt0 (v1 +v2 )2 . 2 2 4 (12) In both cases the kinetic energy before and after the collision are related by an expression of the form Eaf ter = Ebef ore (1 − bγt0 ) where b is a velocity-dependent dimensionless quantity of order unity, and where we have assumed that γt0 ≪ 1. As an order of magnitude estimate it is sufficient to write EN ∼ E0 (1 − γt0 )N . The average energy of a hot granule is kB T1 /2 and in order for energy to flow across the system the energy of the granule that arrives at the cold wall must be greater than kB T2 /2. Consequently, we estimate that a requirement for energy transport is that  1/N T2 γ .1− . (13) T1 We now wish to explore the following specific questions. Is there a finite thermal conductivity for either or both of these models? If so, which model has a higher/lower thermal conductivity? How does the thermal conductivity depend on temperature? On viscosity? On system size? These are the questions we address in the next section. III. 3 E (x 10 ) relevant to our question. Indeed, we can again calculate the post-collision velocities, i i 1 h −γt0 1 h −γt0 − e−γt0 /2 v1 + + e−γt0 /2 v2 , u1 = e e 2 2 i i 1 h −γt0 1 h −γt0 −γt0 /2 u2 = v1 + +e − e−γt0 /2 v2 , e e 2 2 (9) p where t0 = 2π/( 8 − γ 2 ) is again the duration of the collision. It is easy to verify that in this case the momentum of the center of mass decays from v1 + v2 before a collision to NUMERICAL SIMULATIONS 45 40 35 30 25 20 15 10 5 0 -5 0 1 2 3 4 5 6 7 8 9 10 5 time (x 10 ) FIG. 1: (Color online) From top to bottom: energy injection by the hot wall (red), energy absorption by the cold wall (magenta), energy injection by the cold wall (blue), and energy absorption by the hot wall (green), for a momentum nonconserving system of N = 500 particles between walls maintained at Tl = 6 and Tr = 3 and with γ = 0.0002. The dashed lines are the linear regressions of these curves (after the transient). They yield the rate of energy injection/absorption. The initial temperature of the granular gas is T = 1. The low density limit allows us to use an event driven algorithm in our simulations. As indicated earlier, when a granule collides with a wall at temperature T its energy is absorbed by the wall and it acquires a new energy as determined by the velocity distribution given in Eq. (2). The initial temperature of the granular gas is arbitrarily set to T = 1, and we allow the system to arrive at a steady state before beginning our “measurements.” While ideally one would want the temperature gradient in the steady state to be strictly linear as assumed in the linear response theory that leads to the Fourier law, the subtleties encountered in 1d systems are well-known and observed pretty much no matter how the thermalization is implemented [6, 25] and the resulting gradients are only strictly linear away from the boundaries (nonlinear behavior typically sets in near the boundaries). In any case, we can address the question of whether or not there are system size divergences in the transport of heat. That we do in fact achieve a steady state can be seen, for example, in Fig. 1, where we show the energy absorbed and injected by each wall as a function of time for the dissipating momentum model. After a short transient, the rate of energy injection and absorption become constant, as they should in a steady state. We ran tests to ascertain that the initial temperature of the granular gas is not important for this equilibration, that is, we find that the rates of energy injection and absorption are independent of the initial velocity distribution of the granular gas. The momentum-conserving model equili- 4 brates equally well, also independently of initial condition. Henceforth we set the initial temperature of the granular gas to unity. κ (γ) 0.16 0.14 0.12 0.1 J 0.08 0.06 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.01 0.009 5 0.04 10 15 20 -5 γ (x 10 ) 0.02 0 -0.02 0 2 4 6 8 10 12 ∆T FIG. 2: (Color online) Rates of energy transmission as a function of the temperature gradient. In this simulation the viscosity coefficient was set to γ = 0.0002 and the system is composed of 500 granules. The (red) plus signs correspond to the momentum conserving system, while the (blue) stars correspond to the momentum dissipating system. In this figure, the cold wall temperature was set to 3. FIG. 4: (Color online) Thermal conductivity as a function of the viscosity coefficient. In this simulation the temperature of 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 signs correspond to the momentum conserving system, the (blue) stars to the momentum dissipating system. 0.05 0.04 κ (T) J 0.03 0.02 0.019 0.018 0.017 0.016 0.015 0.014 0.013 0.012 0.011 0.01 0.009 0.02 0.01 100 200 300 400 500 600 700 800 900 1000 N 2 4 6 8 10 12 T FIG. 3: (Color online) Thermal conductivity as a function of the temperature. In this simulation the viscosity coefficient was set to γ = 0.0002 and the system is composed of 500 granules. The (red) plus signs correspond to the momentum conserving system, the (blue) stars to the momentum dissipating system. Next, we discuss the dependence of the energy flux on the temperature gradient. The energy flux J = dE/dt was calculated as the slope of the transmitted energy as a function of time, that is [(energy injection by the hot wall - energy absorption by the hot wall) - (energy injection by the cold wall - energy absorption by the cold FIG. 5: (Color online) Rates of energy transmission as a function of the system size. In this simulation the viscosity coefficient was set to γ = 0.0001 and the temperatures of the walls were set to 3 and 6. The (red) plus signs correspond to the momentum conserving system, the (blue) stars to the momentum dissipating system. wall) - (energy dissipated during flow along the chain)] per unit time. As seen in Fig. 2, both models lead to behavior fairly well described by Fourier’s law. Repeating this plot for different temperatures of the cold wall, we obtain the dependence of the thermal conductivity on the temperature. As observed in Fig. 3, κ is an increasing function of T for both models, but it is larger for the momentum dissipating system. On the other hand, for a given temperature, the thermal conductivity decreases (almost linearly) with the viscosity (see Fig. 4). Once again, the dissipating system presents larger values of κ. 5 even when scaled in this way. In any case, we note the very small scale of variation of the ordinate in Fig. 6. 0.0415 0.041 0.0405 J 0.04 IV. CONCLUSIONS 0.0395 0.039 0.0385 0.038 0.0375 100 200 300 400 500 600 700 γN FIG. 6: (Color online) Rates of energy transmission as a function of scaled system size. The temperatures of the walls were set to 3 and 6. The (red) plus signs correspond to the momentum conserving system, the (blue) stars to the momentum dissipating system. Finally, we have analyzed the dependence of the rate of energy transmission on system size, for a fixed temperature gradient and fixed viscosity. This is the crucial test of normal vs anomalous behavior. In Fig. 5 we can clearly see that as the size of the system increases, the rate of energy transmission does not increase with system size. In fact, it decreases. Therefore, in our model the thermal conductivity does not diverge with increasing system size regardless of momentum conservation or dissipation. One may be tempted to think of this behavior entirely as a consequence of the energy dissipation because for bigger systems the number of collisions necessary for the energy to be transmitted from one end of the system to the other increases. Hence more energy is dissipated and less energy arrives at the cold wall. However, the introduction of dissipation changes the dynamics more profoundly. This can be seen in Fig. 6, where we plot the rates of energy transmission against γN , thus taking into account the “simple” effect of energy dissipation [see discussion preceding Eq.(13)]. 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