KINETIC EQUATIONS WITH INTERMOLECULAR POTENTIALS: AVOIDING THE BOLTZMANN ASSUMPTION William Greenberg Dept. of Mathematics, Virginia Tech Blacksburg, Virginia 24060 USA Email: GREENBERG @ VT.EDU ABSTRACT The Boltzmann equation describes the time eolution of a dilute gas, and is the best known transport equation in kinetic theory. Its drawback is that, although it allows for a ariety of long and short range intermolecular potentials, it does not predict non-ideal transport coefficients. To model gas kinetics in denser regimes, the most successful Boltzmann-like equation has been the Enskog equation, which takes into account molecular diameters. Howeer, it does hae the drawback that it does not include intermolecular potentials. We present seeral extentions of the Enskog theory which model intermolecular forces. The first of these includes a piecewise-constant short range potential. The second models longrange forces by coupling the Enskog equation to the electromagnetic field ia a Vlaso collision term. Finally, we introduce discrete elocity models. All of these open new fields for numerical analysis. I. Introduction The fundamental problem of kinetic theory is to describe the properties of a gas in terms of the one-particle distribution function f(x,,t), generally gien by an appropriate eolution equation. This distribution function is defined so that f(x,,t) dx d is the number of molecules which at time t lie in a olume element dx around a point x with elocity in a olume element d around. Its moments then proide fluid-dynamical ariables characterizing the gas. One of the first eolution equations for the distribution function was the Boltzmann equation, first proposed by Ludwig Boltzmann in 872.[] It has the particularly useful feature that its bilinear collision term can model a great ariety of intermolecular potentials. This equation gies the most precise modeling of dilute gases; for example, it is the equation of choice for the precise modeling of atmospheric gases. The Boltzmann equation may be written: (, x t,) + x f (, xt,) = 3 2 = dd ' ε{ f ( x, ', t) f ( x, ', t) f ( xt,, ) f ( x,, t)} K( ε,, t) Here,, and ', ' may be thought of as the pre- and post-collision elocities of two colliding particles with unit ector ε in the direction of '. The kernel K characterizes the interaction potential between the colliding particles. The post-collision elocities are related to the pre-collision elocities by momentum and energy conseration: ' = ε ( ) ε, ' = + ε ( ) ε The Boltzmann equation is obtained by taking a limit as the diameter of molecules anishes while the density is held constant. This is a fundamental assumption of 47 MSAS'2004
Boltzmann, and is why this equation describes a gas of point particles; its equation of state is that of an ideal gas, and yields only triial transport coefficients. Consequently, it can not be expected to proide an adequate description of a dense gas. In 92 Enskog proposed a scheme to surmount this Boltzmann assumption, thereby obtaining a modified Boltzmann-like equation which takes into account the non-zero molecular diameter.[2] In particular, the collision transfer does not take place at a gien point, but rather momentum is transferred upon collision from the center of one molecule to the center of the other, which is at a distance equal to the molecular diameter. The Enskog equation for molecules of diameter σ may be written: ( x, t, ) + f( xt,, ) = x 3 2 = d ' d εθ( V ε) { Y( f, x+ σε) f ( x, ', t) f ( x+ σε, ', t) Y( f, x σε) f( x,, t) f( x σε,, t)} where the kernel Y is a function of the local density at the contact point of the molecules. Here, θ is the characteristic function of the positie half-line, and V =. The collision frequency factor Y, which should represent the pair correlation function corresponding to the system in uniform equilibrium, may be chosen so that the Enskog equation, like the Boltzmann equation, satisfies an H-theorem. The Enskog equation allows for the computation of transport coefficients which, compared with those measured experimentally, are more accurate than the ones proided by the Boltzmann equation. Howeer, it does hae one eident limitation. It describes only the collision of hard spheres; ie., its kernel does not allow for any intermolecular interactions. It is this limitation we wish to address. Three excellent references for both Boltzmann and Enskog theory are [3,4,5]. II. Enskog Square Well Equation A realistic an der Waals molecular potential, as a function of the intermolecular distance, consists of an exponential repulsie inner core and an adjoining attractie potential region, with the potential falling to zero as the intermolecular distance approaches infinity. The Enskog equation may be considered to represent an intermolecular potential which is infinite for x σ and zero for x > σ. As a first approximation to the qualitatie behaior of the an der Walls potential, we hae considered the Enskog equation with an attractie square well potential added to the hard repulsie core. The potential, x < σ φ( x) = Ε, σ < x < R 0 x > R must be introduced at the Liouille leel. The result is the following Enskog Square Well equation: ( x, t, ) + f( xt,, ) = x 48 MSAS'2004
+ + R R = 3 2 d ' d εθ( ε V){ Y( x+ σε) f (, x ',) t f ( x+ σε, ',) t Y( x σε) f(,,) x t f( x σε,,)} t 2 3 2 ( + ) + ), 2 3 2 /2 2 3 2 /2 R d ' d εθ ε V){ Y( x σε f ( x, '', t) f ( x σε, '', t) Y( x σε f( x,, t) f( x σε, t)} dd ' εθ( ε V 2 E ){ Y( x+ σε) f( x, ''', t) f( x+ σε, ''', t) Y( x σε) f( xt,, ) f( x σε,,)} t + d ' d εθ( 2 E ε V){ Y( x+ σε) f(, x ',) t f( x+ σε, ',) t Y( x σε) f(,,) x t f( x σε,,)} t The double and triple primed elocities, representing entrance and exit into the square well, can be computed from the conseration laws. Although this might at first look intractable, it consist simply of four Enskog-type collision operators, corresponding to four `collision processes: entrance into the square well, reflection at the molecular diameter, exit out of the square well, and reflection at the well diameter if the particles, as a result of an intermediate collision, no longer hae the energy to escape the well. In principle, the collision frequency factor Y should be determined by local equilibrium considerations, such as a Mayer cluster expansion. From a numerical point of iew, it would be ery interesting to explore this equation with arious models for Y, for example a monotonic function approaching at low density and infinity at close packing density. Een the case Y, which corresponds to a regime intermediate to dilute and moderately dense, would be of great interest. The Enskog Square Well equation satisfies conseration of mass and momentum, and has a Liapuno fuctional which can be deried in a fashion analogous to that of the Enskog equation. Although energy can not be consered because of the potential well, it satisfies a growth estimate, which has been shown to be sufficient to proe the existence of (weak) global-in-time solutions to the Cauchy problem. [6] Extending the Enskog Square Well Equation to finite range piecewise constant potentials φ( x), which equal Ei for Ri < x < Ri, i=,..., N, presents no technical difficulties. Such an equation will hae 3N+ Enskog-like collision terms, and would proide a numerical model which one might expect to closely emulate a an der Waals potential. III. Vlaso-Enskog Equations Another approach to modeling intermolecular potential effects in dense gases is to add to the hard core repulsion of the Enskog equation the effects of a long range smooth attractie tail. We hae considered two different ideas to accomplish this: coupling a Coulomb potential to the hard-core repulsion, and adding a Fokker-Planck elocity diffusion. Let us write C E (f,f) for the Enskog collision operator (right hand side of the Enskog equation). By adding an extended tail potential Φ(x), which is sufficiently weak to ignore 49 MSAS'2004
two particle correlations as well as effects within the molecular diameter, beginning with the Liouille equation and the BBGKY hierarchy, we hae been able to derie a Vlaso- Enskog equation: (, x t,) + x f(, xt,) = CE( f, f) E(,) xt f(, xt,) which, if Φ(x) is chosen as the Coulomb potential Φ( x) = k x, is coupled to the Poisson equation 3 diext x (, ) = α df( xt,, ). This coupled pair of equations we refer to as the Vlaso Enskog equation. The deriation of the Vlaso Enskog equation neglects elocity correlations. One method of taking into account these effects is the addition of a Fokker-Planck term (or Kolmogoro forward process). Such terms hae been taken in arious forms by different authors. The simplest case is to treat it just as a Laplacian Δ f. Another possibility is to write the process as di (f). From the point of iew of physics, the Laplacian accounts for the thermal background interaction and the diergence arises from dynamical friction forces. Both of these terms, and combinations of them, are referred to as Fokker-Planck terms in the literature. We refer to the system ( x,, t) + f ( x,, t) = C ( f, f ) E( x, t) f ( x,, t) + di{ η f ( x,, t) + ω f ( x,, t)} x E along with the coupled Poisson equation, as the Diffusie Vlaso Enskog equation. We point out that from the iewpoint of either physics or mathematics, the cases η 0 and η =0 can not be inferred from each other. With either type of diffusion term, the system of equations satisfies conseration of mass and momentum, as well as an energy bound of the form 2 2 ddx f ( xt,, ) + dx Ext (, ) = K0 + Kt A Liapuno functional may be deried which includes the term t 0 ds ddx f ( x,, s) For all of these equations, (weak) existence theorems of arious forms hae been proed. [7,8] 2 IV. Discrete Velocity Models Discrete elocity models hae been studied rather extensiely for the Boltzmann equation for more than three decades. Yet the literature is nearly nonexistent for discrete model Enskog equations. We pose here a general framework which will allow the introduction of arious discrete elocity models to Enskog theory. 50 MSAS'2004
Assuming that all particle elocities belong to a finite set { } N i i =, let P ( ε ) be the probability that two particles of elocity i and j encounter at an angle ε and produce post-collision elocities of k and l. Then probability, conseration laws and symmetry would require: P ( ε ) 0 0 P < i j, ε > 0 < k l, ε > 0 if and only if i + j = k + l 2 2 2 2 i + j = k + l P P ( ε) = P ( ε) = lk ( ε ) = P ( ε ) The discrete elocity Enskog equation is then gien by x + + = dε{ Y( x+ σε) P ( ε) f f < ε, > Y( x σε) P f f < ε, > } i i + i = j + B ji k l k l i j i j where f ± = i fi( x ± σε, t) and B + = { ε ε ε =, < ε, 0,, 0} i j > < ε k l >. This model reduces easily to the few known discrete Enskog models which hae appeared in the literature. [9,0,] It is straightforward to check that the equation satisfies conseration of mass, momentum and energy, and has a Liapuno functional t + + Γ () t = dxfilog fi ds dε dx[ fk fi fi f j ] Y( x+ σε) P ( ε) < ε, i j > 2 + i V. Numerical Problems 0 B Numerical implementation of the Boltzmann equation is an area which has attracted considerable attention for seeral decades. Because of the singular nature of the collision term, techniques hae been deeloped which are designed particularly for Boltzmann theory. Most analysts hae approached the problem by adapting Monte Carlo procedures, although some work has been done as well utilizing deterministic methods, ie, discrete ordinates, spherical harmonics, nodal methods, etc. A solid introduction to both approaches may be obtained in reference [2]. 5 MSAS'2004
Although the literature on numerical implementation of the Enskog equation is more scarce, similar strategies hae been applied to obtain numerical solutions of Enskog equations. With reference to the continuous models contained in this paper, the field of numerical analysis is wide open. Indeed, the author knows of no literature which offers a numerical strategy for these equations. The reader is thus kindly inited to consider analysis of these equations. In particular, it would be of great interest to hae estimates on the effect of the potential well in the square well equation. What effect does it hae on clustering, for example? Can one estimate the effects of the Vlaso term and the elocity diffusions in the Enskog-Vlaso models? Of course, for the discrete elocity models, the numerical problem is a great deal simpler. Here, the most immediate interest would be in creating discrete models corresponding to the rich literature of discrete Boltzmann models. We are aware at present of only two such models in the literature [9,0,], but are certain the construction outlined aboe proides a framework for a number of additional models. REFERENCES [] L.Boltzmann, Sitz. Wien. Akad. Wiss. 66, 275-370 (872). [2] D. Enskog, Kinetiske Theorie, Senska Akad. 63 (92). English translation in Kinetic Theory, S. Bruch, ed., ol. 3, Pergamon, New York, 972. [3] C. Cercignani, The Boltzmann Equation and its Applications (Applied Mathematical Sciences ol. 67), Springer-Verlag, Berlin, 988. [4] N. Bellomo, A. Palczewski and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory, World Scientific, Singapore, 988. [5] N. Bellomo, M. Lachowicz, J. Polewczak and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory II, World Scientific, Singapore, 99. [6] W. Greenberg and A. Yao, Jour. Transport Theor. Stat. Phys. 27, 37-50 (997). [7] W. Greenberg and P. Lei, in Proceedings of International Scientific Conference, Russian Academy of Science (Institute for Mathematical Biology), Ter (997). [8] W. Greenberg, in Proceedings of II Congreso International de Matematica Aplicada y Computacional (CIMAC II), to appear. [9] G. Borgioli, V. Gerasimenko, G. Lauro and R. Monaco, Jour. Transport Theor. Stat. Phys. 25, 58-592 (996). [0] G. Borgioli, V Gerasimenko, G. Lauro and R. Monaco, Reports on Mathematical Physics 40, 43-442 (997). [] G. Borgioli, V. Gerasimenko and G. Lauro, Rendiconti del Seminario Matematico dell Uniersita e del Politecnico di Torino, to appear. [2] V. Aristo, Methods of Direct Soing the Boltzmann Equation and Study of NonequilibriumFlows, Kluwer, Amsterdam, 200. 52 MSAS'2004