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The dynamic theory for the inelastically colliding particles
Journal of Molecular Liquids 127 (2006) 84 – 86 www.elsevier.com/locate/molliq
The dynamic theory for the inelastically colliding particles Andrii Sizhuk ⁎, Stanislav Yezhov Theoretical Physics Department, Physical Faculty, Taras Shevchenko National University, Pr. Acad. Glushkova, 2, Bldg. 11, 03022 Kiev, Ukraine Available online 5 May 2006
Abstract In the present work the dynamic system of the particles with model mechanism for the momentum and angular momentum exchange is studied. For hard rough spheres bounded by a wall the evolution equation for the delta functional microscopic density is constructed. In the case of chaotic initial state the kinetic equation for the one-particle distribution is obtained. © 2006 Elsevier B.V. All rights reserved. Keywords: Microscopic equation; Inelastic collision; Non-zero angular momentum; Border conditions
1. Introduction A large number of works is devoted to the kinetics investigation of the classical model system. During the last times much effort has been devoted to achieving a better understanding of the dissipative system behavior (for example in the work [1] it was investigated dissipative system of the rough spheres) and the systems of molecules with internal energy, allowing for exchange of energy between the internal and translation degrees of freedom (for example [2]). The rough sphere model was first introduced for elastically colliding particles, assuming either perfectly smooth particles [3]. And the development of the more exact investigation methods for the more complicated system is actual. This work is aimed to the construction of the evolution equation for the microscopic phase density of model dynamical system taking into account border conditions. The evolution equation can be used for building of the averaged microscopic density. The approximate equations for the macroscopic distribution can be used for the study of the kinetics and hydrodynamics of the respective statistical system taking into account interaction with a plane surface. The evolution equation for the microscopic phase density describing a system of the identical elastic spherical particles with zero intrinsic angular momentum was derived by N. Bogoliubov ⁎ Corresponding author. E-mail addresses: [email protected] (A. Sizhuk), [email protected] (S. Yezhov). 0167-7322/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2006.03.021
[4]. It was shown that the evolution equation for microscopic density in the phase space of coordinates and velocities can be transformed into the Bolzmann–Enskog's Equation. The main feature of the dynamic system considered below is the possibility of the inelastic collisions between the particles. We refer to the collisions with the exchange of momentums and intrinsic angular momentums between the particles as inelastic collisions. In the present work we consider the case of the hard particles: the modulus in torsion and the modulus in tension is infinite large. 2. Microscopic dynamics Let qj = qj (t) is the position of the particle j center of mass, pj = pj (t) is the momentum of the particle j center of mass, Mj = Mj (t) is the angular momentum of the hard rough spherical particle j relative to its center of mass. We consider the system N identical hard rough spheres. Each sphere has three translational and three rotational degrees of freedom. The average density of particles and the time of interaction between the particles are negligibly small in the accepted scheme of classical mechanics; the simultaneous interaction between a three, four and more particles may be disregarded. For the initial values the physical conditions of impenetrability is |qj (t0) − qk (t0)| ≥ a, where a is the particle diameter ( j ≠ k). The particles goes freely up to the first instant in which two of them occur in the point |qj (t) − qk (t)| = a. Here they collide according to some rule Eq. (1) ̂ ; p ; M j ; q ; p ; M k Þ; ðqj*; pj*; M j*; qk*; pk*; M k*Þ ¼ Sdðq j j k k
ð1Þ
A. Sizhuk, S. Yezhov / Journal of Molecular Liquids 127 (2006) 84–86
85
where qj⁎, pj⁎, Mj⁎; qk⁎, pk⁎, Mk⁎, are the phases of the particles j and k after the collision. Then the particles go on up to the instant of the next collision which is performed by the same rule, and so on. We consider the system of the spherical particles which contacts with the planar solid surface. The spheres collide with the wall according to some rule Î: (pj★, Mj★ = Î) ˙ (pj, Mj). In the collisions with the wall the particles changes normal momentum with the surface to opposite orientation: npj = − npj★, where pj★ is momentum of the particle j after collision with the wall and the vector n = (0, 0, 1) is normal to the wall XOY (surface). Using modified the Bogoliubov's technique [4], it can be written that the evolution equation for the microscopic density of the particles j. The evolution equation can be complemented by a member which describes the interaction with the wall:
A p A f ðt; q; p; MÞ ¼ − f ðt; q; p; MÞ At m Aq Z a2 þ dp VdM Vd σhfðpV−pÞ σgjðp−p VÞ σj m f f ðt; q; p*; M *Þf ðt; q þ a σ; p V*; M V*Þ
A p A fj ðt; X Þ ¼ − fj ðt; X Þ At m Aq X þ Tj;k d fj ðt; X Þ þ Lj d fj ðt; X Þ;
where delta function phase density is denoted as:
ð2Þ
In this case, taking into account the relations npj = − npj★ and pj,kσ = − p⁎j,k σ for the hard spheres, after summation over j = 1, …, N we obtain the evolution equation:
− f ðt; q; p; MÞf ðt; q−a σ; pV; M VÞg 1 a þ jnpjd qn− fhðnpÞf ðt; q; p★ ; M ★ Þ m 2 −hð−npÞf ðt; q; p; MÞg; ð7Þ
f ðt; q; p; MÞ ¼
1V kV N k p j
X k
Here fj(t, X) = δ(q − qj(t))δ(p − pj(t))δ(M − Mj(t)) is the distribution function for the particle j in the phase space X = (q, p, M). Every delta functions is three-dimensional. The collisional integral Tj,kd fj(t, X) is defined as Z a2 ̂ Tj;k d fj ¼ hðpj;k d σÞjpj;k d σjdðqk −qj −a σÞfB j;k −1g fj dσ; m ð3Þ ⁎ where Bˆ j,k ˙ fj = δ(q − qj)δ(p − p⁎ j )δ(M − Mj ); pj,k = pj − pk; m is the particle mass, σ is the unit vector, and dσ is the infinitesimal element of a spatial angle. Lj ˙ fj(t, q, p, M) is non-zero in the case of the particle j interaction with the wall within interval Δt and can be presented in the form: Lj d fj ðt; X Þ ¼
1 hð−npj ðtÞ jnpj ðtÞjdðqj ðtÞn−a=2Þfl ̂j ðnÞ−1gd fj ðt; X Þ; m
ð4Þ
where the linear operator lˆj(n) transforms the particle j phase before the collision into the phase after the collision with the wall:
★ l j ðnÞd fj ðt; X Þ ¼ dðq−qj ðtÞ dðp−p★ j ðtÞ dðM−M j ðtÞ : ̂
ð5Þ
3. The microscopic evolution equation We consider the case of the linear operators Ŝ and Î and assume, that the linear operators Ŝ and Î has a reflection operator property: Ŝ2 = 1 and Î 2 = 1. Then using the property of delta function we have: ̂
B j;k dfj ðt; X Þfk ðt; X VÞ ¼ fj ðt; X V*Þfk ðt; X V*Þ; l j ðnÞd fj ðt; q; p; MÞ ¼ dðq−qj ðtÞ dðp★ −pj ðtÞ dðM ★ −M j ðtÞ ; ̂
fk ¼
X
dðq−qk ðtÞÞdðp−pk ðtÞÞdðM−M k ðtÞÞ;
k
ð8Þ
and phases (X⁎; X′⁎) = (q, p⁎, M⁎; q′, M′⁎, p′⁎) can be taken from the work [5] as the particular case of the operator Ŝ: ( 1 2, ,p þ p Vþ ½ðM þ M VÞ σ−, σ ðp−p VÞ σ p* ¼ 1þ, a
;
( 1 a M−,M Vþ ½ σ ðp−pVÞ þ , σ ðM þ M VÞ σ M* ¼ 1þ, 2 ( 1 2, ,p Vþ p− ½ðM þ M VÞ σ þ , σ ðp−p VÞ σ p V* ¼ 1þ, a
( 1 a M V−,M þ ½ σ ðp−p VÞ þ , σ ðM þ M VÞ σ M V* ¼ 1þ, 2
;
;
;
ð9Þ
where (σ(M + M′)) and [(M + M′) × σ] are the scalar multiplication and vector multiplication of vector σ by vector 2 M þ M V; , ¼ ma , J is the particle moment of inertia. 4J When the number of particles under consideration is large, the Eq. (7) with the border conditions in the form (2) can be averaging and written in the case of chaotic initial state for oneparticle probability density, which is the microscopic density averaging in statistical ensemble: h f ðt;Zq; p; MÞi ¼
dX N DðX N ð0ÞÞf ðq; p; M; X N ðX N ð0Þ; tÞÞ;
ð6Þ where (X⁎; X′⁎) = Ŝ(σ) d ((q, p, M);(q′, p′, M′)).
g Þg Þg Þg
Þ
where D (XN(0)) is the adjusted state distribution.
ð10Þ
86
A. Sizhuk, S. Yezhov / Journal of Molecular Liquids 127 (2006) 84–86
In the case of chaotic initial state we expect that the dynamics does not create correlations and have the kinetic equation for the macroscopic density after averaging: A p A h f ðt; MÞi ¼ − hf ðt; q; p; MÞi At m Aq Z a2 þ dp VdM Vd σhfðp V−pÞ σgjðp−p VÞ σj m fh f ðt; q; p*; M *Þihf ðt; q þ a σ; p V*; M V*Þi − h f ðt; q; p; MÞihf ðt; q−a σ; pV; M VÞig 1 a þ jnpjd qn− fhðnpÞh f ðt; q; p★ ; M ★ Þ m 2 − hð−npÞihf ðt; q; p; MÞig; ð11Þ Obtained Eq. (11) without border conditions has the Enskoglike form. 4. Conclusion In the case of an operator, which has the reflection operator property, a physical mechanism for the momentum and angular
momentum exchange between the rough particles is proposed. The microscopic evolution equation can be presented in the form of the Enskog-like Equation. Obtained equation modified for the case of the system bounded by a planar solid surface. In the case of large quantity of the particles obtained result can be used for investigation of the statistical system with prescribed accuracy. As an example by using the exact equation it has built approximation, which describes system kinetics near the equilibrium position. References [1] S. Luding, M. Huthmann, S. McNamara, A. Zippelius, Homogeneous cooling of rough, dissipative particles: theory and simulations, Phys. Rev., E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58 (1998) 3416. [2] G.S. Singh, B. Kumar, Phys. Rev., E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62 (2000) 7927. [3] Chapman, T.G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, London, 1960. [4] N.N. Bogoliubov, N.N. Bogoliubov (young) Introduction to Quantum Statistical Mechanics (in Russian) — Moscow, Science, 1984, pp. 228–245. [5] A. Sizhuk, An kinetic equation for system of inelastically colliding spherical particles, Bull. Univ. Kiev, Ukr., Ser. Phys. Math. (3) (2004) 347–354.
The dynamic theory for the inelastically colliding particles Andrii Sizhuk ⁎, Stanislav Yezhov Theoretical Physics Department, Physical Faculty, Taras Shevchenko National University, Pr. Acad. Glushkova, 2, Bldg. 11, 03022 Kiev, Ukraine Available online 5 May 2006
Abstract In the present work the dynamic system of the particles with model mechanism for the momentum and angular momentum exchange is studied. For hard rough spheres bounded by a wall the evolution equation for the delta functional microscopic density is constructed. In the case of chaotic initial state the kinetic equation for the one-particle distribution is obtained. © 2006 Elsevier B.V. All rights reserved. Keywords: Microscopic equation; Inelastic collision; Non-zero angular momentum; Border conditions
1. Introduction A large number of works is devoted to the kinetics investigation of the classical model system. During the last times much effort has been devoted to achieving a better understanding of the dissipative system behavior (for example in the work [1] it was investigated dissipative system of the rough spheres) and the systems of molecules with internal energy, allowing for exchange of energy between the internal and translation degrees of freedom (for example [2]). The rough sphere model was first introduced for elastically colliding particles, assuming either perfectly smooth particles [3]. And the development of the more exact investigation methods for the more complicated system is actual. This work is aimed to the construction of the evolution equation for the microscopic phase density of model dynamical system taking into account border conditions. The evolution equation can be used for building of the averaged microscopic density. The approximate equations for the macroscopic distribution can be used for the study of the kinetics and hydrodynamics of the respective statistical system taking into account interaction with a plane surface. The evolution equation for the microscopic phase density describing a system of the identical elastic spherical particles with zero intrinsic angular momentum was derived by N. Bogoliubov ⁎ Corresponding author. E-mail addresses: [email protected] (A. Sizhuk), [email protected] (S. Yezhov). 0167-7322/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2006.03.021
[4]. It was shown that the evolution equation for microscopic density in the phase space of coordinates and velocities can be transformed into the Bolzmann–Enskog's Equation. The main feature of the dynamic system considered below is the possibility of the inelastic collisions between the particles. We refer to the collisions with the exchange of momentums and intrinsic angular momentums between the particles as inelastic collisions. In the present work we consider the case of the hard particles: the modulus in torsion and the modulus in tension is infinite large. 2. Microscopic dynamics Let qj = qj (t) is the position of the particle j center of mass, pj = pj (t) is the momentum of the particle j center of mass, Mj = Mj (t) is the angular momentum of the hard rough spherical particle j relative to its center of mass. We consider the system N identical hard rough spheres. Each sphere has three translational and three rotational degrees of freedom. The average density of particles and the time of interaction between the particles are negligibly small in the accepted scheme of classical mechanics; the simultaneous interaction between a three, four and more particles may be disregarded. For the initial values the physical conditions of impenetrability is |qj (t0) − qk (t0)| ≥ a, where a is the particle diameter ( j ≠ k). The particles goes freely up to the first instant in which two of them occur in the point |qj (t) − qk (t)| = a. Here they collide according to some rule Eq. (1) ̂ ; p ; M j ; q ; p ; M k Þ; ðqj*; pj*; M j*; qk*; pk*; M k*Þ ¼ Sdðq j j k k
ð1Þ
A. Sizhuk, S. Yezhov / Journal of Molecular Liquids 127 (2006) 84–86
85
where qj⁎, pj⁎, Mj⁎; qk⁎, pk⁎, Mk⁎, are the phases of the particles j and k after the collision. Then the particles go on up to the instant of the next collision which is performed by the same rule, and so on. We consider the system of the spherical particles which contacts with the planar solid surface. The spheres collide with the wall according to some rule Î: (pj★, Mj★ = Î) ˙ (pj, Mj). In the collisions with the wall the particles changes normal momentum with the surface to opposite orientation: npj = − npj★, where pj★ is momentum of the particle j after collision with the wall and the vector n = (0, 0, 1) is normal to the wall XOY (surface). Using modified the Bogoliubov's technique [4], it can be written that the evolution equation for the microscopic density of the particles j. The evolution equation can be complemented by a member which describes the interaction with the wall:
A p A f ðt; q; p; MÞ ¼ − f ðt; q; p; MÞ At m Aq Z a2 þ dp VdM Vd σhfðpV−pÞ σgjðp−p VÞ σj m f f ðt; q; p*; M *Þf ðt; q þ a σ; p V*; M V*Þ
A p A fj ðt; X Þ ¼ − fj ðt; X Þ At m Aq X þ Tj;k d fj ðt; X Þ þ Lj d fj ðt; X Þ;
where delta function phase density is denoted as:
ð2Þ
In this case, taking into account the relations npj = − npj★ and pj,kσ = − p⁎j,k σ for the hard spheres, after summation over j = 1, …, N we obtain the evolution equation:
− f ðt; q; p; MÞf ðt; q−a σ; pV; M VÞg 1 a þ jnpjd qn− fhðnpÞf ðt; q; p★ ; M ★ Þ m 2 −hð−npÞf ðt; q; p; MÞg; ð7Þ
f ðt; q; p; MÞ ¼
1V kV N k p j
X k
Here fj(t, X) = δ(q − qj(t))δ(p − pj(t))δ(M − Mj(t)) is the distribution function for the particle j in the phase space X = (q, p, M). Every delta functions is three-dimensional. The collisional integral Tj,kd fj(t, X) is defined as Z a2 ̂ Tj;k d fj ¼ hðpj;k d σÞjpj;k d σjdðqk −qj −a σÞfB j;k −1g fj dσ; m ð3Þ ⁎ where Bˆ j,k ˙ fj = δ(q − qj)δ(p − p⁎ j )δ(M − Mj ); pj,k = pj − pk; m is the particle mass, σ is the unit vector, and dσ is the infinitesimal element of a spatial angle. Lj ˙ fj(t, q, p, M) is non-zero in the case of the particle j interaction with the wall within interval Δt and can be presented in the form: Lj d fj ðt; X Þ ¼
1 hð−npj ðtÞ jnpj ðtÞjdðqj ðtÞn−a=2Þfl ̂j ðnÞ−1gd fj ðt; X Þ; m
ð4Þ
where the linear operator lˆj(n) transforms the particle j phase before the collision into the phase after the collision with the wall:
★ l j ðnÞd fj ðt; X Þ ¼ dðq−qj ðtÞ dðp−p★ j ðtÞ dðM−M j ðtÞ : ̂
ð5Þ
3. The microscopic evolution equation We consider the case of the linear operators Ŝ and Î and assume, that the linear operators Ŝ and Î has a reflection operator property: Ŝ2 = 1 and Î 2 = 1. Then using the property of delta function we have: ̂
B j;k dfj ðt; X Þfk ðt; X VÞ ¼ fj ðt; X V*Þfk ðt; X V*Þ; l j ðnÞd fj ðt; q; p; MÞ ¼ dðq−qj ðtÞ dðp★ −pj ðtÞ dðM ★ −M j ðtÞ ; ̂
fk ¼
X
dðq−qk ðtÞÞdðp−pk ðtÞÞdðM−M k ðtÞÞ;
k
ð8Þ
and phases (X⁎; X′⁎) = (q, p⁎, M⁎; q′, M′⁎, p′⁎) can be taken from the work [5] as the particular case of the operator Ŝ: ( 1 2, ,p þ p Vþ ½ðM þ M VÞ σ−, σ ðp−p VÞ σ p* ¼ 1þ, a
;
( 1 a M−,M Vþ ½ σ ðp−pVÞ þ , σ ðM þ M VÞ σ M* ¼ 1þ, 2 ( 1 2, ,p Vþ p− ½ðM þ M VÞ σ þ , σ ðp−p VÞ σ p V* ¼ 1þ, a
( 1 a M V−,M þ ½ σ ðp−p VÞ þ , σ ðM þ M VÞ σ M V* ¼ 1þ, 2
;
;
;
ð9Þ
where (σ(M + M′)) and [(M + M′) × σ] are the scalar multiplication and vector multiplication of vector σ by vector 2 M þ M V; , ¼ ma , J is the particle moment of inertia. 4J When the number of particles under consideration is large, the Eq. (7) with the border conditions in the form (2) can be averaging and written in the case of chaotic initial state for oneparticle probability density, which is the microscopic density averaging in statistical ensemble: h f ðt;Zq; p; MÞi ¼
dX N DðX N ð0ÞÞf ðq; p; M; X N ðX N ð0Þ; tÞÞ;
ð6Þ where (X⁎; X′⁎) = Ŝ(σ) d ((q, p, M);(q′, p′, M′)).
g Þg Þg Þg
Þ
where D (XN(0)) is the adjusted state distribution.
ð10Þ
86
A. Sizhuk, S. Yezhov / Journal of Molecular Liquids 127 (2006) 84–86
In the case of chaotic initial state we expect that the dynamics does not create correlations and have the kinetic equation for the macroscopic density after averaging: A p A h f ðt; MÞi ¼ − hf ðt; q; p; MÞi At m Aq Z a2 þ dp VdM Vd σhfðp V−pÞ σgjðp−p VÞ σj m fh f ðt; q; p*; M *Þihf ðt; q þ a σ; p V*; M V*Þi − h f ðt; q; p; MÞihf ðt; q−a σ; pV; M VÞig 1 a þ jnpjd qn− fhðnpÞh f ðt; q; p★ ; M ★ Þ m 2 − hð−npÞihf ðt; q; p; MÞig; ð11Þ Obtained Eq. (11) without border conditions has the Enskoglike form. 4. Conclusion In the case of an operator, which has the reflection operator property, a physical mechanism for the momentum and angular
momentum exchange between the rough particles is proposed. The microscopic evolution equation can be presented in the form of the Enskog-like Equation. Obtained equation modified for the case of the system bounded by a planar solid surface. In the case of large quantity of the particles obtained result can be used for investigation of the statistical system with prescribed accuracy. As an example by using the exact equation it has built approximation, which describes system kinetics near the equilibrium position. References [1] S. Luding, M. Huthmann, S. McNamara, A. Zippelius, Homogeneous cooling of rough, dissipative particles: theory and simulations, Phys. Rev., E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58 (1998) 3416. [2] G.S. Singh, B. Kumar, Phys. Rev., E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62 (2000) 7927. [3] Chapman, T.G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, London, 1960. [4] N.N. Bogoliubov, N.N. Bogoliubov (young) Introduction to Quantum Statistical Mechanics (in Russian) — Moscow, Science, 1984, pp. 228–245. [5] A. Sizhuk, An kinetic equation for system of inelastically colliding spherical particles, Bull. Univ. Kiev, Ukr., Ser. Phys. Math. (3) (2004) 347–354.