The Solution of the Fokker-Planck Equation Using Lie Groups

Transcription

1 Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 10, HIKARI Ltd, The Solution of the Fokker-Planck Equation Using Lie Groups D. Millan Universidad Nacional de Colombia Departamento de Física Bogotá-Colombia F. Fonseca Universidad Nacional de Colombia Grupo de Ciencia de Materiales y Superficies Departamento de Física Bogotá-Colombia Copyright c 2017 D. Millan and F. Fonseca. This article is distributed under the Creative Commons Attribution License, which permits unrestrikd use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we solve the problem of the Fokker-Planck equation by using the symmetry Lie group method. A good approximation to previous experimental results was obtained for shock waves in Nitrogen and Argon. Keywords: Fokker-Planck equation, Shock Waves, Lie Groups 1 Introduction Shock waves are a common phenomenon in nature and of big importance e.g., in combustion research [1], heat transfer in porous media [2], medicine [3], engineering [4]. On the other hand, symmetry has played a central role in science and is the foundation of Lie group theory. The method consists in finding

2 478 D. Millan and F. Fonseca particular solutions to the differential equations treating them as hypersurfaces. We use infinitesimal coefficients to construct symmetry generators that allow to find infinitesimal invariant solutions under symmetric transformations [5]-[7]. 2 Fokker-Planck equation The Fokker-Planck equation is [8]-[9], ρ(x, v, t) t (v)ρ(x, v, t)) = + ( γ vρ(x, v, t)) m x v + 2 ( γk BT ρ(x, v, t)) m 2 (1) x 2 Using a traffic flow approximation for the velocity, [10]-[11], in eq. (1) ρ(x, v, t) t = (v m(1 ρ ρ m )ρ(x, v, t)) x + ( γ v m m(1 ρ ρ m )ρ(x, v, t)) v + 2 ( γk BT ρ(x, v, t)) m 2 x 2 (2) This is a nonlinear second order partial differential equation in three independent variables, which is intended to be solved in this work using Lie symmetries or Lie groups [5]-[7]. If the terms are rearranged eq. (2) is: Where the contants are: Aρ vv + Bρρ v + Cρ v + Dρρ x + Eρ x ρ t = 0 (3) A = γk BT m 2, B = 2γv m mρ m, C = γv m m, D = 2v m ρ m, E = v m (4) Here k B is the Boltzman constant, γ the drag coefficient, v m and ρ m are the maximum values of velocity and density, respectively, [11]. The infinitesimal coefficients for each basic variable will now be named in the following way: for the position ξ(x, v, t, ρ), the velocity is η(x, v, t, ρ), time is τ(x, v, t, ρ) and the density φ(x, v, t, ρ). The expression for the symmetry generator is [11]: V = ξ(x, v, t, ρ) x + η(x, v, t, ρ) v + τ(x, v, t, ρ) t + φ((x, v, t, ρ)) ρ (5) The second prolongation of the symmetry generator [6]-[11], in eq. (5), is:

3 Solution of the Fokker-Planck equation 479 P r 2 V = ξ x + η v + τ t + φ ρ + φx + φ ν + φ t (6) ρ x ρ ν ρ t +φ xx + φ νν + φ tt + φ xν + φ xt + φ νt ρ xx ρ νν ρ tt ρ xν ρ xt The quantities φ J are given by ρ νt and p φ J = D J Q + ξ i (x, v, t, ρ)ρ J,i (7) i=1 Q(x, v, t, ρ) = φ(x, v, t, ρ) p i=1 ξ i (x, v, t, ρ) ρ x i (8) Where, the subscript J represents each independent variable and indicates the derivative with respect to those variables. ξ i represents the infinitesimal coefficient associated to each of the p independent variables. Also, D J, eq. (7), is the total derivative with respect to J. Q, eqs. (7) and (8), is known as the characteristic equation of the symmetry generator, which means the invariance of the differential equation under the action of the prolongation symmetry generator [6]-[7], [11]. Then, we have: (ξ x + η v + τ t + φ ρ + φx + φ ν + φ t (9) ρ x ρ ν ρ t +φ xx + φ νν + φ tt + φ xν + φ xt + φ νt )(Aρ vv ρ xx ρ νν ρ tt ρ xν ρ xt ρ νt +Bρρ v + Cρ v + Dρρ x + Eρ x ρ t ) = 0 The result of the symmetry criterion is: Aφ vv + (Bρ + C)φ v + (Dρ + E)φ x φ t + (Bρ v + Dρ x )φ = 0 (10) Equations (7) and (8) let us to know the expressions φ j, which are: φ x = D x (φ ξ ρ x η ρ v τ ρ t ) + ξρ xx + ηρ xv + τρ xt (11) φ v = D v (φ ξ ρ x η ρ v τ ρ t ) + ξρ vx + ηρ vv + τρ vt (12)

4 480 D. Millan and F. Fonseca φ t = D t (φ ξ ρ x η ρ v τ ρ t ) + ξρ tx + ηρ tv + vρ tt (13) φ vv = D νν (φ ξ ρ x η ρ v τ ρ t ) + ξρ vvx + ηρ vvv + τρ vvt (14) Replacing them in eq. (10), and organizing as a polynomial of ρ i, the coefficients become a set of coupled system of differential equations for the infinitesimal coefficients. Then, we get: Monomial Coefficient ρ t 2η ν Aτ νν Cτ ν Eτ x + τ t = 0 ρ x Aξ νν +2Eη ν Cξ ν Eξ x +ξ t +Dφ = 0 ρρ x 2Dη ν Bξ ν Dξ x = 0 ρ ν Eη x + Aφ ρν + Cη ν Aη νν + η t + Bφ = 0 ρρ ν Bη ν Dη x = 0 ρ 2 ν Aφ ρρ + 2Cη ρ 2Aη ρν = 0 1 Aφ νν + Cφ ν + Eφ x φ t = 0 ρ xν ρ ν 2Aξ ρ = 0 ρ xν 2Aξ ν = 0 ρ ν ρ x 2Aξ ρν + 2Eη ρ = 0 ρρ ν ρ x 2Dη ρ = 0 ρ x ρ 2 ν Aξ ρρ = 0 ρ ν ρ t 2η ρ + 2Aτ ρν = 0 ρρ 2 ν 2Bη ρ = 0 ρ 3 ν Aη ρρ = 0 ρ ν ρ νt Aτ ρ = 0 ρ νt Aτ ν = 0 ρ 2 νρ t Aτ ρρ = 0 ρ Aφ ν + Dφ x = 0 ρρ t Bτ ν + Dτ x = 0 Table 1: Table of Infinitesimal coefficients, [11]. The solution to this system is the group of infinitesimal coefficients τ(t) = a 1 t + a 2 (15)

5 Solution of the Fokker-Planck equation 481 Figure 1: Density in function of the position for shock waves in Nitrogen gaseous to different Mach s numbers. The straight line is the extension of the largest slope for the density profile with M = 1, [11]. ξ(x, t) = a 1 x a 3 Dt + a 4 (16) v η(x, v, t) = a a B 1 2D x a EB 1 2D t a C 1 2 t a 3Bt + a 5 (17) φ = a 3 (18) Where the constants a i are known as the Lie constants. Then, the symmetry generator, also called symmetry algebra [5]-[7] and [11], is: V = a 1 (x x + t t + (1 2 v + B 2D x EB 2D t C 2 t) v ) + a 2 t +a 3 ( ρ Dt x Bt v ) + a 4 x + a 5 v (19) The lineal independence of the infinitesimal coefficients allows to find particular solutions for each chosen a i. The characteristic equation for the lineal combination of symmetry generators a 4 and a 5 and equation (3), [5]-[7] and [11], become a coupled system of differential equations ρ a 4 x + a ρ 5 v = 0 (20)

6 482 D. Millan and F. Fonseca Figure 2: Thickness of the shock wave as a function of the mach number at 80% of the Nitrogen density. The points correspond to the results in [11]. with a solution given by Aρ vv + Bρρ v + Cρ v + Dρρ x + Eρ x ρ t = 0 (21) ρ = (ae C) 2 + 2c 2 (ad B) (ad B) (22) tanh ( (ae C) 2 + 2c 2 (ad B)(c 3 + ax v ae C )) 2A ad B Where a = a 5 /a 4. Also, c 2 and c 3 are integration constants. The boundary conditions ρ = 0, v = v m when x, and ρ = 1/2, and v = 0 when x = 0 are used. These conditions let the density values to be on the positive side of the plane. Figure (2) plots the density as a function of the position for Nitrogen gas with the following conditions: R 0 = 225x10 12 m (radius of the molecule or Van der Waals), µ = 17.1x10 6 P a.s (dynamic viscosity), m = 46.5x10 27 Kg (mass of the molecule), Γ = 1.6 (ratio of specific heats). The results of the density profiles for different Mach numbers are shown in fig. (1). The thickness corresponds to the region where 80% of the particles are found, and is calculated with the next equation [11]-[13]: L = µ P ( πk BT 2m ) (23)

7 Solution of the Fokker-Planck equation 483 Figure 3: Thickness of the shock wave as a function of the number of mach to 80% of the Argon density. The points correspond to the results in [11]. The values for the Argon, plotted in fig. (3), are: R 0 = 188x10 12 m, µ = 22.9x10 6 P a.s, m = 6.63x10 26 Kg, Γ = 1.6. The a constant is found by iteration until the theoretical curve is on the experimental data: for Nitrogen 1/a = 4.7x10 10 s and for Argon 1/a = 6.8x10 9 s. 3 Conclusions A Fokker-Planck equation was constructed for a shock wave in gases and solved by the Lie group method. The solution was contrasted with the results referenced in [13]. The results, for the Nitrogen shows that the solution matches the experimental results for low Mach numbers, up to Mach 3.5, figure (2), and for Argon up to 2.5, figure (3). Acknowledgements. F. Fonseca acknowledges support by Universidad Nacional de Colombia under Hermes project (32501). References [1] K. Yu. Arefév, A. V. Voronetskii, A. N. Prokhorov, L. S. Yanovskii, Experimental study of the combustion efficiency of two-phase gasification products of energetic boron-containing condensed compositions in a highenthalpy airflow, Combustion, Explosion, and Shock Waves, 53 (2017), no. 3,

8 484 D. Millan and F. Fonseca [2] A. V. Attetkov and E. V. Pilyavskaya, Effect of interfacial heat transfer on the critical conditions of shock-wave initiation of chemical reaction in porous energetic materials, Combustion, Explosion, and Shock Waves, 53 (2017), no. 3, [3] M. Thiel, M. Nieswand, M. Dörffel, Review: The use of shock waves in medicine a tool of the modern OR: an overview of basic physical principles, history and research, Minimally Invasive Therapy and Allied Technol., 9 (2000), no. 3-4, [4] G. Jagadeesh, Industrial applications of shock waves, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 222 (2008), [5] F. Oliveri, Lie Symmetries of Differential Equations: Classical Results and Recent Contributions, Symmetry, 2 (2010), [6] P. J. Olver, Applications of Lie Groups to Differential Equations, New York, Springer-Verlag, [7] A. H. Ortiz, G. F. Jiménez, A. A. Posso, Algunas Soluciones Exactas para la Ecuación Unidimensional de Fokker-Planck Usando Simetrías de Lie, Revista de Matemática de la Universidad de Costa Rica, 22 (2015), [8] M. Scott, Applied Stochastic Processes in Science and Engineering, University of Waterloo, Waterloo, Canada, [9] P. J. García, Introduction to the Theory of Stochastic Processes and Brownian Motion Problems, Lecture notes for a graduate course, Universidad de Zaragoza, [10] R. Borsche, M. Kimathi and Axel Klar, A class of multi-phase traffic theories for microscopic, kinetic and continuum traffic models, Computers and Mathematics with Applications, 64 (2012), [11] D. Millán De la Cruz, Solución del Problema de una Onda de Choque en un Medio Gaseoso por Métodos Estocásticos, Tesis de Maestría en Física, Universidad Nacional de Colombia, Departamento de Física, Bogotá-Colombia, [12] L.Talbot and F. Robben, Measurement of Shock Wave Thickness by the Electron Beam Fluorescence Method, University of California, Berkeley, 1965.

9 Solution of the Fokker-Planck equation 485 [13] M. Linzer, & D. F. Horning, Structure of Shock Fronts in Argon and Nitrogen, Phys. Fluids, 6 (1963), Received: August 16, 2017; Published: September 6, 2017