September  2014, 7(3): 509-529. doi: 10.3934/krm.2014.7.509

Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations

1. 

Department of mathematics, TU Kaiserslautern, and Fraunhofer ITWM Kaiserslautern, 67663 Kaiserslautern, Germany

2. 

Department of mathematics, TU Kaiserslautern, 67663 Kaiserslautern, Germany

3. 

Department of Technomathematics, University of Kaiserslautern, 67663 Kaiserslautern, Germany

Received  October 2013 Revised  February 2014 Published  July 2014

We consider stochastic dynamic systems with state space $\mathbb{R}^n \times \mathbb{S}^{n-1}$ and associated Fokker--Planck equations. Such systems are used to model, for example, fiber dynamics or swarming and pedestrian dynamics with constant individual speed of propagation. Approximate equations, like linear and nonlinear (maximum entropy) moment approximations and linear and nonlinear diffusion approximations are investigated. These approximations are compared to the underlying Fokker--Planck equation with respect to quality measures like the decay rates to equilibrium. The results clearly show the superiority of the maximum entropy approach for this application compared to the simpler linear and diffusion approximations.
Citation: Axel Klar, Florian Schneider, Oliver Tse. Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations. Kinetic & Related Models, 2014, 7 (3) : 509-529. doi: 10.3934/krm.2014.7.509
References:
[1]

F. Andreu, V. Caselles, J. M. Mazon and S. Moll, A diffusion equation intransparent media,, J. Evol. Equ., 7 (2007), 113. doi: 10.1007/s00028-007-0249-3. Google Scholar

[2]

A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors,, J. Math. Phys., 32 (1991), 544. doi: 10.1063/1.529391. Google Scholar

[3]

K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems,, Arch. Ration. Mech. Anal., 199 (2011), 177. doi: 10.1007/s00205-010-0321-y. Google Scholar

[4]

C. Berthon, P. Charrier and B. Dubroca, An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions,, J. Sci. Comput., 31 (2007), 347. doi: 10.1007/s10915-006-9108-6. Google Scholar

[5]

S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy,, Commun. Pure Appl. Math., 60 (2007), 1559. doi: 10.1002/cpa.20195. Google Scholar

[6]

L. L. Bonilla, T. Götz, A. Klar, N. Marheineke and R. Wegener, Hydrodynamic limit of a Fokker-Planck equation describing fiber lay-down processes,, SIAM J. Appl. Math., 68 (2007), 648. doi: 10.1137/070692728. Google Scholar

[7]

L. L. Bonilla, A. Klar and S. Martin, Higher order averaging of linear Fokker-Planck equations with periodic forcing,, SIAM J. Appl. Math., 72 (2012), 1315. doi: 10.1137/11083959X. Google Scholar

[8]

T. A. Brunner and J. P. Holloway, One-dimensional Riemann solvers and the maximum entropy closure,, J. Quant. Spectrosc. Radiat. Transfer, 69 (2001), 543. Google Scholar

[9]

J. A. Carrillo, V. Caselles and S. Moll, On the relativistic heat equation in one space dimension,, Proc. London Math. Soc., 107 (2013), 1395. doi: 10.1112/plms/pdt015. Google Scholar

[10]

I. L. Chern, Long-time effect of relaxation for hyperbolic conservation laws,, Commun. Math. Phys., 172 (1995), 39. doi: 10.1007/BF02104510. Google Scholar

[11]

B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems,, SIAM J. Numer. Anal., 35 (1998), 2440. doi: 10.1137/S0036142997316712. Google Scholar

[12]

J. F. Coulombel, F. Golse and T. Goudon, Diffusion approximation and entropy based moment closure for kinetic equations,, Asymptot. Anal., 45 (2005), 1. Google Scholar

[13]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,, Math. Models Methods Appl. Sci., 18 (2008), 1193. doi: 10.1142/S0218202508003005. Google Scholar

[14]

P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics,, J. Stat. Phys., 152 (2013), 1033. doi: 10.1007/s10955-013-0805-x. Google Scholar

[15]

L. Desvilettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation,, Comm. Pure Appl. Math., 54 (2001), 1. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. Google Scholar

[16]

J. Dolbeault, A. Klar, C. Mouhot and C. Schmeiser, Hypocoercivity and a Fokker-Planck equation for fiber lay-down,, Applied Mathematical Research Express, (2013), 165. Google Scholar

[17]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity,, to appear in TAMS 2014., (2014). Google Scholar

[18]

B. Dubroca and J. L. Feugeas, Entropic moment closure hierarchy for the radiative transfer equation,, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915. doi: 10.1016/S0764-4442(00)87499-6. Google Scholar

[19]

B. Dubroca and A. Klar, Half moment closure for radiative transfer equations,, J. Comput. Phys., 180 (2002), 584. doi: 10.1006/jcph.2002.7106. Google Scholar

[20]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative transfer,, J. Comput. Phys., 218 (2006), 1. doi: 10.1016/j.jcp.2006.01.038. Google Scholar

[21]

M. Grothaus, A. Klar, J. Maringer and P. Stilgenbauer, Geometry, mixing, properties and hypocoercivity of a degenerate diffusion arising in technical textile industry,, , (). Google Scholar

[22]

T. Götz, A. Klar, N. Marheineke and R. Wegener, A stochastic model for the fiber lay-down process in the nonwoven production,, SIAM J. Appl. Math., 67 (2007), 1704. doi: 10.1137/06067715X. Google Scholar

[23]

A. Klar, N. Marheineke and R. Wegener, Hierarchy of mathematical models for production processes of technical textiles,, ZAMM Z. Angew. Math. Mech., 89 (2009), 941. doi: 10.1002/zamm.200900282. Google Scholar

[24]

A. Klar, J. Maringer and R. Wegener, A 3D model for fiber lay-down processes in non-woven production processes,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512500200. Google Scholar

[25]

P. E. Klöden and P. Platen, Numerical Solution of Stochastic Differential Equations,, Springer, (1999). Google Scholar

[26]

A. Klar and O. Tse, An entropy functional and explicit decay rates for a partially dissipative hyperbolic system,, to appear in ZAMM Z. Angew. Math. Mech., (2014). Google Scholar

[27]

C. D. Levermore, Relating Eddington factors to flux limiters,, J. Quant. Spectrosc. Radiat. Transf., 31 (1984), 149. doi: 10.1016/0022-4073(84)90112-2. Google Scholar

[28]

T. Luo, R. Natalini and Z. Xin, Large time behaviour of the solutions to a hydrodynamic model for semiconductors,, SIAM J. Appl. Math., 59 (1999), 810. doi: 10.1137/S0036139996312168. Google Scholar

[29]

G. Papanicolaou, D. Stroock and S. Varadhan, Martingale Approach to Some Limit Theorems,, in Statistical Mechanics and Dynamical Systems (ed. D. Ruelle), (1977). Google Scholar

[30]

H. Risken, The Fokker-Planck Equation,, Springer, (1989). doi: 10.1007/978-3-642-61544-3. Google Scholar

[31]

A. Roth, A. Klar, B. Simeon and E. Zharovsky, A semi-Lagrangian finite volume method for 3-D Fokker-Planck equations associated to stochastic dynamical systems on the sphere,, to appear in J. Sci. Comput., (2014). Google Scholar

[32]

T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[33]

C. Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5. Google Scholar

show all references

References:
[1]

F. Andreu, V. Caselles, J. M. Mazon and S. Moll, A diffusion equation intransparent media,, J. Evol. Equ., 7 (2007), 113. doi: 10.1007/s00028-007-0249-3. Google Scholar

[2]

A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors,, J. Math. Phys., 32 (1991), 544. doi: 10.1063/1.529391. Google Scholar

[3]

K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems,, Arch. Ration. Mech. Anal., 199 (2011), 177. doi: 10.1007/s00205-010-0321-y. Google Scholar

[4]

C. Berthon, P. Charrier and B. Dubroca, An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions,, J. Sci. Comput., 31 (2007), 347. doi: 10.1007/s10915-006-9108-6. Google Scholar

[5]

S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy,, Commun. Pure Appl. Math., 60 (2007), 1559. doi: 10.1002/cpa.20195. Google Scholar

[6]

L. L. Bonilla, T. Götz, A. Klar, N. Marheineke and R. Wegener, Hydrodynamic limit of a Fokker-Planck equation describing fiber lay-down processes,, SIAM J. Appl. Math., 68 (2007), 648. doi: 10.1137/070692728. Google Scholar

[7]

L. L. Bonilla, A. Klar and S. Martin, Higher order averaging of linear Fokker-Planck equations with periodic forcing,, SIAM J. Appl. Math., 72 (2012), 1315. doi: 10.1137/11083959X. Google Scholar

[8]

T. A. Brunner and J. P. Holloway, One-dimensional Riemann solvers and the maximum entropy closure,, J. Quant. Spectrosc. Radiat. Transfer, 69 (2001), 543. Google Scholar

[9]

J. A. Carrillo, V. Caselles and S. Moll, On the relativistic heat equation in one space dimension,, Proc. London Math. Soc., 107 (2013), 1395. doi: 10.1112/plms/pdt015. Google Scholar

[10]

I. L. Chern, Long-time effect of relaxation for hyperbolic conservation laws,, Commun. Math. Phys., 172 (1995), 39. doi: 10.1007/BF02104510. Google Scholar

[11]

B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems,, SIAM J. Numer. Anal., 35 (1998), 2440. doi: 10.1137/S0036142997316712. Google Scholar

[12]

J. F. Coulombel, F. Golse and T. Goudon, Diffusion approximation and entropy based moment closure for kinetic equations,, Asymptot. Anal., 45 (2005), 1. Google Scholar

[13]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,, Math. Models Methods Appl. Sci., 18 (2008), 1193. doi: 10.1142/S0218202508003005. Google Scholar

[14]

P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics,, J. Stat. Phys., 152 (2013), 1033. doi: 10.1007/s10955-013-0805-x. Google Scholar

[15]

L. Desvilettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation,, Comm. Pure Appl. Math., 54 (2001), 1. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q. Google Scholar

[16]

J. Dolbeault, A. Klar, C. Mouhot and C. Schmeiser, Hypocoercivity and a Fokker-Planck equation for fiber lay-down,, Applied Mathematical Research Express, (2013), 165. Google Scholar

[17]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity,, to appear in TAMS 2014., (2014). Google Scholar

[18]

B. Dubroca and J. L. Feugeas, Entropic moment closure hierarchy for the radiative transfer equation,, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915. doi: 10.1016/S0764-4442(00)87499-6. Google Scholar

[19]

B. Dubroca and A. Klar, Half moment closure for radiative transfer equations,, J. Comput. Phys., 180 (2002), 584. doi: 10.1006/jcph.2002.7106. Google Scholar

[20]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative transfer,, J. Comput. Phys., 218 (2006), 1. doi: 10.1016/j.jcp.2006.01.038. Google Scholar

[21]

M. Grothaus, A. Klar, J. Maringer and P. Stilgenbauer, Geometry, mixing, properties and hypocoercivity of a degenerate diffusion arising in technical textile industry,, , (). Google Scholar

[22]

T. Götz, A. Klar, N. Marheineke and R. Wegener, A stochastic model for the fiber lay-down process in the nonwoven production,, SIAM J. Appl. Math., 67 (2007), 1704. doi: 10.1137/06067715X. Google Scholar

[23]

A. Klar, N. Marheineke and R. Wegener, Hierarchy of mathematical models for production processes of technical textiles,, ZAMM Z. Angew. Math. Mech., 89 (2009), 941. doi: 10.1002/zamm.200900282. Google Scholar

[24]

A. Klar, J. Maringer and R. Wegener, A 3D model for fiber lay-down processes in non-woven production processes,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512500200. Google Scholar

[25]

P. E. Klöden and P. Platen, Numerical Solution of Stochastic Differential Equations,, Springer, (1999). Google Scholar

[26]

A. Klar and O. Tse, An entropy functional and explicit decay rates for a partially dissipative hyperbolic system,, to appear in ZAMM Z. Angew. Math. Mech., (2014). Google Scholar

[27]

C. D. Levermore, Relating Eddington factors to flux limiters,, J. Quant. Spectrosc. Radiat. Transf., 31 (1984), 149. doi: 10.1016/0022-4073(84)90112-2. Google Scholar

[28]

T. Luo, R. Natalini and Z. Xin, Large time behaviour of the solutions to a hydrodynamic model for semiconductors,, SIAM J. Appl. Math., 59 (1999), 810. doi: 10.1137/S0036139996312168. Google Scholar

[29]

G. Papanicolaou, D. Stroock and S. Varadhan, Martingale Approach to Some Limit Theorems,, in Statistical Mechanics and Dynamical Systems (ed. D. Ruelle), (1977). Google Scholar

[30]

H. Risken, The Fokker-Planck Equation,, Springer, (1989). doi: 10.1007/978-3-642-61544-3. Google Scholar

[31]

A. Roth, A. Klar, B. Simeon and E. Zharovsky, A semi-Lagrangian finite volume method for 3-D Fokker-Planck equations associated to stochastic dynamical systems on the sphere,, to appear in J. Sci. Comput., (2014). Google Scholar

[32]

T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[33]

C. Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5. Google Scholar

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