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 and 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), 113-143. doi: 10.1007/s00028-007-0249-3.

[2]

A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors, J. Math. Phys., 32 (1991), 544-550. doi: 10.1063/1.529391.

[3]

K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal., 199 (2011), 177-227. doi: 10.1007/s00205-010-0321-y.

[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), 347-389. doi: 10.1007/s10915-006-9108-6.

[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), 1559-1622. doi: 10.1002/cpa.20195.

[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), 648-665. doi: 10.1137/070692728.

[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), 1315-1342. doi: 10.1137/11083959X.

[8]

T. A. Brunner and J. P. Holloway, One-dimensional Riemann solvers and the maximum entropy closure, J. Quant. Spectrosc. Radiat. Transfer, 69 (2001), 543-566.

[9]

J. A. Carrillo, V. Caselles and S. Moll, On the relativistic heat equation in one space dimension, Proc. London Math. Soc., 107 (2013), 1395-1423. doi: 10.1112/plms/pdt015.

[10]

I. L. Chern, Long-time effect of relaxation for hyperbolic conservation laws, Commun. Math. Phys., 172 (1995), 39-55. doi: 10.1007/BF02104510.

[11]

B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712.

[12]

J. F. Coulombel, F. Golse and T. Goudon, Diffusion approximation and entropy based moment closure for kinetic equations, Asymptot. Anal., 45 (2005), 1-39.

[13]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[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), 1033-1068. doi: 10.1007/s10955-013-0805-x.

[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), 1-42. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.

[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), 165-175.

[17]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity, to appear in TAMS 2014.

[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), 915-920. doi: 10.1016/S0764-4442(00)87499-6.

[19]

B. Dubroca and A. Klar, Half moment closure for radiative transfer equations, J. Comput. Phys., 180 (2002), 584-596. doi: 10.1006/jcph.2002.7106.

[20]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative transfer, J. Comput. Phys., 218 (2006), 1-18. doi: 10.1016/j.jcp.2006.01.038.

[21]

M. Grothaus, A. Klar, J. Maringer and P. Stilgenbauer, Geometry, mixing, properties and hypocoercivity of a degenerate diffusion arising in technical textile industry, arXiv:1203.4502.

[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), 1704-1717. doi: 10.1137/06067715X.

[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), 941-961. doi: 10.1002/zamm.200900282.

[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), 1250020, 18 pp. doi: 10.1142/S0218202512500200.

[25]

P. E. Klöden and P. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1999.

[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.

[27]

C. D. Levermore, Relating Eddington factors to flux limiters, J. Quant. Spectrosc. Radiat. Transf., 31 (1984), 149-160. doi: 10.1016/0022-4073(84)90112-2.

[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), 810-830. doi: 10.1137/S0036139996312168.

[29]

G. Papanicolaou, D. Stroock and S. Varadhan, Martingale Approach to Some Limit Theorems, in Statistical Mechanics and Dynamical Systems (ed. D. Ruelle), Duke University Mathematics Series III, Durham, 1977.

[30]

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

[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.

[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), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

[33]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.

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), 113-143. doi: 10.1007/s00028-007-0249-3.

[2]

A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors, J. Math. Phys., 32 (1991), 544-550. doi: 10.1063/1.529391.

[3]

K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal., 199 (2011), 177-227. doi: 10.1007/s00205-010-0321-y.

[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), 347-389. doi: 10.1007/s10915-006-9108-6.

[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), 1559-1622. doi: 10.1002/cpa.20195.

[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), 648-665. doi: 10.1137/070692728.

[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), 1315-1342. doi: 10.1137/11083959X.

[8]

T. A. Brunner and J. P. Holloway, One-dimensional Riemann solvers and the maximum entropy closure, J. Quant. Spectrosc. Radiat. Transfer, 69 (2001), 543-566.

[9]

J. A. Carrillo, V. Caselles and S. Moll, On the relativistic heat equation in one space dimension, Proc. London Math. Soc., 107 (2013), 1395-1423. doi: 10.1112/plms/pdt015.

[10]

I. L. Chern, Long-time effect of relaxation for hyperbolic conservation laws, Commun. Math. Phys., 172 (1995), 39-55. doi: 10.1007/BF02104510.

[11]

B. Cockburn and C. W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712.

[12]

J. F. Coulombel, F. Golse and T. Goudon, Diffusion approximation and entropy based moment closure for kinetic equations, Asymptot. Anal., 45 (2005), 1-39.

[13]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[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), 1033-1068. doi: 10.1007/s10955-013-0805-x.

[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), 1-42. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.

[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), 165-175.

[17]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity, to appear in TAMS 2014.

[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), 915-920. doi: 10.1016/S0764-4442(00)87499-6.

[19]

B. Dubroca and A. Klar, Half moment closure for radiative transfer equations, J. Comput. Phys., 180 (2002), 584-596. doi: 10.1006/jcph.2002.7106.

[20]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative transfer, J. Comput. Phys., 218 (2006), 1-18. doi: 10.1016/j.jcp.2006.01.038.

[21]

M. Grothaus, A. Klar, J. Maringer and P. Stilgenbauer, Geometry, mixing, properties and hypocoercivity of a degenerate diffusion arising in technical textile industry, arXiv:1203.4502.

[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), 1704-1717. doi: 10.1137/06067715X.

[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), 941-961. doi: 10.1002/zamm.200900282.

[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), 1250020, 18 pp. doi: 10.1142/S0218202512500200.

[25]

P. E. Klöden and P. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1999.

[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.

[27]

C. D. Levermore, Relating Eddington factors to flux limiters, J. Quant. Spectrosc. Radiat. Transf., 31 (1984), 149-160. doi: 10.1016/0022-4073(84)90112-2.

[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), 810-830. doi: 10.1137/S0036139996312168.

[29]

G. Papanicolaou, D. Stroock and S. Varadhan, Martingale Approach to Some Limit Theorems, in Statistical Mechanics and Dynamical Systems (ed. D. Ruelle), Duke University Mathematics Series III, Durham, 1977.

[30]

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

[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.

[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), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

[33]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp. doi: 10.1090/S0065-9266-09-00567-5.

[1]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215

[2]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011

[3]

José A. Carrillo, Renjun Duan, Ayman Moussa. Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. Kinetic and Related Models, 2011, 4 (1) : 227-258. doi: 10.3934/krm.2011.4.227

[4]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017

[5]

Jonathan Zinsl. Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2915-2930. doi: 10.3934/dcds.2016.36.2915

[6]

Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic and Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169

[7]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009

[8]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks and Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027

[9]

Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure and Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579

[10]

Lan Luo, Hongjun Yu. Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system. Kinetic and Related Models, 2016, 9 (2) : 393-405. doi: 10.3934/krm.2016.9.393

[11]

Kosuke Ono, Walter A. Strauss. Regular solutions of the Vlasov-Poisson-Fokker-Planck system. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 751-772. doi: 10.3934/dcds.2000.6.751

[12]

Vladimir I. Bogachev, Stanislav V. Shaposhnikov, Alexander Yu. Veretennikov. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3519-3543. doi: 10.3934/dcds.2016.36.3519

[13]

Peter Benner, Tobias Breiten, Carsten Hartmann, Burkhard Schmidt. Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations. Journal of Computational Dynamics, 2020, 7 (1) : 1-33. doi: 10.3934/jcd.2020001

[14]

Roman Shvydkoy. Global hypocoercivity of kinetic Fokker-Planck-Alignment equations. Kinetic and Related Models, 2022, 15 (2) : 213-237. doi: 10.3934/krm.2022005

[15]

Hongjie Dong, Yan Guo, Timur Yastrzhembskiy. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition. Kinetic and Related Models, 2022, 15 (3) : 467-516. doi: 10.3934/krm.2022003

[16]

Anton Arnold, Beatrice Signorello. Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022009

[17]

Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135

[18]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371

[19]

John A. D. Appleby, Alexandra Rodkina, Henri Schurz. Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 667-696. doi: 10.3934/dcdsb.2006.6.667

[20]

Eugenii Shustin. Exponential decay of oscillations in a multidimensional delay differential system. Conference Publications, 2003, 2003 (Special) : 809-816. doi: 10.3934/proc.2003.2003.809

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (167)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]