December 2018, 11(6): 1475-1501. doi: 10.3934/krm.2018058

Linear Boltzmann dynamics in a strip with large reflective obstacles: Stationary state and residence time

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, Ⅰ - 00161, Roma, Italy

* Corresponding author: Alessandro Ciallella

Received  August 2017 Published  June 2018

The presence of obstacles modifies the way in which particles diffuse. In cells, for instance, it is observed that, due to the presence of macromolecules playing the role of obstacles, the mean-square displacement of biomolecules scales as a power law with exponent smaller than one. On the other hand, different situations in grain and pedestrian dynamics in which the presence of an obstacle accelerates the dynamics are known. We focus on the time, called the residence time, needed by particles to cross a strip assuming that the dynamics inside the strip follows the linear Boltzmann dynamics. We find that the residence time is not monotonic with respect to the size and the location of the obstacles, since the obstacle can force those particles that eventually cross the strip to spend a smaller time in the strip itself. We focus on the case of a rectangular strip with two open sides and two reflective sides and we consider reflective obstacles into the strip. We prove that the stationary state of the linear Boltzmann dynamics, in the diffusive regime, converges to the solution of the Laplace equation with Dirichlet boundary conditions on the open sides and homogeneous Neumann boundary conditions on the other sides and on the obstacle boundaries.

Citation: Alessandro Ciallella, Emilio N. M. Cirillo. Linear Boltzmann dynamics in a strip with large reflective obstacles: Stationary state and residence time. Kinetic & Related Models, 2018, 11 (6) : 1475-1501. doi: 10.3934/krm.2018058
References:
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G. AlbiM. BonginiE. Cristiani and D. Kalise, Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710. doi: 10.1137/15M1017016.

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F. Alonso-Marroquin, S. I. Azeezullah, S. A. Galindo-Torres and L. M. Olsen-Kettle, Bottlenecks in granular flow: When does an obstacle increase the flow rate in an hourglass?, Phys. Rev. E, 85 (2012), 020301. doi: 10.1103/PhysRevE.85.020301.

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G. BasileA. NotaF. Pezzotti and M. Pulvirenti, Derivation of the Fick's law for the Lorentz model in a low density regime, Comm. Math. Phys., 336 (2015), 1607-1636. doi: 10.1007/s00220-015-2306-z.

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N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677.

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D. BraessA. Nagurney and T. Wakolbinger, On a paradox of traffic planning, Transportation Science, 39 (2005), 446-450. doi: 10.1287/trsc.1050.0127.

[6]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-1-4419-8524-8.

[7]

A. Ciallella, On the linear Boltzmann transport equation: A Monte Carlo algorithm for stationary solutions and residence times in presence of obstacles, in AIMETA 2017 - Proceedings of the 23rd Conference of the Italian Association of Theoretical and Applied Mechanics, 5 (2017), 952–960.

[8]

A. Ciallella, E. N. M. Cirillo and J. Sohier, Residence time of symmetric random walkers in a strip with large reflective obstacles, Phys. Rev. E, 97 (2018), 052116. doi: 10.1103/PhysRevE.97.052116.

[9]

E. N. M. Cirillo and M. Colangeli, Stationary uphill currents in locally perturbed zero-range processes, Phys. Rev. E, 96 (2017), 052137. doi: 10.1103/PhysRevE.96.052137.

[10]

E. N. M. CirilloM. Colangeli and A. Muntean, Does communication enhance pedestrians transport in the dark?, Comptes Rendus Mecanique, 344 (2016), 19-23. doi: 10.1016/j.crme.2015.09.004.

[11]

E. N. M. Cirillo, O. Krehel, A. Muntean and R. van Santen, Lattice model of reduced jamming by a barrier, Phys. Rev. E, 94 (2016), 042115. doi: 10.1103/PhysRevE.94.042115.

[12]

E. N. M. CirilloO. KrehelA. MunteanR. van Santen and A. Sengar, Residence time estimates for asymmetric simple exclusion dynamics on strips, Phys. A, 442 (2016), 436-457. doi: 10.1016/j.physa.2015.09.037.

[13]

E. N. M. Cirillo and A. Muntean, Can cooperation slow down emergency evacuations?, Comptes Rendus Mécanique, 340 (2012), 625-628. doi: 10.1016/j.crme.2012.09.003.

[14]

E. N. M. Cirillo and A. Muntean, Dynamics of pedestrians in regions with no visibility-a lattice model without exclusion, Phys. A, 392 (2013), 3578-3588. doi: 10.1016/j.physa.2013.04.029.

[15]

E. Cristiani and D. Peri, Handling obstacles in pedestrian simulations: Models and optimization, Appl. Math. Model., 45 (2017), 285-302. doi: 10.1016/j.apm.2016.12.020.

[16]

A. J. Ellery, M. J. Simpson, S. W. McCue and R. E. Baker, Characterizing transport through a crowded environment with different obstacle sizes, The Journal of Chemical Physics, 140 (2014), 054108. doi: 10.1063/1.4864000.

[17]

R. Escobar and A. De La Rosa, Architectural Design for the Survival Optimization of Panicking Fleeing Victims, in Advances in Artificial Life. ECAL 2003 (eds. W. Banzhaf, J. Ziegler, T. Christaller, P. Dittrich, and J. T. Kim), 2801 Springer (2003), 97–106. doi: 10.1007/978-3-540-39432-7_11.

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R. Esposito and M. Pulvirenti, From Particles to Fluids, Hand-Book of Mathematical Fluid Dynamics Vol. Ⅲ, North-Holland, Amsterdam, (2004), 1–82.

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L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[20]

B. W. Fitzgerald, J. T. Padding and R. van Santen, Simple diffusion hopping model with convection, Phys. Rev. E, 95 (2017), 013307. doi: 10.1103/PhysRevE.95.013307.

[21]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.

[22]

D. HelbingL. BuznaA. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transportation Science, 39 (2005), 1-24. doi: 10.1287/trsc.1040.0108.

[23]

D. Helbing, I. Farkas, P. Molnàr and T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations, in Pedestrian and Evacuation Dynamics (eds. M. Schreckenberg and S. D. Sharma), Springer, (2002), 21–58.

[24]

D. HelbingI. J. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023.

[25]

D. HelbingP. MolnárI. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B: Planning and Design, 28 (2001), 361-383. doi: 10.1068/b2697.

[26]

F. Höfling and T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Progr. Phys., 76 (2013), 046602, 50 pp. doi: 10.1088/0034-4885/76/4/046602.

[27]

R. L. Hughes, The flow of human crowds, Annual Review of Fluid Mechanics, 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.

[28]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-1-4757-4317-3.

[29]

M. Matsumoto and T. Nishimura, Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. on Modeling and Computer Simulation, 8 (1998), 3-30. doi: 10.1145/272991.272995.

[30]

M. Matsumoto and T. Nishimura. A Nonempirical Test on the Weight of Pseudorandom Number Generators, in: Monte Carlo and Quasi-Monte Carlo methods 2000 (eds. K. T. Fang, F. J. Hickernel, and H. Niederreiter), Springer-Verlag, (2002), 381–395.

[31]

M. A. MourãoJ. B. Hakim and S. Schnell, Connecting the dots: The effects of macromolecular crowding on cell physiology, Biophysical Journal, 107 (2017), 2761-2766. doi: 10.1016/j.bpj.2014.10.051.

[32]

M. J. Saxton, Anomalous diffusion due to obstacles: A Monte Carlo study, Biophysical Journal, 66 (1994), 394-401. doi: 10.1016/S0006-3495(94)80789-1.

[33]

K. ToP. Y. Lai and H. K. Pak, Jamming of granular flow in a two-dimensional hopper, Phys. Rev. Lett., 86 (2001), 71-74. doi: 10.1103/PhysRevLett.86.71.

[34]

I. Zuriguel, A. Garcimartín, D. Maza, L. A. Pugnaloni and J. M. Pastor, Jamming during the discharge of granular matter from a silo, Phys. Rev. E, 71 (2005), 051303. doi: 10.1103/PhysRevE.71.051303.

[35]

I. Zuriguel, A. Janda, A. Garcimartín, C. Lozano, R. Arévalo and D. Maza, Silo clogging reduction by the presence of an obstacle, Phys. Rev. Lett., 107 (2011), 278001. doi: 10.1103/PhysRevLett.107.278001.

show all references

References:
[1]

G. AlbiM. BonginiE. Cristiani and D. Kalise, Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710. doi: 10.1137/15M1017016.

[2]

F. Alonso-Marroquin, S. I. Azeezullah, S. A. Galindo-Torres and L. M. Olsen-Kettle, Bottlenecks in granular flow: When does an obstacle increase the flow rate in an hourglass?, Phys. Rev. E, 85 (2012), 020301. doi: 10.1103/PhysRevE.85.020301.

[3]

G. BasileA. NotaF. Pezzotti and M. Pulvirenti, Derivation of the Fick's law for the Lorentz model in a low density regime, Comm. Math. Phys., 336 (2015), 1607-1636. doi: 10.1007/s00220-015-2306-z.

[4]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677.

[5]

D. BraessA. Nagurney and T. Wakolbinger, On a paradox of traffic planning, Transportation Science, 39 (2005), 446-450. doi: 10.1287/trsc.1050.0127.

[6]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-1-4419-8524-8.

[7]

A. Ciallella, On the linear Boltzmann transport equation: A Monte Carlo algorithm for stationary solutions and residence times in presence of obstacles, in AIMETA 2017 - Proceedings of the 23rd Conference of the Italian Association of Theoretical and Applied Mechanics, 5 (2017), 952–960.

[8]

A. Ciallella, E. N. M. Cirillo and J. Sohier, Residence time of symmetric random walkers in a strip with large reflective obstacles, Phys. Rev. E, 97 (2018), 052116. doi: 10.1103/PhysRevE.97.052116.

[9]

E. N. M. Cirillo and M. Colangeli, Stationary uphill currents in locally perturbed zero-range processes, Phys. Rev. E, 96 (2017), 052137. doi: 10.1103/PhysRevE.96.052137.

[10]

E. N. M. CirilloM. Colangeli and A. Muntean, Does communication enhance pedestrians transport in the dark?, Comptes Rendus Mecanique, 344 (2016), 19-23. doi: 10.1016/j.crme.2015.09.004.

[11]

E. N. M. Cirillo, O. Krehel, A. Muntean and R. van Santen, Lattice model of reduced jamming by a barrier, Phys. Rev. E, 94 (2016), 042115. doi: 10.1103/PhysRevE.94.042115.

[12]

E. N. M. CirilloO. KrehelA. MunteanR. van Santen and A. Sengar, Residence time estimates for asymmetric simple exclusion dynamics on strips, Phys. A, 442 (2016), 436-457. doi: 10.1016/j.physa.2015.09.037.

[13]

E. N. M. Cirillo and A. Muntean, Can cooperation slow down emergency evacuations?, Comptes Rendus Mécanique, 340 (2012), 625-628. doi: 10.1016/j.crme.2012.09.003.

[14]

E. N. M. Cirillo and A. Muntean, Dynamics of pedestrians in regions with no visibility-a lattice model without exclusion, Phys. A, 392 (2013), 3578-3588. doi: 10.1016/j.physa.2013.04.029.

[15]

E. Cristiani and D. Peri, Handling obstacles in pedestrian simulations: Models and optimization, Appl. Math. Model., 45 (2017), 285-302. doi: 10.1016/j.apm.2016.12.020.

[16]

A. J. Ellery, M. J. Simpson, S. W. McCue and R. E. Baker, Characterizing transport through a crowded environment with different obstacle sizes, The Journal of Chemical Physics, 140 (2014), 054108. doi: 10.1063/1.4864000.

[17]

R. Escobar and A. De La Rosa, Architectural Design for the Survival Optimization of Panicking Fleeing Victims, in Advances in Artificial Life. ECAL 2003 (eds. W. Banzhaf, J. Ziegler, T. Christaller, P. Dittrich, and J. T. Kim), 2801 Springer (2003), 97–106. doi: 10.1007/978-3-540-39432-7_11.

[18]

R. Esposito and M. Pulvirenti, From Particles to Fluids, Hand-Book of Mathematical Fluid Dynamics Vol. Ⅲ, North-Holland, Amsterdam, (2004), 1–82.

[19]

L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[20]

B. W. Fitzgerald, J. T. Padding and R. van Santen, Simple diffusion hopping model with convection, Phys. Rev. E, 95 (2017), 013307. doi: 10.1103/PhysRevE.95.013307.

[21]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.

[22]

D. HelbingL. BuznaA. Johansson and T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transportation Science, 39 (2005), 1-24. doi: 10.1287/trsc.1040.0108.

[23]

D. Helbing, I. Farkas, P. Molnàr and T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations, in Pedestrian and Evacuation Dynamics (eds. M. Schreckenberg and S. D. Sharma), Springer, (2002), 21–58.

[24]

D. HelbingI. J. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. doi: 10.1038/35035023.

[25]

D. HelbingP. MolnárI. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environment and Planning B: Planning and Design, 28 (2001), 361-383. doi: 10.1068/b2697.

[26]

F. Höfling and T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Progr. Phys., 76 (2013), 046602, 50 pp. doi: 10.1088/0034-4885/76/4/046602.

[27]

R. L. Hughes, The flow of human crowds, Annual Review of Fluid Mechanics, 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.

[28]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-1-4757-4317-3.

[29]

M. Matsumoto and T. Nishimura, Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. on Modeling and Computer Simulation, 8 (1998), 3-30. doi: 10.1145/272991.272995.

[30]

M. Matsumoto and T. Nishimura. A Nonempirical Test on the Weight of Pseudorandom Number Generators, in: Monte Carlo and Quasi-Monte Carlo methods 2000 (eds. K. T. Fang, F. J. Hickernel, and H. Niederreiter), Springer-Verlag, (2002), 381–395.

[31]

M. A. MourãoJ. B. Hakim and S. Schnell, Connecting the dots: The effects of macromolecular crowding on cell physiology, Biophysical Journal, 107 (2017), 2761-2766. doi: 10.1016/j.bpj.2014.10.051.

[32]

M. J. Saxton, Anomalous diffusion due to obstacles: A Monte Carlo study, Biophysical Journal, 66 (1994), 394-401. doi: 10.1016/S0006-3495(94)80789-1.

[33]

K. ToP. Y. Lai and H. K. Pak, Jamming of granular flow in a two-dimensional hopper, Phys. Rev. Lett., 86 (2001), 71-74. doi: 10.1103/PhysRevLett.86.71.

[34]

I. Zuriguel, A. Garcimartín, D. Maza, L. A. Pugnaloni and J. M. Pastor, Jamming during the discharge of granular matter from a silo, Phys. Rev. E, 71 (2005), 051303. doi: 10.1103/PhysRevE.71.051303.

[35]

I. Zuriguel, A. Janda, A. Garcimartín, C. Lozano, R. Arévalo and D. Maza, Silo clogging reduction by the presence of an obstacle, Phys. Rev. Lett., 107 (2011), 278001. doi: 10.1103/PhysRevLett.107.278001.

Figure 1.  Domain $\Omega$: strip with large fixed obstacles, where $\partial\Omega_L$ and $\partial\Omega_R$ are the vertical open boundaries and $\partial\Omega_E$ are reflective boundaries.
Figure 2.  Elastic collision with a scatterers: impact parameter $\delta$ and angle of incidence $\alpha$.
Figure 3.  Plot of the simulated solutions $h_{t_m}$ in a $3D$ plot and in a $2D$ plot constructed by averaging on the $x_2$ variable: in dark gray $t_m = 2\cdot 10^{-1}$, in gray $t_m = 10^{-1}$, in light gray $t_m = 2\cdot10^{-2}$. In black (grid and dashed line) the analytic solution $\rho$ of the associated Laplace problem.
Figure 4.  Simulation parameter $t_m = 10^{-2}$: relative error ${|h_{t_m}-\rho|}/{\rho}$.
Figure 5.  Simulation parameter $t_m = 10^{-2}$: on the left in gray the numerical solution $h_{t_m}$ and in black the solution $\rho$ of the associated Laplace problem; on the right the relative error ${|h_{t_m}-\rho|}/{\rho}$. Into the strip there is a square obstacle with side $8\cdot10^{-1}$.
Figure 6.  Simulation parameter $t_m = 10^{-2}$: on the left in gray the numerical solution $h_{t_m}$ and in black the solution $\rho$ of the associated Laplace problem; on the right the relative error ${|h_{t_m}-\rho|}/{\rho}$. In the strip is placed a very thin obstacle with height of $0.8$.
Figure 7.  Simulation parameter $t_m = 10^{-2}$: on the left in gray the numerical solution $h_{t_m}$ and in black the solution $\rho$ of the associated Laplace problem; on the right the relative error ${|h_{t_m}-\rho|}/{\rho}$. In the first line we show the case of two squared obstacles with side $6\cdot 10^{-1}$, in the second one a couple of rectangular obstacles, taller and thinner than the squares.
Figure 8.  Residence time vs. height of a centered rectangular obstacle with fixed width $4\cdot 10^{-2}$ (on the left) and $4 \cdot 10^{-1}$ (on the right). Simulation parameters: $L_1 = 4$, $L_2 = 1$, $t_m = 2\cdot 10^{-2}$, total number of inserted particles $10^8$, the total number of particles exiting through the right boundary varies from $5.3\cdot 10^{5}$ to $3.6\cdot 10^{5}$ (on the left) and from $5.3\cdot 10^{5}$ to $2.1\cdot 10^{5}$ (on the right) depending on the obstacle height. The solid lines represent the value of the residence time measured for the empty strip (no obstacle).
Figure 9.  Residence time vs. height of a centered rectangular obstacle with fixed width $8\cdot 10^{-1}$ (on the left) and $12 \cdot 10^{-1}$ (on the right). Simulation parameters: $L_1 = 4$, $L_2 = 1$, $t_m = 2\cdot 10^{-2}$, total number of inserted particles $10^8$, the total number of particles exiting through the right boundary varies from $5.2\cdot 10^{5}$ to $1.4\cdot 10^{5}$ (on the left) and from $5.2 \cdot 10^{5}$ to $1.1 \cdot 10^{5}$ (on the right) depending on the obstacle height. The solid lines represent the value of the residence time measured for the empty strip (no obstacle).
Figure 10.  Residence time vs. width of a centered rectangular obstacle with fixed height $0.8$ (on the left) and vs. the side length of a centered squared obstacle (on the right). Simulation parameters: $L_1 = 4$, $L_2 = 1$, $t_m = 2\cdot 10^{-2} $, total number of inserted particles $10^{8}$, the total number of particles exiting through the right boundary varies from $4.2 \cdot 10^{5}$ to $1.1\cdot 10^{5}$ (on the left) and from $5.3\cdot 10^5 $ to $1.3\cdot 10^5$ (on the right) depending on the obstacle width. The solid lines represent the value of the residence time measured for the empty strip (no obstacle).
Figure 11.  Residence time vs. position of the center of the obstacle. The obstacle is a square of side length $0.8$ on the left and a rectangle of side lengths $0.04$ and $0.8$ on the right. Simulation parameters: $L_1 = 4$, $L_2 = 1$, $t_m = 2\cdot 10^{-2}$, total number of inserted particles $10^8$, the total number of particles exiting through the right boundary is stable at the order of $2.6\cdot 10^5$ (on the left) and of $4\cdot 10^5$ (on the right) not depending on the obstacle position. The solid lines represent the value of the residence time measured for the empty strip (no obstacle).
Figure 12.  As in the right panel in Figure 8. In the left panel the height of the obstacle is equal to $0.8$. Left panel: the mean time spent by particles crossing the strip in each point of the strip ($0.02\times0.02$ cells have been considered) for the empty strip case (black) and in presence of the obstacle (gray). Right panel: residence time in regions L (circles), C (squares), and R (triangles) in presence of the obstacle (gray) and for the empty strip case (black).
Figure 13.  As in Figure 12 for the geometry in the left panel in Figure 10. In the left panel the width of the obstacle is $2.28$.
Figure 14.  As in Figure 12 for the geometry in the left panel in Figure 11. In the left panel the position of the center of the obstacle is $0.8$.
Figure 15.  Domain $\Lambda$: infinite strip with big fixed obstacles: the whole boundaries of $\Lambda$ is a specular reflective boundary.
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