Motion of a Rigid Body in a Special Lorentz Gas: Loss of Memory Effect

Kai Koike

doi:10.1007/s10955-018-2072-3

Journal of Statistical Physics

August 2018, Volume 172, Issue 3, pp 795–823 | Cite as

Motion of a Rigid Body in a Special Lorentz Gas: Loss of Memory Effect

Authors
Authors and affiliations

Kai Koike

Article

First Online: 01 June 2018

102 Downloads

Abstract

Linear motion of a rigid body in a special kind of Lorentz gas is mathematically analyzed. The rigid body moves against gas drag according to Newton’s equation. The gas model is a special Lorentz gas consisting of gas molecules and background obstacles, which was introduced in Tsuji and Aoki (J Stat Phys 146:620–645, 2012). The specular boundary condition is imposed on the resulting kinetic equation. This study complements the numerical study by Tsuji and Aoki cited above—although the setting in this paper is slightly different from theirs, qualitatively the same asymptotic behavior is proved: The velocity V(t) of the rigid body decays exponentially if the obstacles undergo thermal motion; if the obstacles are motionless, then the velocity V(t) decays algebraically with a rate $t^{-\,5}$ independent of the spatial dimension. This demonstrates the idea that interaction of the molecules with the background obstacles destroys the memory effect due to recollision.

Keywords

Lorentz gas Moving boundary problem Memory effect

This is a preview of subscription content, to check access.

Notes

Acknowledgements

I thank T. Tsuji for his comments on the references.

Appendix: Proof of Theorem 2

The basic strategy is the same as that of Theorem 1: (i) Define an appropriate function space $\mathcal {K}$. (ii) Prove decay estimates of $r_{W}^{\pm }(t)$ given $W\in \mathcal {K}$. (iii) Prove that $V_W$ again belongs to $\mathcal {K}$. (iv) Apply Schauder’s fixed point theorem to show that the map $\mathcal {K}\ni W\mapsto V_W \in \mathcal {K}$ has a fixed point $V\in \mathcal {K}$—this shows the existence part. (v) Prove that any solution (f, V) satisfies ineqs. (90) and that V is decreasing on the interval $[0,\infty )$.

Only steps (i) through (iii) are explained in this section; steps (iv) and (v) can be carried out similarly as in the proof of Theorem 1.

Step (i) is the following:

Definition 2

Let $\gamma >0$. A Lipschitz continuous function $W:[0,\infty )\rightarrow \mathbb {R}$ belongs to $\mathcal {K}=\mathcal {K}(\gamma )$ if $W(0)=\gamma $; W is decreasing on the interval $[0,\infty )$; and satisfies

$$\begin{aligned} \frac{1}{2}\gamma e^{-C_0 t}\le W(t)\le \gamma e^{-C_0 t} \end{aligned}$$

and $|dW(t)/dt|\le 1$.

Let $W\in \mathcal {K}$. Note that $r_{W}^{\pm }(t)\ge 0$ since $W(t)>0$. Moreover, $r_{W}^{+}(t)=0$ since W is decreasing on the interval $[0,\infty )$; therefore, only $r_{W}^{-}(t)$ needs analysis.

Step (ii) is to prove the following:

Proposition 6

Let $\kappa \ge 1$ and $\varepsilon \ge 2\kappa C_0$. If $W\in \mathcal {K}(\gamma )$, then

$$\begin{aligned} 0\le r_{W}^{-}(t)\le C\gamma ^3 \frac{e^{-C_0 t}}{(1+t)^{d+2}} \end{aligned}$$

(91)

for some positive constant C independent of $\gamma $.

Proof

Define $L_t$ by Eq. (46) and let

$$\begin{aligned} L_{t}^{*}=\{ (x,\xi )\in L_t \mid \tau _1 \le t/2 \}, \quad L_{t}^{**}=\{ (x,\xi )\in L_t \mid \tau _1>t/2 \}. \end{aligned}$$

Then $r_{W}^{-}(t)=r_{W,*}^{-}(t)+r_{W,**}^{-}(t)$, where

$$\begin{aligned} r_{W,*}^{-}(t)&=2\int _{L_{t}^{*}}(\xi _1-W(t))^2 (f_0-f_W)\, d\xi dS, \\ r_{W,**}^{-}(t)&=2\int _{L_{t}^{**}}(\xi _1-W(t))^2 (f_0-f_W)\, d\xi dS. \end{aligned}$$

Note first that $(x,\xi )\in L_t$ implies

$$\begin{aligned} 0<\xi _1 -W(t)<\xi _1 \le \frac{1}{t-\tau _1}\int _{\tau _1}^{t}\gamma e^{-C_0 s}\, ds=\gamma e^{-C_0 \tau _1}\frac{1-e^{-C_0(t-\tau _1)}}{C_0(t-\tau _1)}. \end{aligned}$$

Similar arguments as in the proof of Proposition 2 show

$$\begin{aligned} r_{W,*}^{-}(t)&\le C\gamma ^3 w_{\varepsilon ,\kappa ,d}(t)e^{-\frac{\varepsilon }{2\kappa }t}, \\ r_{W,**}^{-}(t)&\le C\gamma ^3 e^{-\frac{3}{2}C_0 t}. \end{aligned}$$

Since $\varepsilon /2\kappa \ge C_0$, these imply ineq. (91). $\square $

This is step (ii). Step (iii) is to prove the following:

Proposition 7

Let $\kappa \ge 1$ and $\varepsilon \ge 2\kappa C_0$. Then $W\in \mathcal {K}=\mathcal {K}(\gamma )$ implies $V_W \in \mathcal {K}$ for sufficiently small $\gamma $.

Proof

Since $r_{W}^{\pm }(t)\ge 0$, the upper bound

$$\begin{aligned} V_W(t)\le \gamma e^{-C_0 t} \end{aligned}$$

(92)

follows from Eq. (18).

To prove the lower bound

$$\begin{aligned} V_W(t)\ge \frac{1}{2}\gamma e^{-C_0 t}, \end{aligned}$$

(93)

I need first to show

$$\begin{aligned} e^{-\int _{0}^{t}K(W(s))\, ds}\ge \frac{3}{4}e^{-C_0 t} \end{aligned}$$

(94)

for sufficiently small $\gamma $: Note first that $D_{0}''(0)=0$ and $D_{0}'''(0)>0$ by Eq. (11); hence for $\gamma \le 1$,

$$\begin{aligned} 0<K(W(s))&=D_0(W(s))/W(s) \\&\le D_{0}'(W(s)) \\&\le C_0+C(W(s))^2 \\&\le C_0+C\gamma ^2 e^{-2C_0 s} \end{aligned}$$

for some positive constant C. This implies

$$\begin{aligned} e^{-\int _{0}^{t}K(W(s))\, ds}\ge e^{-\int _{0}^{t}(C_0+C\gamma ^2 e^{-2C_0 s})\, ds}=e^{-C\gamma ^2}e^{-C_0 t}. \end{aligned}$$

And taking $\gamma $ sufficiently small leads to ineq. (94).

Now ineq. (93) follows from eq. (18), Proposition 6 and ineq.(94) as follows:

$$\begin{aligned} V_W(t)&\ge \frac{3}{4}\gamma e^{-C_0 t}-C\gamma ^3 \int _{0}^{t}e^{-C_0(t-s)}\frac{e^{-C_0 s}}{(1+s)^{d+2}}\, ds \\&=\frac{3}{4}\gamma e^{-C_0 t}-C\gamma ^3 e^{-C_0 t}\int _{0}^{t} \frac{ds}{(1+s)^{d+2}} \\&\ge \frac{1}{2}\gamma e^{-C_0 t} \end{aligned}$$

if $\gamma $ is sufficiently small.

Next, I show that $V_W$ is decreasing on the interval $[0,\infty )$: Since $V_W>0$ (by ineq. (93)) and $r_{W}^{\pm }(t)\ge 0$,

$$\begin{aligned} \frac{dV_W(t)}{dt}=-K(W(t))V_W(t)-r_{W}^{+}(t)-r_{W}^{-}(t)<0 \end{aligned}$$

for $t\ge 0$.

Lastly, since $r_{W}^{+}(t)=0$, $|r_{W}^{-}(t)|\le C\gamma ^3$ (by Proposition 6) and $|V_W(t)|\le \gamma $ (by ineqs. (92) and (93)),

$$\begin{aligned} \left| \frac{dV_W(t)}{dt} \right| \le K(W(t))|V_W(t)|+|r_{W}^{+}(t)|+|r_{W}^{-}(t)|\le C_{\gamma }\gamma +C\gamma ^3 \le 1 \end{aligned}$$

for sufficiently small $\gamma $. $\square $

Steps (iv) and (v) can be carried out similarly to the proof of Theorem 1. This proves Theorem 2.

References

1.
Aoki, K., Cavallaro, G., Marchioro, C., Pulvirenti, M.: On the motion of a body in thermal equilibrium immersed in a perfect gas. ESAIM: M2AN 42, 263–275 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
2.
Aoki, K., Tsuji, T., Cavallaro, G.: Approach to steady motion of a plate moving in a free-molecular gas under a constant external force. Phys. Rev. E 80, 016309 (2009)ADSCrossRefGoogle Scholar
3.
Belmonte, A., Jacobsen, J., Jayaraman, A.: Monotone solutions of a nonautonomous differential equation for a sedimenting sphere. Electron. J. Differ. Equ. 2001, 1–17 (2001)MathSciNetzbMATHGoogle Scholar
4.
Buttà, P., Cavallaro, G., Marchioro, C.: Mathematical Models of Viscous Friction. Springer International Publishing, Basel (2015)CrossRefzbMATHGoogle Scholar
5.
Caprino, S., Marchioro, C., Pulvirenti, M.: Approach to equilibrium in a microscopic model of friction. Commun. Math. Phys. 264, 167–189 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
6.
Caprino, S., Cavallaro, G., Marchioro, C.: On a microscopic model of viscous friction. Math. Models Methods Appl. Sci. 17, 1369–1403 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
7.
Cavallaro, G.: On the motion of a convex body interacting with a perfect gas in the mean-field approximation. Rend. Mat. 27, 123–145 (2007)MathSciNetzbMATHGoogle Scholar
8.
Cavallaro, G., Marchioro, C.: On the motion of an elastic body in a free gas. Rep. Math. Phys. 69, 251–264 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
9.
Cavallaro, G., Marchioro, C., Tsuji, T.: Approach to equilibrium of a rotating sphere in a Stokes flow. Ann. Univ. Ferrara 57, 211–228 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
10.
Chen, X., Strauss, W.: Approach to equilibrium of a body colliding specularly and diffusely with a sea of particles. Arch. Ration. Mech. Anal. 211, 879–910 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
11.
Chen, X., Strauss, W.: Velocity reversal criterion of a body immersed in a sea of particles. Commun. Math. Phys. 338, 139–168 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
12.
Chen, X., Strauss, W.: Convergence to equilibrium of a body moving in a kinetic sea. SIAM J. Math. Anal. 47, 4630–4651 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
13.
Daitche, A.: On the role of the history force for inertial particles in turbulence. J. Fluid Mech. 782, 567–593 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
14.
Daitche, A., Tél, T.: Memory effects are relevant for chaotic advection of inertial particles. Phys. Rev. Lett. 107, 244501 (2011)ADSCrossRefGoogle Scholar
15.
Dechristé, G., Mieussens, L.: Numerical simulation of micro flows with moving obstacles. J. Phys.: Conf. Ser. 362, 012030 (2012)Google Scholar
16.
Fanelli, C., Sisti, F., Stagno, G.V.: Time dependent friction in a free gas. J. Math. Phys. 57, 033501 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
17.
Jin, W., Kleijn, C.R., van Ommen, J.R.: Direct simulation Monte Carlo calculation of rarefied gas drag using an immersed boundary method. AIP Conf. Proc. 1738, 480017 (2016)CrossRefGoogle Scholar
18.
Kobert, M.: Application of the finite pointset method to moving boundary problems for the BGK model of rarefied gas dynamics. Ph.D. thesis, Fachbereich Mathematik, Technische Universität Kaiserslautern (2015)Google Scholar
19.
Koike, K.: Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinet. Relat. Models 11, 441–467 (2018)MathSciNetCrossRefGoogle Scholar
20.
Rader, D.J., Gallis, M.A., Torczynski, J.R.: DSMC moving-boundary algorithms for simulating MEMS geometries with opening and closing gaps. AIP Conf. Proc. 1333, 760 (2011)ADSCrossRefGoogle Scholar
21.
Russo, G., Filbet, F.: Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinet. Relat. Models 2, 231–250 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
22.
Shrestha, S.: Modeling and Simulation of a Moving Rigid Body in a Rarefied Gas. Ph.D. thesis, Fachbereich Mathematik, Technische Universität Kaiserslautern (2015)Google Scholar
23.
Shrestha, S., Tiwari, S., Klar, A.: Comparison of numerical solutions of the Boltzmann and the Navier–Stokes equations for a moving rigid circular body in a micro scaled cavity. Int. J. Adv. Eng. Sci. Appl. Math. 7, 38–50 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
24.
Shrestha, S., Tiwari, S., Klar, A., Hardt, S.: Numerical simulation of a moving rigid body in a rarefied gas. J. Comput. Phys. 292, 239–252 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
25.
Sisti, F., Ricciuti, C.: Effects of concavity on the motion of a body immersed in a Vlasov gas. SIAM J. Math. Anal. 46, 3579–3611 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
26.
Sone, Y.: Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkäuser, Boston (2007)CrossRefzbMATHGoogle Scholar
27.
Tiwari, S., Klar, A., Hardt, S., Donkov, A.: Coupled solution of the Boltzmann and Navier–Stokes equations in gas–liquid two phase flow. Comput. Fluids 71, 283–296 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
28.
Tsuji, T., Aoki, K.: Decay of a linear pendulum in a free-molecular gas and in a special Lorentz gas. J. Stat. Phys. 146, 620–645 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
29.
Tsuji, T., Aoki, K.: Moving boundary problems for a rarefied gas: spatially one-dimensional case. J. Comput. Phys. 250, 574–600 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
30.
Tsuji, T., Aoki, K.: Decay of a linear pendulem in a collisional gas: spatially one-dimensional case. Phys. Rev. E 89, 052129 (2014)ADSCrossRefGoogle Scholar
31.
Tsuji, T., Aoki, K.: Gas motion in a microgap between a stationary plate and a plate oscillating in its normal direction. Microfluid. Nanofluid. 16, 1033–1045 (2014)CrossRefGoogle Scholar
32.
Tsuji, T., Arai, J., Kawano, S.: Slow approach to steady motion of a concave body in a free-molecular gas. Phys. Rev. E 92, 012130 (2015)ADSMathSciNetCrossRefGoogle Scholar
33.
Vázquez, J.L., Zuazua, E.: Large time behavior for a simplified 1D model of fluid-solid interaction. Commun. Partial Differ. Equ. 28, 1–36 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
34.
Versluis, R., Dorsman, R., Thielen, L., Roos, M.E.: Numerical investigation of turbomolecular pumps using the direct simulation Monte Carlo method with moving surfaces. J. Vac. Sci. Technol. A 27, 543–547 (2009)CrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Kai Koike
- 1
- 2
Email authorView author's OrcID profile

1.School of Fundamental Science and Technology, Graduate School of Science and Technology Keio UniversityKohokuku, YokohamaJapan
2.RIKEN Center for Advanced Intelligence ProjectTokyoJapan

Motion of a Rigid Body in a Special Lorentz Gas: Loss of Memory Effect

Abstract

Keywords

Notes

Acknowledgements

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article