Motion of a Body Immersed in a Vlasov System

Abstract

In this chapter we study the motion of a body immersed in a Vlasov system. Such a choice for the medium allows to overcome problems connected to the irregular motion of the body occurring when it interacts with a gas of point particles. On the contrary, in case of a Vlasov system, the motion is expected to be regular. The interaction body/medium is assumed to be hard core, which implies the existence of a stationary motion for any initial data and any intensity of external constant force acting on the body. Moreover, we investigate the asymptotic approach of the body velocity to the limiting one, showing that in case of not self-interacting medium the approach is proportional to an inverse power of time. Such a behavior, surprising for not being exponential as in many viscous friction problems, is due to the recollisions that a single particle of the medium can deliver with the body.

Appendix

We give a derivation of the equation of motion (3.8)–(3.9) in the one-dimensional case, the d-dimensional case following by straightforward modifications. We present two different derivations. In the first one, we will obtain the equation of motion from the time derivative of the total momentum of the system gas+disk in case of null external force, which is conserved along the motion,

$$\displaystyle{ \frac{\mathrm{d}} {\mathrm{d}t}\int _{\mathbb{R}}\!\mathrm{d}x\int _{\mathbb{R}}\!\mathrm{d}v\;vf(x,v;t) +\dot{ V }(t) = 0\;. }$$

(3.124)

Let

$$\displaystyle\begin{array}{rcl} f^{L}(X(t),v;t) =\lim _{ x\rightarrow X(t)^{-}}f(x,v;t)\;,& &{}\end{array}$$

(3.125)

$$\displaystyle\begin{array}{rcl} f^{R}(X(t),v;t) =\lim _{ x\rightarrow X(t)^{+}}f(x,v;t)\;.& &{}\end{array}$$

(3.126)

We calculate the first term in (3.124). Using (3.3) and the fact that

$$\displaystyle{ \lim _{x\rightarrow -\infty }f(x,v;t) =\lim _{x\rightarrow \infty }f(x,v;t) =\rho \left (\frac{\beta } {\pi }\right )^{1/2}\mathrm{e}^{-\beta v^{2} }\;, }$$

(3.127)

we have,

$$\displaystyle\begin{array}{rcl} & & \frac{\mathrm{d}} {\mathrm{d}t}\left [\int _{-\infty }^{X(t)}\!\mathrm{d}x\int _{ \mathbb{R}}\!\mathrm{d}v\;vf(x,v;t) +\int _{ X(t)}^{\infty }\!\mathrm{d}x\int _{ \mathbb{R}}\!\mathrm{d}v\;vf(x,v;t)\right ] \\ & & \quad =\int _{\mathbb{R}}\!\mathrm{d}v\;vf^{L}(X(t),v;t)V (t) -\int _{ \mathbb{R}}\!\mathrm{d}v\;vf^{R}(X(t),v;t)V (t) \\ & & \qquad +\int _{\mathbb{R}}\!\mathrm{d}v\int _{-\infty }^{X(t)}\!\mathrm{d}x\;v(-v\partial _{ x}f(x,v;t)) +\int _{\mathbb{R}}\!\mathrm{d}v\int _{X(t)}^{\infty }\!\mathrm{d}x\;v(-v\partial _{ x}f(x,v;t)) \\ & & \quad =\int _{\mathbb{R}}\!\mathrm{d}v\;vV (t)f^{L}(X(t),v;t) -\int _{ \mathbb{R}}\!\mathrm{d}v\;vV (t)f^{R}(X(t),v;t) \\ & & \qquad -\int _{\mathbb{R}}\!\mathrm{d}v\;v^{2}f^{L}(X(t),v;t) +\int _{ \mathbb{R}}\!\mathrm{d}v\;v^{2}f^{R}(X(t),v;t) \\ & & \quad =\int _{\mathbb{R}}\!\mathrm{d}v\;v(V (t) - v)f^{L}(X(t),v;t) -\int _{ \mathbb{R}}\!\mathrm{d}v\;v(V (t) - v)f^{R}(X(t),v;t)\;.{}\end{array}$$

(3.128)

We consider first the integral involving f ^L, taking into account the fact that

$$\displaystyle\begin{array}{rcl} f^{L}(X(t),v;t)& =& f_{ -}^{L}(X(t),v;t)\,\chi (v \geq V (t)) \\ & & +f_{+}^{L}(X(t),v;t)\,\chi (v < V (t))\;,{}\end{array}$$

(3.129)

with the definition of f _± given in (3.7), since for v ≥ V (t) the velocity v is necessarily a pre-collisional velocity (we are on the left side of the obstacle), while for v < V (t) the velocity v is a post-collisional velocity. Hence,

$$\displaystyle\begin{array}{rcl} \int _{\mathbb{R}}\!\mathrm{d}v\;v(V (t) - v)f^{L}(X(t),v;t)& =& \int _{ -\infty }^{V (t)}\!\mathrm{d}v'\,v'(V (t) - v')f_{ +}^{L}(X(t),v';t) \\ & & +\int _{V (t)}^{\infty }\!\mathrm{d}v\;v(V (t) - v)f_{ -}^{L}(X(t),v;t)\;.{}\end{array}$$

(3.130)

Performing the change of variable $v' = 2V (t) - v$ in the first integral in the right-hand side of (3.130), we have,

$$\displaystyle\begin{array}{rcl} & & \int _{\mathbb{R}}\!\mathrm{d}v\;v(V (t) - v)f^{L}(X(t),v;t) \\ & & \quad = -\int _{\infty }^{V (t)}\!\mathrm{d}v\;(2V (t) - v)(-V (t) + v)f_{ +}^{L}(X(t),2V (t) - v;t) \\ & & \qquad +\int _{ V (t)}^{\infty }\!\mathrm{d}v\;v(V (t) - v)f_{ -}^{L}(X(t),v;t)\;, {}\end{array}$$

(3.131)

and, for the continuity of f ^L along the collisions, by (3.6) it is

$$\displaystyle{ f_{+}^{L}(X(t),2V (t) - v;t) = f_{ -}^{L}(X(t),v;t)\;. }$$

(3.132)

Therefore,

$$\displaystyle\begin{array}{rcl} & & \int _{\mathbb{R}}\!\mathrm{d}v\;v(V (t) - v)f^{L}(X(t),v;t) \\ & & \quad =\int _{ V (t)}^{\infty }\!\mathrm{d}v\;(V (t) - v)(v - 2V (t) + v)f_{ -}^{L}(X(t),v;t) \\ & & \quad = -2\int _{V (t)}^{\infty }\!\mathrm{d}v\;(V (t) - v)^{2}f_{ -}^{L}(X(t),v;t)\;. {}\end{array}$$

(3.133)

The integral with f ^R in (3.128) can be handled in the same way, arriving at

$$\displaystyle\begin{array}{rcl} & & \frac{\mathrm{d}} {\mathrm{d}t}\left [\int _{-\infty }^{X(t)}\!\mathrm{d}x\int _{ \mathbb{R}}\!\mathrm{d}v\;vf(x,v;t) +\int _{ X(t)}^{\infty }\!\mathrm{d}x\int _{ \mathbb{R}}\!\mathrm{d}v\;vf(x,v;t)\right ] \\ & & \quad = 2\int _{-\infty }^{V (t)}\!\mathrm{d}v\;(V (t) - v)^{2}f_{ -}^{R}(X(t),v;t) \\ & & \qquad - 2\int _{V (t)}^{\infty }\!\mathrm{d}v\;(V (t) - v)^{2}f_{ -}^{L}(X(t),v;t)\;, {}\end{array}$$

(3.134)

which is the friction term (3.9) in the one-dimensional case.

We give a second derivation of the model (always in one dimension), computing the momentum exchanged by collisions between gas particles and body. We think at (point-like) light gas particles of mass m and velocity v, hitting a heavy (point-like) body of mass M and velocity V. The law of elastic collision reads,

$$\displaystyle{ V ' = V + \frac{2m} {M + m}(v - V )\;,\qquad v' = V -\frac{M - m} {M + m}(v - V )\;, }$$

(3.135)

where V ′ and v′ are the outgoing velocities. As usual in the mean field limit, we assume the mass of any light particle to be $m = \frac{1} {N} \ll M$, N being the total number of gas particles, so that, by (3.135), we have,

$$\displaystyle{ V ' \approx V + \frac{2} {N\,M}(v - V )\;,\qquad v' \approx 2V - v\;. }$$

(3.136)

We now evaluate the variation of velocity Δ V of the body in the time interval [t, t +Δ t], due to the collisions with the gas particles and the influence of an external constant force E. It is

$$\displaystyle{ \varDelta V = E\varDelta t - \frac{1} {N}\sum _{j\in I^{+}(\varDelta t)} \frac{2} {M}\vert v_{j} - V \vert + \frac{1} {N}\sum _{j\in I^{-}(\varDelta t)} \frac{2} {M}\vert v_{j} - V \vert + h\;, }$$

(3.137)

where h denotes a term o(Δ t) and I ^±(Δ t) denote the indices of the light particles which are colliding from the right v _j  < V and from the left v _j  ≥ V respectively.

We finally apply our mean-field hypothesis by setting

$$\displaystyle{ \frac{1} {N}\sum _{j\in I^{\pm }(\varDelta t)} \frac{2} {M}\vert v_{j} - V \vert =\varDelta t \frac{2} {M}\int \!\mathrm{d}v\;\vert v - V \vert ^{2}f^{\pm }(X,v;t)\;. }$$

(3.138)

Taking the limit Δ t → 0, we obtain Eqs. (3.8)–(3.9). We also set M = 1, M being an irrelevant constant.