7
$\begingroup$

Studying the dynamics of the endpoints of an equilibrium measure (a minimizer of its logarithmic energy in an external field) I ran into the following system of differential equations (which I state for the case of 4 points, for simplicity): let $x_j=x_j(t)$, $j=1, \dots, 4$, be real values dependent on time $t$, all distinct at $t=0$, and satisfying the system $$ \frac{d x_j}{dt} = \frac{m_j}{q'(x_j)}=m_j \prod_{k\neq j} (x_j-x_k)^{-1}, \quad j=1, \dots, 4, $$ where $q(x)=\prod_{j=1}^4 (x-x_j)$ and $q'(x)$ is its derivative with respect to $x$. Here $m_j$ are positive numbers.

My questions (sorry if too elementary or naive) are:

1) is this kind of a system known, does it have any name attached to it?

2) I needed to prove the fact that the interior $x_j$'s collide in a finite time. Does this follow from any general fact in dynamical systems or systems of ODEs?

3) what about the more general situation, when the number of points is $n$ and the right hand sides in the system are rational functions?

Thanks in advance.

$\endgroup$

4 Answers 4

1
$\begingroup$

this is not an answer to your question. hassan aref studied the motion of point vortices.

$\endgroup$
1
  • $\begingroup$ I found a couple of papers by H. Aref on the arxiv, chao-dyn/9907038 might have something I could use, although I have to look more carefully. As you say, not clear that it answers my questions, but it is interesting anyway. Thanks. $\endgroup$
    – Andrei MF
    Mar 18, 2012 at 0:16
1
$\begingroup$

Forgive the brevity - what you are interested in in something called the Osgood condition. Systems of this kind have been studied for a long time, particularly when the $m_i$'s are all equal. If you look at the $n=2$ case and look at $r= (x_1-x_2)$ the problem reduces $r_t =c/r$ and the fact that $c/r$ is not integrable at $r=0$ tells you that you have a finite time collision.

A good starting point is Ruelle's Thermodynamics text.

$\endgroup$
2
  • $\begingroup$ Thanks, very interesting. Could you clarify which Ruelle's book you have in mind? I saw in Amazon "Statistical Mechanics: Rigorous Results", and "Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics". $\endgroup$
    – Andrei MF
    Apr 3, 2012 at 19:01
  • $\begingroup$ If $c>0$ you don't have a collision for $t > 0$, only for $t < 0$. $\endgroup$ Apr 29, 2012 at 4:54
1
$\begingroup$

Assume $x_1(0) < x_2(0) < x_3(0) < x_4(0)$. Note that $\dfrac{dx_2}{dt}$ and $\dfrac{dx_4}{dt}$ are positive and the other two are negative. So for $t > 0$ (and before the collision), $x_1 < x_1(0) < x_2(0) < x_2 < x_3 < x_3(0) < x_4(0) < x_4$. Now $\dfrac{dx_4}{dt} \le \dfrac{m_4}{(x_4-x_3(0))^3}$. Solving the differential equation obtained by making this an equality, we find that $x_4(t) \le x_3(0) + ((x_4(0) - x_3(0))^4 + 4 m_4 t)^{1/4}$. Call the right side $B_4(t)$. Similarly $x_1(t) \ge B_1(t) = x_2(0) - ((x_2(0) - x_1(0))^4 + 4 m_1 t)^{1/4}$.

So $$ \dfrac{dx_2}{dt} > \frac{m_2}{(x_3(0) - B_1(t))(B_4(t) - x_2(0))(x_3 - x_2)}$$ $$ \dfrac{dx_3}{dt} < \frac{m_3}{(x_3(0) - B_1(t))(B_4(t) - x_2(0))(x_2 - x_3)}$$ $$ (x_3 - x_2) \dfrac{d}{dt} (x_3 - x_2) < - \dfrac{m_3+m_2}{(x_3(0) - B_1(t))(B_4(t) - x_2(0))} $$ Thus a collision will occur by time $T$ if $$ \int_0^T \dfrac{ dt}{(x_3(0) -B_1(t))(B_4(t) - x_2(0))} > \frac{(x_3(0)-x_2(0))^2}{2(m_2 + m_3)} $$ The integral of the left side from $0$ to $\infty$ is infinite, since $x_3(0) - B_1(t)$ and $B_4(t) - x_2(0)$ only grow like $t^{1/4}$ as $t \to \infty$.

So there will always be a collision in finite time.

$\endgroup$
0
0
$\begingroup$

Oops! There was an error in my argument.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.