# simple physical system involving convergence to equilibrium at decreasing speed

I want to teach an economic model to my students (Solow, 1956), using a simple "device", or physical system that I can take to my lectures. The system must have these three properties:

1. There is one equilibrium, which is stable (e.g. the bottom inside an empty sphere).
2. Variable (say, $x$) converges over time to it (e.g. a ball inside the sphere with friction on its inner surface will end up at the equilibrium).
3. The speed of convergence is directly related to the distance between $x$ to that equilibrium. In other words, the speed decreases the closer $x$ is to the equilibrium. Unfortunately, the example of a ball inside a sphere does not fulfill this condition, because the ball oscilates around the equilibrium before reaching it.

No particular speed of convergence is needed (in fact, in the Solow model it takes infinite time to reach the equilibrium). The key is point 3, that the movement towards it depends directly on its distance.

I'm looking for a system as simple as the ball inside a sphere (which I could easily replicate with basic tools) but where the three conditions hold. I just can't think of anything like this.

• Could you be precise about your concept of "inversely related"? If you just want the speed of approach to equilibrium to decrease proportionately to the distance from equilibrium, then exponential relaxation, i.e. $\mathrm{d}x/\mathrm{d}t = -\gamma (x - x_{\mathrm{eq}})$ does the trick. This behaviour describes an enormous number of simple physical systems. – Mark Mitchison Jan 22 '15 at 16:24
• I mean I want an example of something I can physically (i.e. manually, materialistically) have, experiment with, touch, like my example of a ball and sphere (ping-pong ball and a salad bowl). – luchonacho Jan 22 '15 at 22:20

## 1 Answer

Any system has a stable equilibrium point will obey the kind of law that you are after. This is actually obvious with a little thought, since in order for the system to come to equilibrium it must slow down as it approaches the equilibrium point. To give a simple mechanical example, imagine sliding a wooden block across a surface. Friction between the block and the surface will slow the block down until it stops.

Assume that a block of mass $m$ initially at $x=0$ is given an initial velocity $v_0$ (by you pushing it and then letting go). When the block is in motion, friction is the only force acting on it. The friction force is proportional to the velocity of the block, with coefficient of proportionality $\gamma$. Then Newton's law $F = ma$ tells you that $$m \ddot{x} = -\gamma \dot{x},$$ where dots denote derivatives with respect to time, and $m$ is the mass of the block. This can be integrated to give $$\dot{x} = v_0 - \frac{\gamma}{m} x.$$ If you define the equilibrium position $x_{\mathrm{eq}} = mv_0/\gamma$, the equation can be rewritten $$\dot{x} = -\frac{\gamma}{m}(x-x_{\mathrm{eq}}). \qquad \qquad (1)$$ This equation tells you that as the distance to the equilibrium point $(x-x_{\mathrm{eq}})$ gets smaller, the block slows down ($\dot{x}$ decreases) until the block reaches the equilibrium point and stops. The actual solution to the equation is $$x(t) = x_{\mathrm{eq}}(1-e^{-\gamma t/m} ).$$

All systems which have a stable equilibrium ultimately obey an equation like (1) when they are sufficiently close to the equilibrium point. The example of a ping-pong ball in a salad bowl is similar, although the maths is a bit more complicated.

• In your example friction is the only force in play but there are infinite stable equilibriums. In the salad bowl example, its evident that the ball accelerates as it goes towards the bottom of the bowl due to gravity and its only the friction that makes the ball reach lower levels every round until it rest still in the bottom. So it's not quite the same example. Even more, your example depends crucially on the initial speed. In my model, speed of convergence is fully endogenous. Your position is the only determinant of the speed at which you converge to the equilibrium. – luchonacho Jan 23 '15 at 11:28
• @luchonacho I don't know what you're looking for because you haven't put any details in the question. You asked for a problem where "speed decreases the closer it is to the equilibrium". This is one. I suggest that you edit your question to add some mathematical detail and then you might get some answers that are useful to you. – Mark Mitchison Jan 23 '15 at 11:53
• I will do. Your example fulfills point 2 and 3 of my list but not 1. I need the three of them to hold. – luchonacho Jan 23 '15 at 14:11
• @luchonacho You are aware that your example of a ping-pong ball in a salad bowl satisfies all of your conditions, right? ...assuming I actually understood what you are after now. There must be some acceleration towards the equilibrium, otherwise it is not a stable equilibrium, or it is not unique. The point is that when you get close enough, the acceleration term becomes negligible compared to the friction term and so the system slows down – Mark Mitchison Jan 23 '15 at 14:13
• Oh my, you are right. I actually meant directly (as in the brackets in point 3). Fixing that. – luchonacho Jan 23 '15 at 14:17