Jtdylktuy

I worked through much of 3.3.1 (Laser Threshold) in Strogatz's Nonlinear Dynamics and Chaos, but I'm struggling to understand the adiabatic elimination he does and when it's allowable.

We have a system modeling a laser where $n$ is the number of photons in the laser and $N$ is the number of excited atoms. The equations are:

$$dot n= GnN-kn$$
$$dot N= -GnN-fN+p$$

where $G$, $k$, $f$, and $p$ are various control parameters.

To convert it from a one-dimensional system, we make the 'quasi-static' approximation $dot N approx 0$, which Strogatz says represents "$N$ relaxing more rapidly than $n$".

This approximation is the part I'm confused about:

a) If $dot Napprox0$, do we assume that $N$ is constant? Or are these different assumptions? How can $dot Napprox0$ be true when $dot N$ has the constant, non-zero $p$ term?

b) In the 4th part of the question, we are asked to find the range of parameters for which this approximation is acceptable. I tried the 'dimensionless' groups approach from earlier in the book, but that led to a dead-end. Is there a good introduction to when adiabatic elimination is allowed that isn't in the context of complex Quantum Mechanics?

As for the "a)" part - I searched for an answer to this question as well. I think I'm having a similar issue with Strogatz's book where he just assumes too much without explaining it properly.

Adiabatic elimination assumes two timescales exist in the differential equation - one "fast" (here followed by $N$) and one "slow" (followed by $n$). In other words $N$ changes very, very fast compared to $n$ - so fast you could almost view $n$ as constant when considering changes in $N$.

Let's say we're most interested in values of $n$, rather than $N$ and try to analyze the evolution of the equation. We can picture that at the beginning both $dot{N} neq 0$ and $dot{n} neq 0$. However, very quickly (given our timescale - we are interested in $n$) $N$ reaches some kind of equilibrium ($dot{N} = 0$). During this short time $n$ almost stays constant. Then the value of $n$ changes a little bit and so $N$ has to reach a new equilibrium for this value of $n$. This happens so quickly that it seems almost instant in our timescale and we again have $dot{N} = 0$. This is where the sentence from Strogatz's book comes in - We can say that evolution of $N$ is slaved to that of $n$. We can say that we're seeing only $dot{N} = 0$.

You could imagine driving an indestrucible car into a slowly moving concrete wall on a runway. Assume both wall and your car are moving in the same direction. At time $t_0$, the wall is in the middle of the runway, moving 1 m/h and you're at the beginning, driving 230 km/h. After a very short time you hit the wall and then continue to move 1 m/h, despite pushing the pedal to the metal. Your movement is entirely controlled by movement of the concrete wall.