Adiabatic approximation for joint probability distribution

[nigels]

Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic approximation to a joint probability distribution of two ever-fluctuating spatial variables. A search on the web shows that this method is primarily found in quantum physics, a field I'm wholly ignorant of. Is there any document/textbook one would recommend that can explain the technique to me given my background? Of course, it'd also be fantastic if someone can clarify the concept for me themselves here on the board.

THANKS!

 

[Stephen Tashi]

I speculate that you won't get a useful answer until you pose a specific problem.

The combination of the term "adiabatic" with "joint probability distribution" is so curious that I couldn't resist looking it up on the web. The simple explanation that I found ( http://www.google.com/url?sa=t&rct=...sg=AFQjCNGjHL56nLD-LA26oZ3ectFJFwCt0Q&cad=rja ) makes the technique sound disappointing. If you have a differential equation involving time with variables that vary quickly with time and some variables that only vary slowly with time, then treat the slow varying ones as constants, get the answer, put the time dependence back by making the slow varying variables functions of time again. Is that all it amounts to?

 

[nigels]

Hi Stephen,

What you provided is exactly the explanation I need! And yes, the technique makes sense in this case since the model is assuming one of the two spatial variables (movement of the animal) to be diffusive and the other one (territory border of said animal) subdiffusive. The link you've provided, albeit heavily quantum physicsy, is quite helpful in painting an intuitive picture of the effect of relative time scale on the solution. Thank you so much for the wonderful help! It has made a significant difference.

Cheers.