I Runaway Solutions: Unstable System Implications

[srallaba]

TL;DR Summary

What is a run away solution?

I was asked to show that a system has run away solutions, the implications of which are that it is inherently unstable.

 

[berkeman]

Welcome to the PF.

Did you really mean to post this in the Quantum forum? Are you asking specifically about some quantum system? If so, can you post links to such systems?

If not, are you just asking about Stability Criteria for systems in general?

https://en.wikipedia.org/wiki/Stability

 

[srallaba]

Thanks for the response!

Sorry for the spam if it is. But I am not getting the kinds of answers on a generic forum. I specifically want to know how a 'run away' solution is different from a stable solution. Does 'run away' imply that there are multiple solutions to the system? Or that none exist( like log(0))

 

[vanhees71]

To find an answer yourself and also to get an answer from others you must tell yourself and the others clearly what you are talking about, i.e., you have to clearly state your question. This is more than half way to the answer!

 

[Swamp Thing]

I was asked to show...

... by whom, and in what course, about what kind of system? -- That kind of context might help other members to target your problem a lot better. Stability is a multidisciplinary topic where context would help narrow down to what you need.

 

[bhobba]

I do not know quite what you are after, but since you asked in the quantum forum I will give you an example from classical EM that requires QM to rectify. Its part of issues with classical physics that points to it not being the whole story eg Black Body radiation.

Its called the Lorentz-Dirac equation:
https://arxiv.org/abs/gr-qc/9912045
It has acausal runaway solutions as detailed in the paper. The issue lies in taking the election as a point particle. In QM we do not have point particles, but rather excitation's in a so called Electron-Positron Field that permeates everywhere. Its part of what is called Quantum Field Theory which also takes into account relativity - its not generally pointed out but ordinary QM with the Schrodinger Equation etc is not relativistic - in fact as shown in Chapter 3 of Ballentine - QM - A Modern Development they can be derived from the Galilean Transformation and its associated symmetries, in particular that probabilities of quantum observation do not depend on velocity. If you want to study the proof devote a whole weekend to it - its a bit detailed and tricky. As a deep question to think about how does this affect non-locality as per Bell's Theorem? We have a thread going on about that right now - but that is just bye the bye - its quite deep.

The bottom line here is we need a deeper theory than Maxwell's Equations (specifically QED) to resolve it. A good paper to start that journey (but remember its just a start) is the following, which also shows how a deeper theory resolves a puzzle:
http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf
Thanks
Bill

 

[vanhees71]

I found the split of the field in retarded and advanced parts always confusing. At the end everything should be retarded in classical electrodynamics due to causality. Indeed one can treat the radiation-reaction problem, as far as it is treatable at all, without the use of advanced fields, i.e., using only retarded fields. This makes this unsolved (and in my opinion unsolvable) problem of classical relativistic charged-point-particle dynamics, at least a bit more consistent:

https://doi.org/10.1103/PhysRevD.1.1572

 

[srallaba]

It has acausal runaway solutions as detailed in the paper.

Great. Thanks a lot. I see the line " Even though the applied force is constant, the acceleration grows exponentially ...". So a solution is termed as a run away solution if it results in an exponential number of possibilities. In other words, a run away solution is also an intractable solution. Thanks

 

[srallaba]

... by whom, and in what course, about what kind of system? -- That kind of context might help other members to target your problem a lot better. Stability is a multidisciplinary topic where context would help narrow down to what you need.

Sorry for the confusion. I just wanted what 'run away' implies.

 

[Demystifier]

At the end everything should be retarded in classical electrodynamics due to causality.

Causality in this sense is not a fundamental microscopic law, but an emergent macroscopic law. It is closely related to the 2nd law of thermodynamics.

 

[vanhees71]

This is a common misunderstanding of the 2nd law. It does not introduce an arrow of time but just confirms that the "thermodynamic arrow of time" is the same as the "causal arrow of time", which is (usually tacitly) put into all dynamical laws of physics. In the classical proof of the H-theorem by Boltzmann it enters in the derivation of the Boltzmann equation at the moment, where you make the "molecular-chaos ansatz" to truncate the BBGKY hierarchy, and their you use the "causal arrow of time".