Chaos and Ergodicity

[@GeoffryGifari]

that "painting" of the initial angles and the time it takes for the second pendulum to flip over seems to contain a pattern. Maybe we can extract physics from seeing how the "painting" morphs as we change the system parameters (masses and pendulum lengths)

[@GeoffryGifari]

From that swinging atwood machine example, it seems that with one mathematical form of the hamiltonian, the system can be chaotic or not depending on the parameter (mass)!

[@GeoffryGifari]

how would classical chaotic systems behave under time-reversal transformation? can we "tell the difference" between time running forwards and backwards?

[@GeoffryGifari]

given a system of coupled equations of motion, can we tell if the trajectory will end up being chaotic?

[@GeoffryGifari]

Given a chaotic hamiltonian, can we derive the lyapunov exponent?

[@GeoffryGifari]

and how do we know whether or not an equation of motion (or a system of) has a closed-form solution?

[@insightfool]

Thanks for this great video! In the universe of ergodicity explanations there appear to be two worlds. The statistical one, and the physics based one. Never the two shall meet, it seems. I am looking for an explanation of ergodicity which bridges the two. For example, how can one map the statistical example of coin flips to the physical example of triple pendulums? Setting aside the fact that one is discrete and the other continuous, I feel like there should be a more general way to explain ergodicity such that the two analogies make sense together.

[@insightfool]

Ah! Post note. I think I made sense of it! In the case of classical physics "systems". A single process is described as ergodic if it can trace the all steps (events) that a collection of events (in the statistical case) might encounter through apparently separate processes.

[@drmitchellsphysicschannel2955]

Yes, in statistical mechanics we rely on the fact that the long-time average of observables is the same as the ensemble average. However, note that this ergodicity can break down across a phase transition leading to spontaneous symmetry breaking (there's a whole lecture on that in the QCMT course!).

[@insightfool]

@@drmitchellsphysicschannel2955 But I wonder about the edge cases of time partitioned and aggregate (spatially) partitions systems (which I guess is considered in the case of macro->micro scale Boltzmann thermodynamics? If symmetry breaking has something to say about that, then it's really interesting!

[@jibranbhat211]

Could you kindly upload a lecture on Quantum chaos

[@danielkonstantinovsky108]

I have a hunch that as you add more and more weights to the pendulum, the chaos will dissipate gradually. Is this true, or it the true behavior the exact opposite?

[@drmitchellsphysicschannel2955]

The behaviour may be less chaotic, but nothing dissipates in these closed systems -- the forces are all conservative. There will still be a chaotic parameter regime even with higher mass however