I will quote this quote from the wikipedia page "ultimate fate of of universe " under section of heat death
"Over infinite time, there would be a spontaneous entropy decrease by the Poincaré recurrence theorem, thermal fluctuations,[14][15] and the fluctuation theorem."
What exactly does this mean? How does a Poincare theorem somehow lead to a new Big Bang happening in an empty universe?
Simple. What is it?
If the theorem states that a system will return to initial conditions given a finite time, does this defy entropy? In the degenerate era of the universe, nothing reacts with anything, entropy will be gone, does this mean that there is a sort of mortality to the Second law?
The Poincaré recurrence theorem states that certain systems, specifically systems where energy is conserved (such as our universe), will eventually return to a state very close to the initial state. However, the second law of thermodynamics states that a system where energy is conserved will never return to the initial state or experience an increase in order.
This is a contradiction. P Λ (¬P). Both statements cannot be true under any circumstances.
Wikipedia, unfortunately, is very vague on the subject, stating that "The most typical argument is that for thermodynamical systems like an ideal gas in a box, recurrence time is so large that for all practical purposes it is infinite." Does this mean that the second law isn't really true, but simply a good approximation of the behaviour of systems? Or is there something else I'm missing here?
I just watched this Numberphile video and got confused.
My understanding was that at some point in the far future the universe would experience heat death and everything would just become very boring for the rest of eternity.
But eve before that, the space between galaxies will at some point grow faster than the speed of light so it will become impossible for matter from one galaxy to reach another galaxy.
Poincaré recurrence theorem seems to completely contradict this. What's going on?
inspired by /u/djasdalabala's post
this could possibly be the largest number ever physically related to our universe
so, considering all the possible worlds of the Many-worlds interpretation, how long would the Poincaré Recurrence Time be for all of the universes to start repeating the same particle arrangements in the exact same order as the first time?
now /u/djasdalabala used the limitation of having the unobservable universe be the theoretically minimum size of 23 trillion light years, but we're not gonna do that here.
instead, we'll be focusing on the three universe sizes mentioned on the Googology wiki: the size of the observable universe, the size of a Linde-type super-inflationary universe and the theoretically maximum largest size for a finite universe (10^10^10^122 Mpc.)
what would the Poincaré Reurrence Time be for all of those universes? by that i mean, what would be the time required for all of those universes to get back to the starting point at the same time?
The Poincaré recurrence theorem basically states that distinct dynamical systems will return to a their initial state or somewhat close to the initial state.
Andrei Linde's cosmic inflation models our universe may a Poincaré recurrence time within 10^10^10^10^10^1.1 Planck times. One Planck time is 5.391×10^-44 seconds.
In de-Sitter space, thermodynamical systems of the Poincaré recurrence time is exponentially large in the Boltzmann entropy of the system. However, Kolmogorov-Sinai entropy and could theoretically, be considerably shorter. If there is anyone doing interdisciplinary research between thermodynamics and cosmology, please chime in on this because I'm not an expert in that arena.
However, when considering quantum fluctuations, the Casimir-Polder experiments, quantum tunneling and quantum vacua, it is argued that de-Sitter space is meta-stable. The quantum breaking time and a meta-stable vacua may cause inflationary bubble nucleation.
What do I think about this?
As a person who is in the nascent stages of pursuing interdisciplinary research between materials chemistry, electrical engineering and quantum cosmology, I stress the importance of apparatuses, experiments and observations.
I infer that the transition-edge-sensor bolometric detectors on the Probe of Inflation and Cosmic Origins (PICO for short) would be able to lend some evidence for or against the hypothesis that a Poincaré recurrence time may occur within 10^10^10^10^10^1.1 Planck times (this is on the basis of stochastic inflationary models).
PICO will attempt to detect the primordial gravitational waves that emerged from cosmic inflation and within a confidence interval of 5σ. 5σ is a confidence level of 99.99994% within a standard deviation.
...
I'm not a STEM type so please correct any errors I made in describing these two phenomena.
Parallel universes - One of Stephen Hawking's final projects was an attempt to map out a way with which we could probe alternate, parallel universes. This I imagine could pose a significant challenge to universalist religions with the implication that there are parallel "people" out there who may not have their own Jesus equivalent.
Poincaré recurrence - This is a theorem that any self-contained system will after a certain amount of time get arbitrarily close to its condition now, and it could apply if Earth is part of any self-contained system. This in effect means that, if applicable to Earth, there are an infinite number of incarnations of /u/19djafoij02 out there. IDK if this is technically an afterlife or reincarnation, but it flatly contradicts a lot of (Abrahamic) religious doctrines.
I apologise if this has been asked before, but I didn't quite know how to search for it (I found this thread on /r/Physics, though).
In this Numberphile video around 1:26, Dr. Tony Padilla states that due to the deck of cards being finite in size, eventually dealing a royal flush is "guaranteed". I understand that the probability of any event with non-zero probability happening at least once goes towards infinity, as the number of trials goes to infinity, but it doesn't seem to be mathematically guaranteed. There will be some repetition, at least after C(52,5) trials (I think), that seems to be evident from the pigeonhole principle, but for any specific outcome that's not so clear to me.
(I also have an intuition that if it is guaranteed to happen in an infinite number of trials, then the event will still occur after a finite number of trials. But then couldn't you "discard" the remaining infinite number of trials? They haven't happened yet, so they don't seem to be able to affect the inevitability of that one royal flush. I suspect that this isn't correct, though, and in that case I would also appreciate to be corrected. I can't quite get my head around the infinite trials.)
Although, in the video Dr. Padilla states around 2:24 that, at least regarding Poincaré recurrence, the system in question will return to a state arbitrarily close to the initial state. I don't know if that has anything to do with it.
This is OC creepypasta; treat it as such.
Ever since watching the Watch For Rolling Rocks in 0.5x A Presses video on YouTube, I've been interested in Super Mario 64 and the progress of the A Button Challenge. In short, Pannenkoek2012 is trying to beat Super Mario 64 by pressing the A button (which makes Mario jump, among other things) as few times as possible. He's down to 30 or so A presses right now, and most of those come from the Tick Tock Clock level—which he has numerous videos of research on.
One of the unique properties of the level is that its activity changes depending on when you enter the painting—the minute hand of the clock painting rotates, so if you enter when it's pointing at 12, the inside of the clock will be frozen and the platforms won't move. They'll move slowly at 3, quickly at 9, and jitter randomly at 6. In one of Pannen's videos, he'd shown that there was no way to get clock parts to stay reasonably still during the random setting, not without manipulating the random number generator. There were just too many possible states of the level. He'd even shared a Javascript program that simulated the clock.
After fiddling with that program for a few hours, I actually found a sequence of actions that caused the “random” clock to loop after 16 days. This has to do with something called the Poincaré recurrence time in physics—random systems will eventually, eventually, loop. A very basic estimate for Tick Tock Clock is that it loops after 10^153 years, which is longer than it will take for the universe to reach heat death. So I got pretty lucky in finding one of the short loops. Just to see it in action, and maybe make a video, I fired up my ROM of SM64, made Mario fling himself around and kill a bob-omb as per the sequence, then put the emulator on fast-forward for the rest of the week.
When I came back after the fast-forward should have caused the clock to loop, the ROM was instead just playing the cheerful demos and demo music, as if I had gotten a Game Over somehow. The other bob-omb probably knocked Mario off the stage; whatever. I re-selected my save file, went back through the castle, and back towards Tick Tock Clock so I could try this again, with Mario in a safer spot.
This was where it got 'new'. The minute hand had fallen off the painting, and rested on the raised platform in front of it. I jumped onto the platform, and Mario did a short animation where he picked up the minute hand and placed it back on the painting
... keep reading on reddit ➡Two recent examples of advanced solved math problems and their proofs' method are Andrew Wiles' proof of Fermat's Last Theorem, using advanced applications of elliptic curves and highly specialized theorems; and Grigori Perelman's proof of the Poincaré Conjecture, using the Riemannian Metric, modifications of Ricci Flow, and (apparently) not-too-exotic applications of manifolds.
My summaries above of the proofs' main methods are probably too general. I'm wondering about the topic in general, so here are a few questions I've sussed out to try to get at the core of what I'm trying to learn more about:
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Can these, or other highly advanced math problems, be solved using highly different methods and approaches? (I would, of course, still expect a proof of a topology problem to be achieved using primarily tools from the field of topology.)
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Are the problems too advanced and specialized for highly different proofs to be meaningfully produced? In other words, is there a limit as to how "different" such alternate proofs can end up being?
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Is it ever useful to even try to tackle these kinds of problems from two highly unrelated directions?
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And catch-all: Is there anything else fundamental to this issue that I overlooked or that would be interesting to know?
Thanks for all of your detailed insight!
I have read some about the Poincaré recurrence theorem, and it left me very much confused, especially the examples given. If you have all particles that make up a gas in one corner of the room, in a matter of seconds the pressure will reach an equilibrium. Due to Poincaré recurrence, in an almost arbitrarily long time, the room will return to the state with all particles in one corner of the room.
But how? Why would the particles do that? If the volume of the room is constant, and the amount of "stuff" in the room is constant, pressure should also be constant. I don't see where time comes in as a factor in this at all.
Also, how would such an event look like? Would particles all of a sudden "decide" to not obey Boyle's law (among others), and spontaneously rush into the corner of the room? Would they be driven by some force against the pressure of the gas already in the corner? Could anyone clear it up for me?
Sorry if I misinterpreted the whole concept, my understanding in this field is very poor.
Hi all,
I am a Control and Automation Engineering student, and I am currently taking Nonlinear Control Systems course. A particular problem gave me a hard time, finally I managed to solve it. It was a HW question so my professor didn't help me at the time. After the deadline I went back to his office and we discussed the question in detail and he has shown me how he had solved it. However, I was not satisfied with his solution. At this point I should mention that, I have found the same (his solution) method in MIT's published notes. So i might be a bit presumptuous to say I am not satisfied with the solution. The reasoning behind it is that, an arbitrarily wide set called M is to be defined and then it will be inspected with respect to the theorem. However, origin is an equilibrium point for the dynamical system, so that equilibrium point's 'stability' (the linearized system matrix's eigenvalues) has to be inspected for the use of the theorem. That linearization was, however, extremely difficult. So, both MIT and my professor define the set M without the origin. I, however managed to solve the uncertainties at the origin and -in my humble opinion- have come up with a larger solution.
Now, here comes my question: have you ever seen a paper/section in a book, where uncertainties at the equilibrium points of nonlinear system are inspected. More specifically, is there any work regarding this matter and the Poincaré-Bendixson Theorem (the theorem which is used to prove there exists at least one periodic orbit in a given set M, if M holds for certain criteria) ? I have been, and still am, looking for such a paper/work because if there isn't one I'm planning on maybe writing a paper on it. One thing i should mention is that, with my solution M is wider but more importantly it includes an equilibrium point. Regardless of its stability, the nonlinear system has a constant point of operation which increases its controllability. So, not excluding the origin an solving the system for a larger M has practical benefits. I would also like to ask you, how do you feel about this?
TL;DR: Are there any papers/book sections regarding the uncertainties in linearization while using the Poincaré-Bendixson Theorem? Any research regarding the inspection of equilibrium points while using the Poincaré-Bendixson Theorem?
Hey everyone.
Currently I am comfortable with basic recurrence relations, algorithms, and problems. But I am completely lost with everything when it comes to working out the correct answers for these 2 problems. I have both the problems, and thorough answers for each, but I am still completely lost on what's happening, what we are doing, and WHY.
If anyone has the free time, could you please go through these and explain the big picture as to exactly what's happening and why like I'm 5? Could someone eli5 these problems and answers?
-Problem 1: https://imgur.com/JdvAqcI. Problem 1 answer: https://imgur.com/a/kvPJTGI
-Problem 2: https://imgur.com/a/ERkusgd. Problem 2 answer: https://imgur.com/a/bsSMqip
Bonus points for anyone that can point me to study material to help me learn the concepts better. Sorry to be such a pain and thank you so much for the help in advance!!
100 years ago, George Pólya proved that if you take a random walk in 1D or 2D, you have probability 1 of eventually returning to the origin. But in 3D, you will most likely never return. Here's a quick video I made about the intuition behind this surprising theorem: www.youtube.com/watch?v=byvEzyFgv44
I am currently reading Emmy Noether's Wonderful Theorem, an introductory book by Neuenschwander. It is a "historical survey" on mathematical physics containing nearly 2 equations per page. It is not an academic "textbook" though, so it is still readable and enjoyable. It wraps the equations in historical anecdotes within the development of physics in centuries past.
I would like to find a book, of similiar spirit, which introduces the reader to the Wigner classification of particles in quantum mechanics based on aspects of the Poincaré group. I understand that it is possible that such a book doesn't exist because the reader must speak the language of algebras of Lie groups. I'm all googled out on this topic, and wikipedia only references hefty tomes. I'm not sure where to start, really.
I appreciate your thoughts or feedback on this topic.
(EDIT: I prefer the book have at least some equations.)
I am currently searching for the first proof using Stirling Approximation and enumerating the paths to obtain that the probability that a one dimensional random walker returns to the point is 1 (i.e. it recurrent). I have seen numerous sites go over how the probability = ^(2n)Cn 4^n which approximates to 1/sqrt(pi * n), which diverges. But it would be of great help if someone could point me to who first proved the recurrence of 1 dimensional random walks with this method
inspired by /u/djasdalabala's post
this could possibly be the largest number ever physically related to our universe
so, considering all the possible worlds of the Many-worlds interpretation, how long would the Poincaré Recurrence Time be for all of the universes to start repeating the same particle arrangements in the exact same order as the first time?
now /u/djasdalabala used the limitation of having the unobservable universe be the theoretically minimum size of 23 trillion light years, but we're not gonna do that here.
instead, we'll be focusing on the three universe sizes mentioned on the Googology wiki: the size of the observable universe, the size of a Linde-type super-inflationary universe and the theoretically maximum largest size for a finite universe (10^10^10^122 Mpc.)
what would the Poincaré Reurrence Time be for all of those universes? by that i mean, what would be the time required for all of those universes to get back to the starting point at the same time?
