Processes which propagate faster than exponentially
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From: Gottfried Helms on 21 Jan 2010 12:25 Am 15.01.2010 20:48 schrieb I.N. Galidakis: > > Excellent. So is there then general agreement that there are no *naturally > occuring* (excluding Rod's examples on Wiki which I am not sure they qualify as > *natural*) processes in nature which propagate faster than exponential? What about the (expected) processes near big-bang (just re-read a book about this and your question came into mind). In the models the time-frame for significant changes increases extremely and also the rate of change changes extremely with the time. I'd propose to look here for fast propagation. Perhaps there is something like loop-back or iteration? Gottfried Helms
From: jbriggs444 on 21 Jan 2010 15:06 On Jan 15, 10:51 pm, Frisbieinstein <patmpow...(a)gmail.com> wrote: > On Jan 16, 12:54 am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > > > > > On Jan 15, 11:14 am, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > > > > Apologies for the crosspost, but this is related to many areas. Is anyone aware > > > of any physical/chemical/nuclear processes which propagate at rates faster than > > > exponential? > > > > From my search so far, it appears that the fastest processes available, like > > > cancer and viruses in biology, and nuclear explosions and supernova explosions > > > in physics all propagate at most exponentially. > > > Some processes are too fast to even have a decent way to categorize > > the rate. > > > Take, for instance, the chemical core of a nuclear device. The pieces > > are set off simultaneously so that the reaction need not progress from > > a single point of ignition. The limit on the reaction rate is the > > number of detonators used and the precision with which they can be set > > off. Rather than being a log, a cube root, a square root or linear in > > the reactant size, the reaction time can be held constant. > > > That's without considering Thiotimoline, a substance which, when > > purified by repeated resublimation has a solubility reaction rate that > > goes endochronic. > > Thiotimoline is a fictitious chemical compound conceived by science > fiction author Isaac Asimov. Indeed. I had hoped that the references to repeated resublimation and endochronicity would make the tongue-in-cheek nature of the remark clear, even to those fortunate few who can still look forward to reading the article below for the first time. A. Asimov: "The Endochronic Properties of Resublimated Thiotimoline", The Journal of Astounding Science Fiction, March, 1948.
From: jbriggs444 on 21 Jan 2010 15:52 On Jan 15, 10:49 pm, Frisbieinstein <patmpow...(a)gmail.com> wrote: > On Jan 16, 4:42 am, Marvin the Martian <mar...(a)ontomars.org> wrote: > > > On Fri, 15 Jan 2010 18:14:25 +0200, I.N. Galidakis wrote: > > > Apologies for the crosspost, but this is related to many areas. Is > > > anyone aware of any physical/chemical/nuclear processes which propagate > > > at rates faster than exponential? > > > > From my search so far, it appears that the fastest processes available, > > > like cancer and viruses in biology, and nuclear explosions and supernova > > > explosions in physics all propagate at most exponentially. > > > > Many thanks, > > > Google "Taylor series". Any real function can be approximated by a series > > of exponentials. Thus, your question makes no sense. > > An infinite sum of exponentials can increase superexponentially. A > Taylor series is infinite. A Taylor series is not a sum of exponentials. It is a sum of polynomials. As I recall, a not-identically zero, real-valued function may have a derivitive of zero to all degrees. Its Taylor series then has not one damned thing to do with its value except at zero.
From: jbriggs444 on 21 Jan 2010 16:30 On Jan 18, 8:11 am, Andrew Usher <k_over_hb...(a)yahoo.com> wrote: > On Jan 15, 10:14 am, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > > > Apologies for the crosspost, but this is related to many areas. Is anyone aware > > of any physical/chemical/nuclear processes which propagate at rates faster than > > exponential? > > First, this is not really a mathematical question. Of course equations > may be defined that grow arbitrarily rapidly. > > Second, any exponential growth process in the real world can only > maintain such growth for a short time, and this would apply even more > to super-exponential processes. > > Third, if one requires only super-exponential growth _in time_ > (there's really no such thing as even exponential growth in space), > there's an obvious example: any exothermic chemical chain reaction. > Since the growth would be exponential if temperature were constant, > but temperature is also increasing rapidly, the progress of the whole > process is faster than exponential (until the concentration of > reactive particles has reached its peak). > > Andrew Usher Suppose you have 10 miles of oil in the Alaskan pipeline bearing down on a closed valve at a rate of one meter per second buffered by a 2 meter air bubble at 1 atmosphere directly adjacent to the valve. (Some idiot closed the valve abruptly) Assume that all relevant safety devices have been disabled and no other air bubbles exist. I think that the rate of increase of air pressure with respect to time and with respect to distance are both super-exponential (or are best modelled by a curve that is super-exponential) right up to the point where the pipe breaks.
From: Andrew Usher on 21 Jan 2010 19:23
On Jan 21, 3:30 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote: > Suppose you have 10 miles of oil in the Alaskan pipeline bearing down > on a closed valve at a rate of one meter per second buffered by a 2 > meter air bubble at 1 atmosphere directly adjacent to the valve. > (Some idiot closed the valve abruptly) > > Assume that all relevant safety devices have been disabled and no > other air bubbles exist. > > I think that the rate of increase of air pressure with respect to time > and with respect to distance are both super-exponential (or are best > modelled by a curve that is super-exponential) right up to the point > where the pipe breaks. Yes, this is another example. One needs only to formulate the diff.eq. to see that this is super-exponential in the ideal case. If the oil has infinite motivating force and the pipe infinite strength, we have P = 1/(t'-t) where t' is when the bubble is crushed. This is equivalent to P' = P^2 which clearly grows faster than the exponential P' = P . Andrew Usher |