Processes which propagate faster than exponentially
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From: T.H. Ray on 16 Jan 2010 02:48 Ioannis 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, > -- The more I think about it, the more interesting this question becomes. It brings into doubt, actually, what we mean by "physical." It occurred to me while driving through the mountains the other day, as I checked the accuracy of my speedometer against the mile markers, that the metric function (the correspondence of the gauge to the increasing number of mile markers)is an arbitrary accounting; i.e., I can take a side excursion off the metric (as I in fact did) and either I would have to reset my calculation at the next marker or account for the side trip as a segment of the continuous metric. In the latter case, however, the perturbation caused by the side trip has a zero point beginning and end, allowing that I left and entered the highway at the same point--or a zero plus n component to add or subtract to the perturbation. Point is, the rate of growth of the function may be constant, but the rate of information growth, i.e., the description of the state of a point on the metric at time t as t ---> T is not. Even setting T at infinity and allowing exponential acceleration on the metric, you are correct--the relativistic limit (speed of light in vacuuo)will limit propagation of a physical process by the principle that disallows communication between particles at greater than light speed. IOW, the point particle on a metric, unless massless, cannot attain infinite acceleration, whatever the value of the exponent. _However_, the rate of _information_ growth in the system may certainly exceed the speed of light This means that the communication among (massles) particles is nonlocal. A (non-relativistic) quantum system is timeless; a measurement on a quantum state is unitary. Some researchers (e.g., Markopoulou, Goldfarb, and I-- independently and from different perspectives) have given time a role identical to that of information, with interesting results. Relevant to your question, the physical aspects of time have more significance than spatial measurements in physics. My key idea hangs on an outright physical definition of time: n-dimensional infinitely orientable metric on self avoiding random walk. This allows an extradimensional theory of physical processes in a complex of Riemann surfaces, and supports string theory as a quantum field theory. One may then ask, in your context, can time propagate faster than exponentially? My answer is no--time is dissipative and rectilinear past the four-dimension horizon. This comports with relativity in the 3 + 1 dimension domain, where time is a simple parameter of reversible trajectory. Tom > Ioannis >
From: rabid_fan on 16 Jan 2010 19:21 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? > The question concerns natural processes. Thus, we must ask: How does a natural process produce an exponential rate? An exponential rate arises according to the model where the rate is proportional to the amount of substance: dx/dt = k * x If we assume that x = 1 at t = 0, the solution becomes: x = exp(k*t) So to find processes that would be faster than exponential (if they exist) we can create models where the rate is proportional to quantities greater than the linear amount, i.e.: dx/dt = k * x^2, with x(0)=1 The solution is x = 1/(1-k*t) which increases faster than exponential. dx/dt = k * x^3, with x(0)=1 The solution is x = 1/sqrt(1-2k*t) which increases faster than exponential. dx/dt = k * exp(x), with x(0)=1 The solution is x = ln(1/e-k*t) which increases faster than exponential. We can easily create these models that all lead to a faster rate than the exponential. Whether or not they actually exist in the natural world is another story.
From: I.N. Galidakis on 17 Jan 2010 14:20 rabid_fan 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? >> > > The question concerns natural processes. Thus, we must ask: > How does a natural process produce an exponential rate? > > An exponential rate arises according to the model where > the rate is proportional to the amount of substance: > > dx/dt = k * x > > If we assume that x = 1 at t = 0, the solution becomes: > > x = exp(k*t) > > So to find processes that would be faster than exponential > (if they exist) we can create models where the rate is > proportional to quantities greater than the linear amount, > i.e.: > > dx/dt = k * x^2, with x(0)=1 > > The solution is x = 1/(1-k*t) which increases faster than > exponential. > > dx/dt = k * x^3, with x(0)=1 > > The solution is x = 1/sqrt(1-2k*t) which increases faster than > exponential. > > dx/dt = k * exp(x), with x(0)=1 > > The solution is x = ln(1/e-k*t) which increases faster than > exponential. > > We can easily create these models that all lead to a faster > rate than the exponential. Whether or not they actually exist > in the natural world is another story. In view of your and Tom's response, I think it's a good idea to make this distinction (natural vs man made), because otherwise it seems to me there's no limit. case and point: Consider the following example (I am chopping the newsgroups a little because if the physicists hear of ordinals they will start yelling obscenities :-) Let w be the first uncountable ordinal and [a->b->c] be Conway's chained arrow notation. Suppose that through some ingenious counting trick, I can "count" to w in finite time. So let me "count", as follows (mono-spaced font required for the following): 1, 2, 3, ..., w, 2*w, 3*w, ... w*w = w^2 = [w->2->1], w^3, ..., w^w = w^^2 = [w->2->2], w^^3, ..., w^^w = w^^^2 = [w->2->3], w^^^3, ..., w^^^w = w^^^^2 = [w->2->4] w^^^^3, ..., w^^^^w = [w->2->5], [w->2->6], ..., [w->2->w], ..., [w->3->w], ..., [w->w->w] = ([3-w's with arrows]) [4-w's with arrows] ..., [w-w's =w^2 with arrows] ..., etc. Can anyone estimate how "fast" is the rate of my "counting"? :-) Looks to me like it's not only infinite, but many-many multiples of infinity. I think this process beats the heck out of the super-double-exponential. But is it "natural"? -- Ioannis
From: T.H. Ray on 17 Jan 2010 05:43 Ioannis wrote > rabid_fan 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? > >> > > > > The question concerns natural processes. Thus, we > must ask: > > How does a natural process produce an exponential > rate? > > > > An exponential rate arises according to the model > where > > the rate is proportional to the amount of > substance: > > > > dx/dt = k * x > > > > If we assume that x = 1 at t = 0, the solution > becomes: > > > > x = exp(k*t) > > > > So to find processes that would be faster than > exponential > > (if they exist) we can create models where the rate > is > > proportional to quantities greater than the linear > amount, > > i.e.: > > > > dx/dt = k * x^2, with x(0)=1 > > > > The solution is x = 1/(1-k*t) which increases > faster than > > exponential. > > > > dx/dt = k * x^3, with x(0)=1 > > > > The solution is x = 1/sqrt(1-2k*t) which increases > faster than > > exponential. > > > > dx/dt = k * exp(x), with x(0)=1 > > > > The solution is x = ln(1/e-k*t) which increases > faster than > > exponential. > > > > We can easily create these models that all lead to > a faster > > rate than the exponential. Whether or not they > actually exist > > in the natural world is another story. > > In view of your and Tom's response, I think it's a > good idea to make this > distinction (natural vs man made), because otherwise > it seems to me there's no > limit. case and point: Consider the following example > (I am chopping the > newsgroups a little because if the physicists hear of > ordinals they will start > yelling obscenities :-) > > Let w be the first uncountable ordinal and [a->b->c] > be Conway's chained arrow > notation. Suppose that through some ingenious > counting trick, I can "count" to w > in finite time. So let me "count", as follows > (mono-spaced font required for the > following): > > 1, 2, 3, ..., w, > 2*w, > 3*w, > ... > w*w = w^2 = [w->2->1], > w^3, > ..., > w^w = w^^2 = [w->2->2], > w^^3, > ..., > w^^w = w^^^2 = [w->2->3], > w^^^3, > ..., > w^^^w = w^^^^2 = > w^^^w = w^^^^2 = [w->2->4] > w^^^^3, > ..., > w^^^^w = > w^^^^w = [w->2->5], > > > > > > > > [w->2->6], > > ..., > > > > > > > > [w->2->w], > > ..., > > > > > > > > [w->3->w], > > ..., > > > > > > > [w->w->w] > [w->w->w] = ([3-w's > [w->w->w] = ([3-w's with > arrows]) > > > > > > > > > > > > > > > > > [4-w's > [4-w's with > arrows] > > > > > > > > > > > > > > > ..., > > > > > > > > > > > > > > > > > [w-w's > [w-w's =w^2 > [w-w's =w^2 with > arrows] > > > > > > > > > > > > > > > ..., > > > > > > > > > > > > > > > etc. > > Can anyone estimate how "fast" is the rate of my > "counting"? :-) > > Looks to me like it's not only infinite, but > many-many multiples of infinity. I > think this process beats the heck out of the > super-double-exponential. But is it > "natural"? > -- Is it counting and natural, not natural and counting, counting and not natural, or not counting and natural? Consider, if one allows that counting is infinitely orientable, there is always a fourfold relationship between natural processes and arithmetic -- because the limit of the complete algebra is the four dimensional complex plane limit. Therefore, in two dimensional complex terms, counting is not natural -- because measured results are always real, from the origin. It seems to me that the rate of counting is compelled to be nonlocal, in physical terms, which implies that "fastest" is indeed infinite, because the measure of time interval in such a case is zero. Tom > Ioannis >
From: Andrew Usher on 18 Jan 2010 08:11
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 |