Processes which propagate faster than exponentially

I.N. Galidakis wrote:
Dann Corbit wrote:
In article ,
says...

"I.N. Galidakis" writes:
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.

Anything which goes from 0 to anything strictly positive?

E.g., I had zero apples now I have one apple. The rate of increase was
infinite.

The Dirac delta function was invented for a reason (it pops up in all
sorts of interesting places), and it has infinite slope.

I have a jar with ten marbles of one color in it (say 'red').
If I pull out all the marbles how many different color sequences can I
get:
1.

Now, I add ten new marbles of a different color (for example, 'blue').
If I pull out all the marbles one at a time after shaking how many
different color sequences can I get:
Essentially, we have 2^20 in binary different color combinations.

Now, I add ten new marbles of a different color (for example, 'yellow').
If I pull out all the marbles how many different color sequences can I
get:
Essentially, we have 3^30 in ternary different color combinations.

If we continue adding new marbles of different colors in this manner,
the number of possible color combinations we get grows faster than
exponential, because both the base and the exponent are increasing.

Isn't that the sequence a(n)=n^{n+10}?

Maple reports:
f:=x-x^(x+10);
g:=x-exp(exp(x));

then
limit(f(x)/g(x),x=infinity);

0

so it looks to me like SUB-double-exponential.

Sorry, I just had a bout with severe stupidity. The sequence looks like:

a(n)=n^{10*n}, (NOT n^{n+10}) for which Maple gives:

f:=x-x^(10*x);
g:=x-exp(exp(x));

limit(g(x)/f(x),x=infinity);

0

so looks like it's actually SUPER-double-exponential.
--
Ioannis

On Jan 16, 4:42*am, Marvin the Martian 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.

On Jan 16, 12:54*am, jbriggs444 wrote:
On Jan 15, 11:14*am, "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.

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.

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.

rabid_fan ) writes:
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.

Missing minus sign, I think.


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.

Second-order rate equations are commmon enough in chemistry and
third-order rates are not unknown. If only one reactant is involved or the
initial concentrations of all reactants are equal, the simple expressions
you give for these cases follow automatically (to appropriate levels of
approximation as always).

--John Park

On Jan 15, 10:14*am, "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?

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

Tom Potter wrote:

Considering that no one commented on my
observation that the Weibull function
is far superior to the exponential function in modeling reality,
[snip crap]

Dog turd, cat turd, Potty turd - all turds.

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm

On Jan 15, 10:51*pm, Frisbieinstein wrote:
On Jan 16, 12:54*am, jbriggs444 wrote:





On Jan 15, 11:14*am, "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.

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.

On Jan 15, 10:49*pm, Frisbieinstein wrote:
On Jan 16, 4:42*am, Marvin the Martian 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.

On Jan 18, 8:11*am, Andrew Usher wrote:
On Jan 15, 10:14*am, "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?

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.