| Pages:
1
2
3 |
watson.fawkes
International Hazard
Posts: 2793
Registered: 16-8-2008
Member Is Offline
Mood: No Mood
|
|
Quote: Originally posted by woelen  | A branching chain reaction is a chain reaction, where one of the products in the chain of reactions catalyses the conversion of one of the initial
reactants.
[...]
The type of equations for a branching chain reaction lead to non-linear equations, exhibiting super-exponential behavior. |
A chain reaction always has an expanding propagation step with more than one unit of output per unit input. Rapid growth in a chain
reaction occurs when the expansion rate is greater than the extinction rate. They can also be modeled with a first-order, linear differential
equation. Chain reactions exhibit exponential growth, not super-exponential. Another physical example with a similar mathematical model is the
ionization shower created by an energetic charged particle.
Now these differential equations are generally in more than one variable, not the single variable ones most commonly encountered early in calculus
courses. "It's complicated" is not at all the same "it's nonlinear". Multidimensional linear equations can have rather surprising behavior, and these
include sudden state changes. Large classes of reaction rates can be modeled with continuous-time Markov chains, and it takes some argument to claim that this class of equations cannot model the system in question.
|
|
|
PHILOU Zrealone
International Hazard
Posts: 2893
Registered: 20-5-2002
Location: Brussel
Member Is Offline
Mood: Bis-diazo-dinitro-hydroquinonic
|
|
Good examples of chain reactions are:
H2 + Cl2 --> 2 HCl
2 H2 + O2 --> 2 H2O
In a chain reaction there are 3 phases:
-Initiation
-Propagation
-Termination
If the propagation reaction is fast enough or induces more and more reactive species then the reaction can be explosive.
Sometimes initiation of the reaction happens because of:
-the glass surface
-the temperature
-the pressure
-the light
Hydrogen and oxygen can be made to explode at ambient T° in a glass tube, simply by increasing slowly pressure of the gas mix in the tube.
Delay reaction are typical of a mix of many reducer-oxydiser couples:
All are temperature and concentration dependant:
-Formol + HNO3
-Aceton/propanon + HNO3
-Glycerol/propantriol + KMnO4
Being such a mix I stil think that the mix of NH2OH.HCl and NaClO3 would form a mix that is spontaneously explosive with a given delay because it
would form NH2OH.HClO3 what is very close and related to NH4Cl and NaClO3 mix what forms the infamous NH4ClO3.
@Woelen
In your videos in real time, the reaction is very fast and starts between two frames.
In your slow motion it is funny to see the reaction starts in the bottom right and goes up to the left like an explosion.
Also the crystals seems to take rocket propelling behaviour making funny loopings
Very noticeable are the NOx colour in the exces hydroxylamonium movie and the Br2 droplets in the excess bromate movie.
Very nice videos!
[Edited on 9-11-2013 by PHILOU Zrealone]
PH Z (PHILOU Zrealone)
"Physic is all what never works; Chemistry is all what stinks and explodes!"-"Life that deadly disease, sexually transmitted."(W.Allen)
|
|
|
woelen
Super Administrator
Posts: 7977
Registered: 20-8-2005
Location: Netherlands
Member Is Offline
Mood: interested
|
|
I do not agree with you, watson.fawkes. The description of this type of reaction definitely is non-linear and I perfectly know the difference between
complicated and non-linear! Multidimensional linear equations have nothing really surprising for me (at least the time-invariant ones which can be
described as dx/dt = Ax, where x is the state vector and A is a matrix), simply determine the eigenvalues and you know the characteristics of the
solution.
Actually, nearly all chemical reactions must be described by non-linear equations, because you have products of concentrations in the equations. Only
if a concentration can be considered (nearly) constant, behavior may be approximated by linear equations, but for adequate explanation of branched
chain reactions (and also of the related oscillating and chaotic reactions) you need to have a non-linear model.
I know that the state-vector of the system I am studying is multi-dimensional. It is both non-linear and multi-dimensional and that can be quite an
interesting situation. From dimension 3 and up the system can exhibit chaotic behavior, but even from dimension 1 you can get clock-like behavior.
In the meantime I have done some more reseach already and have done some math. The basic idea behind the dynamics of the system I have is the behavior
of the system dx/dt = x*x (or somewhat more generic: dx/dt = x^a, with a > 1). Such systems have solutions which remain less than 1 for a long time
and within one second they flash from less than 1 to infinity.
Of course, this is an oversimplification of the system described here, but it captures the non-thermal explosion perfectly. In reality, there are not
unlimited amounts of reactants, and the equation is more complicated, but the basis is there. I'll try to get a better set of differential equations
and keep the order of the system as low as possible (reaction steps which proceed very fast can be regarded as quasi-static and only add algebraic
equations, not additional differential equations).
What is special about this chain reaction is not the fact that it is a chain, but the fact that one of the products, produced in the chain enormously
speeds up the initial reaction. It catalyses the reaction. The catalyst is bromide ion and another catalyst is H(+) ion. I tried this, by adding
NaBrO3 to a solution of the hydroxylammonium salt to which some KBr is added as well. The addition of KBr has a marked effect on the induction time.
It becomes much shorter. Addition of H(+) (e.g. a single drop of 2M H2SO4) before adding the NaBrO3 also has a strong effect. When this is done, then
a violent reaction starts nearly immediately after adding the NaBrO3, but the reaction is less spectacular. It looks more like a normal violent
runaway.
Temperature also has a strong effect. As I wrote before, when I handwarm the solution before adding NaBrO3, then the induction period is much shorter
than when I use cold tap water (which is appr. 15 C this time of the year).
[Edited on 9-11-13 by woelen]
|
|
|
blogfast25
International Hazard
Posts: 10562
Registered: 3-2-2008
Location: Neverland
Member Is Offline
Mood: No Mood
|
|
Woelen:
Very interesting that you were able to prove bromide and H<sub>3</sub>O<sup>+</sup> (you didn't mean H<sup>+</sup>
literally, right? There is water present) have a catalytic effect.
Re the equations, I'll reserve judgement until someone comes up with the most correct set (I assume some simplifying assumptions will have to be made)
but I agree that unless some reagent concentrations are large with respect to others (and thus quasi-constant), most equations governing the system
will be non-linear. They have to be, relying on non-linear reaction equilibrium constants as they do.
Have you got any links up on your website to these vids?
|
|
|
woelen
Super Administrator
Posts: 7977
Registered: 20-8-2005
Location: Netherlands
Member Is Offline
Mood: interested
|
|
@blogfast25: I wrote H(+), but of course this is a simplification. Even H3O(+) is a simplification, albeit a much better one already.
I have not yet made a webpage about this reaction. I made the videos, so that I can show some results already, but the math must be much more
rigorously established. I also want to show a simulation, which shows behavior, similar to the real reaction.
|
|
|
deltaH
Dangerous source of unreferenced speculation
Posts: 1663
Registered: 30-9-2013
Location: South Africa
Member Is Offline
Mood: Heavily protonated
|
|
woelen, I'm curious, have you tried this reaction with an iodate as oxidant by chance?
[Edited on 10-11-2013 by deltaH]
|
|
|
watson.fawkes
International Hazard
Posts: 2793
Registered: 16-8-2008
Member Is Offline
Mood: No Mood
|
|
I spoke too hastily above about the reaction rate equations being linear. I was principally objecting to the hyperbole involved in describing them
with phrases such as "highly non-linear". Too inflamed to think through the situation, I posted too soon. In a mathematical sense, reaction rate
equations are anything but highly non-linear. They are, indeed, about as simple as its possible to be and not be linear. "Highly nonlinear" would be
equations that include terms such as (∂f/∂y)2, but there's no hint of that here.
Generically, reaction rate equations are in a class of equations called "semi-linear". This is one class of equations in a sequence: linear -->
semi-linear --> quasi-linear --> nonlinear. (The buzzwords are all there for searching, but it's some serious mathematics behind them.)
Semi-linear equations are defined for PDE's, not just the simple temporal ODE's we've been considering here. (Simple because we assume no spatial
difference in concentrations of reagents.) These are first order (only the first derivative) and the derivative coefficients are constant.
Furthermore, the nonlinear terms are simple polynomials in the concentration coefficients, not arbitrary functions. And in most cases, the equations
are even component-wise linear, say, under the assumption of dilute solutes.
First-order semi-linear equations have solutions by the so-called "method of characteristics". This method applies to PDE's of arbitrary dimension, so
it can handle proper chemical engineering problems, where concentrations vary with position. The essence of the technique is find a coordinate
transform that "straightens out" some of the coordinates. The characteristic curves are well-behaved and it's straightforward to get numerically
stable computational solutions. The point I'm making is that these reaction rate equations are mathematically relatively simple even in the PDE case,
much less the ODE case.
"Super-exponential" is another phrase I object to. The fastest that any solutions grow (locally) is bounded by an exponential of some polynomial
function. That's in distinction to shock formation, where you do get super-exponential growth at the shock in the neighborhood where the solutions
ends. As I understand it, this class of reaction rate equations here cannot form shocks, though I'd need to sit down and prove it to be sure. (Shock
formation is at the center of the Clay prize about the Navier-Stokes problem; it's not a particularly well-understood area of mathematics.)
|
|
|
woelen
Super Administrator
Posts: 7977
Registered: 20-8-2005
Location: Netherlands
Member Is Offline
Mood: interested
|
|
| Quote: | | The fastest that any solutions grow (locally) is bounded by an exponential of some polynomial function. |
Local analysis is not the solution to problems like this. You need global analysis. Actually, global analysis also is a term from
mathematics, it is about describing dynamical systems with (possibly) non-euclidian state space (sometimes quite complicated manifolds), but I do not
specifically refer to that branch of mathematics. The use of differential geometry is not needed for solving this kind of problems.
The term "highly non-linear" refers to the type of solutions, needed to describe systems like this. Even a simple first order non-linear equation like
dx/dt = a*x + x*x already has quite interesting behavior for certain initial conditions and certain values of a, which I would describe as "highly
non-linear".
|
|
|
blogfast25
International Hazard
Posts: 10562
Registered: 3-2-2008
Location: Neverland
Member Is Offline
Mood: No Mood
|
|
| Quote: Originally posted by woelen | Even a simple first order non-linear equation like dx/dt = a*x + x*x already has quite interesting behavior for certain initial conditions and
certain values of a, which I would describe as "highly non-linear".
|
Do you mean: dx/dt = ax + x2 ? Like x(t)' = ax + x2 ?
t = (1/a) ln[x/(x +a)] + C (C constant)
Yup, that could get interesting for some a.
Or am I missing some notation here?
[Edited on 10-11-2013 by blogfast25]
|
|
|
woelen
Super Administrator
Posts: 7977
Registered: 20-8-2005
Location: Netherlands
Member Is Offline
Mood: interested
|
|
Yes, I meant what you write. I am a software nerd and I am spoiled so much that I use * for multiplication, even in non-software contexts  . On the other hand, software engineers are fond of notations, using simple ASCII
characters only on a single line, for even the most draconian mathematical expressions 
Let me rewrite the thing: dx/dt = a*x + x^2 
[Edited on 10-11-13 by woelen]
|
|
|
watson.fawkes
International Hazard
Posts: 2793
Registered: 16-8-2008
Member Is Offline
Mood: No Mood
|
|
| Quote: Originally posted by woelen | | Even a simple first order non-linear equation like dx/dt = a*x + x*x already has quite interesting behavior for certain initial conditions and certain
values of a, which I would describe as "highly non-linear". |
Well, then, I will just continue to protest and
view this use of language with disdain. "It's complicated" is just not the same thing as "highly non-linear".
There are highly non-linear equations out there, with far more structure, such as the recursion operators in certain integrable systems such as the
KdV (Korteweg-deVries) equation that map solutions to other solutions. Such operators devolve to trivial (identity) operators in simpler situations.
Even here, though, inverse scattering transforms can linearize these equations, in a certain sense, so they are not as non-linear as others out there.
In short, non-linearity is not about the external behavior of the solutions, but about internal structure. Chemical rate equations are just barely
non-linear.
|
|
|
Pok
potassium Prometheus
Posts: 176
Registered: 5-12-2010
Member Is Offline
|
|
Fantastic videos, woelen!! I first watched the slow motion video with these wonderful "bubble rockets". I did't imagine that in real
time it happens soooo suddenly.
The reaction reminds me of a similar experiment which I did. The colour change and the sudden evolution of gas:
<iframe sandbox frameborder="0" width="480" height="270" src="http://www.dailymotion.com/embed/video/x14acdf"></iframe><br />
It's part of this experiment: a luminol clock reaction. Unfortunately, this reaction is not nearly as "shocking" as your one:
<iframe sandbox frameborder="0" width="480" height="270" src="http://www.dailymotion.com/embed/video/x149yl0"></iframe><br />
[Edited on 11-11-2013 by Pok]
|
|
|
deltaH
Dangerous source of unreferenced speculation
Posts: 1663
Registered: 30-9-2013
Location: South Africa
Member Is Offline
Mood: Heavily protonated
|
|
I'm sorry, those luminol clock reaction's are simply too beautiful for words...
[Edited on 11-11-2013 by deltaH]
|
|
|
woelen
Super Administrator
Posts: 7977
Registered: 20-8-2005
Location: Netherlands
Member Is Offline
Mood: interested
|
|
I found an interesting paper, which describes how one can determine in general when a dynamical system, described by polynomial differential
equations, shows "blow-up" in its solution:
http://www.math.bme.hu/~csikja/files/v26-43.pdf
This is exactly the type of phenomenon which is observed in this reaction. I also made some progress in my own determination of suitable equations:
There are two rate-determining steps in the reaction:
1) BrO3(-) + NH3OH(+) + H(+) --> HBrO2 + NH2(OH)2(+) (slow reaction)
2) BrO3(-) + Br(-) + 2H(+) --> HBrO2 + HOBr (quite fast reaction, but not instantaneous)
The first reaction has rate [BrO3(-)]*[NH3OH(+)]*K, where K can be written as K = k*[H(+)]/(A + B*[H(+)]), with k, A, and B constants.
The second reaction has rate c*[BrO3(-)]*[Br(-)]*[H(+)]², with c being some constant.
Further reaction from HOBr and HOBr2 with excess hydroxylamine to bromide and N2O are very very fast.
The intermediate species NH2(OH)2(+) decomposes to N2O and water and H(+). This reaction also is very fast.
The reason for the complicated order for H(+) in equation (2) is because of the presence of a very fast equilibrium reaction between NH2OH+H(+) and
NH3OH(+). Such equilibria lead to algebraic (non-differential) constraints.
I have references to papers for all of these reactions, unfortunately I do not have them with me here. Lateron in subsequent posts, I will post the
references, so that other can check my findings.
[Edited on 11-11-13 by woelen]
|
|
|
deltaH
Dangerous source of unreferenced speculation
Posts: 1663
Registered: 30-9-2013
Location: South Africa
Member Is Offline
Mood: Heavily protonated
|
|
I have some humble questions about these kinetics woelen.
Firstly, I am confused about this statement that two reactions are rate limiting, by definition can there not be only one? Do you mean to say that two
are candidates and it's unclear which actually is?
Secondly, these reactions do not appear elementary steps as written, is it not therefore incorrect to call either a rate limiting step (the true rate
limiting step being an elementary reaction step in a sequence of steps which those equations describe)?
Thanks.
[Edited on 11-11-2013 by deltaH]
|
|
|
woelen
Super Administrator
Posts: 7977
Registered: 20-8-2005
Location: Netherlands
Member Is Offline
Mood: interested
|
|
My wording indeed is not entirely clear. I meant to say that there are two steps in the reaction which add dynamics on such a timescale, that it
cannot be described as an instantaneous reaction. These reactions lead to differential equations, while all other steps in the total net reaction lead
to algebraic equations (examples of the latter are the equilibrium equations for acid-base reactions, involving the Kz of acids).
The first reaction seems indeed to be a really elementary step. It is described in two of the papers I used in my research and in both the rate is
described as c*[BrO3(-)]*[Br(-)]*[H(+)]². This fits the stoichiometry of the reactants, used up in the reaction, the coefficients are 1, 1, 2 and
these are exactly the exponents in the reaction rate.
The second reaction is somewhat more involved. There is interaction with an acid-base equilibrium between NH3OH(+) and NH2OH+H(+). This leads to the
somewhat more complicated form, presented in my previous post, due to some algebraic constraints on the concentration of H(+). Precise details will
follow in my write-up.
There are many more steps, the total net reaction is bromate plus hydroxylammonium ion gives bromide plus N2O plus water plus acid. Because all these
steps use up hydroxylamine and finally produce bromide ions, H(+) ions and N2O, there is a positive feedback to the two rates, given above. Especially
the formation of bromide leads to a spectacular speedup. This is something I confirmed with a little experiment in which I added bromide before adding
bromate. The experiment of kmno4 also is instructive. He added bromate to the spent solution of hydroxylamine (which contains excess hydroxylamine)
and when the bromate is added to this, then there immediately is a violent reaction. This is because a lot of bromide and acid is present in the spent
liquid and then there is immediate fast production of HBrO2 and HOBr, which in turn rapidly react with excess NH2OH.
[Edited on 11-11-13 by woelen]
|
|
|
deltaH
Dangerous source of unreferenced speculation
Posts: 1663
Registered: 30-9-2013
Location: South Africa
Member Is Offline
Mood: Heavily protonated
|
|
Thanks.
I think there is some confusion here with nomenclature then and wording.
To help clear it up, I'll paraphrase the IUPAC definition that an elementary reaction step can only consist of one transition state.
As a result, they are very simple things, generally of the form A + B => C or D => E + F. Your equation has three reactants, which tells me that
it is a combination of elementary steps as written.
In other words, all three reagents do not need to simultaneously collide to form the two products by a single reaction step (and don't, the
probability of this happening statistically is insignificant).
Much more likely, two need to collide to form some hidden intermediate and then this collides with your third reactant to form the products, or even
more steps than this, but you get the picture.
I was also under the impression (maybe wrong) that only an elementary step could be a rate limiting step, i.e. the slowest step causes all others to
be assumed to be at equilibrium.
Links:
http://en.wikipedia.org/wiki/Elementary_reaction
[Edited on 11-11-2013 by deltaH]
|
|
|
woelen
Super Administrator
Posts: 7977
Registered: 20-8-2005
Location: Netherlands
Member Is Offline
Mood: interested
|
|
| Quote: | | [...]the slowest step causes all others to be assumed to be at equilibrium.[...] |
This cannot be true, because if this indeed were the case, then every chemical reaction would behave like a first order system (only one differential
equation, all other equations being algebraic, because all other steps are in equilibrium).
What I have understood about elementary reactions is that they can involve more than 2 species. The chance that a certain species is in a certain
small piece of space is proportional to the concentration of that species. Assuming independent motion of all species, this is true for all involved
species. So, the chance that all required species for a reaction are in a certain small volume of space is the product of all concentrations. If N
molecules (or ions) of the same species are needed, then the product for that species is simply its concentration to the power N. So, the chance of
three entitities (or in my case, even 4 entities) is not negligible, although at low concentrations of all entities it can be very low. If, however,
the species are reactive, then the constant k for the reaction may be high and this may compensate.
From experimental data it is known that the rate of the reaction between bromate, bromide and H(+) ions is as described in my previous post and this
is in perfect agreement with what I described above. I can only imagine a process where all 4 ions are close to each other in a small space, but if
you see another mechanism, which could lead to the same rate exponents (1 for bromate, 1 for bromide, 2 for H(+)), then I would like to know that. For
the discussion and the explanation of the observed behavior, however, this is not important. What is important is the observed dynamics from
experimental data and the equations describing this dynamics accurately. It is nice to know the precise physics behind this, but if this is not
possible/unknown, then I still can do the math and try to describe the total reaction. It then becomes more descriptive, but that is a valid approach
in natural sciences.
|
|
|
deltaH
Dangerous source of unreferenced speculation
Posts: 1663
Registered: 30-9-2013
Location: South Africa
Member Is Offline
Mood: Heavily protonated
|
|
Okay, I concede that describing chemical kinetics by short posts is a nightmare! Ok there is too much to have to describe here and I am too poor a
describer to pull this off concisely.
[Edited on 11-11-2013 by deltaH]
|
|
|
PHILOU Zrealone
International Hazard
Posts: 2893
Registered: 20-5-2002
Location: Brussel
Member Is Offline
Mood: Bis-diazo-dinitro-hydroquinonic
|
|
| Quote: Originally posted by deltaH | woelen, I'm curious, have you tried this reaction with an iodate as oxidant by chance?
[Edited on 10-11-2013 by deltaH] |
See post of Woelen posted on 7-11-2013 at 21:11 in this tread
PH Z (PHILOU Zrealone)
"Physic is all what never works; Chemistry is all what stinks and explodes!"-"Life that deadly disease, sexually transmitted."(W.Allen)
|
|
|
watson.fawkes
International Hazard
Posts: 2793
Registered: 16-8-2008
Member Is Offline
Mood: No Mood
|
|
| Quote: Originally posted by woelen | | I found an interesting paper, which describes how one can determine in general when a dynamical system, described by polynomial differential
equations, shows "blow-up" in its solution: |
"Blow up" is essentially the same thing that I referred to above
as shock formation (the detailed differences don't matter for the present topic). There's a rather useful section for the present discussion therein:
| Quote: | | Suppose we have a mass conserving reaction endowed with mass action type kinetics. Then, no solution of the induced kinetic differential equation with
nonnegative initial condition blows up. |
Thus, given a particular class of equations that looks a lot like
the right class of equations to model a system, we may well find that it contains behaviors that don't actually occur. In other words, given a class
of first-order semi-linear differential equation that varies from linearity only by a quadratic form (this is the class of equations at the start of
this paper), we see that some blow up and some do not. Making a further restriction (see the quote) eliminates the blow up. What we should conclude is
that the unrestricted class of equations is the wrong class to model the phenomenon of interest, even though it looked like a good candidate at the
outset. Such is the progress of science.
Differential equations are chock full of this kind of thing. Things that seems close to each other turn out not to be very similar at all. Overall,
the reason for this is that there are many incompatible notions of "close to". Mathematically, "close to" is formalized as a topology. Closeness in,
say, ordinary 3-dimensional space is (mostly) unique, but utterly non-unique in infinite dimensional spaces, and even more so in spaces with more than
one kind of infinite extent such as differential equations. If anyone would like a taste of this field, just take a gander at the Wikipedia page on jet bundles; jet bundles are the appropriate structure in which to define coordinate-invariant differential equations (the only kind expected to
model physical reality). That page is mostly definitions; it doesn't even get to topological issues.
|
|
|
deltaH
Dangerous source of unreferenced speculation
Posts: 1663
Registered: 30-9-2013
Location: South Africa
Member Is Offline
Mood: Heavily protonated
|
|
aw... missed that, thanks PHILOU.
|
|
|
PHILOU Zrealone
International Hazard
Posts: 2893
Registered: 20-5-2002
Location: Brussel
Member Is Offline
Mood: Bis-diazo-dinitro-hydroquinonic
|
|
| Quote: Originally posted by woelen | | I tried the NaClO3 as well, but this is not interesting at all. It does not react and this is exactly what I expected. My excuse for not mentioning
that. I considered it common knowledge that chlorate ion is sluggish in aqueous solution and only acts as serious aqueous oxidizer at low pH or at
elevated temperatures, which cannot be achieved in aqueous solution at normal pressure. |
I think that water solutions of HONH3Cl are relatively acidic just like solution of NH4Cl are.
NH4Cl is already enough acidic to ensure strong oxydising properties of chlorate mixed with it in water solution and favourise the slow NCl3
formation...
In the case of HONH3(+) it must be a stronger acid than NH4(+), just like H2N-NH3(+) is.
PH Z (PHILOU Zrealone)
"Physic is all what never works; Chemistry is all what stinks and explodes!"-"Life that deadly disease, sexually transmitted."(W.Allen)
|
|
|
woelen
Super Administrator
Posts: 7977
Registered: 20-8-2005
Location: Netherlands
Member Is Offline
Mood: interested
|
|
| Quote: Originally posted by watson.fawkes | | Thus, given a particular class of equations that looks a lot like the right class of equations to model a system, we may well find that it contains
behaviors that don't actually occur. In other words, given a class of first-order semi-linear differential equation that varies from linearity only by
a quadratic form (this is the class of equations at the start of this paper), we see that some blow up and some do not. Making a further restriction
(see the quote) eliminates the blow up. What we should conclude is that the unrestricted class of equations is the wrong class to model the phenomenon
of interest, even though it looked like a good candidate at the outset. Such is the progress of science. |
Here you have a very important point! The type of equations, you mention indeed cannot show blow-up, because they describe the reaction of finite
amounts of reagents. In my experiments, the very violent reaction looks like a blow-up, but it is not. Although it is very violent, it is not
infinitely violent and does not involve infinite amounts of reactants. The total amount of each of the elements in the system is constant.
So, I need to search for strong splike-like behavior instead of true blow-up. Another approach may be to use approximate systems, where blow-up is an
approximation of a very strong spike, but if such a solution is obtained, then one knows that it is not a true description of reality, but only a
(coarse) approximation.
Thanks for providing this piece of insight!
|
|
|
blogfast25
International Hazard
Posts: 10562
Registered: 3-2-2008
Location: Neverland
Member Is Offline
Mood: No Mood
|
|
Like Delta, I'm sceptical about elementary reactions that require collisions between more than two species.
|
|
|
| Pages:
1
2
3 |
![[*]](images/xpblue/default_icon.gif){kind=icon}
