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System described by probabilities

by aaaa202
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aaaa202
#1
Mar25-14, 10:12 AM
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Imagine some kind of system, where you have at t=0 N single atoms (a gas). Now in a later instant dt there is a certain probability that 2 atoms will have collided and formed a 2-atom unit. Similarly dt after this event there is a certain probability that this 2-atom unit has either collided with another atom to form a 3-atom unit or a probability that it has decayed back to 2 single atoms. From this should it be possible to find the probability at time t, that an n-atom unit has been formed.
Do you guys know of any theoretical work that describes a dynamic system like this?
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UltrafastPED
#2
Mar25-14, 11:07 AM
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PF Gold
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Is there also a probability for an atom leaving an existing unit?

You can study systems like this with very simple computer simulations; try Mathematica.
Nugatory
#3
Mar25-14, 11:25 AM
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Quote Quote by aaaa202 View Post
Imagine some kind of system, where you have at t=0 N single atoms (a gas). Now in a later instant dt there is a certain probability that 2 atoms will have collided and formed a 2-atom unit. Similarly dt after this event there is a certain probability that this 2-atom unit has either collided with another atom to form a 3-atom unit or a probability that it has decayed back to 2 single atoms. From this should it be possible to find the probability at time t, that an n-atom unit has been formed.
Do you guys know of any theoretical work that describes a dynamic system like this?
A fair case could be made that this post would be better off in the General Math section, but I'll let the mentors decide that...

If you google around for "Markov chain", "random walk", "Poisson process" you will find plenty of relevant theory. In fact, you'll find so much that you'll probably conclude that if you just need results, simulating as UltrafastPED suggests is the way to go.

aaaa202
#4
Mar25-14, 11:33 AM
P: 1,005
System described by probabilities

Yeh well simulation might be the way to go in the end but I would like some basic theoretical understand of a process like this. I tried writing up some equations, but I don't think they made much sense.
Basically I said starting from t=0 in a time step dt there will be a certain probability that a 2atom unit has formed. Then in the next time step there is a probability that this atom detaches, stays, or another atom attaches etc. etc.
This left me with some iterated expressions for P(2 atom unit, dt), P(2 atom unit, 2dt), P(2 atom unit, 3dt), P(3 atom unit, dt), P(3 atom unit, 2dt), P(3 atom unit, 3dt). Is this related to markov chains?
It seems a problem that I need to choosing the timesteps infinitesimal, I don't in general end up with an integral or something like that.
I'm sorry if this is confusing to read, I am trying to get some intuition.
Do you have any good reading suggestions for a process like this?
UltrafastPED
#5
Mar25-14, 01:48 PM
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Readup on Markov chains: http://www.dartmouth.edu/~chance/tea.../Chapter11.pdf
aaaa202
#6
Mar25-14, 03:59 PM
P: 1,005
I have been thinking a bit more. Really this system must be some kind of "coupled" poisson process system (at least that is what I think the probabilistic proces describing e.g. radioactive decay is called).
Let us look at one atom in a gas that can interact with other atoms to form a dimer, trimer etc and let us try to find a general expression for the probability that at the time t it is a monomer, dimer, trimer etc.
Now at t=0 it is a single atom i.e. a monomer and if it were so that when the atom reacted to form a dimer it stayed that, the probability at time t that it is still a monomer would then be:
P(1,t) = exp(-λt) (the 1 signals that this is the probability for the atom being a monomer at time t)
And similarly the probability that it would be a dimer would be:
P(2,t) = 1-exp(-λt)
This is analogous to radioactive decay. Of course this is not strictly true for this system since, when it has formed a dimer there is a finite probability that it can decay back to a monomer or form a trimer. The problem is, how do I account for such terms in an overall description, where I want to find a general expression for the probability that at time t, the atom is an n-mer?
256bits
#7
Mar26-14, 06:32 AM
P: 1,407
Chemical equilibrium where the reactants and products have both forward and backward reactions. You might cruze through some of that literature to see how chemists deal with rates of reactions.

With species A, B, and C, the reaction would be,
A [itex]\Leftrightarrow[/itex] B [itex]\Leftrightarrow[/itex] C

Depending upon concentrations of reactants or products and other variables such as temperature or pressure, the reaction would have a greater tendency to proceed left or right.


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