# System described by probabilities

• aaaa202
In summary: The rates in each direction could be different. There would be an equilibrium constant.In summary, a theoretical understanding of a dynamic system where single atoms collide and form units of varying size is possible through the use of concepts such as Markov chains, random walks, and Poisson processes. The system can also be studied through simple computer simulations. The process is similar to chemical equilibrium reactions and can be described using equilibrium constants and rates of reactions.
aaaa202
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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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.

aaaa202 said:
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.

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?

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?

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 $\Leftrightarrow$ B $\Leftrightarrow$ 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.

## 1. What is a system described by probabilities?

A system described by probabilities is a mathematical model used to represent real-world systems that involve randomness or uncertainty. It is based on the theory of probability, which assigns a numerical value to the likelihood of different outcomes occurring.

## 2. How is a system described by probabilities used in science?

A system described by probabilities is used in science to make predictions and analyze data in situations where the outcome is uncertain. It is commonly used in fields such as physics, biology, and economics to model complex systems and make informed decisions based on the probabilities of different outcomes.

## 3. What are the key components of a system described by probabilities?

The key components of a system described by probabilities include a set of possible outcomes, a set of probabilities associated with each outcome, and a set of rules or equations that describe how the probabilities change over time or under different conditions.

## 4. Can a system described by probabilities accurately predict the future?

No, a system described by probabilities cannot accurately predict the future. It can only provide an estimate of the likelihood of different outcomes based on the available data and assumptions. The actual outcome may differ from the predicted outcome due to the inherent randomness and unpredictability of real-world systems.

## 5. How is the accuracy of a system described by probabilities evaluated?

The accuracy of a system described by probabilities is evaluated by comparing the predicted outcomes to the actual outcomes and calculating the margin of error. The smaller the margin of error, the more accurate the model is considered to be. Additionally, the model can be refined and improved by incorporating new data and adjusting the underlying assumptions.

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