Parametrizing a Pair Distribution Function

[Korbid]

Hi!,

If we have a bidimensional system of N particles.
How can i parametrize a radial distribution function g(r) in another single variable ?
This variable is the "time to collision", which we define as the duration of time for which two particles could continue at their current velocities before colliding.

Thank you!

 

[eljose79]

Hello!

That's a great question. To parametrize a radial distribution function (RDF) in terms of the time to collision, we first need to understand the relationship between the RDF and the time to collision.

The RDF is a measure of the probability of finding a particle at a certain distance from another particle. It is typically calculated by dividing the number of particles found in a shell of width dr at a distance r from a reference particle by the total number of particles and the volume of the shell. This gives us a probability distribution function that describes the spatial arrangement of particles in the system.

On the other hand, the time to collision is a measure of how long it will take for two particles to collide given their current velocities and positions. This is dependent on the distance between the particles as well as their velocities.

To parametrize the RDF in terms of the time to collision, we can use the following equation:

g(t) = g(r(t)) * v(t)

where g(t) is the RDF as a function of time, g(r(t)) is the RDF as a function of distance, and v(t) is the probability of collision per unit time.

By substituting t = r/v, we can rewrite the equation as:

g(t) = g(r(t)) * v(r(t))/r(t)

This gives us a new RDF that is parametrized in terms of the time to collision. However, it is important to note that this new RDF will only be valid for a specific set of initial conditions, such as particle velocities and positions.

I hope this helps! Let me know if you have any other questions.