# Distribution function for specific 1D problem

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## Main Question or Discussion Point

Hello!
Maybe someone will be able to suggest something about the following quite simple problem:

1D problem on axis "X". Particle moves only along "X" axis and starts its motion from X=0. However, when "X<0" particle disappears. Particle is influenced by some kind of force in such way that we know only the density probability function of particle velocity F[v]. In the case, when the average velocity (can be found from F[v]) is negative, we can conclude that any particle after some time eventually will disappear, since the probability to go to negative "X" is bigger.

The question: if we have infinite number of such particles (and as a result infinite number of trajectories that ends on X<0), what is distribution function for these particles on the axis "X"?

Thank you very much for any suggestions and relevant to this problem sources.

andrewkirk
Homework Helper
Gold Member
1. what are the initial positions of all the particles? This could be given as a distribution, for example a uniform distribution on [0,1].
2. what, if any, are the dependencies between
(a) velocities of particle p at time s and at time t ($s\neq t$)
(b) velocities of particle p and particle q at time t ($p\neq q$)
(c) velocities of particle p at time s and particle q at time t ($p\neq q$ and $s\neq t$)

For the problem to make any physical sense the velocity of a particle would need to be a continuous function of time, which requires strong dependency in (a). A Wiener Process (Brownian Motion with drift) could be a suitable process for the velocity of a particle. That requires specification of two parameters for each particle $p$:
I. instantaneous drift $\mu_p$ of velocity
II. instantaneous volatility $\sigma_p$ of velocity

A simple model would have $\mu_p=\mu\forall p$ and $\sigma_p=\sigma\forall p$ and all particles' velocity processes independent of one another.

Sorry that I didn't formulate the problem clear enough from the beginning.
I am interested in the simplest case, when particles are independent and velocity is a continuous function of time - (a) we know only the probability to have some velocity at the next time moment (from the probability density function of particle velocity F[v]).

To imagine the problem in different way, we can think that we do experiment:
we put the first particle at X=0 and record it trajectory. When the first particle eventually disappears (since the probability to move to negative X bigger), we put the second particle and do the same procedure. Each particle will have its own trajectory. If we average all these trajectories, we obtain the most probable trajectory - that says what is the probability to find the particle on some interval.

Or similarly, we can think that on the X<0 there is a wall that does not allow particle to move to negative value of X (particle doesn't bounce from the wall, just stay at X=0 if the force is directed to negative direction of X). And, assuming that from the beginning of that motion passed long time (initial condition will stop play the role, the particle mostly will be near the wall, even if it started far away), what is the probability density function for this particle on the axis X? What is the probability to find particle at some interval on the axis X in any fixed time?