Model free collision aggregation for the computation of escape distributions

TL;DR

A particle undergoing collisions in a space-time domain is considered and a method to sample its escape time, space and direction from the domain is proposed and shown to be an efficient method to sample the escape distribution of the particle.

Abstract

Motivated by a heat radiative transport equation, we consider a particle undergoing collisions in a space-time domain and propose a method to sample its escape time, space and direction from the domain. The first step of the procedure is an estimation of how many elementary collisions is safe to take before chances of exiting the domain are too high; then these collisions are aggregated into a single movement. The method does not use any model nor any particular regime of parameters. We give theoretical results both under the normal approximation and without it and test the method on some benchmarks from the literature. The results confirm the theoretical predictions and show that the proposal is an efficient method to sample the escape distribution of the particle.

Figures (9)

• Figure 1: An illustration of the time dynamics and escape space-time distribution for a particle starting at $x_0$, $\mathcal{D}=\mathcal{D}_2$. The abscissas represent the position of the particle and the ordinates the time. For each remarkable point we give its $x,t$ coordinates, for instance $A$ has $x=0$ and $t=0$ and represents the left extremity of the segment at $t=0$ while $C$ is the left extremity at the final time. The time to next collision is an exponential random variable of average $1/\sigma$ i.e., distributed $Exp(\sigma)$. The joint escape distribution $\mathcal{E}(\sigma,\mathcal{X},T,x_0)$ collects the location $x^\star$, time $t^\star$ and direction $a^\star$ when the boundary $x=0$, $x=L$ or $t=T$ is reached for the first time (i.e., particle hits $AB$, $BC$ or $CD$). The simulation is stopped when this happens. The colored histograms are artist views of each conditional escape distributions: the magenta/orange is the histogram of the escape time $t^\star$ given that particle escaped through the left/right while the green is the histogram of escape position $x^\star$ for particles that did not escape before time $T$ or, equivalently, the escape is due to time $T$ being totally consumed.
• Figure 2: Examples of trajectories for the 1D $S_2$ test case with $L=0.01cm$, $T_f=4\times 10^4fs$, $v=3.0\cdot 10^{-5} fs/ cm$, $\sigma \in \{10^{-2}$, $10$, $10^{4}$, $10^{6}\}$ ($cm^{-1}$), $x_0=L/2$, initial direction $+1$ (oriented towards right). Each row is a single trajectory, each trajectory corresponding to a different value of $\sigma$. First column plots the particle evolution $x(t)$ with ordinate axis being the time and abscissas the location in the $[0,L]$ segment. Second column displays the histogram of the $n_c$ values (cf algorithm \ref{['algo:accelerated']}.) The third column gives information on the total steps taken and total number of collisions treated. The abscissas are the number of collisions and the ordinate axis is time in femtoseconds. The particle in the first row has an exit direction $a^\star=+1$ because no collision takes place. The particles in the last three rows have a exit direction $a^\star$ that will be drawn at random from $\{-1,1\}$ because in all cases exit is due to the fact that time was consumed.
• Figure 3: The escape laws corresponding to figure \ref{['fig:1D_traj']}. Same convention for the signification of the rows is used. For the middle column the abscissas are time values expressed in femtoseconds; for all other plots the abscissas are $x$ values in the segment $[0,L]$. Each column is a conditional distribution: in the first column are drawn histograms of $t^\star$ for escape points with $x(t^\star)=0$ (escape through the left extremity of the domain), second column draws histograms of $x^\star=x(T)$ for escape points with with $t=T$ (escape because time is consumed) and third histograms of $t^\star$ for escape points with $x(t^\star)=L$ (escape through the right extremity of the domain). The probability of escape for each alternative is given above the plot. When a histogram is void there is no escape for the alternative the histogram is supposed to represent. For instance, in the third row, first column one expects to see the histogram of escape time $t^\star$ conditioned by the fact that escape occurred through the left side i.e. $x(t^\star)=0$ (before time $T$ and before reaching the right side). But this never happens because in this case $\sigma$ is too large, many collisions occur and the particle does not have the time to reach the left side (the right side neither in fact) before time $T$.
• Figure 4: Results for the setting in section \ref{['sec:onedsn']} for $n_\mathcal{N}=300$, $N=100$. The histograms of the escape distributions $t,x,a$ are plotted as follows: first column are histograms of the time values $t^\star$, second column the positions $x^\star$ and third the directions $a^\star$. Each row is a conditional distribution: the first row are escape points with $x^\star(t^\star)=0$ (escape through left side), second row with $t^\star=T$ (escape due to total time $T$ being consumed) and third with $x^\star(t^\star)=L$ (escape through the right side).
• Figure 5: One example trajectory corresponding to results in figure \ref{['fig:1D_sn_laws']}, i.e. $n_\mathcal{N}=300$, $N=100$. Left plot: the trajectory $x(t)$ of the particle, $y$ axis is the time. Exit direction is $a^\star=-1$. Middle plot: the distribution of the number of collisions that have been aggregated. Right plot: the number of collisions with time in the $y$ axis. Many are equal to $1$ but some go as high as $600-800$.

Theorems & Definitions (10)

• Remark 1
• Proposition 2
• proof
• Proposition 3
• Remark 4
• proof
• Corollary 5
• proof
• Proposition 6
• proof