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From Galactic Clusters to Plasmas in a Single Monte Carlo: Branching Paths Statistics for Poisson-Vlasov/Boltzmann

Daniel Yaacoub, Gaëtan Brunetto, Stéphane Blanco, Richard Fournier, Gerjan Hagelaar

Abstract

Recent advances have allowed to tackle path-space probabilistic representations of mesoscopic Boltzmann transport nonlinearly coupled to a sub-model of the force-field by step forward approaches in terms of continuous branching stochastic processes. In this work, path-space probabilistic representations of free-space Poisson-Vlasov and Poisson-Boltzmann systems are exhibited. This yields novel propagator representations and opens new routes for efficient and reference simulations by use of new branching backward Monte Carlo algorithms. Subsequent statistical estimator are benchmarked on gravitational clusters and plasmas dynamics.

From Galactic Clusters to Plasmas in a Single Monte Carlo: Branching Paths Statistics for Poisson-Vlasov/Boltzmann

Abstract

Recent advances have allowed to tackle path-space probabilistic representations of mesoscopic Boltzmann transport nonlinearly coupled to a sub-model of the force-field by step forward approaches in terms of continuous branching stochastic processes. In this work, path-space probabilistic representations of free-space Poisson-Vlasov and Poisson-Boltzmann systems are exhibited. This yields novel propagator representations and opens new routes for efficient and reference simulations by use of new branching backward Monte Carlo algorithms. Subsequent statistical estimator are benchmarked on gravitational clusters and plasmas dynamics.

Paper Structure

This paper contains 19 sections, 44 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. context.
  3. mathematical model.
  4. numerical simulation methods.
  5. need for physical clarity and computational feasability.
  6. Branching Path-space probabilistic representation
  7. Monte Carlo method and statistical estimates
  8. Monte Carlo method.
  9. Distribution function path sampling.
  10. Force field path sampling.
  11. Results and discussions
  12. Unstationary collisional ion-neutral gas.
  13. Plasma relaxation.
  14. conclusion
  15. Analytical solutions for the relaxing plasma configuration.
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Temporal profile of the distribution function at the phase-space probe position $(\mathbf{r}_\text{obs},\mathbf{c}_\text{obs})=(0.01,0.1,0.1,1,10)$. Branching Backward Monte Carlo estimations are computed by use of samples $N=1\times 10^4$ for $\sigma=5\times 10^{-1}$ [m], $\varepsilon_\text{o}=1\times 10^{-3}$ [F.m$^{-1}$], $e=1$ [C], $m=1$ [kg], $\nu_\text{a}=5\times 10^1$ [Hz], $\nu_\text{d}=5\times 10^1$ [Hz], $k_\text{B}=1$, $T=1\times 10^2$ [K], $\delta s=2\times 10^{-3}$ [s] and $\alpha=2$ [m$^{-3}$].
  • Figure 2: Spatial profiles of the distribution function at the phase-space probe position $(\mathbf{r}_\text{obs},\mathbf{c}_\text{obs})=(r_{x,\text{obs}},0.1,0.1,1,10)$. Branching Backward Monte Carlo estimations are computed by use of $N=1\times 10^4$ samples for $\sigma=5\times 10^{-1}$ [m], $\varepsilon_\text{o}=1\times 10^{-3}$ [F.m$^{-1}$], $e=1$ [C], $m=1$ [kg], $\nu_\text{a}=5\times 10^1$ [Hz], $\nu_\text{d}=5\times 10^1$ [Hz], $k_\text{B}=1$, $T=1\times 10^2$, $\alpha=2$ [m$^{-3}$]. Spatial profiles are computed for $t_\text{obs}=1\times 10^{-3}$ [s], $t_\text{obs}=5\times 10^{-3}$ [s] and $t_\text{obs}=1\times 10^{-2}$ [s]
  • Figure 3: Temporal profile of the electron distribution function $f_e$ at the phase-space probe position $(\mathbf{r}_\text{obs},\mathbf{c}_\text{obs})=(1\times 10^{-2},1\times 10^{-2},1\times 10^{-1},1\times 10^{2},1\times 10^{2},1)$. Branching Backward Monte Carlo estimations are computed by use of $N=1\times 10^4$ samples for $\sigma_i=1\times 10^{-4}$ [m], $\rho_\text{o}=1\times 10^{11}$ [m$^{-3}$], $\sigma_e=1\times 10^{-1}$ [m], $T_i=0.05\times1,160\times 10^4$ [K], $T_e=6\times 1,160\times 10^4$ [K], $k_\text{B}=1.380\times 10^{-23}$, $\varepsilon_\text{o}=8,854\times 10^{-12}$ [F.m$^{-1}$], $e=1,602\times 10^{-19}$ [C], $m_i=6,6\times 10^{-26}$ [kg], $m_e=9,109\times 10^{-31}$ [kg], $\delta s=7\times 10^{-7}$ [s], $\nu_\text{d}=1\times 10^5$ [Hz], $\nu_\text{a}=4\times 10^5$ [Hz], $\rho_\infty=1\times 10^{10}$.
  • Figure 4: Spatial profile of the electron distribution function $f_e$ at the phase-space probe position $(\mathbf{r}_\text{obs},\mathbf{c}_\text{obs})=(r_{x,\text{obs}},1\times 10^{-2},1\times 10^{-1},1\times 10^{2},1\times 10^{2},1)$. Branching Backward Monte Carlo estimations are computed by use of $N=1\times 10^4$ samples for $\sigma_i=1\times 10^{-4}$ [m], $\rho_\text{o}=1\times 10^{11}$ [m$^{-3}$], $\sigma_e=1\times 10^{-1}$ [m], $T_i=0.05\times1,160\times 10^4$ [K], $T_e=6\times 1,160\times 10^4$ [K], $k_\text{B}=1.380\times 10^{-23}$, $\varepsilon_\text{o}=8,854\times 10^{-12}$ [F.m$^{-1}$], $e=1,602\times 10^{-19}$ [C], $m_i=6,6\times 10^{-26}$ [kg], $m_e=9,109\times 10^{-31}$ [kg], $\delta s=3\times 10^{-8}$ [s], $\nu_\text{d}=1\times 10^5$ [Hz], $\nu_\text{a}=4\times 10^5$ [Hz], $\rho_\infty=1\times 10^{10}$, and $t_\text{obs}=3\times 10^{-7}$ [s].