Space-resolved transport properties of the thermalizing particle ensemble via Monte Carlo simulations

Understanding the thermalization process of mono-energetic particle beams in gases is fundamental for various applications in plasma physics. A statistical model is introduced and analyzed through Monte Carlo simulations. The simulations are initialized with a delta-function impulse of a non-interacting particle beam colliding with a gas in an infinite domain at a finite temperature. Spatially-resolved profiles of the thermalizing particles, including their average kinetic energy, reveal spatial variations during their evolution. The overall energy balance over time reveals that the local kinetic energy near the center of mass of the thermalizing particles is lower than the thermal energy of the gas, a phenomenon referred to as ‘diffusive cooling’. At the periphery of the particle swarm, the local kinetic energy exceeds the thermal energy, resulting in ‘diffusive heating’. Previous studies have mostly examined these phenomena separately and in confined spaces, such as those observed in the Cavalleri experiment. These effects are explored in an unbounded gas. Calculated quasi-stationary, spatially-resolved profiles in an unbounded gas are compared with stationary profiles observed in confined systems between two infinite planes with perfect absorption. The effective diffusion coefficient, derived from the diffusion equation used in the Cavalleri model, is shown to align with the flux value of the transverse diffusion coefficient predicted by swarm theory. Additionally, it was observed that certain thermalized particles exhibit higher kinetic energy than their initial values at both the front and tail edges of the beam, marking an unexpected transitional phenomenon in the evolution of the beam swarm.

Graphical abstract

The graphical abstracts show two images:

Figure A presents the quasi-stationary, spatially-resolved profile of the ion dissipated power due to elastic collisions, PD(z) in an unlimited space. Within the range from -1σz to +1σz , the local values of PD(z) are negative, while beyond 1σz , PD(z) becomes positive. Here σz represents the standard deviation of the spatial distribution of the ions along their initial velocity direction. Since approximately 68% of thermalizing particles fall within the -1σz to +1σz range of the Gaussian distribution, this indicates that, during thermalization, 68% of the particles experience collisional heating, while 38% of them undergo collisional cooling.

Quasi-stationary spatially-resolved ion dissipated power density and ion number density

Figure B depicts the effective diffusion coefficient, derived from solving the Boltzmann equation that models the Cavalleri diffusion experiment (involving particle diffusion in a gas between two fully absorbing parallel planes). This effective diffusion coefficient is lower than the thermal diffusion values. It essentially represents the flux of the transverse diffusion coefficient. These findings are consistent with the understanding that, in swarm experiments, bulk transport coefficients are measured, whereas the corresponding flux values cannot be directly observed. However, these flux values can be calculated from the distribution function by solving the Boltzmann equation or via direct Monte Carlo simulations.

Time relaxation of the diffusion coefficient, D, in unbounded space, as well as in confined space: bulk D_L^B and flux D_L^F values of the longitudinal diffusion coefficient, together with the corresponding transverse diffusion coefficients D_T^B and D_T^B.