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Register a new userThe Maxwell-Boltzmann Velocity Distribution Function in Detail
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Despite its importance, in the introductory disciplines of exact science courses, the demonstration of the Maxwell-Boltzmann velocity distribution law is not explained, only its final equation is shown. In order to fill this deficiency, in this work we try to show in detail, in a very didactic way, the demonstration of such a law. For this, the kinetic theory of gases is initially introduced. The good agreement of the Maxwell-Boltzmann velocity distribution law with experimental results and its applicability limit is also presented.
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We provide the most stringent constraint to date on possible deviations from the usually-assumed Maxwell-Boltzmann (MB) velocity distribution for nuclei in the Big-Bang plasma. The impact of non-extensive Tsallis statistics on thermonuclear reaction rates involved in standard models of Big-Bang Nucleosynthesis (BBN) has been investigated. We find that the non-extensive parameter $q$ may deviate by, at most, $|delta q|$=6$times$10$^{-4}$ from unity for BBN predictions to be consistent with observed primordial abundances; $q$=1 represents the classical Boltzmann-Gibbs statistics. This constraint arises primarily from the {em super}sensitivity of endothermic rates on the value of $q$, which is found for the first time. As such, the implications of non-extensive statistics in other astrophysical environments should be explored. This may offer new insight into the nucleosynthesis of heavy elements.
Nuclear reactions in stars occur between nuclei in the high-energy tail of the energy distribution and are sensitive to possible deviations from the standard equilibrium thermal-energy distribution, the well-known Maxwell-Boltzmann Distribution (textsf{MBD}). In a previous paper published in Physics Letters 441B(1998)291, DeglInnocenti {it et al}. made strong constrains on such deviations with the detailed helioseismic information of the solar structure. With a small deviation parameterized with a factor exp$[{-delta (E/kT)^2}]$, it was shown $delta$ restricted between -0.005 and +0.002. These constrains have been carefully reexamined in the present work. We find that a normalization factor was missed in the previous modified textsf{MBD}. In this work, the normalization factor $c$ is calculated as a function of $delta$. It shows the factor $c$ is almost unity within the range 0$< delta leq$0.002, which supports the previous conclusion. However, it demonstrates that $delta$ cannot take a negative value from the normalization point of view. As a result, a stronger constraint on $delta$ is defined as 0$leq delta leq$0.002. The astrophysical implication on the solar neutrino fluxes is simply discussed based on a positive $delta$ value of 0.003. The reduction of the $^7$Be and $^8$B neutrino fluxes expected from the modified textsf{MBD} can possibly shed alternative light on the solar neutrino problem. In addition, the resonant reaction rates for the $^{14}$N($p$,$gamma$)$^{15}$O reaction are calculated with a standard textsf{MBD} and a modified textsf{MBD}, respectively. It shows that the rates are quite sensitive even to a very small $delta$. This work demonstrates the importance and necessity of experimental verification or test of the well-known textsf{MBD} at high temperatures.
The current Big-Bang Nucleosynthesis (BBN) model has been constructed based on a nuclear reaction network operating with thermal reactivities of Maxwell-Boltzmann (MB) distribution plasma. However, does the classical MB distribution still hold for the extremely high-temperature (in order of 10$^9$ K) plasma involved in the Big-Bang environment? In this work, we have investigated the impact of non-extensive Tsallis statistics (in $q$-Guassian distribution) on the thermonuclear reaction rates. We show for the first time that the reverse rates are extremely sensitive to the non-extensive $q$ parameter. Such sensitivity does not allow a large deviation of non-extensive distribution from the usual MB distribution. With a newly developed BBN code, the impact of primordial light-element abundances on $q$ values has been studied by utilizing the most recent BBN cosmological parameters and the available nuclear cross-section data. For the first time, we have accurately verified the microscopic MB distribution with the macroscopic BBN theory and bservation. By comparing the recent observed primordial abundances with our predictions, only a tiny deviation of $pm$6$times$10$^{-4}$ at most can be allowed for the MB distribution. However, validity of the classical statistics needs to be studied further for the self-gravitating stars and binaries of high-density environment, with the extreme sensitivity of reverse rate on $q$ found here.
The nature of the velocity distribution of a driven granular gas, though well studied, is unknown as to whether it is universal or not, and if universal what it is. We determine the tails of the steady state velocity distribution of a driven inelastic Maxwell gas, which is a simple model of a granular gas where the rate of collision between particles is independent of the separation as well as the relative velocity. We show that the steady state velocity distribution is non-universal and depends strongly on the nature of driving. The asymptotic behavior of the velocity distribution are shown to be identical to that of a non-interacting model where the collisions between particles are ignored. For diffusive driving, where collisions with the wall are modelled by an additive noise, the tails of the velocity distribution is universal only if the noise distribution decays faster than exponential.
The mechanism of heating for hot, dilute, and turbulent plasmas represents a long-standing problem in space physics, whose implications concern both near-Earth environments and astrophysical systems. In order to explore the possible role of interparticle collisions, simulations of plasma turbulence -- in both collisionless and weakly collisional regimes -- have been compared by adopting Eulerian Hybrid Boltzmann-Maxwell simulations, being proton-proton collisions explicitly introduced through the nonlinear Dougherty operator. Although collisions do not significantly influence the statistical characteristics of the turbulence, they dissipate nonthermal features in the proton distribution function and suppress the enstrophy/entropy cascade in the velocity space, damping the spectral transfer toward large Hermite modes. This enstrophy dissipation is particularly effective in regions where the plasma distribution function is strongly distorted, suggesting that collisional effects are enhanced by fine velocity-space structures. A qualitative connection between the turbulent energy cascade in fluids and the enstrophy cascade in plasmas has been established, opening a new path to the understanding of astrophysical plasma turbulence
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