Bayesian derivation of plasma equilibrium distribution function for tokamak scenarios and the associated Landau collision operator

Abstract

A class of parametric distribution functions has been proposed in [C. DiTroia, Plasma Physics and Controlled Fusion, 54, (2012)] as equilibrium distribution functions (EDFs) for charged particles in fusion plasmas, representing supra-thermal particles in anisotropic equilibria for Neutral Beam Injection, Ion Cyclotron Heating scenarios. Moreover, the EDFs can also represent nearly isotropic equilibria for Slowing-Down alpha particles and core thermal plasma populations. These EDFs depend on constants of motion (COMs). Assuming an axisymmetric system with no equilibrium electric field, the EDF depends on the toroidal canonical momentum P φ , the kinetic energy w and the magnetic moment µ. In the present work, the EDFs are obtained from first principles and general hypothesis. The derivation is probabilistic and makes use of the Bayes' Theorem. The bayesian argument allows us to describe how far from the prior probability distribution function (pdf), e.g. Maxwellian, the plasma is, based on the information obtained from magnetic moment and GC velocity pdf. Once the general functional form of the EDF has been settled, it is shown how to associate a Landau collision operator and a Fokker-Planck equation that ensures the system relaxation towards the proposed EDF.

Toroidal Alfvén eigenmode stability in JET internal transport barrier afterglow experiments

“…In order to de-couple numerical convergence of heating codes from stability calculations while retaining finite orbitwidth and anisotropic effects, physically motivated parametric distribution functions derived in [35,36] were adopted…”

Non-Perturbative Guiding Center and Stochastic Gyrocenter Transformations: Gyro-Phase Is the <i>Kaluza-Klein</i> 5<sup>th</sup> Dimension also for Reconciling General Relativity with Quantum Mechanics

“…if λ = −ψ p /F . The latter equation, that can be written as an eigenvalue equation for the Shafranov operator, was already obtained but wrongly written in [6] (see [18] for details).…”

“…In practice,R/G = R/Ĝ, as if we are considering flat the space described by the canonical coordinates ε and µ. It is worth noticing that although in five dimensions, all the quantities can depend also on ε and µ, e.g the distribution function f m is always the distribution of masses in the whole extended phase space and it surely depends on ε and/or µ if it describes an equilibrium [18]. Even if the action is the same, now |ĝ| should be decomposed into |ĝ| = |g|J P , where |g| is the square root of the absolute value of the determinant of the metric tensorg ab , andJ P is the jacobian, not specified here, for measuring the density of states for assigned ε and µ.…”

Toroidal Alfvén eigenmodes observed in low power JET deuterium–tritium plasmas

“…Fits of the Monte Carlo generated distribution function itself (e.g. [5]) retain finite orbit and anisotropic effects. However, this method smooths over important features of the distribution function to mitigate the impact of noise on the stability calculation [6].…”