Max Krook

Bio: Max Krook is an academic researcher from Harvard University. The author has contributed to research in topics: Boltzmann equation & Boltzmann distribution. The author has an hindex of 3, co-authored 3 publications receiving 410 citations.

Exact solutions of the Boltzmann equation

Abstract: The nonlinear Boltzmann equation for the relaxation to equilibrium of a homogeneous one‐component gas, is considered for a class of collision models. The models are characterized by elastic cross sections inversely proportional to the relative speed, but with arbitrary dependence on center‐of‐mass scattering angle. The Boltzmann equation is solved exactly for a particular family of physically interesting initial distributions. The distribution functions are of the similarity form and consist of the product of a Maxwell function with ’’time‐dependent temperature’’ and a linear function of v2.

Formation of Maxwellian Tails

Abstract: Using two models, we study the relaxation to a Maxwell distribution in the context of classical kinetic theory For the first model, an exact solution of the nonlinear Boltzmann equation is derived For the second model, an asymptotic solution exhibits the remarkable feature of a transient tail population sometimes much larger than the equilibrium Maxwell distribution This phenomenon may be of importance for calculating rates of fast chemical reactions and for controlled thermonuclear fusion

Direct Simulation Scheme Derived from the Boltzmann Equation. I. Monocomponent Gases

Abstract: The stochastic law that prescribes the velocities of simulated molecules after a small time increment was derived from the Boltzmann equation. The scheme to determine the velocity of a molecule after a small time increment is divided into three steps. The first step gives the collision probability of the molecule without specifying its collision partner. The second step gives a conditional probability distribution. If the molecule is accepted in the first step as a colliding molecule, its collision partner is sampled from this probability distribution. The last step gives a probability density from which the direction of the relative velocity after collision is sampled, and hence the step gives the post-collision velocity of the molecule. It is shown that the use of the present simulation scheme gives an exact solution of the Boltzmann equation.

Invariant Manifolds for Physical and Chemical Kinetics

Abstract: Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- Newton Method with Incomplete Linearization.- Quasi-chemical Representation.- Hydrodynamics from Grad's Equations: Exact Solutions.- Relaxation Methods.- Method of Invariant Grids.- Method of Natural Projector.- Geometry of Irreversibility: The Film of Nonequilibrium States.- Slow Invariant Manifolds for Open Systems.- Estimation of Dimension of Attractors.- Accuracy Estimation and Post-Processing.- Conclusion.

Monte Carlo simulation of ion motion in drift tubes

Abstract: The motion of a swarm of ions in a uniform electric field is studied by simulating the motion of a single ion through many collisions with neutral atoms in order to obtain the drift velocity, average energy, and velocity distribution for the ions. For K+ ions in He at low field strengths, the results agree well with the solutions of the Boltzmann equation by Kumar and Robson; and for K+ in Ar at all field strengths, the computed mobilities demonstrate that the Viehland–Mason moment method can give useful results, especially if carried through to third order. The velocity distributions computed for O+ ions in He and Ar are used in the accompanying paper by Albritton et al. to analyze drift tube measurements of O+ reaction rates. Significant deviations from the Maxwell–Boltzmann form have been found and are seen to have important effects in that application. Velocity distributions have also been obtained for Li+ in He. The sensitivity of ionic mobilities to changes in the ion–atom interaction potential is examined with particular reference to K+ ions in Ar.