Numerical Solution And Its Application Of Linear Langvein Equation

In1908Paul Langevin researched the molecular dynamics theory and devel-oped the particle stochastic motion (Brownian motion) theory which was derivedby Einstein at first, put forward the trajectory equation of a single Brownian par-ticles motion, it was named as Langevin equation. And nowadays the Langevinequation, in mathematics, physics, and biological chemistry had been widely used.The Langevin equation was a kind of important stochastic diferential equations inmathematics. Its numerical algorithm was based on stochastic analysis and numer-ical solution of diferential equation. Langevin equation had been researched anddeveloped in the field of natural science and engineering, but because of the existingof the random variables in the equation, constructing the numerical algorithm ofthe equation, and simulating of the practical problem based on the equation hadcertain difculties.This paper mainly in view of researching numerical algorithm of the Langevinequation, analyzed the structure of the Langevin equation and Verlet speed method,as well as the Monte Carlo method of the Langevin equation. Through studyingthe collision mechanism of the Brownian particle motion, the Langevin equationwas deduced, at the same time researched efectiveness of the Langevin dynamicsequation in the molecular dynamics simulation. Using the computer simulation ofrandom movement of the Brown particles, the feasibility of simulation to Brownianmotion by the Langevin equation was proved. Studied and analyzed applicationof the Langevin equation in the Car Parrinello molecular dynamics (CPMD), andrun instances through the PWSCF software, including CPMD simulation to somecommon molecular, and simulation of the Langevin molecular dynamics under dif-ferent temperature. In molecular dynamics simulation at the high temperature, thesimulation of the Langevin molecular dynamics could get a better numerical results.