计算物理ComputationalPhysics计算物理 (8).pdf
Computational physiCsMolecular dynamics simulations General behavior of a classical system The Verlet algorithm Structure of atomic clustersMany-body systemsatomclusterprotein moleculea drop of waterSun-Earth-Moongalaxy In quantum mechanics:Hydrogen atom:one electron and one proton Analytical solutions for eigen-energies and eigen-wavefunctions.Helium atom:two electrons and a nucleus No exact analytical solution.A system of a large number of interacting objects is the so-called many-body system.Too many?Statistical mechanics!Many-body systemsGeneral behavior of a classical system The molecular dynamics solves the dynamics of a classical many-body system described by the Hamiltonian.EK:kinetic energy;EP:potential energymi,ri,and pi are the mass,position vector,and momentum of the ith particleV(ri j)and U(ri)are the corresponding interaction energy and external potential energy From Hamiltons principle,the position vector and momentum satisfy for the ith particle in the system.To simplify the notation,R:all the coordinates(r1,r2,.,rN)G:all the accelerations(f1/m1,f2/m2,.,fN/mN).Rewrite Newtons equations:We can also apply the three-point formula to the velocity The Verlet algorithm for a classical many-body system is:with t=kt.The Verlet algorithm can be started if the first two positions R0 and R1 of the particles are given.If only the initial position R0 and initial velocity V0 are given,we need to figure out R1 before we can start the recursion.A common practice is to treat the force during the first time interval 0,t as a constant,and then to apply the kinematic equation to obtain where G0 is the acceleration vector evaluated at the initial configuration R0.The position R1 can be improved by carrying out the Taylor expansion to higher-order terms if the accuracy in the first two points is critical.We can also replace G0 with the average(G0+G1)/2,with G1 evaluated at R1.This procedure can be iterated several times before starting the algorithm for the velocity V1 and the next position R2.Halleys cometEdmond Halley1656-1742Predicted the re-appearance of comet in 1758.Observed in Dec.25,1758前613年，春秋“秋七月，有星孛入于北斗”。前240年，史记始皇本纪“始皇七年，彗星先出东方，见北方；五月见西方，十六日”1910-1986-2061 The gravitational potential:where r is the distance between the comet and the Sun,M and m are the masses of the Sun and comet,respectively.G is the gravitational constant.Henry Cavendish1731-1810 Using the center-of-mass coordinate system for the two-body system,the dynamics of the comet is governed by:with the reduced mass:We can take the farthest point(aphelion)as the starting point,and then we can easily obtain the comets whole orbit with the Verlet algorithm.Two conservations:the total energy and the angular momentum.The motion of the comet in the xy plane:x0=rmax=5.28x1012 m,vx0=0 y0=0,vy0=vmin=9.13x102 m/s.Let us apply the Verlet algorithm to this problem.where the time-step index is given in parentheses as superscripts.Then we have The acceleration components are given by with r2=x2+y2 and k=GM.We can use more specific units in the numerical calculations,for example,76 years as the time unit and the semimajor axis of the orbital a=2.68x1012 m as the length unit.Then we have rmax=1.97,vmin=0.816,and k=39.5.Code example 7.1.Halley.cpp To determine the structure and dynamics of a cluster consisting N atoms that interact with each other through the Lennard-Jones potential where r is the distance between the two atoms,and e and s are the system dependent parameters.Structure of atomic clusters The force exerted on the ith atom is:The Verlet algorithm:We can then simulate the structure and dynamics of the cluster starting from a given initial position and velocity for each particle.N bodies=N x 1 body?Philip Warren Anderson(1923-2020)Nobel Prize in Physics(1977)Phase transition-from solid to liquidTotal energy:Average kinetic energy:=3/2NkBTFor each simulation,we can calculate the (thus temperature).We can tune the temperature to see what will happen.Examples-animations Solid-Liquid-Gas Growth of a cluster Growth dynamics at the droplet-nanowire interface A powerful toolLAMMPS:Large-scale Atomic/Molecular Massively Parallel Simulatorby Sandia National Labortories,USAhttp:/lammps.sandia.gov/Free software,distributed under the terms of the GNU General Public License.Homework Use the molecular dynamics simulation to simulate the one-dimensional lattice vibration and analyze its spectrum.
