N-body simulations. Phys 750 Lecture 10

Transcription

1 N-body simulations Phys 750 Lecture 10

2 N-body simulations Forward integration in time of strongly coupled classical particles or (rigid-body composite objects) dissipation Brownian motion Very general class of second-order ODE: r = r apple r apple r 5 + ë (2) (r, r )+ <, ë Referred to as N-body simulation or molecular dynamics in various scientific communities arbitrary forces (3) (r, r, r )+5

3 Broad applicability Useful in many contexts and over vastly different scales galaxy formation virus classical water

4 N-body simulations Classical systems of many particles, dissipationless and interacting via central forces, have a Hamiltonian of the form = < ( ) Momentum variables are conjugate to the position variables ( 2DN dimensional phase space) r i p i = m i v i Interactions determined by the pair potential U(r)

5 N-body simulations The set of second-order differential equations maps to a set of coupled first-order equations: Ür i = )H )p i = p i m i = v i Üp i = )H )r i = jëi )U(r ij ) )r i = jëi U (r ij )Çr ij net force exerted by the other N 1 particles

6 N-body simulations Special considerations: For long-range forces all particle pairs interact strongly with one another at best, N(N 1) 2 scaling Repulsive interactions require a container (via walls or periodic boundary conditions) Time step must be adjusted such that the length scale v t can resolve the spatial structure of U(r)

7 Finite simulation cell Typical setup: finite L L L periodic cell serves as a representative sample of space How to judge distance for the L purpose of computing long range forces? L L

8 Finite simulation cell Periodicity is equivalent to tiling all space with copies of the same cell Forces arise from interactions with each of an infinite number of image particles

9 Finite simulation cell Approximating the infinite summation in 1D: F e (r) = 1X n= 1 F (r + nl) = + F (r L)+F (r)+f(r + L)+ F (min{ r,l r }) closest-image distance intra-cell separation L<r<L r

10 Finite simulation cell Summation has an F e (r) exact form for EMlike and gravity-like interactions F (r) = 1 r 2 F (min{ r,l r }) F (r) F e (r) =F (chord(r)) r sin( r/l) /L chord length

11 Finite simulation cell F (chord(r)) closest-image F e (r) approximation increasingly good as interactions become short-ranged; e.g., F (min{ r,l r }) F (r) F (r) = 1 r 7

12 Finite simulation cell r i =(x i,y i,z i ) Example distance measure r j =(x j,y j,z j ) in a 3D orthogonal cell d i, j = min{ Dx,L Dx } 2 +min{ Dy,L Dy } 2 j L +min{ Dz,L Dz } 2 1/2 i Dx = x i Dy = y i x j y j L L Dz = z i z j

13 Lennard-Jones Potential U(r) Model of Van der Waal s r 0 =2 1/6 attraction between neutral atoms plus an inner core repulsion U(r) =4 r 12 6 r Equilibrium point U (r 0 )=0 separates the attractive and repulsive regions

14 Choice of Time Step U(r) 1 two particles at rest at separation distance

15 Choice of Time Step U(r) artificial energy gain particle penetrated well beyond the classical turning point!

16 Adaptive Time Step t/3 t/3 Allow time step to vary dynamically within the simulation t/3 t Check convergence by comparing evolution of two otherwise identical copies of the system

17 Range Truncation If interactions are finite-range, there are only a small number of forces acting on each particle (unless the particles have coalesced) Algorithm now scales as 1 2 N i=1 N (i) nearby N(N 1) 2 Can we impose a cutoff?

18 Range Truncation E.g., Lennard-Jones tail is very weak beyond r c U(r) U (r)

19 Range Truncation Care must be taken not to introduce unphysical forces 10 8 U(r)θ(r c r) U (r)θ(r c r) + U(r)δ(r c r) discontinuous, impulsive force

20 Range Truncation Ũ(r) U(r) U(r c )+(r r c )U (r c ) (r c r) F (r) = Ũ (r) = U (r)+u (r c) (r c r) 10 8 ~ U(r) ~ U (r)

21 Ewald summation Naive truncation is not possible in the case of truly longranged interactions Ewald summation is a useful Fourier space trick E.g., consider non-self electrostatic interactions between particles in all image cells: U Coulomb = 1 2 X i,j X X q i q j (1 ri rj,nl) r i r j + nl n2z 3 0 q i q j 1 2 i,j,n r i,j,n

22 Ewald summation Exact rewriting in terms of a background constant, a direct space sum, and a dual space sum: U 0 = U 1 = X p i q 2 i 0 X erfc(r i,j,n / 0 ) 0q i q j i,j,n U 2 = 1 2 L 3 X i,j q i q j r i,j,n X m2z 3 m6=0 exponentially real lattice decaying envelope exp ( m ) 2 +2 im (r i r j ) m 2 reciprocal lattice

23 Ewald summation Safe to truncate the sum in U 1 at separations above 0 Order sum over particle indices in can be N 2 evaluated as two order U 2 = 1 2 L 3 X m6=0 N exp sums: U 2 ( m ) 2 m 2 S(m) 2 S(m) = X j q j exp(2 im r j ) structure factor

24 Barnes-Hut tree dynamic data structure: recursive subdivision of space into quads containing at most one particle The centre-of-mass position is calculated at each tree level and passed up the linked-list hierarchy

25 Barnes-Hut tree Assume that a particle cluster sufficiently far away can be represented by a single representative particle at the centreof-mass (neglects tidal forces) d l Distance judged by the size of the opening angle l/d < c Scales as N log N

26 Parallel Computation Simulate on many CPUs simultaneously Each machine updates the particles in a sub-region of the full space 8 3 Position information passed as messages passed around the ring

27 Parallel Computation Because of message-passing overhead, the algorithm scales somewhat worse than N(N : Each CPU is numbered cyclically n =0, 1,...,N CPU 1 Loop for each time step: Node n computes forces exerted by its own particles Repeat n 1 times: 1)/2N CPU Node n sends positions to node n + 1 and receives positions from node n 1 Uses received positions to compute forces on its own particles Each node updates its own positions and velocities

28 Neutral territory methods R R R R traditional half-shell neutral territory message passing overhead 1 2 R2 +2R 2R