1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 133

c. IfU(x) =mω^2 x^2 /2, find the exactly conserved Hamiltonian.

Hint: Assume the exactly conserved Hamiltonian takes the form

H ̃(x,p; ∆t) =a(∆t)p^2 +b(∆t)x^2 ,

and determine a specific choice for the unknown coefficientsaandb.

d. Write a program that implements this algorithm and verify that it exactly

conserves your Hamiltonian for part c and that the true Hamiltonian

remains stable for a suitably chosen small time step.

3.6. A single particle moving in one dimension is subject to a potential ofthe form

U(x) =

1

2

m

(

ω^2 + Ω^2

)

x^2 ,

where Ω≪ω. The forces associated with this potential have two time scales,

Ffast=−mω^2 xandFslow=−mΩ^2 x. Consider integrating this system for

one time step ∆tusing the propagator factorization scheme in eqn. (3.11.6),

whereiLfastis the full Liouville operator for the fast oscillator.

a. The action of the operator exp(iLfast∆t) on the phase space vector (x,p)

can be evaluated analytically as in eqn. (3.11.8). Using this fact, show

that the phase space evolution can be written in the form

(

x(∆t)

p(∆t)

)

= A(ω,Ω,∆t)

(

x(0)

p(0)

)

,

where A(ω,Ω,∆t) is a 2×2 matrix. Derive the explicit form of this matrix.

b. Show that det(A) = 1.

c. Show that, depending on ∆t, the eigenvalues of A are either complex

conjugate pairs such that− 2 <Tr(A) < 2, or both real, such that

|Tr(A)|≥2.

d. Discuss the numerical implication of the choice ∆t=π/ω. This time step

is known as a resonant time step (Schlicket al., 1998; Maet al., 2003)

and indicates that large step in the RESPA algorithm is fundamentally

limited.

3.7. A single particle moving in one dimension is subject to a potential ofthe form

U(x) =

1

2

mω^2 x^2 +

g

4

x^4.

Choosingm= 1,ω= 1,g= 0.1,x(0) = 0, andp(0) = 1, write a program

that implements the RESPA algorithm for this problem. If the small time

stepδtis chosen to be 0.01, how large can the big time step ∆tbe chosen for

accurate integration? Compare the RESPA trajectory to a single time step

trajectory using a very small time step. Use your program to verify that the

RESPA algorithm is globally second order.