1549380323-Statistical Mechanics Theory and Molecular Simulation

132 Microcanonical ensemble

as completely rigid, with internal bond lengthsdOHanddHH, so that the

constraints are:

|rO−rH 1 |^2 −d^2 OH= 0

|rO−rH 2 |^2 −d^2 OH= 0

|rH 1 −rH 2 |^2 −d^2 HH= 0.

a. Derive the constrained equations of motion for the three atomsin the

molecule in terms of undetermined Lagrange multipliers.

b. Assume that the equations of motion are integrated numerically using the

velocity Verlet algorithm. Derive a 3×3 matrix equation that can be used

to solve for the multipliers in the SHAKE step.

c. Devise an iterative procedure for solving your matrix equation based on

a linearization of the equation.

d. Derive a 3×3 matrix equation that can be used to solve for the multipliers

in RATTLE step. Show that this equation can be solved analytically

without iteration.

3.4. A one-dimensional harmonic oscillator of massmand frequencyωis described

by the Hamiltonian

H=

p^2

2 m

+

1

2

mω^2 x^2.

For the phase space function a(x,p) =p^2 , prove that the microcanonical

ensemble average〈a〉and the time average

̄a=

1

T

∫T

0

dt a(x(t),p(t))

are equal. Here,T= 2π/ωis one period of the motion.

3.5. Consider a single particle moving in one dimension with a Hamiltonian ofthe

formH=p^2 / 2 m+U(x), and consider factorizing the propagator exp(iL∆t)

according to the following Trotter scheme:

exp(iL∆t)≈exp

(

∆t

2

p

m

∂

∂x

)

exp

(

∆tF(x)

∂

∂p

)

exp

(

∆t

2

p

m

∂

∂x

)

.

a. Derive the finite-difference equations determiningx(∆t) andp(∆t) for

this factorization. This algorithm is known as theposition Verletalgo-

rithm (Tuckermanet al., 1992).

b. From the matrix of partial derivatives

J =





∂x(∆t)

∂x(0)

∂x(∆t)

∂p(0)

∂p(∆t)

∂x(0)

∂p(∆t)

∂p(0)



,

show that the algorithm is measure-preserving and symplectic.