1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 209

p ̇η 1 =

p^2

m

−kT

p ̇η 2 =

p^4

3 m^2

−(kT)^2.

a. Show that these equations of motion are non-Hamiltonian.

b. Show that the equations of motion conserve the following energy:

H′=

p^2

2 m

+U(q) +

p^2 η 1

2 Q 1

+

p^2 η 2

2 Q 2

+kT(η 1 +η 2 ).

c. Use the non-Hamiltonian formalism of Section 4.9 to show that these

equations of motion generate the canonical distribution in the physical

HamiltonianH=p^2 / 2 m+U(q).

∗d. These equations of motion are designed to control the fluctuations in

the first two moments of the Maxwell-Boltzmann distributionP(p)∝

exp(−βp^2 / 2 m). A set of equations of motion designed to fix an arbitrary

numberMof these moments is

q ̇=

p

m

p ̇=F(q)−

∑M

n=1

∑n

k=1

pηn

Qn

(kT)n−k

Ck− 1

p^2 k−^1

mk−^1

η ̇n=

[

(kT)n−^1 +

∑n

k=2

(kT)n−k

Ck− 2

(

p^2

m

)k− 1 ]

pηn

Qn

p ̇ηn=

1

Cn− 1

(

p^2

m

)n

−(kT)n,

whereCn=

∏n

k=1(1 + 2k) andC^0 ≡1. These equations were first intro-

duced by Liu and Tuckerman (who also introduced versions of thesefor

N-particle systems) (Liu and Tuckerman, 2000). Show that these equa-

tions conserve the energy

H′=

p^2

2 m

+U(q) +

∑M

n=1

p^2 ηn

2 Qn

+kT

∑M

n=1

ηn

and therefore that they generate a canonical distribution in the Hamilto-

nianH=p^2 / 2 m+U(q).

4.3. a. Consider the Nos ́e–Hoover equations for a single free particle of massm
moving in one spatial dimension. The equations of motion are

p ̇=−

pη

Q

p, η ̇=

pη

Q

, p ̇η=

p^2

m

−kT.