1549380323-Statistical Mechanics Theory and Molecular Simulation

184 Canonical ensemble

H ̃N=

(

HN(r,s,p,ps)−H

(0)

N

)

s

=

(N

∑

i=1

p^2 i

2 mis^2

+U(r 1 ,...,rN) +

p^2 s

2 Q

+gkTlns−H(0)N

)

s, (4.8.15)

which is known as theNos ́e–Poincar ́eHamiltonian. The proof that the microcanonical
ensemble in this Hamiltonian is equivalent to a canonical distribution in the physical
HamiltonianH(r,p) follows a procedure similar to that used for the Nos ́e Hamiltonian
and, therefore, will be left as an exercise at the end of the chapter. Note that the
parameterg=dNin this case. Eqn. (4.8.15) generates the following set of equations
of motion:

r ̇i=

pi

mis

p ̇i=−s

∂U

∂ri

s ̇=

sps

Q

p ̇s=

∑N

i=1

p^2 i

mis^2

−gkT−∆HN(r,s,p,ps), (4.8.16)

where

∆HN(r,s,p,ps) =

∑N

i=1

p^2 i

2 mis^2

+U(r 1 ,...,rN) +

p^2 s

2 Q

+gkTlns−H

(0)

N

=HN(r,s,p,ps)−H(0)N. (4.8.17)

Eqns. (4.8.16) possess the correct intrinsic definition of time and can, therefore, be
used directly in a molecular dynamics calculation. Moreover, becausethe equations
of motion are Hamiltonian and, hence, manifestly symplectic, integration algorithms
such as those introduced in Chapter 3, can be employed with minor modifications as
discussed by Bond,et al.(1999).
The disadvantage of adhering to a strictly Hamiltonian structure is that a measure
of flexibility in the design of molecular dynamics algorithms for specific purposes is
lost. In fact, there is no particular reason, apart from the purelymathematical, that a
Hamiltonian structure must be preserved when seeking to develop molecular dynamics
methods whose purpose is to sample an ensemble. Therefore, in theremainder of this
chapter, we will focus on techniques that employ non-Hamiltonian equations of motion.
We will illustrate how the freedom to stray outside the tight Hamiltonian framework
allows a wider variety of algorithms to be created.