1549380323-Statistical Mechanics Theory and Molecular Simulation

202 Canonical ensemble

part of the propagator acts on the outside but with the small time step. We de-
note the schemes in eqn. (4.11.20) and (4.11.21) as XO-RESPA (eXtended-system
Outer RESPA) and XI-RESPA (eXtended-system Inner RESPA), respectively (Mar-
tynaet al., 1996).
The next point we address concerns the use of Nos ́e–Hoover chains with holonomic
constraints. Constraints were discussed in Section 1.9 in the context of Lagrangian
mechanics, and numerical procedures for imposing them within a given integration
algorithm were presented in Section 3.9. Recall that the numerical procedure employed
involved the imposition of the constraint conditions

σk(r 1 ,...,rN) = 0 k= 1,...,Nc (4.11.22)

and their first derivatives with respect to time

∑N

i=1

∇iσk·r ̇i=

∑N

i=1

∇iσk·

pi

mi

= 0. (4.11.23)

Note that the time derivatives above are linear in the velocities or momenta. Thus, the
velocities or momenta can be multiplied by any arbitrary constant, and eqn. (4.11.23)
will still be satisfied. Since the factorization in eqn. (4.11.17) only scales the particle
momenta in each application, when all particles involved in a common constraint are
coupled to the same thermostat, their velocities will be scaled in exactly the same way
by the thermostat operators because

exp

[

−

δα

2

pη 1

Q 1

pi·

∂

∂pi

]

pi=piexp

[

−

δα

2

pη 1

Q 1

]

,

which preserves eqn. (4.11.23).

4.12 The isokinetic ensemble: A simple variant of the canonical

ensemble

Extended phase space methods are not unique in their ability to generate canoni-
cal distributions in molecular dynamics calculations. In this section, we will discuss
an alternative approach known as theisokinetic ensemble. As the name implies, the
isokinetic ensemble is one in which the total kinetic energy of a systemis maintained
at a constant value. It is, therefore, described by a partition function of the form

Q(N,V,T,K) =

K 0

N!h^3 N

∫

dNp

∫

D(V)

dNrδ

(N

∑

i=1

p^2 i

2 mi

−K

)

e−βU(r^1 ,...,rN),

(4.12.1)
whereKis preset value of the kinetic energy, andK 0 is an arbitrary constant having
units of energy. Eqn. (4.12.1) indicates that while the momenta are constrained to a
spherical hypersurface of constant kinetic energy, the position-dependent part of the
distribution is canonical. Since this is the most important part of the distribution for
the calculation of equilibrium properties, the fact that the momentum distribution is