Integrating Nos ́e–Hoover chains 197no longer exist in the system, a fact which leads to a simplification of the proof that
the canonical distribution is generated. In fact, it is possible to take this idea one step
further and couple a Nos ́e–Hoover chain toeach Cartesian degree of freedomin the
system, for a total ofdNheat baths. Such a scheme is known colloquially as “massive”
thermostatting and was shown by Tobiaset al.(1993) to lead to very rapid thermal-
ization of a protein in aqueous solution. Such multiple thermostattingconstructs are
not easily achieved within the Hamiltonian framework of the Nos ́e andNos ́e-Poincar ́e
approaches.
4.11 Integrating the Nos ́e–Hoover chain equations
Numerical integrators for non-Hamiltonian systems such as the Nos ́e–Hoover chain
equations can be derived using the Liouville operator formalism developed in Sec-
tion 3.10 (Martynaet al., 1996). However, certain subtleties arise due to the generalized
Liouville theorem in eqn. (4.9.8) and, therefore, the subject meritssome discussion. Re-
call that for a Hamiltonian system, any numerical integration algorithm must preserve
the symplectic property, in which case, it will also conserve the phase space volume.
For non-Hamiltonian systems, there is no clear analog of the symplectic property.
Nevertheless, the existence of a generalized Liouville theorem, eqn. (4.9.8), provides
us with a minimal requirement that numerical solvers for non-Hamiltonian systems
should satisfy, specifically, the preservation of the measure
√
g(x)dx. Integrators that
fail to obey the generalized Liouville theorem cannot be guaranteedto generate correct
distributions. Therefore, in devising numerical solvers for non-Hamiltonian systems,
care must be taken to ensure that they are measure-preserving(Ezra, 2007).
Keeping in mind the generalized Liouville theorem, let us now develop an integrator
for the Nos ́e–Hoover chain equations. Despite the fact that eqns. (4.10.1) are non-
Hamiltonian, they can be expressed as an operator equation just as in the Hamiltonian
case. Indeed, a general non-Hamiltonian system
̇x =ξ(x) (4.11.1)can always be expressed as
̇x =iLx, (4.11.2)
where
iL=ξ(x)·∇x. (4.11.3)
Note that we are considering systems with no explicit time dependence, although the
Liouville operator formalism can be extended to systems with explicit time depen-
dence (Suzuki, 1992). The Liouville operator corresponding to eqns. (4.10.1) can be
written as
iL=iLNHC+iL 1 +iL 2 , (4.11.4)
where
iL 1 =∑N
i=1pi
mi·
∂
∂ri