1549380323-Statistical Mechanics Theory and Molecular Simulation

480 The Feynman path integral

u ̇k=

pk

m′k

p ̇k=−mkω^2 Puk−

1

P

∂U

∂uk

−

pηk, 1

Q 1

pk

η ̇k,γ=

pηk,γ

Qk

p ̇ηk, 1 =

p^2 k

m′k

−kT−

pηk, 2

Qk

pηk, 1

p ̇ηk,γ=

[

p^2 ηk,γ− 1

Qk

−kT

]

−

pηk,γ+1

Qk

pηk,γ γ= 2,...,M− 1

p ̇ηk,M=

[

p^2 ηk,M− 1

Qk

−kT

]

, (12.6.14)

whereγindexes the thermostat chain elements. WhenωP is the highest frequency
in the system, the optimal choice for the parametersQ 1 ,...,QPareQ 1 =kTτ^2 and
Qk=kT/ω^2 Pfork= 2,...,P. Hereτis a characteristic time scale of the corresponding
classical system. Since each staging variable has its own thermostat of lengthM, the
dimensionality of the thermostat phase space is 2MP, which is considerably larger
than the physical phase space! Luckily, with the exception of simple“toy” problems,
the computational overhead of “massive” thermostatting is low relative to that of a
force calculation in a complex system. Moreover, the massive thermostatting method is
rapidly convergent, particularly when integrated using a multiple timescale algorithm
such as the RESPA method of Section 3.11. The staging transformation is simple to
implement and because of the recursive relations in eqns. (12.6.6) and (12.6.13), it
scales linearly withP.
As an interesting alternative to the staging transformation, it is also possible to use
the normal modes of the cyclic chain (Tuckermanet al., 1993; Cao and Voth, 1994b).
The normal mode transformation can be derived straightforwardly from a Fourier
expansion of the periodic path

xk=

∑P

l=1

ale^2 πi(k−1)(l−1)/P. (12.6.15)

The complex expansion coefficientsalare then used to construct a transformation to
a set of normal mode variablesu 1 ,...,uPvia

u 1 =a 1 , uP=a(P+2)/ 2

u 2 k− 2 = Re(ak), u 2 k− 1 = Im(ak). (12.6.16)

The normal-mode transformation can also be constructed as follows: 1) Generate the
matrixAij= 2δij−δi,j− 1 −δi,j+1,i,j= 1,...,P, with the path periodicity condi-
tionsAi 0 =AiP,Ai,P+1=Ai 1 ; 2) Diagonalize the matrix and save the eigenvalues