1549380323-Statistical Mechanics Theory and Molecular Simulation

Numerical evaluation 489

εvir

εvir

εvir

<ε >

vir

<ε >

vir

<ε >

vir

σ

σ

σ

Steps

Steps

Steps

Steps

Steps

Steps

Block size

Block size

Block size

Fig. 12.13Left column: Instantaneous virial estimator. Middle column: Cumulative average
of virial estimator. Right column: Error bar as a function ofblock size. Top row: Path-integral
molecular dynamics with no variable transformations. Middle row: Path-integral molecular
dynamics with staging transformation. Bottom row: Stagingpath-integral Monte Carlo with
j= 80. All energies are in units of ̄hω(reprinted with permission from Tuckermanet al. J.
Chem. Phys. 99 , 2796 (1993), copyright, American Institute of Physics).

and forNparticles inddimensions, the generalization of the virial estimator is

εvir({r(1),...,r(P)}) =

dNkT

2

+

1

P

∑P

k=1

∑N

i=1

1

2

(

r(ik)−r(ic)

)

·

∂U

∂r(ik)

+

1

P

∑P

k=1

U

(

r( 1 k),...,r(Nk)

)

, (12.6.41)

wherer(ic)is the centroid of particlei. Similarly, by applying the path-integral virial
theorem to the pressure estimator in eqn. (12.5.12), one can derive a virial pressure
estimator