1549380323-Statistical Mechanics Theory and Molecular Simulation

Numerical evaluation 487

(^01) ́ 105 2 ́ 105
Steps
-30
-15
0
15
30
eprim
(^01) ́ 105 2 ́ 105
Steps
-30
-15
0
15
30
eprim
(^01) ́ 105 2 ́ 105
Steps
-30
-15
0
15
30
eprim
(^01) ́ 105 2 ́ 105
Steps
-30
-15
0
15
30
evir
(^01) ́ 105 2 ́ 105
Steps
-30
-15
0
15
30
evir
(^01) ́ 105 2 ́ 105
Steps
-30
-15
0
15
30
evir
P= (^32) P= (^64) P= 128
P= 32 P= 64 P= 128
Fig. 12.12Instantaneous fluctuations (grey) and cumulative averages(black) of the primi-
tive (top row) and virial (bottom row) energy estimators fora harmonic oscillator simulated
with staging path-integral molecular dynamics usingP= 32 (left column),P= 64 (middle
column), andP= 128 (right column) beads.
We now proceed to prove the theorem. First, we define a functionα(x 1 ,...,xP) as
α(x 1 ,...,xP) =

1

2

mω^2 P

∑P

k=1

(xk−xk+1)^2. (12.6.35)

Note that the effective potential in eqn. (12.2.28) can now be written as

φ(x 1 ,...,xP) =α(x 1 ,...,xP) +

1

P

∑P

k=1

U(xk)

≡α(x 1 ,...,xP) +γ(x 1 ,....,xP). (12.6.36)

Recalling the discussion of Euler’s theorem in Section 6.2, the functionα(x 1 ,...,xP)
is a homogeneous function of degree 2. Hence, applying Euler’s theorem, we can write
α(x 1 ,...,xP) as

α(x 1 ,...,xP) =

1

2

∑P

k=1

xk

∂α

∂xk

. (12.6.37)

Now consider the average〈α〉fover the finite-P path-integral distributionfof eqn.
(12.3.9), which we can write as