1549380323-Statistical Mechanics Theory and Molecular Simulation

474 The Feynman path integral

×e−βφ

(

x(1) 1 ,...,x( 1 P),x(1) 2 ,...,x( 2 P)

)[

perm

(

A ̃

)]

(12.5.7)

for bosons. The matrixA is defined to be ̃ A ̃ij=Aij/Aiiwith

Aij= exp

[

−

1

2

βmω^2 P

(

x(iP)−x(1)j

) 2 ]

, (12.5.8)

and det and perm denote the determinant and permanent of the matrixA ̃, respectively.
In this two-particle example, A andA are 2 ̃ ×2 matrices. ForN-particle systems, they
areN×Nmatrices, giving rise toN! terms from the determinant or permanent.
The presence of the determinant or permanent in eqns. (12.5.6) and (12.5.7), respec-
tively, can be treated as objects to be averaged over direct paths; alternatively, they can
be added as additional terms inφin the form−(1/β) ln det(A) or ̃ −(1/β) ln perm(A). ̃
When exchange effects are important, the fermion case becomes particularly prob-
lematic, as the determinant is composed of the difference of two terms that are large
and similar in magnitude. Hence, the determinant becomes a small difference of two
large numbers, which is very difficult to converge. This problem is known as theFermi
sign problem, which is only exacerbated in a system ofNfermions where det(A) is ̃
the difference of two sums each containingN!/2 terms. Consequently, when det(A) ̃
is absorbed intoφ, it exhibits large fluctuations, which are numerically problematic.
The sign problem does not exist for bosons, sinceA and its permanent are positive ̃
definite, which means that numerical calculations for bosonic systems are tractable.
Techniques for treating bosonic systems are discussed in detail, for example, in the
review by Ceperley (1995). More recently, Boothet al.introduced a novel scheme for
approaching the many-fermion problem (Boothet al., 2009).
Now let us suppose that exchange terms can be safely ignored, which is the case of
Boltzmann statistics discussed in Section 11.3. In this limit, the path integral reduces
to a sum over independent particle paths. Consider the Hamiltonian of anN-particle
system inddimensions of the standard form

Hˆ=

∑N

i=1

ˆp^2 i

2 mi

+U(ˆr 1 ,...,ˆrN). (12.5.9)

As the limit of a discrete path integral, each particle will be characterized by a path in
ddimensions specified by pointsr(1)i ,...,r(iP), and the path integral for the partition
function takes the form

Q(N,V,T) = lim

P→∞

∏N

i=1

(

miP

2 πβ ̄h^2

)dP/ 2 ∫ ∏N

i=1

dr(1)i ···dr(iP)

×exp

{

−

∑P

k=1

[N

∑

i=1

miP

2 β ̄h^2

(

r(ik+1)−r(ik)

) 2

+

β

P

U

(

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

)

]}

r(iP+1)=r(1)i

(12.5.10),

where we now letiindex the particles andkindex the imaginary-time intervals. Eqn.
(12.5.10) can also be written as adN-dimensional functional integral: