Page 181 -

168 Density matrices and path integrals

particles:

ρ N-part ({x 1 ,... , x N }, {x ,..., x }, ∆ τ )

1
N
;
N pair

: free : ρ ({x k , x l }, {x , x }, ∆ τ )
l
k

ρ (x k , x , ∆ τ ) . (3.59)
k
ρ free (x k , x , ∆ τ ) ρ free (x l , x , ∆ τ )

k=1 k<l k l

prob. that paths k and l do not collide
Fortwo particles, this is the same as eqn (3.56), and it is exact.For
N particles, eqn (3.59) remains correct under the conditionthat wecan
treat the collisionprobabilities forany pair ofparticles as independent of
those for other pairs.This condition was already discussed in the context
ofthe virial expansionforclassical hard spheres (see Subsection2.2.2).
Itis justified at low densityorathigh temperature.In the first case (low
density) paths rarely collide, so that the paths interfere verylittle.Inthe
second case (∆ τ corresponding to high temperature), the path ofparticle
k does notmoveaway far fromthe position x k x , and the interference

k
of paths is again limited.Because ofthe relation ∆ τ = β/N, wecan
always find an appropriate value of N for which the N-density matrix in
eqn (3.59) is essentially exact.The representation of the density matrix
in eqn (3.59) combines elements ofa high-temperature expansionand
ofa low-density expansion.Itis sometimes called a Wigner–Kirkwood
expansion.
Inall practical cases, the values of N that must be used are much
smaller than the number oftime slices needed in the naiveapproach of
Subsection 3.4.1 (see Pollock and Ceperley(1984), Krauth (1996)).
3.5 Geometry of paths

Quantum statistical mechanics can be formulated in terms ofrandom
paths in space and imaginary time.This is the path-integral approach
that westarted to discuss in Sections 3.3 and 3.4, and for which we
havebarely scratched the surface. Rather than continue with quantum
statistics as it is shaped by path integrals, weanalyze in this sectionthe
shapes of the paths themselves.This will lead usto new sampling al-
gorithms using Fourier transformationmethods.The geometryofpaths
also provides an example of the profound connections between classical
statistical mechanics and quantumphysics, because randompaths do
notappear in quantumphysics alone.They can describe cracks in ho-
mogeneousmedia (such as a wall), interfaces between different media
(such as between air and oil in a suspension)orelse between different
phases of the same medium (such as the regions of a magnet with differ-
ent magnetizations). These interfaces are often very rough.They then
resemble the paths of quantumphysics, and can be described byvery
similar methods.This will be the subject of Subsection 3.5.3.