The previous expression for the thermal density matrix cannot be evaluated analyti- cally for a generic form of the potential and, in practical applications, it is necessary to go back to the discretized form (2.55). However, in this case the error due to the primitive factorization remains finite and the number M of slices must be large enough in order to obtain an accurate representation of the density matrix. It is however possible to go beyond the primitive approximation and obtain an expression for the density matrix that converges more rapidly with M . In this section we will introduce a possible choice of approximated expression for the discretized density matrix.
Let us first introduce a notation that will be useful to this aim. In absence of interactions among the particles the continuous path density matrix (2.73) reduces to ρ0(R, R0; β) = Z R(0)=R R(β)=R0 DR exp " − Z β 0 " | ˙R|2 4λ # dτ # (2.74) that represents the thermal density matrix of an ideal gas. For classical systems, it is common to distinguish into ideal and excess contributions to the thermodynamic quantities: in quantum systems this distinction is no more valid. However, we can still introduce an “excess” density matrix ρex(R, R0; β), by factorizing the total
density matrix as follows ρ(R, R0; β) = ρ0(R, R 0; β) ρ0(R, R0; β) ρ(R, R0; β) = ρ0(R, R0; β)ρex(R, R0; β) (2.75) where ρex(R, R0; β) = ρ(R, R0; β) ρ0(R, R0; β) = R R(0)=R R(β)=R0 DR e− Rβ 0 » | ˙R|2 4λ +V(R) – dτ R R(0)=R R(β)=R0 DR e−R0β | ˙R|2 4λ dτ (2.76)
Unlike the classical case, our definition of the excess density matrix contains terms arising from the kinetic energy. However, it is possible to factorize the integrand in the numerator of (2.76) into the product exp[−Rβ
0 | ˙R|2/(4λ)dτ ]exp[− Rβ 0 V(R)dτ ] and to obtain ρex(R, R0; β) = Z R(0)=R R(β)=R0 DR w(R, R0; β)e− Rβ 0 V(R)dτ = D e− Rβ 0 V(R)dτ E FP (2.77)
where the ratio w(R, R0; β) = exp[−R0β| ˙R|2/(4λ)dτ ]/ρ0(R, R0; β) can be interpreted
as the probability associated to a free particle path starting in R and ending in R0, and we introduced the symbol h· · · iFP, meaning an average over the free particle paths with fixed end points. Expression (2.77) is still exact, since we have simply recast in a different way terms arising from the continuous representation of the density matrix.
However, we can use (2.77) to derive for the density matrix of systems with pair interactions (described through the potential v(r)) an approximation accurate and
fast to compute: the pair density matrix approximation [64, 75].
Suppose that the total potential energy of the system under consideration can be written as the sum of pairwise terms
V(R) =X
i<j
v(Rij) (2.78)
where Rij = Ri− Rj are here three dimensional vectors, representing the vectorial
distance between the particle i and the particle j. Formula (2.77) in this case becomes ρex(R, R0; β) = D e− Rβ 0 dτ V(R(τ )) E FP= * Y i<j exp − Z β 0 dτ v(Rij(τ )) + FP = * Y i<j eij + FP (2.79)
having defined eij = exp
h −Rβ
0 dτ v(Rij(τ ))
i
. At low temperature, since the paths are spread in the configuration space, many body contributions cannot be neglected and the factors eij cannot be considered independent from each other. However
as the temperature is increased the paths shrink and we can assume that the eij
becomes less correlated4. Under this assumption we can invert the product and the average in (2.79), obtaining ρex(R, R0; β) ≈ Y i<j exp − Z β 0 dτ v(Rij(τ )) FPij (2.80)
in which each factor represents the excess density matrix of a pair of particles, exact since it is just the continuous representation of the density matrix written for a system of two particles. The two body density matrix can be computed without introducing further approximations, apart to solve numerical issues. A technique used to obtain the pair density matrix for a pair of particles, is the squaring method, developed by Storer [76] and described in detail in Ref. [64].
To conclude, let us compare now the pair density matrix with the primitive expres- sion. At very high temperature, as β → 0, we can approximate the τ integral in (2.80) as the product of β times the average of the potential: −β(v(Rij(0)) + v(Rij(β)))/2,
recovering the primitive approximation (2.67). At finite temperature, this form of the density matrix is instead more accurate than the primitive approximation, since it contains, on average, informations about the interactions along the entire path and not only at the extremes.
2.4.3 Path sampling
Once obtained an explicit expression for the thermal density matrix, we need to define an algorithm to sample the path configuration space. A simple approach to
4being the value of τ determined by convergence issues, as the temperature increases, β decreases
and so the number of slices of which the paths are made. As a consequence the average distance among paths of different particles increases and the correlations among different beads are reduced.
generate new paths is the Lévy reconstruction method [77], a recursive procedure to build a Brownian path of the desired number of slices between two fixed end points. Let us assume that R(t = 0) = R0 and R(t = β) = RM are fixed and that
M is an even integer, M = 2n. Let us call τ the time length of each slice, such that β = M τ = 2nτ .
The first step of the Lévy algorithm reside in the random sampling of the mid point RM/2 as
RM/2 = R
0+ Rβ
2 + ηβ (2.81)
where ηβ is a gaussian random number of zero mean and variance σβ =
√ λβ. Together with the fixed end points, RM/2 will define two intervals, [R0, RM/2] and
[RM/2, RM], of time lenght β/2. The next step of the Lévy algorithm consist in the
application of (2.81) to each one of these new intervals, adjusting the variance of the random number to the new interval time length β/2, to obtain their middle points. The resulting two new points are given by
RM/4 =
R0+ RM/2
2 + ηβ/2 and R3M/4=
RM/2+ RM
2 + ηβ/2 Generalizing, at the lthstep, 2l−1 new points are sampled from a gaussian of suitable variance. The procedure must be repeated n times, until the desired number of slices is reached, as illustrated in Figure 2.2.
For free particles the Lévy construction gives the exact sampling, since it samples the coordinates from a gaussian distribution of right variance. To prove that, we can apply the Lévy construction to a chain of M = 3 beads. In this case, two edges R0 and R2 are fixed and one must sample only the central point R1. For non
interacting particles the correct path probability associated to the point R1 is given
by the product of two gaussians
Π(R1|R0, R2) ∝ e−
(R1−R0)2
4λτ e−(R2−R1) 2
4λτ (2.82)
that we can expand separating the part that depends on R1
Π(R1|R0, R2) ∝ e− (2R21−2R0R1−2R2R1) 4λτ e− (R22+R21) 4λτ = e− (R22+R20−2R2)¯ 4λτ e− (R1− ¯R)2 2λτ = Ae−(R1− ¯2λτR)2 (2.83)
where A = exp[−(R22+ R20− 2 ¯R2)/(4λτ )] and we have introduced the symbol ¯R to indicate the midpoint of the path: ¯R = (R1+ R2)/2. The distribution of the point
R1 is then a gaussian centered in ¯R with variance η =
√
λτ , as assumed in the Lévy construction.
In the presence of interaction this is no longer true and, in order to correct the sampling accounting of the particle interactions, it is necessary to add an acceptance test at the end of the path construction. However, if the path displacements are too large, frequent rejections of the attempted moves can occur and needlessly waste computational resources. The sampling can be performed more efficiently by adopting other algorithms. We choose the bisection algorithm [78], which can be viewed as a minor modification of the Lévy method. The difference consists in
applying the acceptance test at each step of the path reconstruction (i.e. each time that new points of the path are added) rather than at the end of the procedure. Only if the new positions are accepted the path reconstruction proceed to the next step, otherwise the procedure is repeated from the beginning, discarding also the steps previously accepted.
The advantage of the bisection method with respect to the Lévy reconstruction relies on the fact that the application at each step of the acceptance test can help in discarding in advance the less favorable move, without waiting that the building procedure has arrived at the end, saving in this way large amounts of computer time. The reason may be better understood looking at Figure 2.2. At the lth step of the Lévy reconstruction 2l−1 beads sampled together and than 2l−1 density
matrices must be computed at once. The major part of the computational effort required by the algorithm is then spent in the more advanced steps of the algorithm. However, for those steps the intervals on which one applies the Lévy rule (2.81) have small time length and the new points cannot be too far from the midpoints of the respective intervals: the new positions have then high probability to be accepted. On the other hand, in the first step of the algorithm a single coordinate must be sampled. The trial move has the largest variance, ητ =
√
λτ , and is hence more likely to be source of rejection of the proposed path; on the other hand, it is also the cheapest from the point of view of the computational resources required, since it involves the calculation of only one density matrix. As a consequence if this move is discarded since the beginning the waste of computer time is quite limited, while if it is accepted, a noticeable jump forward in the path sampling is achieved. In order to guarantee that the detailed balance condition is overall satisfied, a modification of the acceptance test, accounting for the previous rejections, is required [78].
Figure 2.2. Illustration of the Lévy construction of the path, for M = 8 slices. The three
construction steps are showed from top to bottom. To begin, all the links of the old path are deleted, leaving only the endpoints R0 and R8unchanged. Then the construction starts:
first, the point R4 is sampled according to (2.81), then other two points, R2 and R6, are
obtained. Finally, with the sampling of the last four points (R1, R3, R5 and R7) the path
construction is completed. In the bisection method the path building follows the same steps, apart for the fact that an acceptance test is applied each time a new set of points is sampled, rather than only at the end of the construction.