The PIM expression for the thermal Wigner density

Let us begin by inserting a resolution of the identity in the coordinate basis to rewrite equation 3.3as:

W(q, p) = 1 In the exact same way as we described in the previous Chapter, the matrix elements in the expression above can be represented in path integral form, via a symmetric Trotter break up. In fact, repeating the steps that lead to the expression of an off-diagonal element of the density matrix, we obtain:

and similarly we have:

, and with boundary conditions y0 =x0 =r0,yν =q+ ∆rν/2and yν =q−∆rν/2. For equations 3.5 and 3.6, the expression above becomes exact in the limitν→ ∞(ν here is the number of beads, previously denoted asP). We then introduce the change of variables:

r_λ = x_λ+y_λ

2 (3.7)

∆r_λ =x_λ−y_λ (3.8)

note thatλ = 0,· · · , ν and with the definition above, rν =q and ∆r0 = 0. In the new variables, equation 3.4becomes:

W(q, p) = C_ν² W(r,∆r, q)as the product of exponentials in the curly bracket on lines one,two and three of the equation 3.9. To set the stage for the Monte Carlo scheme we intend to apply, it is convenient to express the Wigner density as an expectation value. To that end, we begin by using the identity:

W(q^∗, p^∗) =

Introducing the δ functions in the expression above is not strictly necessary and most of the considerations that follow could be modified to avoid it. However, writ-ing the Wigner density as in equation3.11allows us to proceed in complete analogy with previous work done with PIM [8, 12] to first introduce the cumulant expan-sion mentioned before and detailed below, and then to construct the Monte Carlo algorithm. This simplifies somewhat the discussion and, perhaps more importantly, makes it possible to rely on the performance of the sampling algorithm introduced in those references when discussing the scaling properties of the method. To set the stage for our developments, note thatW(r,∆r, q) is a positive definite quantity which, barring a normalization factorN, can be interpreted as the joint probability density for all coordinate variables {r,∆r, q}. (At this stage, it is in particular the inclusion of q in the set of variables that simplifies manipulations.) Let us indicate this density as:

ρ(r,∆r) =W(r,∆r)/N (3.12)

with a supplementary simplification due to the fact that q = rν, we have now We can express the density above in terms of marginal ρm(r) and conditional ρ_c(∆r|r) probabilities defined as::

ρ_m(r) =

Using these notations we can rewrite our Wigner density as:

W(q^∗, p^∗) = NC_ν² Let us now consider more in detail the term in square brackets in the expression above. Its logarithm is, by definition, the generating function of the random variable

∆r_ν with respect to ρ_c(∆r|r) [8]. The integral over the ∆r variables can then be formally expressed as:

Z

d∆r e^~ip∆rνρc(∆r|r) =e^−E(p,r) (3.17)

with

E(p,r) =−

∞

X

n=1

(−ip/~)ⁿ

n! h∆rⁿ_νi^c_ρ

c(∆r|r) (3.18)

where h∆r_νⁿi^c_ρ

c(∆r|r) is the nth cumulant. Since, as it can be seen from the defi-nition of ρ(r,∆r), ρ_c(∆r|r) =ρ_c(−∆r|r), only even order terms in equation 3.18 are non zero. Thus, when the cumulant series converges, E(p,r) is a real function and the exponential in equation 3.17is positive definite. This implies, see equation 3.19 below, that also the Wigner density is positive definite. It is however known that there are systems (e.g. double well potential at low temperature) that have a non positive definite Wigner density. For such systems, the writing above fails because the cumulant series does not converge and our method cannot be applied.

Importantly however, this convergence can always be tested numerically (an ana-lytical assessment of the convergence of the cumulant series, on the other hand, is non trivial for generic potentials) ensuring control of the suitability and accuracy of the method. In the following subsection we discuss an alternative scheme that is applicable also for non positive definite Wigner densities. Substituting the cumulant form of the∆r integral in equation 3.16, we obtain:

W(q^∗, p^∗) = NC_ν² 2π~Z

Z dp

Z

dr ρm(r)e^−E(p,r)δ(rν−q^∗)δ(p−p^∗) (3.19)

By reverse engineering the steps in equations 3.9-3.17it is not difficult to see that:

NC_ν² 2π~Z

Z dp

Z

dr ρ_m(r)e^−E(p,r)= 1 (3.20) thus:

W(q^∗, p^∗) =hδ(q−q^∗)δ(p−p^∗)iP (3.21) where the average is over the following quantity:

P(r, p) = NC_ν²

2π~Zρ_m(r)e^−E(p,r) (3.22) P(r, p) = ρ_m(r)e^−E(p,r)

R dpR

drρ_m(r)e^−E(p,r) (3.23) Due to the fact that E(p,r) is a real function and the exponential in equation 3.17 is positive definite, P(r, p) fulfills all the conditions to be defined as a probability density.

We will see in the next section how to deal with such a probability density from an algorithmic point of view.

Equation 3.19 can be most effectively read as the definition of an histogram and calculated by generating a set of values of (r_ν, p) distributed according to the probability density introduced above. This probability density is, however, unusual since it contains two factors (ρm(r) and e^−E(p,r)) that are, in general, not known analytically. This poses a problem for the numerical sampling of P for example via the standard Monte Carlo schemes such as the standard Metropolis scheme [64]. Indeed, in the Metropolis algorithm, the probability density needs to be known analytically in order to sample it. Notably, in general this probability density does not factor in a term depending only on coordinates and one depending only on momenta indicating that, in contrast to the classical distribution but also to other available approximate approaches, our expression of the Wigner density can account for correlation among these degrees of freedom. Perhaps even more strikingly compared to the classical case, since for more than one degree of freedom the cumulant expansion allows for coupling among different momenta, this distribution can also manifest correlation among the momenta. We shall see in the Results section that these features can be physically relevant.