We first consider the average particle number 〈 N 〉 . When Δ ( N ) µ = µ − µ ( N ) c is small and L$_{t}$ is large we can write We will demonstrate later that our data fits very well to this s imple one parameter fit and we are able to extract µ ( N ) c very accurately for all L ≤ 20 for a variety of values of β . Next we discuss the number susceptibility κ = L$_{t}$/L $^{2}$〈 N $^{2}$〉 . We now obtain The value of µ ( N ) c is the same as obtained from the average particle number. So this observable has no new free parameters. In the effective quantum mechanical description the superfl uid density is given by where is an operator in the Hilbert space made up of the conserved current operator J$_{i}$ ( x$_{1}$, x$_{2}$ ) in the direction i at the site with coordinates ( x$_{1}$, x$_{2}$ ) . Note the sum is over the surface perpendicular to the direction of the current. Since it is a conserved current it does not matter which surface one chooses. Now if we introduce a complete set of energy eigenstates we get First we note that 〈 n, k | O | n $^{′}$k$^{′}$〉 ∝ δ$_{nn}$ ′ , since the current operator commutes with the particle number operator and hence does not change the particle number. Further, as before we assume 9 ρ$_{s}$ = 1 Z ∑ n,k e − ( E ( n ) k − nµ ) L$_{t}$ ∑ n $^{′}$k ′ |〈 n, k | O | n $^{′}$k$^{′}$〉 | 2 ( 1 − e − ( E ( n ′ ) k ′ − E ( n ) k − ( n $^{′}$− n ) µ ) L$_{t}$ ) E ( n $^{′}$) k ′ − E ( n ) k − ( n ′ − n ) µ (16) O = 1 L ∑ x$_{2}$ J$_{1}$ ( x$_{1}$, x$_{2}$ ) (15) ρ$_{s}$ = 1 Z ∫ L$_{t}$ 0 dt Tr ( e − ( L$_{t}$ − t ) $^{H}$O $^{†}$e− $^{t H}$O ) (14) 〈 N $^{2}$〉 = N 2 + ( N + 1) $^{2}$eΔ ( N ) µ L$_{t}$ 1 + e Δ ( N ) µ L$_{t}$ (13) B. Number Susceptibility C. Superfluid Density A. Particle Number 〈 N 〉 = N + ( N + 1)e Δ ( N ) µ L$_{t}$ 1 + e Δ ( N ) µ L$_{t}$ (12)