Finite size scaling

The relatively small number of particles in the field of view has other ramifications. Because the various configurational temperature definitions involve different approximations of order $1/N$, we might expect their results to differ from each other and from the actual thermodynamic temperature accordingly. Imaging a substantially larger region of the sample is not feasible. On the other hand, deliberately sub-sampling the field of view allows us to probe the dependence on sample size, for which we can extrapolate the configurational temperatures to the thermodynamic limit.

The data in Fig. 6 were obtained by calculating the configurational temperature with the $640 \times 480~\unit{pixel}^2$ field of view reduced by borders of 20, 40, 60, 80, 100, 120, 140, 160 and 180 pixels. The interaction range in this system is $R = 2\sigma = 14.9~\unit{pixel}$. Reducing the number of particles in this manner substantially changes the estimated temperatures, thereby confirming the importance of finite-size scaling. We fit these results to polynomials in $1/N$ with statistical weighting estimated from the area and the interaction range. These weighted fits describe the data very well, so that the extrapolation to $1/N=0$ should yield meaningful estimates for the configurational temperatures in the thermodynamic limit. Indeed, the extrapolated results, $T_{conF} = 0.965$, $T_{con1} = 0.956$ and $T_{con2} = 0.976$, agree quite well with each other, and all appear to be consistent with unity. Confirming thermodynamic consistency, however, requires us to assess the configurational temperatures' sensitivity to errors in $u(r)$.

Figure 6: Finite size scaling of $T_{conF}$, $T_{con1}$ and $T_{con2}$ for the data in Fig. 5, with area-weighted fits to second-, second- and third-order polynomials, respectively.