free energy variation with twist by, ${\cal F}(\Delta_\mu)\equiv -T\ln P(\Delta_\mu)+{\rm constant}$. The helicity modulus $\Upsilon_\mu$ in direction $\hat \mu$ is then defined in terms of the curvature of ${\cal F}(\Delta_\mu)$ at its minimum $\Delta_{\mu 0}$, $$\Upsilon_\mu\equiv {L_\mu\over L_\nu L_\sigma}\left.{\partial^2{\cal F}\over\partial\Delta_\mu^2}\right|_{\Delta_\mu=\Delta_{\mu 0}}\enspace,
\label{eUps}$$ where $\mu,\nu,\sigma$ are a permutation of $x,y,z$. Note, unlike pure systems, for random systems it is not generally true that $\Delta_{\mu 0}=0$. When ${\cal F}(\Delta_\mu)$ varies with $\Delta_\mu$, and so the system is sensitive to the boundary conditions, we have $\Upsilon_\mu>0$ and the system possesses superconducting phase coherence. When ${\cal F}(\Delta_\mu)$ is flat and independent of $\Delta_\mu$, $\Upsilon_\mu=0$ and superconducting phase coherence is lost. The vanishing of $\Upsilon_\mu$ thus is a signature of the superconducting transition.

At a second order phase transition $T_{\rm c}$, the temperature and system size dependence of $\Upsilon_\mu$ is expected to obey a critical scaling law. Since the applied magnetic field singles out a special direction, there is the possibility that this scaling may be [*anisotropic*]{}. In such a case, the expected scaling law for $\Upsilon_\mu(T,L,L_z)$ is, $${L_\nu L_\sigma\over L_\mu}\Upsilon_\mu (T,L,L_z) = u_\mu(tL^{1/\nu},L_z/L^\zeta)\enspace,$$ where $t\equiv T-T_{\rm c}$, $u$ is the scaling function, $\nu$ is the correlation length critical exponent, and $\zeta$ is the anisotropy 