  We study, both numerically and analytically, the finite size scaling of the
fidelity susceptibility \chi_{J} with respect to the charge or spin current in
one-dimensional lattice models, and relate it to the low-frequency behavior of
the corresponding conductivity. It is shown that in gapless systems with open
boundary conditions the leading dependence on the system size L stems from the
singular part of the conductivity and is quadratic, with a universal form
\chi_{J}= 7KL^2 \zeta(3)/2\pi^4 where K is the Luttinger liquid parameter. In
contrast to that, for periodic boundary conditions the leading system size
dependence is directly connected with the regular part of the conductivity
(giving alternative possibility to study low frequency behavior of the regular
part of conductivity) and is subquadratic, \chi_{J} \propto L^\gamma(K), (with
a K dependent constant \gamma) in most situations linear, \gamma=1. For open
boundary conditions, we also study another current-related quantity, the
fidelity susceptibility to the lattice tilt \chi_{P} and show that it scales as
the quartic power of the system size, \chi_{P}=31KL^4 \zeta(5)/8 u^2 \pi^6,
where u is the sound velocity. We comment on the behavior of the current
fidelity susceptibility in gapped phases, particularly in the topologically
ordered Haldane state.
