  The modelling of emission spectra of molecules seen in interstellar clouds
requires the knowledge of collisional rate coefficients. Among the commonly
observed species, N$_2$H$^+$ is of particular interest since it was shown to be
a good probe of the physical conditions of cold molecular clouds. Thus, we have
calculated hyperfine-structure resolved excitation rate coefficients of
N$_2$H$^+$(X$^1\Sigma^+$) by H$_2(j=0)$, the most abundant collisional partner
in the cold interstellar medium. The calculations are based on a new potential
energy surface, obtained from highly correlated {\it ab initio} calculations.
State-to-state rate coefficients between the first hyperfine levels were
calculated, for temperatures ranging from 5 K to 70 K. By comparison with
previously published N$_2$H$^+$-He rate coefficients, we found significant
differences which cannot be reproduced by a simple scaling relationship. As a
first application, we also performed radiative transfer calculations, for
physical conditions typical of cold molecular clouds. We found that the
simulated line intensities significantly increase when using the new H$_2$ rate
coefficients, by comparison with the predictions based on the He rate
coefficients. In particular, we revisited the modelling of the N$_2$H$^+$
emission in the LDN 183 core, using the new collisional data, and found that
all three of the density, gas kinetic temperature and N$_2$H$^+$ abundance had
to be revised.
