  Dynamics of a one-dimensional system of Brownian particles with short-range
repulsive interaction (diameter sigma) is studied with a liquid-theoretical
approach. The mean square displacement, the two-particle displacement
correlation, and the overlap-density-based generalized susceptibility are
calculated analytically by way of the Lagrangian correlation of the
interparticulate space, instead of the Eulerian correlation of density that is
commonly used in the standard mode-coupling theory. In regard to the mean
square displacement, the linear analysis reproduces the established result on
the asymptotic subdiffusive behavior of the system. A finite-time correction is
given by incorporating the effect of entropic nonlinearity with a Lagrangian
version of mode-coupling theory. The notorious difficulty in derivation of the
mode-coupling theory concerning violation of the fluctuation-dissipation
theorem is found to disappear by virtue of the Lagrangian description. The
Lagrangian description also facilitates analytical calculation of four-point
correlations in the space-time, such as the two-particle displacement
correlation. The two-particle displacement correlation, which is asymptotically
self-similar in the space-time, illustrates how the cage effect confines each
particle within a short radius on one hand and creates collective motion of
numerous particles on the other hand. As the time elapses, the correlation
length grows unlimitedly, and the generalized susceptibility based on the
overlap density converges to a finite value which is an increasing function of
the density. The distribution function behind these dynamical four-point
correlations and its extension to three-dimensional cases, respecting the
tensorial character of the two-particle displacement correlation, are also
discussed.
