  We construct a time-dependent solution in vacuum string field theory and
investigate whether the solution can be regarded as a rolling tachyon solution.
First, compactifying one space direction on a circle of radius R, we construct
a space-dependent solution given as an infinite number of *-products of a
string field with center-of-mass momentum dependence of the form e^{-b p^2/4}.
Our time-dependent solution is obtained by an inverse Wick rotation of the
compactified space direction. We focus on one particular component field of the
solution, which takes the form of the partition function of a Coulomb system on
a circle with temperature R^2. Analyzing this component field both analytically
and numerically using Monte Carlo simulation, we find that the parameter b in
the solution must be set equal to zero for the solution to approach a finite
value in the large time limit x^0\to\infty. We also explore the possibility
that the self-dual radius R=\sqrt{\alpha'} is a phase transition point of our
Coulomb system.
