  We present a new exact method to numerically compute the thermodynamical
properties of an interacting Bose gas in the canonical ensemble. As in our
previous paper (Phys. Rev. A, 63 023606 (2001)), we write the density operator
$\rho$ as an average of Hartree dyadics $\ketbra{N:\phi_1}{N:\phi_2}$ and we
find stochastic evolution equations for the wave functions $\phi_{1,2}$ such
that the exact imaginary-time evolution of $\rho$ is recovered after average
over noise. In this way, the thermal equilibrium density operator can be
obtained for any temperature $T$. The method is then applied to study the
thermodynamical properties of a homogeneous one-dimensional $N$-boson system:
although Bose-Einstein condensation can not occur in the thermodynamical limit,
a macroscopic occupation of the lowest mode of a finite system is observed at
sufficiently low temperatures. If $k_B T \gg \mu$, the main effect of
interactions is to suppress density fluctuations and to reduce their
correlation length. Different effects such as a spatial antibunching of the
atoms are predicted for the opposite $k_B T\leq \mu$ regime. Our exact
stochastic calculations have been compared to existing approximate theories.
