  We develop a highly efficient method to numerically simulate thermal
fluctuations and correlations in non-relativistic continuous bosonic
one-dimensional systems. The method is suitable for arbitrary local
interactions as long as the system remains dynamically stable. We start by
proving the equivalence of describing the systems through the transfer matrix
formalism and a Fokker-Planck equation for a distribution evolving in space.
The Fokker-Planck equation is known to be equivalent to a stochastic
differential (It\={o}) equation. The latter is very suitable for computer
simulations, allowing the calculation of any desired correlation function. As
an illustration, we apply our method to the case of two tunnel-coupled
quasi-condensates of bosonic atoms. The results are compared to the predictions
of the sine-Gordon model for which we develop analytic expressions directly
from the transfer matrix formalism.
