  We discuss in detail the experimental investigation of thermally induced
fluctuations of the relative phase between two weakly coupled Bose-Einstein
condensates. In analogy to superconducting Josephson junctions, the weak
coupling originates from a tunneling process through a potential barrier which
is obtained by trapping the condensates in an optical double-well potential.
The observed fluctuations of the relative phase are in quantitative agreement
with a many body two mode model at finite temperature. The agreement
demonstrates the possibility of using the phase fluctuation measurements in a
bosonic Josephson junction as a primary thermometer. This new method allows for
measuring temperatures far below the critical temperature where standard
methods based on time of flight measurements fail. We employ this new
thermometer to probe the heat capacity of a degenerate Bose gas as a function
of temperature.
