  In this paper, we investigate the dynamics of the Bloch vector of a single
two-level atom which interacts with a single quantized electromagnetic field
mode according to the Jaynes-Cummings model, where the field is initially
prepared in a thermal state. The time evolution of the Bloch vector S(t) seems
to be in complete disorder because of the thermal distribution of the initial
state of the field. Both the norm and the direction of S(t) oscillate hard and
their periods seem infinite. We observe that the trajectory of the time
evolution of S(t) in the two- or three-dimensional space does not form a closed
path. To remove the fast frequency oscillation from the trajectory, we take the
time-average of the Bloch vector S(t). We examine the histogram of
{S_{z}(n\Delta t)|n=0,1,...,N} for small \Delta t and large N. It represents an
absolute value of a derivative of the inverse function of S_{z}(t). (When the
inverse function of y=S_{z}(t) is a multi-valued function, the histogram
represents a summation of the absolute values of its derivatives at points
whose real parts are equal to y on the Riemann surface.) We examine the
dependence of the variance of the histogram on the temperature of the field. We
estimate the lower bound of the entanglement between the atom and the field.
