  Using a new, increased dataset of 7 QSOs from VLT/UVES observations combined
with one QSO from the literature, the minimum Doppler parameters as a function
of neutral hydrogen column density N(HI), b_c(N), of the Lyman-alpha forest has
been derived at three redshifts z = 2.1, 3.3 and 3.8. When incompleteness of
the number of the observed lines towards lower N(HI) is accounted for, the
derived slopes of b_c(N), (Gamma-1), are consistent with no-redshift evolution
with an indication of lower value at z = 3.3, while b_c(N) at a fixed column
density N(HI) = 10^(13.6) cm^(-2), b_c(13.6), increases as z decreases.
Assuming a QSO-dominated UV background, the slope of the equation of state
(gamma-1) shows no z-evolution within large uncertainties and the temperature
at the mean density, T_0, decreases as z decreases at three redshift ranges.
There is a large fluctuation of (Gamma-1) and b_c(13.6) even at the similar
redshifts, in particular at z = 3.3 and 3.8. The lower (Gamma-1) and higher
b_c(13.6) values at z ~ 3.1 and 3.6 compared to ones at z ~ 3.4 and 3.9 are
caused by a lack of lower-N(HI) and lower-b lines, probably due to the HeII
reionization. This result suggests that an impact from the HeII reionization on
the forest might be mainly on the lower-N(HI) forest. We find some forest
clouds with a high ratio of SiIV column density to CIV column density at z <
2.5, although the bulk of the forest clouds shows lower N(SiIV)/N(CIV). This
high N(SiIV)/N(CIV) at z < 2.5 suggests that some forest clouds are exposed to
a soft UV background.
