  The theory of current transport in a narrow superconducting channel
accounting for thermal fluctuations is developed. These fluctuations result in
the appearance of small but finite dissipation in the sample. The value of
corresponding voltage is found as the function of temperature (close to
transition temperature) and arbitrary bias current. It is demonstrated that the
value of the activation energy (exponential factor in the Arrenius law) when
current approaches to the critical one is proportional to (1-J/Jc)^(5/4). This
result is in concordance with the one for the affine phenomenon of the
Josephson current decay due to the thermal phase fluctuations, where the
activation energy proportional (1-J/J_c)^(3/2)(the difference in the exponents
is related to the additional current dependence of the order parameter). Found
dependence of the activation energy on current explains the enormous
discrepancy between the theoretically predicted before and the experimentally
observed broadening of the resistive transition.
