  We present systematic numerical simulations for directed polymers at finite
temperatures in 1+1 and 2+1 dimensions. The transverse fluctuations and free
energy fluctuations tend to the strong coupling limit at any temperature in
both 1+1 and 2+1 dimensions for long time t. Two different definitions for
energy fluctuations at finite temperatures, which are the ensemble energy
fluctuations and the internal energy fluctuations, are investigated. Apart from
zero temperature, the behavior of the energy fluctuations and the free energy
fluctuations for directed polymers is shown to be different. At finite
temperatures, the ensemble energy fluctuations in both 1+1 and 2+1 dimensions
and internal energy fluctuations in 1+1 dimensions scale as t^{1/2} where the
free energy fluctuations in 1+1 dimensions and 2+1 dimensions scale as t^{1/3}
and t^{0.2} respectively. As a consequence of that the specific heat in both
1+1 and 2+1 dimensions scales as t and the entropy fluctuations in 1+1
dimensions scale as t^{1/2} at any finite temperature.
