Numerical Simulations for directed polymers in 1+1 and 2+1 dimensions show that the transverse fluctuations and free energy fluctuations tend to the strong coupling limit: Δx∝tν,ΔF∝tω (ν=2/3,ω=1/3 for d=1+1; ν≈0.6, ω≈0.2 for d=2+1) at any finite temperature as their behaviors at zero temperature for the long length t of polymer. At finite temperatures, as a result of that the specific heat in both 1+1 dimensions and 2+1 dimensions scales as t and the entropy fluctuations in 1+1 dimensions scale as t1/2 , the ensemble energy fluctuations in both 1+1 and 2+1 dimensions and the internal energy fluctuations in 1+1 dimensions are more pronounced than the free energy fluctuations and scale as t1/2. There is an indication of a phase transition in 2+1 dimensions. It is plausible that the entropy fluctuations and the internal fluctuations per unit length polymers for t→∞ will become zero value at high temperatures from nonzero values at low temperatures at some temperature T=TC where the entropy fluctuation reaches maximum.