  Thermodynamics and transport properties of a dissipative particle in a
tight-binding model are studied through specific heat and optical conductivity.
A weak coupling theory is constituted to study the crossover behavior between
the low-temperature region and the high-temperature region analytically. We
found that coherent part around zero frequency in the optical conductivity
disappears for 0<s<2, where s is an exponent of a spectral function of the
environment. Detailed calculation is performed for ohmic damping (s=1). In this
case, the specific heat shows an unusual $T$-linear behavior at low
temperatures, which indicates that the environment strongly influences the
particle motion, and changes the low-energy states of the dissipative particle.
The optical conductivity \sigma(\omega) takes a non-Drude form even at zero
temperature, and the high-frequency side behaves as \omega^(2K-2), where K is a
dimensionless damping strength. The high frequency side of the optical
conductivity is independent of temperatures, while the low frequency side
depends on the temperature, and behaves as T^(2K-2) at high temperatures. We
also comment on the application of this model to the description of incoherent
motion in correlated electron systems.
