  Understanding how much energy is needed and dissipated as heat for a given
computational system and for a given program is a physically interesting and
practically important problem. However, the thermodynamic costs of
computational systems are only partially understood. In this paper, we focus on
a specific logic gate, the CMOS NAND gate, operating in the sub-threshold
region and analyze the dissipated heat from two aspects. One is the general
Landauer bound, which is the change in entropy of the computational system, and
the other is a cost that depends on the difference between the initial and
steady-state distributions of the system. We find that the general Landauer
bound is the same order for different inputs to the gate, but that the another
cost has partially different order due to the difference between the initial
and steady-state distributions over output logical states. We also investigate
the interplay between the costs, time scale, and reliability of the process and
find that for different inputs, there is not always a trade-off between
reliability and dissipation of computations.
