Lower limit to power consumption

• Energy required only when information is destroyed. Hence perform all calculations in a reversible way. No energy required.

  Information loss is irreversible, hence increases Entropy $S$, which has to be removed. Entropy normally removed via heat flow $Q$ to environment (at temperature $T$). $S=Q/T$ therefore $Q=TS$.

• Entropy $S = \log(\Omega)\ k$.

  $\Omega$ number of configurations of the computer.

  Boltzman's constant $k=1.380658\ 10^{-23}$ Joules per degree Kelvin.

• If a bit's value is known $\Omega=1$ and so $S=0$.
  If a bit's value has been lost then $\Omega=2$ and so $S=\log(2)\ k$.

  Electrical energy required $W = Q = TS \ge \log(2)\ k T$

  Energy consumption $\ge \log(2)\ k T$ per bit of information destroyed.