\label{sec:LP}
I calculate, using a Hamiltonian model of thermal reservoir and the bit itself, that the average work done to reset a bit with initial probability \(p_1=p_0=1/2\) is \(\langle W\rangle \geq T\ln2\) \cite{Piechocinska_2000}.
When the initial probabilities are arbitrary, with \(p_1=\gamma\) and \(p_2=1-\gamma\), I derived the following equation which seems wrong. \[\beta\langle W\rangle \geq \int_0^\infty\alpha\ln\alpha\;dx\,dp + \ln2 - \frac{1}{2}\gamma\ln\gamma - \frac{1}{2}(1-\gamma)\ln(1-\gamma),\] where \(\alpha\) is the initial probability distribution in \((x,p)\) for \(p_1=p_0=1/2\). But the answer I want is \[\beta\langle W\rangle \geq -\gamma\ln\gamma - (1-\gamma)\ln(1-\gamma),\] i.e. lose the first two terms and the factor of 1/2.
To do:
Correct derivation of LP for arbitrary initial probabilities.