  In this paper two connected Szilard single molecule engines (with different
temperature) model of Maxwell's demon are used to demonstrate and analysis the
logical compare operation. The logical and physical complexity of compare
operations are both showed to be kTln2. Then this limit was used to prove the
time complexity lower bound of sorting problem. It confirmed the proposed way
to measure the complexity of a problem, provided another evidence of the
equivalence between information theoretical and thermodynamic entropies.
