The physical limit of logical compare operation

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 " > " 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. PACS number: 89.70.-a Ⅰ. INTRODUCTION There are already many ways of measuring complexity in the field of complex systems. These measures of complexity are divided into four categories by Lloyd [1]. Computational complexity is a particularly important objective to us in complex systems [2]. Certain amount of resources is necessary in order to solve any given problems. For instance, space (memory), time and energy are all essential to run an algorithm on a computer. We proposed a definition to measure the complexity of a problem: the minimum energy required to solve it [3]. This measure was based on entropy reduction (negentropy) between the entropy before the problem was solved and after, which is interesting because the definition of information is also negentropy. As computers are firstly physical systems: the laws of physics dictate what they can and cannot do [4,5]. Or in other words, information is physical [6]. The physics of information told us everything in the universe is doing computation. Physical objects might be thought of as being made of information [7]. The position and velocity of an atom in a gas register information. As pointed out by Fredkin and Toffoli [8,9], each atomic collision performs AND, OR, NOT, or COPY operations on suitably defined input and output bits. In this way, the second law of thermodynamics is a statement about information processing: the underlying physical dynamics of the universe preserve bits and prevent their number from decreasing [7]. Discussion on the relationship between energy and information can be traced back to the Maxwell demon which is proposed by Maxwell in 1867. The demon is a construct that can distinguish the velocities of individual gas molecules and then separate hot and cold molecules into two domains of a container, after that the two domains will have different temperature [10]. The result seems to contradict the second law of thermodynamics. Maxwell demon can be expressed by entropy which is …