Conditional statistical physical properties in two-joint complex systems having long-range interactions

We propose the sum and the difference of the normalized velocity of two-joint systems to describe its long-range interaction. It is found that the conditional probability distribution function (CPDF) of the normalized velocity between two-joint systems can be derived. The analytical CPDF needs only six parameters for arbitrary two-joint systems. Two typical currency exchange databases, i.e., EUR/USD and GBP/USD, which collect the minutely opening exchange prices from 1 January 1999 to 31 December 2011, are adopted as example. By calculating the CPDF in the currency exchange databases, it is shown that all of the results are well described by our theory. We also use the analytical CPDF to calculate the entropy of two-joint systems, it is found that the entropy of two-joint systems is less than the sum of entropy of each system in the two currency exchange databases. It means that some information of two-joint systems may overlap. We must important to note that the results presented here do not need to know the form of interaction of two-joint systems, and the analytical CPDF is a general expression. It is valid not only for current exchange systems, but also helpful for investigate other conditional statistical properties in any two-joint long-range interactions complex systems.