Conditional statistical properties of the complex systems having long-range interactions

TL;DR

This paper introduces a new analytical framework using the concept of available force to analyze the conditional statistical properties of complex systems with long-range interactions, demonstrated on currency exchange data.

Contribution

It proposes a novel analytical form for the CPDF based on available force, applicable to various complex systems with long-range interactions or memory.

Findings

The analytical CPDF fits well with currency exchange data.

Conditional expectation of velocity is non-monotonic, variance is monotonic.

The method is general for complex systems with long-range interactions.

Abstract

A new concept of the available force is proposed to investigate the performance of the complex systems having long-range interactions. Since the covariance of average velocity in double time interval and available force equals zero, it is possible to calculate the conditional probability distribution function (CPDF) within the systems. It is found that the asymmetric CPDF of the velocity between two adjacent time intervals can be derived from the symmetrical CPDF between the available force and the double time interval velocity. 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 examples. It is found that the analytical CPDF needs only six parameters for an arbitrary system. By calculating the CPDF in the currency exchange databases, it is shown that the results are well fitted by our analytical expression. The analytical CPDF can be also used to calculate the conditional expectation and the conditional variance of velocity. Interestingly, the two databases show that the conditional expectation of the velocity between two adjacent time intervals is not monotonic, while the conditional variance tends to monotonic. All of these results are well described by our theory. It is worthwhile to note that the analytical CPDF is a general expression. It is valid not only for current exchange systems but also for any complex systems having long-range interactions and/or long-duration memory.