  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.
