Search

In this paper, we attempt to derive the expression of ensemble average internal energy in long-range interaction complex system. Further, the Shannon entropy hypothesis is used to derive the probability distribution function of energy. It is worth mentioning that the probability distribution function of energy can be equivalent to the q-Gaussian distribution given by Tsallis based on nonextensive entropy. In order to verify the practical significance of this model, it is applied to the older subject of income system. The classic income distribution is two-stage, the most recognized low-income distribution is the exponential form, and the high-income distribution is the recognized Pareto power law distribution. The probability distribution can explain the entire distribution of United States income data. In addition, the internal energy, entropy and temperature of the United States income system can be calculated, and the economic crisis in the United States in recent years can be presented. It is believed that the model will be further improved and extended to other areas.

We constructed a model that evolved from a non-equilibrium state to an equilibrium state. The model only needs two basic coefficients, including self-similar coefficients and non-equilibrium coefficients. The coefficients of the model can be obtained directly from real data. The model can well represent the phenomena of anomalous diffusion, non-Gaussian distribution and so on in non-equilibrium complex systems. It is important to mention that the model is a global iterative model that can exhibit statistical physical properties on multiple time scales, as well as exhibiting long-term memory properties. This model may be used as a basic model for the research of various non-equilibrium relaxation process phenomena., Comment: 5pages,3figures

Considering the interactions of two arbitrary particles, we obtain an internal energy expression of the complex system having long-range interactions. Based on the postulate of "equal-probability principle" for all microstates, the probability distribution function of the system is derived, and consequently, some main statistical physical properties of the system are revealed. It is important to find that the properties of a long-range interaction system are very closely dependent on the interaction coefficient and particle number., Comment: 14pages, 2 figures

Based on the probability distribution observed in complex systems and an assumption that the probability distributions of complex systems satisfy a new generalized multiplication, it is proved that the statistical theory of complex systems can be established in the analogous extensive framework., Comment: 11pages

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., Comment: 18pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1305.0090

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., Comment: 21 pages, 4figures

A new concept of the available force in long-range interaction complex systems is proposed. The relationship between the available force in different time intervals and the interaction parameters of complex systems is described. It is found that when the interaction parameters satisfy a determined condition, the trajectory that the velocity is divergent but the displacement is convergent can be well described and that the long-range interaction, anomalous diffusion, and q-Gaussian type distribution of complex systems can also be well described by the interaction parameters in different cases. In addition, by utilizing the velocity of time series randomly and analyzing its probability distribution of displacement, it is explained that when there exists the long-range interaction in complex systems, the fat-tail distributions will exhibit. The results obtained show that the relationship between the available force and the interaction parameters may be used to investigate the statistical physical properties in long-range interaction complex systems., Comment: 19pages,6figures