# Theory of nonequilibrium phenomena based on the BBGKY hierarchy. I. small deviations from equilbrium

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## Abstract

The BBGKY hierarchy is expanded in a series with respect to the small parameter\(\varepsilon = {\sigma \mathord{\left/ {\vphantom {\sigma \mathcal{L}}} \right. \kern-\nulldelimiterspace} \mathcal{L}}\), where σ is the diameter of the particles, and\(\mathcal{L}\) is a characteristic macroscopic length (for example, the diameter of the system). Since neither σ nor\(\mathcal{L}\) occurs explicitly in the equations of the hierarchy, a preliminary step consists of separation from the distribution functions\(\mathcal{G}_{(l)} \) of short-range components that vary over distances of order σ and long-range components that vary over distances of order\(\mathcal{L}\). By a transition to dimensionless variables, terms of zeroth and first order in ε in the hierarchy are separated, this making it possible to perform the expansion with respect to ε. It is shown that in the zeroth order in ε the BBGKY hierarchy determines a state of local equilibrium that for any matter density can be described by a Maxwell distribution “with shift.” The higher terms of the series in ε describe the deviations from local equilibrium. At the same time, the long-range correlations that always arise in nonequilibrium systems are described by the balance equations for mass, momentum, and energy, which are also a consequence of the BBGKY hierarchy, whereas the short-range correlations are described by the equations for\(\mathcal{G}_{(l)} \) obtained from the same hierarchy by expanding\(\mathcal{G}_{(l)} \) in a series with respect to ε.

## Keywords

Small Deviation Balance Equation Small Parameter Matter Density Zeroth Order## Preview

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## References

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