Theory of nonequilibrium phenomena based on the BBGKI hierarchy. I. Small deviation from equilibrium

The BBGKY hierarchy is expanded in a series with respect to the small parameter $\varepsilon =\sigma / \mathcal L$, where $\sigma$ is the diameter of the particles, and $\mathcal L$ is a characteristic macroscopic length (for example, the diameter of the system). Since neither $\sigma$, 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 $\sigma$ 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 $\varepsilon$ in the hierarchy are separated, this making it possible to perform the expansion with respect to $\varepsilon$. It is shown that in the zeroth order in $\varepsilon$ 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 $\varepsilon$ 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 $\varepsilon$.