# Modelling liquids like water with BBGKY-hierarchy

The BBGKY hierarchy is a well-known useful possibility to derive kinetic equations for gases and Plasma. The N-particle System is reduced to few-particle Systems by Integration over many Phase space coordinates. Can this method be useful to at least approximately describe liquids with several intermolecular interactions including electrostatic interaction and hydrogen Bonding? In Plasma Physics, the 2-particle Liouville equation (Phase space Density $$f(x_1,p_1,x_2,p_2,t)$$) that is coupled to a term including the 3-particle particle Phase space Density $$f(x_1,p_1,x_2,p_2,x_3,p_3,t)$$ is considered. A crucial step is that the statistical cumulant Expansions are performed:

$$f(x_1,p_1,x_2,p_2,x_3,p_3,t) = f(x_1,p_1,t)f(x_2,p_2,t)f(x_3,p_3,t)+f(x_1,p_1,t)g(x_2,p_2,x_3,p_3,t)+f(x_2,p_2,t)g(x_1,p_1,x_3,p_3,t)+f(x_3,p_3,t)g(x_1,p_1,x_2,p_2,t)+g(x_1,p_1,x_2,p_2,x_3,p_3,t)$$

$$f(x_1,p_1,x_2,p_2,t)=f(x_1,p_1,t)f(x_2,p_2,t)+g(x_1,p_1,x_2,p_2,t)$$.

The function $$g$$ denotes a correlation function, if it is a function of $$m$$ particles, it is depicting the $$m$$-th statistical cumulant.

In Plasma physics, the third cumulant is neglected (I refer to Landau theory) to have a closed hierarchy of equations. But what is with the third cumulant if I want to describe liquid state fluids like water with statistical physics? Do I Need an extra equation for the third cumulant while dropping the next higher order cumulant for Hierarchy closure? Or do I Need even higher cumulants for sufficiently accurate description of microscopic behavior?

Is there a General rule that given the material, fluid, System, or whatever that tells me which statistical cumulant order I have to incorporate in hierarchy and what I could safely Neglect?

I think it may have to do with the characteristic structure of the radial Distribution function. For water, this function is distinct from the function for a real gas, but have similarities with it. So do I even Need the equation for the third order cumulant $$g(x_1,p_1,x_2,p_2,x_3,p_3,t)$$?

• Have you been reading into how to construct Fluid theory from kinetic theory? – AtmosphericPrisonEscape Feb 8 at 13:58
• Yes, I also know how hydrodynamic equations can be derived from kinetic equations. – kryomaxim Feb 8 at 14:16
• Have you tried looking at studies like aip.scitation.org/doi/10.1063/1.861545 ? – honeste_vivere Feb 8 at 14:41
• I saw an article here: sciencedirect.com/science/article/pii/037843719500124P that says that significance of higher order correlations increase with increasing Density of the System. Moreover it is stated that if we have non-hard-sphere potential (which is important for liquids), then higher order correlations are of more importance. – kryomaxim Feb 8 at 15:27