A Non-equilibrium Statistical Mechanics of Liquids

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Summary
Do I need triple and higher correlation functions in this equations to make non-Equilibrium kinetic models that describe liquids?
Molecular Transport equations for Liquids are harder to compute than that for gases, because intermolecular interactions are far more important in liquids. A System of equations for particle Distribution function and the correlation functions (BBGKY-Hierarchy) is used in General. For gases, it is possible to derive e.g. the Boltzmann equation from the BBGKY-hierarchy. The structure of this hierarchy is given by

d/dt f_n + L_n f_n = C_n(f_(n+1))

with n-particle Distribution function f_n, n-particle Liouville Operator L_n and the n-particle collision Operator C_n that depends on the next higher Partition function f_(n+1). We can do an Expansion for the n-particle Distribution in Terms of correlation functions, e.g. f_2(x_1,x_2) = f_1(x_1) f_1(x_2)+g_2(x_1,x_2) for a two-body correlation function g_2 and Phase-space coordinates x. After some calculus, we can obtain equation of Motion for every correlation function.

Frequently I have heard that in liquids, the thermal kinetic Energy (that is kT) of a molecule is of similar magnitude as the interparticle interaction Energy. Also the collision Operator scales with particle number density times the effective volume of a particle (that is the volume it occupies and the volume, where it can attract other particles); in liquids this quantity cannot be assumed as a small perturbation. Some People say because of These reasons the hierarchy would couple to third and higher order correlations; this would make the calculations extremely complicated.

But can we develop a kinetic equation valid for liquids without incorporating higher correlation functions? When we say g_3 = 0 we have a closed set of equations. The Terms that account for intermolecular interactions would be

- A mean field force term, where the potential is averaged over the particle Distribution function (see also Vlasov equation, Hartree-Fock method, ...)
- A 2-body correlation term that couples to the inter-particle potential, but also accounts for the Motion of particles in an effective potential

One note to the 2-body correlation term: There will be a linear equation for g_2 in the form

A*g_2 = S

where S is the source of correlation dependent on Partition function and A is an effective Liouville Operator containing also the interparticle potential. The Operator A can, due to effective potential, Shield the particle from getting correlations with other particles. Would a Theory up to 2-body correlations predict characteristic Parameters of a simple liquid like viscosity, thermal conductivity, … with sufficient accuracy? Or are many-body correlations mandatory for having not too large Deviation of transport quantities from the experimental values?
 
Start with this book:


But I would say the history of theories using only 2-body correlations for liquids is not great. Sometimes things work, sometimes they don't, and I don't think people have had much success predicting ahead of time which it's going to be.
 
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Dear Mr. Grayce,

thank you very much for your Reply! In which cases a kinetic model that incorporates only 2-body correlations is sufficient for liquids? Is it sufficient for Special types of liquids?

And models that have 3-body correlations, but neglecting (or only roughly modeling) 4-body and higher order correlations? Will they have much better accuracy or are These still inaccurate, too?

A Reply I would highly acknowledge.
 

DrDu

Science Advisor
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I just wanted to point out that one of the simplest and best studied liquids is the "Fermi liquid" for which a Boltzmann equation can be derived and inclusion of only two body interactions is a good approximation.
 

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