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Hydrodynamic equations
We are
interested in the non-equilibrium behavior of the system. Hence it will be
convenient to re-frame the equation in the form of hydrodynamic equations by
taking moments of the original equation with respect to velocities. Let us
define the moments as

(A-7)

(A-8)

(A-9)

(A-10)

(A-11)

(A-12)

(A-13)

Integrating eqn. 3.6
with respect to velocities gives the continuity equation in phase space as

(A-14)

The momentum equation can be obtained by multiplying eqn. 3.6 by or and then integrating over all velocities. The result can be written as

(A-15)

(A-16)

where

, , and can be interpreted as stress tensors in the momentum equations. They
are composed of two parts

(A-17)

(A-18)

(A-19)

(A-20)

Here the 's are the kinetic contributions given by eqn. 3.10-3.13. The
's are the contribution due to the potential energy of inter-particle
interaction. They are defined by

(A-21)

(A-22)

(A-23)

(A-24)

It should remembered that all these moments are dependent on the pair
although the notation does not make it evident. Wherever necessary,
the dependence will be made explicit.

One could also go ahead and write the equation for evolution of temperature
and energy density. But we will assume an isothermal system and hence ignore the
temperature field.

The case of is quite special and we will write the equations for and for future reference

0

(A-25)

(A-26)

In the next chapter, we will try to adapt this theory for the problem of
no-slip.

Shreyas Mandre 2002-09-23