The shear viscosity of a system measures is resistance to
flow. A simple flow field can be established in a system
by placing it between two plates and then pulling the plates
apart in opposite directions. Such a force is called a
*shear force*, and the rate at which the plates are pulled
apart is the shear rate.
A set of microscopic equations of motion for
generating shear flow is

where is a parameter known as the shear rate. These equations have the conserved quantity

The physical picture of this dynamical system corresponds to the presence of a velocity flow field shown in the figure.

The flow field points in the direction and increases with
increasing *y*-value. Thus, layers of a fluid, for example, will slow
past each other, creating an anisotropy in the system. From the
conserved quantity, one can see that the momentum of a particle is
the value of plus the contribution from the field evaluated
at the position of the particle

and in equilibrium. Here, *V* is the volume of the
system. By analogy, one can write down an estimator for the
pressure tensor :

and

where is a unit vector in the direction, . This (nine-component) pressure tensor gives information about spatial anisotropies in the system that give rise to off-diagonal pressure tensor components. The isotropic pressure can be recovered from

which is just 1/3 of the trace of the pressure tensor. While most systems
have diagonal pressure tensors due to spatial isotropy, the application of
a shear force according to the above scheme gives rise to a nonzero value
for the *xy* component of the pressure tensor . In fact,
is related to the velocity flow field by a relation of the
form

where the coefficient is known as the *shear viscosity* and
is an example of a *transport coefficient*.
Solving for we find

where is the nonequilibrium average of the pressure tensor estimator using the above dynamical equations of motion.

Let us apply the linear response formula to the calculation of the
nonequilibrium average of the *xy* component of the pressure tensor.
We make the following identifications:

Thus, the dissipative flux becomes

According to the linear response formula,

so that the shear viscosity becomes

Recall that means average of a canonical distribution with . It is straightforward to show that for an equilibrium canonical distribution function. Finally, taking the limit that in the above expression gives the result

which is a relation between a transport coefficient, in this case, the shear
viscosity coefficient, and the integral of an equilibrium time correlation function.
Relations of this type are known as *Green-Kubo* relations. Thus, we have
expressed a new kind of thermodynamic quantity to an equilibrium time
correlation function, which, in this case, is an autocorrelation function of
the *xy* component of the pressure tensor.

Thu Apr 13 13:07:24 EDT 2000