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The shear viscosity

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

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where tex2html_wrap_inline841 is a parameter known as the shear rate. These equations have the conserved quantity

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The physical picture of this dynamical system corresponds to the presence of a velocity flow field tex2html_wrap_inline843 shown in the figure.

The flow field points in the tex2html_wrap_inline845 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 tex2html_wrap_inline849 plus the contribution from the field evaluated at the position of the particle

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   figure265
Figure 2:

Such an applied external shearing force will create an asymmetry in the internal pressure. In order to describe this asymmetry, we need an analog of the internal pressure that contains a dependence on specific spatial directions. Such a quantity is known as the pressure tensor and can be defined analogously to the isotropic pressure P that we encountered earlier in the course. Recall that an estimator for the pressure was

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and tex2html_wrap_inline853 in equilibrium. Here, V is the volume of the system. By analogy, one can write down an estimator for the pressure tensor tex2html_wrap_inline857 :

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and

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where tex2html_wrap_inline859 is a unit vector in the tex2html_wrap_inline861 direction, tex2html_wrap_inline863 . 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

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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 tex2html_wrap_inline867 . In fact, tex2html_wrap_inline867 is related to the velocity flow field by a relation of the form

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where the coefficient tex2html_wrap_inline871 is known as the shear viscosity and is an example of a transport coefficient. Solving for tex2html_wrap_inline871 we find

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where tex2html_wrap_inline875 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:

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Thus, the dissipative flux tex2html_wrap_inline879 becomes

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According to the linear response formula,

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so that the shear viscosity becomes

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Recall that tex2html_wrap_inline881 means average of a canonical distribution with tex2html_wrap_inline883 . It is straightforward to show that tex2html_wrap_inline885 for an equilibrium canonical distribution function. Finally, taking the limit that tex2html_wrap_inline839 in the above expression gives the result

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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.


next up previous
Next: The diffusion constant Up: Time correlation functions and Previous: Time correlation functions and

Mark Tuckerman
Thu Apr 13 13:07:24 EDT 2000