One of the striking properties of superfluids is the ability to
flow without friction. This fact allows us to define the
normal fluid density as the fraction of liquid which is
carried along by the walls if they are set in motion. For example
[17,18],
consider the liquid inside a long tube (see. Fig 2.1), which
was at rest at time and was then adiabatically
accelerated up to time (for instance with the exponential
law
with infinitesimal
). The normal component can be defined through the
momentum density
at
Figure 2.1:
Long tube filled with superfluid
(2.25)
and the superfluid density as the difference between the total
density and
(2.26)
The effect of the perturbation caused by the moving walls is given
by the energy
(2.27)
where is the momentum density and is the
external velocity field. The linear response is given by the Kubo
formula
(2.28)
where
(2.29)
For uniform systems the response function depends
only on the difference of its arguments
.
The static susceptibility is defined as
(2.30)
or in terms of Fourier components
(2.31)
At time the linear response function satisfies the equation
(2.32)
Since is a second rank tensor, it can be decomposed
into the sum of longitudinal and transverse components
(2.33)
Let us consider first the transverse response. Due to the
rotational invariance of it is enough to examine the
response in an arbitrary direction, for example , that is the
momentum response due to an imposed velocity field in the
direction .
Suppose first that the velocity field is created by dragging the
walls of an indefinitely long pipe (see Fig. 2.2) and
the cross section of the pipe tends to infinity. This arrangement
corresponds to the limiting procedure , followed by , . Then, the part of the system responding to the
shear force is defined as the normal fluid. Carrying out the
limiting procedure
gives
(2.34)
Figure 2.2:
Illustration of the transverse response. Only the normal
component is dragged in direction
Next suppose that the pipe of infinite radius is constrained by two
plates, normal to the axis, with separation between the plates
approaching infinity, as illustrated in Fig. 2.3. In
this case the entire fluid
responds to the
external probe. This arrangement corresponds to the limiting
procedure , , followed by . The
result of the limiting procedure is
(2.35)
Figure 2.3:
Illustration to the longitudinal response. Both superfluid
and normal components are pushed in the direction