At the critical point of a second order phase transition the correlation length
associated with the order parameter correlation function diverges.
Other thermodynamic quantities show singular behavior near the critical
point. When the system is confined and its size is limited by a confining length L,
at the bulk critical point the correlation length remains finite and this
can influence not only local properties but also global properties of
the system. The singularities in global properties, such as specific
heat become rounded by finite-size effects.
The finite-size scaling theory asserts that close enough to the critical point,
these global properties obey a scaling law and there exist a universal scaling
functions characteristic of the confining geometry and the condition imposed
by the system boundaries.
The superfluid transition in liquid helium for several reasons is an ideal
situation to test the finite size scaling theory. These studies are of
fundamental importance for the theory of phase transitions and for
understanding the process of taking the continuum limit of observable
which are averages over a fluctuating field (such quantum field theories).
In the case of pure systems such as liquid helium, impressive experimental
work has approached the lambda transition with sub-nano-Kelvin
accuaracy. In this case the specific heat of systems of macroscopic
size (up 100 microns) are seen to be influenced by such finite-size
effects. There are recent attempts to develop a new technology the so-called
nanotechnology where one can imagine situations where one could benefit
from the understanding of the role of the finite-size effects. The reason
is that in some cases of the systems used for such applications
one uses materials which are near criticality for example, near the
During the last several years our group has made a significant effort
to study confined helium and finite-size scaling in various confined
geometries. We have made a number of predictions for the
scaling functions which are in agreement with experiments which
were carried out later under microgravity conditions.
(see .J. A. Lipa et al, Phys. Rev. Lett. 84, 4894 (2000)).