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

metal-insulator transition.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)).