Phase transitions whose first derivatives of the free energy are continuous are
referred to as Continuous Transitions. Their thermodynamic properties are
determined by a single length scale called the correlation length. This
correlation length diverges at the transition temperature.
The correlation length describes the spatial extent of fluctuations about the
average. It is the distance over which deviations from the average of
some quantity can be correlated. An example is the density fluctuations in a
gas in thermal equilibrium. There could be an area with a slightly higher
density then the average, like a droplet of near liquid density in the gas.
Although there will be a distribution of droplet sizes, there is a well defined
average droplet size. Roughly speaking this average size is the correlation
length. A more abstract example is the distance fluctuations in the energy
density of liquid helium are correlated.
refers to the physics associated with a critical point. Typically some
intrinsic length in the system diverges as a power-law as the system
approaches this point. The region over which this power-law behavior is obeyed
is called the "critical region", and the power-law exponents are called
"critical exponents". Interestingly, these critical exponents are not
necessarily different for different systems that demonstrate critical
behavior. This is know as Universality. Systems which are members of the same
Universality Class all share the same critical exponents. This allows one to
infer behaviour about one system from studies performed on a different system
in the same universality class.
Finite size effects are deviations from the thermodynamic limit, caused by
systems having finite sizes. In everyday situations these deviations are to
small to observe. However, situations can arise when the dimensions of a system
are comparable to length scales associated with thermodynamic responses. In
these situations finite size effects become much more apparent.
When two separate thermodynamic systems are placed in close contact with each other the responses near the interface are altered. In superfluids and superconductors this has traditionally been explained by the order parameter of one system, a wave function, not terminating at the interface but decaying as it enters the neighboring system. This decay is an exponential decay on the length scale of the correlation length. Recently there have been measurements of proximity effects over much larger distances which cannot be explained with the traditional understanding.
Scaling theories say that under suitable conditions a finite system can be
described by scaling functions which depend on only one variable. They predict
that all the data
from various confinements will collapse onto a universal locus.
This locus will depend only on how many dimensions are confined and the
universality class of the system being studied.
At 2.1768K liquid Helium undergoes a continuous transition from a normal liquid
to a superfluid. The normal liquid behaves much like water or any other liquid
you may be familiar with. The superfluid phase behaves much more interestingly.
Among its properties are a near zero viscosity and infinite thermal
conductivity. The former allows the liquid to flow through the tiniest of
cracks, while the later implies that superfluid Helium will not boil.
We use Helium in our research for various reasons. It is in the same
universality class as two dimensional ferromagnetic systems as well as
superconducting systems, both of which are of great interest for industrial
applications. Also it is very easy to have extremely pure samples of helium.
At the temperatures where helium goes superfluid almost all impurities will be
frozen out, this makes it much easier to measure, accurately and reproducibly,
quantities throughout the critical region.
The Thermodynamic Limit refers to systems where the dimensions of the system
are much larger than any length scale associated with thermodynamic responses.
In this limit knowledge of the shape of a sample are not relevant. Formally
the thermodynamic limit is the limit in which the number of particles N and the
volume of the system V become so large they are considered infinite, but the
particle density N/V remains finite. Systems in this limit are often referred
to as "bulk" systems.
This is the temperature at which the system undergoes a phase transition. For
liquid helium, this is the transition between normal liquid and superfluid.
Universality refers to the idea that two different systems can share the exact
same critical exponents. Systems that share critical exponents are said to be
members of the same "Universality Class".
A universality class is a group of systems that share the same critical
exponents. Members of a class have three things in common, Symmetry in their
Hamiltonians, dimensionality, and the ranges of forces. Our system, liquid
helium near the superfluid transition, is a member of the "XY" universality
class along with two dimensional ferromagnetic systems and superconductors.
Wednesday, 24-Aug-2011 15:52:24 EDT