In summary, free energy is an important concept in thermodynamics that is used to describe the energy of a system that is not completely isolated from its environment. It is denoted by ##F## and is related to the internal energy ##U## through a Legendre transform. At a constant temperature, the free energy is minimized and can be used to determine the work a system can perform.
What is the free energy and why is it so important? I realize there's different types but what is the underlying concept? Is a general principle of CM systems to attempt to minimise free energy?
"Free energy" is not a specific concept in condensed matter physics, but in thermodynamics in general.
You might be familiar with the concept of "internal energy", mostly denoted by ##U##. This is the total energy contained within a system that is isolated from the environment: the volume is constant, no heat or work can be transferred to the system, the particle number is constant.
This is simple to understand, but not very interesting overall, as real systems never are fully isolated from the environment.
Now, the next step is to allow for heat exchange between the system and the environment: then the temperature ##T## of the system is held constant, as well as the volume and the particle number. Then the "free energy" ##F## is the interesting energy-like quantity to look at. It is connected to ##U## via a Legendre transform ##F=U-TS##, where ##T## is the fixed temperature and ##S## is the entropy of the system. At fixed temperature ##T##, the free energy ##F## is minimized for a system in thermodynamic quilibrium. In (quasistatic) thermodynamic processes, the difference in free energy to the minimum free energy is the amount of work the system may perform. If left to itself, the free energy will then again tend to its minimum.