## Energy can be minimized in different waysThe expression for the free energy of a system is:
As a system equilibrates it searches for the set of configurations that allows it to lower it's free energy the most. To minimize 1. The system can minimize its potential energy by placing each particle at the minimum of the potential energy surface created by the rest of the system. In practice, this often translates in placing a particle at a specified distance from its nearest neighbors and allowing small vibrations about that position. Typically, at low temperatures, At this point the system can only minimize free energy by maximizing entropy. |

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## Entropy can be maximized by crystallizing!Entropy is often naively misidentified as a measure of disorder. If this were the case then entropy maximization could hardly drive the transition from a disordered liquid to an ordered crystal! In reality, entropy is related to Ω, the number of possible configurations a system can assume, by:
_{B} is Boltzmann's constant)So how does crystallization increase Ω? Consider a system of hard spheres that are random close-packed at Φ |

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## The packing fraction of a tetrahedronVolume of a regular tetrahedron of edge 2R isV_{tet} = 2/3 R^{3}√2Dihedral angle of a tetrahedron is θ = cos^{-1}(1/3) ~ 70.5^{ o} ~ 0.3918πArea of a sphere subtented by a tetrahedron with a vertex at the center of sphere is A = (3x0.3918-1)R^{2}πVolume of the section of a sphere inside a regular tetrahedron, with a vertex at the sphere center is V_{sph} = AR/3 = 0.05847R^{3}Tetrahdedral packing fraction is 4V
_{sph}/V_{tet} ≈ 77.94% |

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## Theorists like spin glassesA spin glass is another glassy system that is frequently studied, often from a theoretical or computational point of view. Immagine a collection of spins (or compass needles) placed on a regular lattice. Here the disorder necessary to glassy dynamics is not in the spatial structure but in the randomness of the interactions between spins. Spin glass models can vary substantially in their details but they always require a mix of ferro- and antiferromagnetic interactions between spins, with spin pairs preferring an aligned or anti-aligned orientation respectively. At high temperatures, the spins do not feel the constraint of the interaction with their neighbors as much and are free to flip. As the temperature is lowered, the thermal energy
To learn more about spin glasses you can go and talk to Stefan Boettcher or visit his website. |