Phase Transition: Determine Order of Transitions

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The order of phase transitions can be determined by analyzing the continuity of the Gibbs function. A first-order transition occurs when the Gibbs function is continuous, but its slope is discontinuous, indicating latent heat, such as in melting. In contrast, a second-order transition is characterized by both the Gibbs function and its first derivative being continuous, while the second derivative is discontinuous. The conductor-superconductor transition in liquid helium is noted as a known example of a second-order transition. Recent developments in thermodynamics have expanded the understanding of phase transitions, including concepts like the 'lambda transition' and renormalization.
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How does one determine the order of phase transitions?
 
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By the degree of continuity in the Gibbs function. If the Gibbs function is continuous, but its slope is discontinuous (say due to the latent heat of melting), the transition is first-order. If both the Gibbs function and it's derivative is continuous, but the second derivative is discontinuous, the phase change is a second-order. AFAIK, the conductor-superconductor transition in liquid He is the only known second-order transition, but I don't have any really up-to-date references.

There's also a 'lambda transition', in which some of the thermodynamics properties (specific heat, for example) diverges. The resolution of that led to renormalization and scaling concepts.

The concept of a phase transition has really expanded over the past 10 years or so- some of what I wrote may be out of date.
 

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