How does nucleation work in cosmology?

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Rodrigo Olivera
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In this article https://en.wikipedia.org/wiki/False_vacuum#Bubble_nucleation it says that "In the theoretical physics of the false vacuum, the system moves to a lower energy state – either the true vacuum, or another, lower energy vacuum – through a process known as bubble nucleation." What is nucleation and what does mean a nucleated bubble? This thread is focused in inflationary cosmology, in the escenario of the universe after a Big Freeze and how quantum tunneling leads a false vacuum to become a new "Big Bang".
 
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It takes something like 7 years of studies after senior high school level before someone can be able to actually calculate that kind of things (and most people never are). An example of a more easily understood nucleation process is a supersaturated salt solution that has more salt dissolved in it than is possible in thermodynamical equilibrium. If you add a little "seed" crystal of salt in such a solution, it will act as a nucleating agent and cause sudden precipitation of lots of salt. The vacuum nucleation things are based on exactly the same concepts of entropy and thermodynamical equilibrium, but are more difficult to handle theoretically.