thermodynamics - How many particles is needed to observe a phase transition?

This is a question that was rised when we were discussing "what is melting actually". How many particles you need to form a liquid or solid. I have some remarks to point out what I want to know.

Q: How many particles is needed to observe a phase transition?

A:

1. just one, phase transition is not a collective phenomena.


2. 1 million, Ising model in 1000x1000 lattice produces a phase transition.


3. $N_a$, phase transition is a thermodynamic concept and you always need to be in TD limit.


4. The question is dependent on the process. For instance, in phenomena A you need N particles, but in B you need M particles.


5. The question is stupid, you should ask X instead.

I would like to know which arguments are true and which are false (and why). Also, how many particles you need to form a liquid from solid (one particle with Gibbs energy above some limit X is a liquid?).

Answer

Option 3. An equilibrium phase transition is a non-analytic point of the thermodynamic free-energy. For a finite number of particles, the free-energy is always analytic. So you cant get a phase transition. Kardar discusses this point.