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In the past decade, a literary phrase "No man's land" has been flooded in thescientific papers. The expression is used to describe a meta-stable region inthe phase-diagram that cannot be accessed by experiments. It has been claimedbased on the molecular dynamics (MD) simulation that there is a critical point,or the second critical point (SCP), in the "no man's land," and it has createda big dispute in the field of science. It is proved in the present paper thatthe hypothesis of SCP is completely against the rigorous theorem ofthermodynamics, referred as the Gibbs phase rule. The reason why thesimulations have found SCP erroneously is merely because the method violatesthe requirement which all the statistical-mechanics treatments should satisfyto reproduce the thermodynamics. That is the thermodynamic limit. It isclarified what is the identity of the "liquid-liquid phase transition" and SCPin pure liquids, discovered by the simulations and by some experiments. Inorder to explain the physics of liquid-liquid phase transition observedexperimentally in single component liquids, a new concept is proposed.

Does the second critical-point of water really exist in nature?