Quantum to Classical Transition in the Stochastic Hydrodynamic Analogy: The Explanation of the Lindemann Relation and the Analogies Between the Maximum of Density at He Lambda Point and that One at Water-Ice Phase Transition

In the present paper the gas, liquid and solid phases made of structureless particles, are visited to the light of the quantum stochastic hydrodynamic analogy (SQHA). The SQHA shows that the quantum behavior is maintained on a distance shorter than the theory-defined quantum correlation length (lc). When, the physical length of the problem is larger than lc, the model shows that the quantum (potential) interactions may have a finite range of interaction maintaining the non-local behavior on a finite distance “quantum non-locality length” lq. The present work shows that when the mean molecular distance is larger than the quantum non-locality length we have a “classical” phases (gas and van der Waals liquids) while when the mean molecular distance becomes smaller than lq or than lc we have phases such as a solid crystal or a superfluid one, respectively, that show quantum characteristics. The model agrees with Lindemann empirical law (and explains it), for the mean square deviation of atom from the equilibrium position at melting point of crystal, and shows a connection between the maximum density at the He lambda point and that one near the water-ice solidification point.