Stochastically-Induced Quantum-to-Classical Transition: The Lindemann Relation, Maximum Density at the He Lambda Point and Water-Ice Transition

In the present paper, by using the quantum stochastic hydrodynamic analogy (SQHA), the transition
between gas, liquid and solid phases, made of structureless particles, have been analyzed. The
interest for the quantum hydrodynamic analogy (QHA) has been recently growing by its strict relation
with the Schrödinger mechanics. The SQHA shows that the quantum behaviour is maintained on a
distance shorter than the theory-defined quantum correlation length (c). When the physical length of
the problem is larger than c, the model shows that the quantum (potential) interactions may have a
finite range of interaction maintaining the non-local behaviour on a finite distance “quantum nonlocality
length” q (with q >c ). The present work shows the realization of “classical” phases (gas
and van der Waals liquids), when the mean molecular distance is larger than the quantum non-locality
length q. On the other hand, when the mean molecular distance becomes smaller than q or than c
phases transitions such as to solid crystal or to superfluid appear, respectively. The model shows that
the quantum character of the matter emerges as a consequence of the random noise suppression
generated by the quantum potential below the induced noise correlation length. The model explains
the Lindemann empirical law about the mean square deviation of atoms 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 at the water-ice solidification point. The SQHA shows that both the linearity of the
particle interaction and the reduction of amplitude of stochastic fluctuations elicited the emergence of
quantum behavior. The SQHA model also shows that the nonlinear behavior of physical forces, other
than to play an important role in the establishing of thermodynamic equilibrium, is a necessary
condition to pass from the quantum to the classical phases and that fluctuation alone are not
sufficient.