  A rigorous theory of liquid-crystal transitions is developed starting from
the Liouville equation. The starting point is an all-atom description and a set
of order parameter field variables that are shown to evolve slowly via Newton's
equations. The separation of timescales between that of atomic collisions and
the order parameter fields enables the derivation of rigorous equations for
stochastic order parameter field dynamics. When the fields provide a measure of
the spatial profile of the probability of molecular position, orientation, and
internal structure, a theory of liquid-crystal transitions emerges. The theory
uses the all-atom/continuum approach developed earlier to obtain a functional
generalization of the Smoluchowski equation wherein key atomic details are
embedded. The equivalent non-local Langevin equations are derived and
computational aspects are discussed. The theory enables simulations that are
much less computationally intensive than molecular dynamics and thus does not
require oversimplification of the system's constituent components. The
equations obtained do not include factors that require calibration and can thus
be applicable to various phase transitions which overcomes the limitations of
phenomenological field models. The relation of the theory to phenomenological
descriptions of Nematic and Smectic phase transitions, and the possible
existence of other types of transitions involving intermolecular structural
parameters are discussed.

Soft Condensed Matter;Statistical Mechanics