A measure theoretical approach to non-uniform phases of matter with no long-range spatial order

In this paper, the development of a mathematical method is presented to explore spatially non-uniform phases with no long-range order in mathematical models of first order phase transitions. We use essential results regarding the concentration of measure phenomenon to re-formulate partial differential equations for probability measures, which extends the concept of analytical solutions to random fields. A stochastic solution of an equation is such a non-singular probability measure, according to which the random variable is almost surely a solution to the equation. The general concept is applied for continuum theories, where the concept of symmetry breaking is extended to probability measures. The concept is practicable and predictive for non-local continuum mean-field theories of first order phase transitions. The results suggest that symmetry breaking must be present in stochastic stationary points of the energy on the level of the probability measure. This is in agreement with the observation that amorphous solid structures preserve local ordering.