Markov-type evolution of materials into a polar state.

Assembling polar building blocks into a solid material by a Markov-chain process of unidirectional growth principally results in a metastable state that shows effects of macroscopic polarity. Stochastic polarity formation can be described by probabilities for the attachment of building blocks to a surface. Because of the polar symmetry of the building blocks, there is a fundamental difference in the probabilities for attaching them "tip-first" or "back-first" to growth sites at a surface. A difference in the corresponding probabilities drives the evolution of a vectorial property through a gain in configurational entropy. Examples from the mechanical, the crystalline and the biological world demonstrate growth-induced macroscopic polarity. In crystals, growth upon centrosymmetric seeds can produce twinned crystals with a "sectorwise" pyroelectric effect. Polarity formation in connective tissues is explained by a Markov-chain mechanism, which drives the self-assembly of collagen fibril segments. An unified stochastic growth model brings up a general concept for the formation of materials with polar properties.