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The Legendre transformation is one of the ways to rewrite the theory by small number of selected variables. It leads naturally to the generalized action functional called the effective action and everything starts from this generating functional. It is applied to various problems of physics ranging from the relativistic field theory to the condensed matter physics. The main techniques used here are the on-shell expansion and the inversion method. The scheme presented in this article is thus a generating functional formalism of the many particle systems.

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Bio-inspired functional materials have received much attention for their potential to provide sustainable and advanced materials. The lotus effect has proven to be one of the most remarkable biomimetic effects since it was discovered by Barthlott. A superhydrophobic surface with the ability to bounce water droplets is the origin of the self-cleaning mechanism that keeps the surface clean by removing dust using water droplets moving with momentum. We have developed a crystal growth technique (CGT) of photochromic diarylethenes over the past decade, and from this, we fabricated a surface structure that closely resembles the natural lotus leaf’s characteristic of controlling the Laplace pressure and clarified the importance of the double roughness structure of the surface. The bouncing ability is also discussed in terms of the characteristic size of the double roughness structure theoretically. Moreover, this work clarifies the exquisiteness of the double roughness structure of the leaf. We also show that the CGT is a versatile technique with the potential to fabricate desired structured surfaces.