Linear theory of random textures of 3He-A in aerogel

Abstract

Spacial variation of the orbital part of the order parameter of $^3$He-A in aerogel is represented as a random walk of the unit vector $\mathbf{l}$ in a field of random anisotropy produced by the strands of aerogel. For a range of distances, where variation of $\mathbf{l}$ is small in comparison with its absolute value correlation function of directions of $\mathbf{l}(\mathbf{r})$ is expressed in terms of the correlation function of the random anisotropy field. With simplifying assumptions about this correlation function a spatial dependence of the average variation $\langle\delta\mathbf{l}^2\rangle$ is found analytically for isotropic and axially anisotropic aerogels. Average projections of $\mathbf{l}$ on the axes of anisotropy are expressed in terms of characteristic parameters of the problem. Within the "model of random cylinders" numerical estimations of characteristic length for disruption of the long-range order and of the critical anisotropy for restoration of this order are made and compared with other estimations .