We investigate how the interaction of particles mediated by an elastic deformation of a nematic liquid crystal is influenced by the initial deformation of the director field. To this end, we calculate the interaction energy between particles in a nematic cell with hybrid boundary conditions, homeotropic on the surface of one confining plate and planar on the other. We find an analytic form of the interaction energy in the case of weak anchoring on the surface of the particle. This interaction energy sensitively depends on the position of the two particles as well as the interparticle distance and can be nonmonotonic with a minimum in its landscape. This nontrivial energy landscape might lead to a chainlike superstructure of particles.

Stochastic dynamics in the energy representation is used as a method to represent nonequilibrium Brownian-like systems. It is shown that the equation of motion for the energy of such systems can be taken in the form of the Langevin equation with multiplicative noise. Properties of the steady states are examined by solving the Fokker-Planck equation for the energy distribution functions. The generalized integral fluctuation theorem is deduced for the systems characterized by the shifted probability flux operator. From this theorem, a number of entropy and fluctuation relations such as the Evans-Searles fluctuation theorem, the Hatano-Sasa identity, and the Jarzynski's equality are derived.

Colloidal suspension show interesting phenomena which are mediated by the solvent. Recently, colloids in nematic liquid crystal phase used as anisotropic solvent have been investigated experimentally and have inspired theoretical work[1-3]. In relation to these work, effective pair potential between spherical colloidal particles suspended in nematic liquid crystals has been suggested[4,5]. The interparticle interaction can cause nontrivial collective behavior that results in formation of new spatial structures in colloidal particle systems. We determine these structures by particle dynamics simulations using these effective pair-interaction.

We obtain the analytical solutions of the classical and quantum cosmological models with exponential dependence of the fundamental scalar-field potential in homogeneous and inhomogeneous cases. At the homogeneous scalar field, we get the equation for the description of the dynamics of the scale factor for any type of Universe. The solution of such equation gives linear time dependence of the scale factor. At the inhomogeneous change of the scalar field, we find the new solution describing the formation of the Universe.