Geometrical phases and symmetries in dissipative systems

Shear-induced transitions in colloidal and polymeric liquids

This review reflects different aspects of wide current studies of the phenomena related to the shear-induced structure transformation in various complex liquids. Experimental data, being the basis of this discussion, were obtained for polymeric liquids (melts, blends, solutions) and different dispersions (colloidal solutions, suspensions, emulsions). The general initial input of shearing is the creation of inhomogeneities which can continue to remain as separate domains, become the nuclei of new phases, or become diffuse, leading to phase separation. The following effects are discussed: diffusion-induced phase separation, phase transitions occurring mainly due to the deformation-driven orientation of polymer chains and worm-like micelles that results in the formation of a liquid-crystal state, as well as self-assembly effects. It can be stressed that the appearance of regular structures takes place in systems that can coexist in different concentrations or phase states at the same stress or shear rate. This is linked with the existence of two-value points on flow curves (part of a flow curve with negative slope) or transient regimes of deformation that lead to instability of the flow. The described experimental facts are briefly discussed on the basis of the application of different constitutive molecular or phenomenological rheological models.

Few-Mode Geometric Description of a Driven-Dissipative Phase Transition in an Open Quantum System

By example of the nonlinear Kerr mode driven by a laser, we show that hysteresis phenomena in systems featuring a driven-dissipative phase transition can be accurately described in terms of just two collective, dissipative Liouvillian eigenmodes. The key quantities are just two components of a non-Abelian geometric connection, even though a single parameter is driven. This powerful geometric approach considerably simplifies the description of driven-dissipative phase transitions, extending the range of computationally accessible parameter regimes, and providing a new starting point for both experimental studies and analytical insights.

Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory

Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and, in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems has opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturbation theory to analytically calculate the Hannay angle for the van der Pol oscillator's limit cycle trajectory when its parameters-the strength of the nonlinearity and the frequency of the linear part-evolve cyclically and adiabatically. For this van der Pol oscillator, we also numerically calculate the corresponding geometric phase and establish its equivalence with the Hannay angle.