Dynamical systems symmetry breaking

In the MPC theory, the problem is not even posed. One starts defining the purely mathematical concept of dynamical system without any reference to a representation of reality. (The baker s transformation or the Bernoulli shift are obvious examples.) From here on, one proves mathematically the existence of a class of abstract dynamical systems (K-flows) that are intrinsically stochastic —that is, that possess precise mathematical properties (including a temporal symmetry breaking that can be revealed by a change of representation). 

MSN.171.1. Prigogine and T. Petrosky, Laws of nature, probability and time symmetry breaking, in Generalized Functions, Operator Theory and Dynamical Systems, I. Antoniou and G. Lumer, eds.. Chapman Hall, London, pp. 99-110, 1999. 

BIFURCATIONS AND SYMMETRY BREAKING IN FAR-FROM-EQUILIBRIUM SYSTEMS TOWARD A DYNAMICS OF COMPLEXITY... 

This simple example also displays the importance of nonlinear dynamics. The unsealed elasticities are constant only for a linear system and hence result in the same behavior if a linear system is stable, it will remain stable when boundary conditions change, and if it is unstable, it will remain unstable. Under nonlinear equations, however, the values of elasticity depend on the system state, which varies with the boundary conditions. This may lead to a transition from a stable to unstable system, known as bifurcation or symmetry breaking. This means that nonlinearity causes a variety of new behaviors in a system. 

Tamura R, Takahashi H, Fujimoto D, Ushio T (2007) Mechanism and Scope of Preferential Enrichment, a Symmetry-Breaking Enantiomeric Resolution Phenomenon. 269 53-82 Thar J, Reckien W, Kirchner B (2007) Car-Parrinello Molecular Dynamics Simulations and Biological Systems. 268 133-171... 

Rate events are fluctuations and statistical averaging requires a large number of them. If the time scale of averaging is long compared with the amplification of the fluctuations, symmetry breaking occms and one enantiomer dominates. This view is in line with mathematical analysis which shows that macroscopic behavior derived from collective dynamics of microscopic components cannot be modeled using spatially continuous density functions. One needs to take into account the actual individual/discrete character of the microscopic components of the system. 

Effects due to the geometric phase (GP) have been reported by Joubert-Doriol, Ryabinkin and Izmaylov.In particular they report on symmetry breaking and spatial localisation, and on GP effects studied with the multi-dimensional LVQ model. In the first study by Ryabinkin and Izmaylov the ground state dynamics is considered of a two-state system approximated by (a) a Hamiltonian of a two-state Cl model, (b) the Born-Oppenheimer (BO) model and (c) a BO model augmented with an explicit GP dependence in the kinetic energy operator. It is demonstrated that... 

---