Dynamical system theory dissipative systems

Prigogine thus arrived at the idea that the description of unstable dynamical systems requires an extension of dynamics. As a result, the temporal symmetry of the theory is broken. Irreversibility appears as a result of the scission of the unitary group describing the evolution into two dissipative, nonunitary semigroups. 

Such fluctuations of the photon flux, emitted from a molecule, have been predicted to be due to cooperative effects (21.22). The theory is based on an idea of Prigogine and coworkers (s. e.g.(22)) who treated the irreversible part of a physical process by transforming the wavefunctions of a dissipative system into another space using a "dynamical" non-unitary representation D = exp(-iVT /fI) with a "star-Hermitian" time operator 3 and V describing the interaction of a relevant local system Hq, e.g. the complex chromophore, and the total system H, i.e. our crystal. In the new representation y>=D Y no additional time dependence is introduced, dD/dt = 0, any expectation value of an operator M=DMD should be unchanged = M> and the total Hamiltonian is transformed by 1T=DHD 1 = Hq to the local system Hamiltonian (21.22). To describe the time development in the new representation, the electron density... 

Willems, J. C. Dissipative Dynamical Systems, Part I General Theory," Arch, Rational Mech. Anal., 45, 321-350 (1974). 

On the theoretical physics side, the Kolmogorov-Arnold-Moser (KAM) theory for conservative dynamical systems describes how the continuous trajectories of a particle break up into a chaotic sea of randomly disconnected points. Furthermore, the strange attractors of dissipative dynamical systems have a fractal dimension in phase space. Both these developments in classical dynamics—KAM theory and strange attractors—emphasize the importance of nonanalytic functions in the description of the evolution of deterministic nonlinear dynamical systems. We do not discuss the details of such dynamical systems herein, but refer the reader to a number of excellent books on the... 

Voth, G. A., Chandler, D., MiUer, W. H. (1989). Rigorous formulation of quantum transition state theory and its dynamical corrections. J. Chem. Phys. 91,7749-7760. Weiss, U. (1993). Quantum Dissipative Systems, World Scientific, Singapore. 

ABSTRACT We present a dynamical scheme for biological systems. We use methods and techniques of quantum field theory since our analysis is at a microscopic molecular level. Davydov solitons on biomolecular chains and coherent electric dipole waves are described as collective dynamical modes. Electric polarization waves predicted by Frohlich are identified with the Goldstone massless modes of the theory with spontaneous breakdown of the dipole-rotational symmetry. Self-organization, dissipativity, and stability of biological systems appear as observable manifestations of the microscopic quantum dynamics. 

Differential equations of pure mechanical systems generate transformation groups for which the Lebesgue measure is invariant this statement is called the Liouville theorem. Major results of the modern theory of dynamic systems are connected with physical sciences, mostly with mechanics. Differential equations of physics may refer to particle or planetary motions described by ordinary differential equations, or to wave motion described by partial differential equations. Dissipative effects are neglected in all these systems, and so the emphasis is on conservative or Hamiltonian systems. 

Bearing in mind the above agreements, the spin dynamics of superfluid helium-3 can be formulated within the usual Hamiltonian mechanics, and it appears to be part of the general theory of dynamical systems on orbits of Lie groups. In what follows we ignore the physically important dissipation effects (there may, in fact, be taken into account) and examine a conservative Hamiltonian system, referred to as Leggett equations, which is given by ... 

Decoherence is an essential concept appearing in a system in which a quantum subsystem contacts classical subsystem(s) in one way or another. As is widely recognized, the SET cannot describe this dynamics since there is no mechanism in it to switch off the electronic coherence along the nuclear path. The decoherence problem is critically important not only in our nonadiabatic dynamics but in other contemporary science such as spin-Boson dynamics in quantum computation theory and more extensively a quantum theory in open (dissipative) systems [147]. The decoherence problem is also critical to chaos induced by nonadiabatic djmamics [136, 137,182, 453, 454]. Therefore, in the rest of this section, we pay deeper attention to the aspect of the effect of electronic state decoherence strongly coupled with the relevant nuclear motion. A review about the notion of decoherence related to quantum mechanical measmement theory is found in the papers by Rossky et al. [53]. 

Many theories on the nonlinear dynamics of dissipative systems are based on the first-order ordinary differential equations... 

The key quantity in quantum dissipative dynamics is the reduced system density operator, ps(t) = trBPT(0> Ihe bath-subspace trace over the total composite density operator. It is worth mentioning here that the harmonic bath described above assumes rather Gaussian statistics for thermal bath influence. Realistic anharmonic environments usually do obey Gaussian statistics in the thermodynamic mean field limit. For general treatment of nonperturbative and non-Markovian quantum dissipation systems, HEOM formalism has now emerged as a standard theory. It is discussed in the next section. 

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

A chemical relaxation technique that measures the magnitude and time dependence of fluctuations in the concentrations of reactants. If a system is at thermodynamic equilibrium, individual reactant and product molecules within a volume element will undergo excursions from the homogeneous concentration behavior expected on the basis of exactly matching forward and reverse reaction rates. The magnitudes of such excursions, their frequency of occurrence, and the rates of their dissipation are rich sources of dynamic information on the underlying chemical and physical processes. The experimental techniques and theory used in concentration correlation analysis provide rate constants, molecular transport coefficients, and equilibrium constants. Magde" has provided a particularly lucid description of concentration correlation analysis. See Correlation Function... 

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