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A general theory of electronic transitions in condensed media is developed which describes in particular various possible chemical reactions in condensed media. The approximations used are : the Born-Oppenheimer adiabatic approximation, the Condon approximation to describe the dependence of the matrix elements of the electron transition, and the harmonic approximation to describe the nuclear motion in the initial and final electronic states. In the framework of these approximations general expressions are obtained for the rate constants of the corresponding reactions. I n a certain approximation these constants can be reduced to the form given by the Eyring theory of rate processes. In the other region the expressions for the rate constants are obtained which are of an essentially different character and are interpreted in terms of the theory of the multiphonon transitions. Reaetions are considered which occur only with a conformation transition of the system as well as those which are accompanied by conformation transitions and by breaking of a chemical bond.M rapMoHmecKoe n p a 6 n a w e~~e AJIR onkicaHm pEimeHiafi RAep B HCXOAHOM M Teopm ~~C O J I I~T H~I X cIcopocTefi p e a~~~f i 3 i i p~~r a . B zpyrofi 06JIaCTH a~a s e~~f i
An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary, and also a circular scatterer in the interior of the disk. We investigate stability properties of some periodic orbits in annular billiards in which the scatterer is touching or close to the boundary. We analytically show that there exist linearly stable periodic orbits of arbitrary period for scatterers with decreasing radii that are located near the boundary of the disk. As the position of the scatterer moves away from a symmetry line of a periodic orbit, the stability of periodic orbits changes from elliptic to hyperbolic, corresponding to a saddle-center bifurcation. When the scatterer is tangent to the boundary, the periodic orbit is parabolic. We prove that slightly changing the reflection angle of the orbit in the tangential situation leads to the existence of KAM islands. Thus we show that there exists a decreasing to zero sequence of open intervals of scatterer radii, along which the billiard table is not ergodic.A billiard is a dynamical system where a point particle moves with constant velocity inside a domain and experiences elastic collisions with the boundary of the domain. The shape of the boundary determines the dynamics of the billiard. Billiards in a disk on a plane are completely integrable, while annular billiard tables consisting of a particle confined between two nonconcentric disks generically display mixed phase space due to a family of regular orbits that never touch the scatterer. Billiard models find applications in a variety of problems in statistical 1 , classical and quantum 2 physics. In this paper, we consider annular billiard tables formed of a small circular scatterer placed in the interior of a unit circle; this is a popular geometry for microwave billiard experiments 3 . Circular boundaries allow us to analytically examine linear and nonlinear stability of some periodic orbits. Depending on the parameters of the problem, we find that there exist linearly stable orbits of arbitrarily large period. We show the existence of a saddle-center bifurcation as the parameters vary, corresponding to a change of stability from linearly elliptic to saddle type. Placing the scatterer tangentially to the external circle creates a cusp that is a source of singularities in the billiard. We use KAM theory to establish that in the cusp case, the periodic orbits are nonlinearly stable.
Articles you may be interested inComment on: Diffusion theory of multidimensional activated rate processes: The role of anisotropy J. Chem. Phys. 95, 1424 (1991); 10.1063/1.461126 Diffusion theory of multidimensional activated rate processes: The role of anisotropy J. Chem. Phys. 90, 1141 (1989); 10.1063/1.456169 Does reaction path curvature play a role in the diffusion theory of multidimensional activated rate processes? J. Chem. Phys. 88, 4765 (1988); 10.1063/1.454689 Theory of rate processes at metal surfaces. II. The role of substrate electronic excitationsThe influence of the nonequilibrium degrees of freedom of the electron-nuclear system on the rate of relaxation processes is investigated. The electron-nuclear system is modeled by two potential energy hypersurfaces-electronic terms, plus the perturbation causing transitions between these terms. A harmonic approximation with displaced equilibrium positions and with identical frequencies is assumed for a description of the terms. The equations describing rate processes in such a system are derived. These equations determine the change in time not only of mean occupation numbers of electronic levels but of the amplitudes of nuclear vibrations as well. The dependence of the rate constants on the amplitudes of nonequilibrium excited vibrations is investigated under various assumptions about nonequilibrium of the system. The possibility of excitation of coherent nuclear vibrations in the course of nonradiative transition and in the presence of a constant input on the upper electronic level is investigated as well. The conditions under which such coherent vibrations may arise are obtained, and the stationary amplitude of these vibrations is derived. Attempts are made to apply the theory for a description of enzyme catalysis.
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