Eigenvalue equations evolution times

Indeed, one can analyze In the same manner the evolution of the system under consideration under conditions of reversibility of all of the elementary reactions in scheme (3.30). Unfortunately, in this situation the analytic solution of the eigenvalue equation in respect to parameter X appears unreasonably awkward. However, if the kinetic irreversibility of both nonlinear steps are a priori assumed, it is easy to find stationary valued (Y, Z ), and we come to the preceding oscillating solution. At the same time, near thermodynamic equilibrium (i.e., at R aa P), there exits only a sole and stable stationary state of the system with (Y Z R). 

Time evolution of this correlation function is ruled by the eigenvalues of operator T(0), which are determined by the equation... 

The matrix eigenvalues are analyzed for the stationary state in respect to the concentrations of aU independent intermediates. In the thermody namic representation of kinetic equations, independent variables that describe the system evolution in time are not the concentrations but ther modynamic rushes A of the intermediates. Hence, the analysis of the sta bihty criterion in terms of thermodynamic variables needs an inspection of eigenvalues of matrix M p = However, the specific form of the... 

The original equation of motion preserves the trace of the density matrix. This implies that the time evolution eigenvalues sum to unity, and hence the growing state found above must be compensated by a state going negative. This is easily found to be given by... 

The stability that we mentioned before refers to the evolution of the deviation vector ( ) = x (t) — x(l) between the perturbed solution x (t) and the periodic orbit x(t) at the same time t. If ( ) is bounded, then the periodic orbit is stable. In this case two particles, one on the periodic orbit x t) and the other on the perturbed orbit x (t), that start close to each other at t = 0, would always stay close. A necessary condition is that all the eigenvalues of the monodromy matrix be on the unit circle in the complex plane. However, in a Hamiltonian system this condition is not enough for stability, because there is only one eigenvector corresponding to the double unit eigenvalue and consequently a secular term always appears in the general solution, as can be seen from Equation (49). We remark that this secular term appears if the vector of initial deviation (0) = a (0) — s(0) has a component along the direction /2(C)). 

Dirac notation (p. 19) time evolution equation (p. 20) eigenfunction (p. 21) eigenvalue problem (p. 21) stationary state (p. 22) measurement (p. 22) mean value of an operator (p. 24) spin angular momentum (p. 25) spin coordinate (p. 26)... 

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