Chaotic approximation

Show that the equality of adsorption and desorption rates for dissociating molecules, derived in the mean field and chaotic approximations for interacting the nearest neighbors, do not satisfy the equations of isotherms in similar approximations (this means the absence of a self-consistency between description of the equilibrium and dynamic characteristics of the system). Check out, that the discussed self-consistency property is fulfilled for equations in the quasi-chemical approximation. 

Shirts R B and Reinhardt W P 1982 Approximate constants of motion for classically chaotic vibrational dynamics vague tori, semiclassical quantization, and classical intramolecular energy flow J. Cham. Phys. 77 5204-17... 

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

Statistically, in a high-pressure region, an ion will be struck by neutral molecules randomly from all angles. The ion receives as many collisions from behind as in front and as many collisions from one side as from the other. Therefore, it can be expected that the overall forward motion of the ion will be maintained but that the trajectory will be chaotic and similar to Brownian motion (Figure 49.4b). Overall, the ion trajectory can be expected to be approximately along the line of its initial velocity direction, since it is still influenced by the applied potential difference V. 

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

Next, consider statistical properties of the eigenfunctions of the chaotic two-dimensional quantum billiards in hard wall approximation... 

While the conditions 1,2 can be verified approximately by simulation, proving the condition 3 is very difficult. Note that in many studies of chaotic behavior of a CSTR, only the conditions 1,2 are verified, which does not imply chaotic d3mamics, from a rigorous point of view. Nevertheless, the fulfillment of conditions 1,2, can be enough to assure the long time chaotic behavior i.e. that the chaotic motion is not transitory. From the global bifurcations and catastrophe theory other chaotic behavior can be considered throughout the disappearance of a saddle-node fixed point [10], [19], [26]. 

In an isotropic composite (chaotic distribution of fibre direction) E is given by = 0.2 Epar + 0.8 transv The first term follows from the answer to Question 19, the second one from 20, and the result is = 0.2 21 + 0.8 4.9 = 8.1 GPa. This value, therefore, approximates the highest attainable value with 25 vol% of glass... 

Many kinds of molecular systems pumped by a strong laser light show chaotic dynamics. Indeed, in a semiclassical model of a multiphoton excitation on molecular vibration, chaos was discovered by Ackerhalt et al. [85] and theoretically and numerically investigated in detail [86,87]. Moreover, the equations of motion that describe a rotating molecule in a laser field can exhibit a chaotic behavior and have been applied in the classical case of a rigid-rotator approximation [87,88]. 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

This formula was first derived in ref. 6 when calculating the kinetics of donor luminescence decay in the presence of the randomly, i.e. chaotically, located acceptors under the condition n N and on the assumption of the resonance exchange mechanism of energy transfer. Similar equations were later used for the analysis of experimental data on the kinetics of electron tunneling reactions obtained under conditions of the chaotic distribution of the reagents and at n < N. As a rule, only the first term of the exponent in eqn. (23) has been taken into account, which is equivalent to employing the previously mentioned (see Sect. 2.1) stepwise approximation of the function 0(R,t) = exp[- 1V(jR)(]. In this case, one obtains... 

Ya.B. s more recent papers have been devoted to the study of nonlinear problems. In 1966 Ya.B. turned his attention to the stabilizing effect of accelerated motion through a hot mixture of a boundary of intersection of two flame fronts, convex in the direction of propagation, and proposed an approximate model of a steady cellular flame. G. I. Sivashinsky, on the basis of this work, proposed a nonlinear model equation of thermodiffusional instability which describes the development of perturbations of a bent flame in time and, together with J. M. Michelson, studied its solution near the stability boundary Le = Lecrit. It was shown numerically that the flat flame is transformed into a three-dimensional cellular one with a non-steady, chaotically pulsating structure. The formation of a two-dimensional cellular structure was also the subject of a numerical investigation by A. P. Aldushin, S. G. Kasparyan and K. G. Shkadinskii, who obtained steady flames in a wider parameter interval. 

Various approximation of this model (e.g. those of molecular splitting, pseudo-chemical, chaotic and molecular field approximations) were reported and analyzed by Tovbin and Fedyanin [88, 89]. In particular, the pseudochemical approximation for chemisorption kinetics is of the form... 

Fig. 8.7. The desorption rates K/W for T = 840K, calculated for the chaotic (1), polynomial (2), and quasi-chemical (3) approximations [111], where V = = ) symbols...

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