Markov approximation systems

Importantly, the value of the results gained in the present section is not limited to the application to actual systems. Eq. (4.2.11) for the GF in the Markov approximation and the development of the perturbation theory for the Pauli equation which describes many physical systems satisfactorily have a rather general character. An effective use of the approaches proposed could be exemplified by tackling the problem on the rates of transitions of a particle between locally bound subsystems. The description of the spectrum of the latter considered in Ref. 135 by means of quantum-mechanical GF can easily be reformulated in terms of the GF of the Pauli equation. 

Figure 23. Radial segment density profile through a cross-section of a highly curved spherical vesicle. The origin is at r = 0, and the vesicle radius is very small, i.e. approximately r = 25 (in units of segment sizes). The head-group units, the hydrocarbons of the tails and the ends of the hydrocarbon tails are indicated. Calculations were done on a slightly more simplified system of DPPC molecules in the RIS scheme method (third-order Markov approximation), i.e. without the anisotropic field contributions...

The RC is an ideal system to test theoretical ideas (memory effect, coherence effect, etc.), fundamental approximations (isolated line approximation, Markov approximation, etc.), and techniques (generalized linear response theory, Forster-Dexter theory, Marcus theory, etc.) for treating ultrafast phenomena. As mentioned above, this ideality is mainly due to the fact that the electronic energy level spacing in RC is small (typically from 200 to 1500 cm-1), and the interactions between these electronic states are weak. 

Although a theoretical approach has been desecrated as to how one can apply the generalized coupled master equations to deal with ultrafast radiationless transitions taking place in molecular systems, there are several problems and limitations to the approach. For example, the number of the vibrational modes is limited to less than six for numerical calculations. This is simply just because of the limitation of the computational resources. If the efficient parallelization can be realized to the generalized coupled master equations, the limitation of the number of the modes can be relaxed. In the present approach, the Markov approximation to the interaction between the molecule and the heat bath mode has been employed. If the time scale of the ultrashort measurements becomes close to the characteristic time of the correlation time of the heat bath mode, the Markov approximation cannot be applicable. In this case, the so-called non-Markov treatment should be used. This, in turn, leads to a more computationally demanding task. Thus, it is desirable to develop a new theoretical approach that allows a more efficient algorithm for the computation of the non-Markov kernels. Another problem is related to the modeling of the interaction between the molecule and the heat bath mode. In our model, the heat bath mode is treated as... 

In conclusion, in this section we have proved that the Markov approximation requires some caution. The Markov approximation may be incompatible with the quantum mechanical nature of the system under study. It leads to the Pauli master equation, and thus it is compatible with the classical picture of a particle randomly jumping from one site to another, a property conflicting, however, with the rigorous quantum mechanical treatment, which yields Anderson localization. 

The Markov approximation, in which we assume that the correlation time of the EM field is much shorter than the timescale of radiation processes of the bare systems. This approximation is equivalent to the white-noise (broadband) description of the EM field modes, and allows us to replace p(f - x) by p(f). 

The procedure above has not in any sense derived the macroscopic relaxation equations only some formal conditions have been stated under which the structures of the microscopic and macroscopic equations become the same. One crucial point, which certainly deserves further comment, is the physical basis of the Markov approximation. This approximation removes the memory effects from (5.5) so that the structures of the microscopic and macroscopic equations become similar. For this approximation to be useful, the memory kernel must decay much more rapidly than the density fields. The projected time evolution will guarantee that this is the case, provided these fields decay much more slowly than other variables in the system. 

In a chemically reacting fluid, where the number of molecules of a given species is no longer conserved, the small k limit is not sufficient to guarantee this time-scale separation. For the Markov approximation to be useful, we must also require that the chemical reaction be slow. We can make this discussion more precise by using the same methods that have been applied in the past to justify this point in other contexts. We consider, for simplicity, the case of a spatially homogeneous system and take the fc-+0 limit of (5.4) at the outset to obtain... 

In many systems comprising a large number of particles, even though a detailed quantum treatment of all degrees of freedom is not necessary, there may exist subsets that have to be treated quantum mechanically under the influence of the rest of the system. If the typical timescales between system and bath dynamics are very different, Markovian models of quantum dissipation can successfully mimic the influence of the bath onto the system dynamics [2]. However, in the femtosecond regime studied with ultrashort laser pulses, the so-called Markov approximation is not generally valid [3]. Furthermore, very often the bath operators are assumed to be of a special form (harmonic, for instance) which are sometimes not realistic enough. 

Contributions from terms like Cs pyare smaU and ignored in the summation. Starting from the von Neumann-Liouville equation, a reduced density matrix for the system mode is derived using the time-dependent perturbation theory after tracing over the bath degrees of freedom. The commonly employed Markov approximation is avoided and no assumption of a separation in timescales between system and bath mode relaxation is invoked in this theory [84]. The ultimate result for the evolution of the ground state vibrational population is... 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

The Fokker-Planck equation is a special type of master equation, which is often used as an approximation to the actual equation or as a model for more general Markov processes. Its elegant mathematical properties should not obscure the fact that its application in physical situations requires a physical justification, which is not always obvious, in particular not in nonlinear systems. 

The difference between the Markov model lineshapes and those from the Smoluchowski model is particularly pronounced when the diffusion coefficient is of the order of the quadrupole coupling constant. In the limit of large diffusion coefficients, the two models converge, and in the limit of low diffusion coefficients, the spectra are dominated by small-amplitude oscillations within potential wells, which can be approximately modelled by a suitable Markov model. This work strongly suggests that there could well be cases where analysis of powder pattern lineshapes with a Markov model leads to a fit between experimental and simulated spectra but where the fit model does not necessarily describe the true dynamics in the system. 

Chvosta et alP consider another case where the work probability distribution function can be determined. They study a two energy-level system, modelled as a stochastic, Markovian process, where the transition rates and energies depend on time. Like the previous examples it provides an exact model that can be used to assist in identifying the accuracy of approximate, numerical studies. Ge and Qian extended the stochastic derivation for a Markovian chain to a inhomogeneous Markov chain. 

This is the Markovian memory-less approximation to the Master Equation. In this approximation, the effective time evolution operator becomes independent of t and the integral may be extended to infinity. It is also consistent to assume that the system lost memory of the initial state of the reservoir, whatever this was. In the limit when Uq is calculated in perturbation theory and pq(0) = 0, we obtain the conventional Born-Markov time evolution which has a long and successful history. 

Approximate models Steady-state distributions and partuneters ace known for many stochastic processes e.g., queueing, inventory, Markov chains. These results ctm be used to approximate the simulation model. For example, a service system can be approximated by a Markovian queue to determine the expected number of customers in the system. This value can be used to set the initial number of customers in the system for the simulation, rather them using the (convenient) initial condition of an empty system. Chapter 81 of the Handbook is a good source of approximations. Even cruder approximations, such as replacing a random quantity by its expectation, can also be used. 

Several examples of NMR studies of copolymers that exhibit Bernoullian sequence distributions but arise from non-Bernoullian mechanisms have been reported. Komoroski and Schockcor [11], for example, have characterised a range of commercial vinyl chloride (VC)/vinylidene chloride (VDC) copolymers using carbon-13 NMR spectroscopy. Although these polymers were prepared to high conversion, the monomer feed was continuously adjusted to maintain a constant comonomer composition. Full triad sequence distributions were determined for each sample. These were then compared with distributions calculated using Bernoullian and first-order Markov statistics the better match was observed with the former. Independent studies on the variation of copolymer composition with feed composition have indicated that the VDC/VC system exhibits terminal model behaviour, with reactivity ratios = 3.2 and = 0.3 [12]. As the product of these reactivity ratios is close to unity, sequence distributions that are approximately Bernoullian are expected. 

Contents A Historical Introduction. - Probability Concepts. -Markov Processes. - The Ito Calculus and Stochastic Differential Equations. - The Fokker-Planck Equatioa - Approximation Methods for Diffusion Processes. - Master Equations and Jump Processes. - Spatially Distributed Systems. - Bistability, Metastability, and Escape Problems. - Quantum Mechanical tokov Processes. - References. - Bibliogr hy. - Symbol Index. - Author Index. - Subject Index. 

The problems of randomly excited nonlinear systems are diverse, the majority of which must be solved by some suitable approximate procedures. Possibility for mathematically exact solutions does exist however. It exists only when random excitations are independent at any two instants of time, and the system response, represented as a vector in a state space, is a Markov vector. In this case the probability density of the system response satisfies a parabolic par-... 

Our numerical self-consistent field (SCF) calculations are based on the model developed by Scheutjens and Fleer. In this treatment, the phase behavior of polymer systems is modeled by combining Markov chain statistics with a mean field approximation for the free energy. The equations in this lattice model are solved numerically and self-consistendy. The self-consistent potential is a function of the polymer segment density distribution and the Flory-Huggins interaction parameters, or... 

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