Fokker-Planck quantum equation

Analytical solutions of quantum Fokker-Planck equations such as Eq. (63) are known only in special cases. Thus, some special methods have been developed to obtain approximate solutions. One of them is the statistical moment method, based on the fact that the equation for the probability density generates an infinite hierarchic set of equations for the statistical moments and vice versa. 

The s terms in Eq. (80) contribute only the term E,2 in Eq. (97). Thus, the term represents the quantum diffusional. v-terms in the Fokker-Planck equation. The other terms in Eqs. (93)-(100) originate in the drift terms of the Fokker-Planck equation. The terms B12 and C in Eqs. (93)-(94) play the role of feedback terms that pump quantum fluctuations into the classical Bloembergen equations. If the s terms in Eq. (80) do not appear (the classical case), the term in Eq. (97) does not appear, either. In this case the subset (95)—(100) with zero initial conditions has zero solutions and in consequence leads to the first truncation [171]. 

S. Zhang and E. Poliak (2003) Quantum dynamics for dissipative systems A numerical study of the Wigner-Fokker-Planck equation. J. Chem. Phys. 118, p. 4357... 

We perform concrete calculations in the complex P-representation [Drummond 1980 McNeil 1983] in the frame of both probability distribution functions and stochastic equations for the complex c-number variables. We follow the standard procedures of quantum optics to eliminate the reservoir operators and to obtain a master equation for the density operator of the modes. The master equation is then transformed into a Fokker-Planck equation for the P-quasiprobability distribution function. In particular, for an ordinary NOPO and in the case of high cavity losses for the pump mode (73 7), if in the operational regime the pump depletion effects are involved, this approach yields... 

It is impossible to do justice within the limited extent of one chapter to all theoretical developments. For example, I will omit methods relating to the Fokker-Planck equation representation of the dynamics. This includes the method of adiabatic elimination discussed extensively in Ref. 33 or the approach based on the Rayleigh quotient, developed by Talkner (34,35). There are a number of reviews, monographs, and special journal issues devoted to the theory of activated rate processes (5,13,14,36-40), the interested reader is urged to consult them. I will also omit any quantum theory of activated rate processes. The thread which connects the material presented in this chapter will be the use of the Hamiltonian equivalent form of the STGLE and more general forms to derive the classical theory of activated rate processes. 

Through many enduring classic texts, such as Haken s Synergetics and Information and Self-Organization, Gardiner s Handbook of Stochastic Methods, Risken s The Fokker Planck-Equation or Haake s Quantum Signatures of Chaos, the series has made, and continues to make, important contributions to shaping the foundations of the field. 

Despite the fact that the exact j or TZ can be formally expressed in terms of an infinite series expansion, its evaluation, however, amounts to solve the total composite system of infinite degrees of freedom. In practice, one often has to exploit weak system-bath interaction approximations and the resulting COP [Eq. (1.2)] and POP [Eq. (1.3)] of QDT become nonequivalent due to the different approximation schemes to the partial consideration of higher order contributions. It is further noticed that in many conventional used QDT, such as the generalized quantum master equation, Bloch-Redfield theory and Fokker-Planck equations, there involve not only... 

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. 

F.6.4.2. Lineshape Models. The Mossbauer lineshape can be influenced by all relaxation modes of the Fokker-Planck equation (see Section D.3). Because the relative importance of these modes depends on their population, it should be necessary to know both the eigenvalues of Brown s equation and the amplitudes of the associated modes. In fact, to determine the lineshape, it is necessary to connect the dynamics of the stochastic vector m given by Brown s equation with the quantum dynamics of the nuclear spin. This necessitates the use of superoperator Fokker-Planck equations and, to our knowledge, the problem has not yet been completely solved. 

The parameters Cq, C, Ce, and are model constants and need to be specified [30,31]. The same goes for the pressure dilatation term lid [31,32]. The transport equations for all of the SGS moments are readily obtained by integration of this Fokker-Planck equation. This provides a complete statistical description of turbulence. The idea is to find methods that could take advantage of quantum resources in order to speed up these calculations, at least polyno-mially in the number of variables. Because of the size of the problem typically considered, such a speedup could transform the way these problems are treated in engineering providing solutions to problems many orders of magnitude faster than are possible with classical computers. 

The three-pulse EOM-PMA can be formulated not only in terms of density matrices and master equations but also in terms of wavefunctions and Schrodinger equations [29]. The EOM-PMA can therefore be straightforwardly incorporated into computer programs which provide the time evolution of the density matrix or the wavefunction of material systems. Besides the multilevel Redlield theory, the EOM-PMA can be combined with the Lindblad master equation [49], the surrogate Hamiltonian approach [49], the stochastic Liouville equation [18], the quantum Fokker-Planck equation [18], and the density matrix [50] or the wavefunction [14] multiconfigurational time-dependent Hartree (MCTDH) methods. When using the... 

Quantum mechanics and statistical mechanisms did not change. Starting from the basic equations - Schrodinger, Langevin, Fokker-Planck - one has developped very good methods of solution as the simulation methods which take into account most of the observed features and, perhaps, some imaginary ones. 

In this section, which should also be viewed as an introduction to the much harder quantum mechanical problems, we wiU briefly outline some of the essential properties of the dynamical phase space density, going back to what we already treated in this chapter, and show how those ideas can be cast in the form called the classical Liouville equation and the Fokker-Planck equation. 

So far aU we have written about are classical problems, but there are definite quantum aspects to the proton transfer reaction and, in addition, to the interaction of systems with light and dissipation mechanisms. Vibrations cannot always be treated classically either. At the outset, it should be stated that, unlike the classical description, which allows a relatively simple mechanism for dissipation by means of the Fokker-Planck equation, quantum mechanics does not allow such a possibihty. In addition, aU treatments based on a mixed description of a classical and a quantum system are fundamentally flawed [29]. In the remainder of this chapter, we nevertheless introduce a possible description that looks formaUy the same in classical and quantum mechanics and has some features that make it possible to introduce an elementary mechanism of decay to the equilibrium state. And, equaUy important, it gives a mechanism for averaging over the strongly coupled vibrational modes, as weU as a unified description of the interaction with light. 

Therefore, decay to a final equilibrium state should be inherent in a proper dynamical description of a system. It was indicated at the end of Section 9.10 that in classical systems this decay is well described by the formalism of the Fokker- Planck equation, which itself was shown to be an extension of the classical Liouville equation. In view of the similarities between the classical and quantum Liouville equations, it seems a natural question to ask whether it is also possible to find an extension of the quantum Liouville equation that lets any initial density operator decay to the equilibrium density operator for that system. 

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