Generalized Liouville equation

Recall from the section on non-Hamiltonian dynamics that the phase space distribution function satisfies a generalized Liouville equation ... 

APPENDIX 2 Geometric Derivation of the Generalized Liouville Equation... 

A generalization of Liouville s equation was presented in [10]. Applying modern techniques in differential geometry [13], as they are applied to dynamical systems [14-16], gives rise to the generalized Liouville equation of the form [17] ... 

It is important to note that it is possible to construct a distribution function that satisfies the general Liouville equation, (65), while not generating the particular phase space distribution function corresponding to the given d3mamical system. To illustrate, consider a distribution function that is constructed from a product of an arbitrary subset of the -functions of the conservation laws. This distribution function would also satisfy the generalized Liouville equation and be of the form ... 

Steeb, W. H. (1979). Generalized Liouville equation, entropy and dynamic systems containing limit cycles. Physica, 95A, 181-90. 

Now, we can show an important property of the generalized Liouville equation. If we write... 

The generalized Liouville equation states the conservation of the total number of systems in an ensmble. If U = (q, p), this conservation law can be written... 

The density operator r(<) is a Hermitian and positive function of time, and satisfies the generalized Liouville-von Neumann (LvN) equation(47, 45)... 

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

Section II deals with the general formalism of Prigogine and his co-workers. Starting from the Liouville equation, we derive an exact transport equation for the one-particle distribution function of an arbitrary fluid subject to a weak external field. This equation is valid in the so-called "thermodynamic limit , i.e. when the number of particles N —> oo, the volume of the system 2-> oo, with Nj 2 = C finite. As a by-product, we obtain very easily a formulation for the equilibrium pair distribution function of the fluid as well as a general expression for the conductivity tensor. 

After these technical preliminaries, we may consider the solution of the Liouville equation (10). However, we shall not discuss the most general situation but we shall limit ourselves to the special case where ... 

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

Remark. From the linear integro-differential equation for P(y, t) we have derived a nonlinear equation for y(t). Thus the essentially linear master equation may well correspond to a physical process that in the laboratory would be regarded as a nonlinear phenomenon inasmuch as its macroscopic equation is nonlinear. This is not paradoxical provided one bears in mind that the distinction between linear and nonlinear is defined for equations. It is wrong to apply it to a physical phenonemon, unless one has agreed upon a specific mathematical description of it. Newton s equations for the motion of the planets are nonlinear, but the Liouville equation of the solar system is linear. This connection between linear and nonlinear equations is not a matter of approximation the linear Liouville equation is rigorously equivalent with the nonlinear equations of motion of the particles. Generally any linear partial... 

Having treated in the previous two sections linear stochastic differential equations we now return to the general case (1.1). Just as normal differential equations, it can be translated into a linear equation by going to the associated Liouville equation. To do this we temporarily take a single realization y(t) of Y(t) and consider the non-stochastic equation... 

The evolution of the molecule is described by the generalized Liouville-von Neumann equation [35, 36]... 

Partitioning technique refers to the division of data into isolated sections and it was put into successful practice in connection with matrix operations. Lowdin, in his pioneering studies, [21, 22] developed standard finite dimensional formulas into general operator transformations, including treatments appropriate for both the bound state and the continuous part of the spectrum, see also details in later appendices. Complementary generalizations to resonance-type problems were initiated in Ref. [23], and simple variational formulations were demonstrated in Refs. [24,25]. Note that analogous forms were derived for the Liouville equation [26, 27] and further developed in connection with a retarded-advanced subdynamics formulation [28]. 

Abstract. In this chapter we discuss approaches to solving quantum dynamics in the condensed phase based on the quantum-classical Liouville method. Several representations of the quantum-classical Liouville equation (QCLE) of motion have been investigated and subsequently simulated. We discuss the benefits and limitations of these approaches. By making further approximations to the QCLE, we show that standard approaches to this problem, i.e., mean-field and surface-hopping methods, can be derived. The computation of transport coefficients, such as chemical rate constants, represent an important class of problems where the QCL method is applicable. We present a general quantum-classical expression for a time-dependent transport coefficient which incorporates the full system s initial quantum equilibrium structure. As an example of the formalism, the computation of a reaction rate coefficient for a simple reactive model is presented. These results are compared to illuminate the similarities and differences between various approaches discussed in this chapter. 

To take the quantum-classical limit of this general expression for the transport coefficient we partition the system into a subsystem and bath and use the notation 1Z = (r, R), V = (p, P) and X = (r, R,p, P) where the lower case symbols refer to the subsystem and the upper case symbols refer to the bath. To make connection with surface-hopping representations of the quantum-classical Liouville equation [4], we first observe that AW(X 1) can be written as... 

In this chapter, the ultrafast radiationless transition processes are treated theoretically. The method employed is based on the density matrix method, and specifically, a generalized linear response theory is developed by applying the projection operator technique on the Liouville equation so that non-equilibrium cases can be handled properly. The ultrafast molecular... 

In view of the fact that the correlation function for the random force, as given by Eq. [16], is a Dirac 8 function, the strict Langevin equation (Eq. [15]) is not amenable to computer simulation. In order to circumvent the above difficulty, it is convenient to describe the motion of the fictitious particle by the generalized Langevin equation. The generalized Langevin equation, which can be derived from the Liouville equation (11), along with the supplementary conditions are... 

The crisis of the GME method is closely related to the crisis in the density matrix approach to wave-function collapse. We shall see that in the Poisson case the processes making the statistical density matrix become diagonal in the basis set of the measured variable and can be safely interpreted as generators of wave function collapse, thereby justifying the widely accepted conviction that quantum mechanics does not need either correction or generalization. In the non-Poisson case, this equivalence is lost, and, while the CTRW perspective yields correct results, no theoretical tool, based on density, exists yet to make the time evolution of a contracted Liouville equation, classical or quantum, reproduce them. 

In this subsection we illustrate the attempt made in Ref. 59 to derive a generalized diffusion equation from the Liouville approach described in Section III. We are addressing the apparently simple problem of establishing a density equation corresponding to the simple diffusion equation... 

It is now well understood that this is not an approximation rather it is a way to force an equation with infinite memory to become compatible with Levy diffusion. The assumption (152) makes it possible for us to get rid of the time convolution nature of the generalized diffusion equation (133). At the same time, this key relation replaces the correlation function (t) with its second-order derivative and, as a consequence of Eq. (147), with the waiting time distribution /( ). This fact is very important. In fact, any Liouville-like approach makes the correlation function 3F(f) enter into play. The CTRW is a perspective resting on trajectories and consequently on /(f). Establishing a connection between the two pictures implies the conversion of 4> (f) into j/(r), or vice versa. Here, this conversion has been realized paying the price of altering the physics of the generalized diffusion equation (133). 

This is a plausible way to prove that Eq. (162) is the diffusion equation that applies to the gaussian condition. It is important to point out that a more satisfactory derivation of this exact result can be obtained by using the Zwanzig projection approach of Section III [67,68]. Thus, Eq. (162) as well as Eq. (133) must be considered as generalized diffusion equations compatible with a Liouville origin. 

In Section IV we have shown that the CTRW can be thought of as being the solution of an equation of motion with the same time convoluted structure as the GME derived from a Liouville or Liouville-like approach by means of a contraction on the irrelevant degrees of freedom. However, this formal equivalence may not imply an equivalence of physical meanings. This review affords the information necessary to establish this important point. Let us consider, for instance, the diffusion process dx/dt = widely discussed in this chapter. From within the density method treatment, this equation has been proven to generate two possible generalized diffusion equations, which are written here again for the reader s convenience. 

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