Reduced Liouville equation

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. 

In the foregoing discussion of the Brownian motion method, the ensemble averages are all constructed from an ensemble of replica systems of the subset of h molecules, the behavior of each replica having been time-smoothed over the interval r. However, in a steady-state transport process dfiN)/dt = 0 at every point in phase space, where f N) is the instantaneous phase density of the N molecules. In principle, at least, it should thus be possible to express the steady-state pressure tensor and the mass and heat currents in terms of ensemble averages constructed without preliminary time-averaging in the replica systems. Thus it is desirable to examine the possibility of obtaining solutions to the reduced Liouville equation, Eq. 8, without preliminary timeaveraging. 

It should be noted that the left-hand side of Eq. [58] is identical to that of the reduced Liouville equation.Indeed, several theories have been developed that obtain Eq. [58] from the reduced Liouville equation. Eollow-ing the standard Smoluchowski expansion of the full time-dependent Fokker-Planck equation, it can be shown that, for large y, the following model is obtained for the probability density at steady state ... 

As will be shown in subsequent chapters, the solution to the Liouville equation for the N -particle density function p r is the basis for determination of the equilibrium and nonequilibrium properties of matter. However, because of the large number of dimensions (6N), solutions to the Liouville equation represent formidable problems Fortunately, as will be shown in Chaps. 4 and 5, the equilibrium and nonequilibrium properties of matter can usually be expressed in terms of lower-ordered or reduced density functions. For example, the thermodynamic and transport properties of dilute gases can be expressed in terms of the two-molecule density function /02(ri,r2, pi,p2,f). In Chap. 3 we will examine the particular forms of the reduced Liouville equation. 

Reduced Density Functions and the Reduced Liouville Equation 55... 

Equation (3.24) is the reduced Liouville equation for pairwise additive interaction forces. Note that this is an integro-differential equation, where the evolution of the fs distribution depends on the next higher-order fs+i distribution. This is known as the BBGKY hierarchy (named after its originators Bogoliubov, Bom, Green, Kirkwood, and Yvon see the Further Reading section at the end of this chapter). 

It suffices to say that extensions of the reduced Liouville equation to higher orders, necessary for dense gases and liquids, become extremely cumbersome. Approximate solutions to the reduced Liouville equation for nonequilibrium dense gas and liquid systems will be considered in more detail in Chap. 6. In the next chapter, however, we will show that exact analytical solutions to the Liouville equation, and its reduced forms, are indeed possible for systems at equilibrium. 

Now, to obtain an entropy conservation equation, we can work with Eq. (5.9) modified to include the time dependence in a itself, or it is somewhat easier to work directly with the reduced Liouville equation, Eq. (3.20) or (3.24), for pairwise additive systems we choose the latter representation. 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

In order to nondimensionalize the reduced Liouville equation, we introduce the following dimensionless variables denoted by an... 

Interlude 6.1 What Exactly is Scaling In general, the reduced Liouville equation represents a complex, multidimensional partial differential equation that must be solved subject to the boundary and initial conditions of a particular problem. In many cases, however, a number of terms in this equation are negligible, small, or insignificant and, thus, they can be discarded... 

Regular Perturbative Expansion of the Reduced Liouville Equation for Dense Gases and Liquids... 

As noted in the previous chapter, to close the transport equations an expression for two-molecule density function /2(r, r, p, p, t) is needed. Approximate solutions to the reduced Liouville equation for the case s = 2 are therefore sought. Of course, if we were able to obtain the complete or exact solution to the function /2 from the reduced Liouville equation, for any given S3rstem in a nonequilibrium state, then the local spatial and temporal thermodynamic state functions could be obtained directly from their basic definitions (at least for pairwise additive systems) and the solution via the transport equations becomes unnecessary or superfluous. Unfortunately, complete or exact solutions to the reduced Liouville equation for nonequilibrium systems are extremely difficult to obtain, even by numerical means, so that the transport route is often our only recourse. 

Due to complexity of the real world, all QDT descriptions involve practically certain approximations or models. As theoretical construction is concerned, the infiuence functional path integral formulation of QDT may by far be the best [4]. The main obstacle of path integral formulation is however its formidable numerical implementation to multilevel systems. Alternative approach to QDT formulation is the reduced Liouville equation for p t). The formally exact reduced Liouville equation can in principle be constructed via Nakajima-Zwanzig-Mori projection operator techniques [5-14], resulting in general two prescriptions. One is the so-called chronological ordering prescription (COP), characterized by a time-ordered memory dissipation superoperator 7(t, r) and read as... 

Alternative prescription, which is also often called partial ordering prescription (POP), is characterized by a time-local dissipation superoperator 7 .(t) and the resulting reduced Liouville equation reads as... 

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