Liouville equation derivation

In the Liouville equation derivation of the Boltzmann equation, one is forced to follow the streaming of the particles backward in time. Here one would say that since particles 1 and 3 appear to have collided in the past, the Boltzmann equation treats this collision as if it had actually taken place. 

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

Liouville s equation, derivation of Boltzmann s equation from, 41 Littlewood, J. E., 388 Lobachevskies method, 79,85 Local methods of solution of equations, 78... 

We shall use the projection operator method to derive the Pauli master equation. With the Liouville equation, we separate the Liouville operator into two parts ... 

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. 

In order to make clear that this theory is not derived from the Liouville equation, we use here a notation different from the rest of the paper. 

Of course, as was shown in Section V-A, this latter expression may also be derived starting from the hydrodynamical equations for the pair distribution and the Poisson equation it is also the final result of the theories developed independently by Falken-hagen and Ebeling,9 and by Friedman 12-13 in these two approaches, the starting point is a Liouville equation for the system of ions with an ad hoc stochastic term describing the interactions with the solvent. 

The equation (6) for s = 1 connects fx to /2, which is itself connected to /3. The ensemble of equations (6) constitutes the hierarchy derived independently by Bogolubov, Bom, Green, Kirkwood, and Yvon. This hierarchy is equivalent to the Liouville equation and to try to solve it is equivalent to studying the trajectories of 1023 particles whose phases at the initial instant are known. 

Taking the logarithmic derivative of (A.36) with respect to r, multiplying on both sides by <0 S T 0>, and using finally the Liouville equation (2), we have (see A.33) ... 

Finally, from the Liouville equation for 3 we derive the equations of... 

In the cause of logical clarity, it is unfortunate that the equations of transport are so often derived from the Boltzmann integro-differential equation. Their derivation from the Liouville equation is a straightforward exercise in -dimensional calculus,... 

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. 

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... 

Kadanoff and Swift have considered that the time evolution of the state is described by the Liouville equation. They also wrote down conservation equations for the number, momentum, and energy density similar to the ones given by Eqs. (1)—(3). The only difference was that in the treatment of Kadanoff and Swift the densities and currents are operators. The time derivative of the densities are replaced by the commutator of the respective density operator and the Liouville operator, L (as the Liouville operator governs the time evolution). The suffix op in the following equations stands for operator ... 

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. 

The quantum-classical Liouville equation can be derived by formally expanding the operator on the right side of Eq. (6) to O(h0). One may justify [4] such an expansion for systems where the masses of particles in the environment are much greater than those of the subsystem, M > tn. In this case the small parameter in the theory is p = (m/Mj1/2. This factor emerges in the equation of motion quite naturally through a scaling of the variables motivated... 

We now show how the mean field equations can be derived as an approximation to the quantum-classical Liouville equation (8) [9]. The Hamiltonian... 

Q. Shi and E. Geva. A derivation of the mixed quantum-classical Liouville equation from the influence functional formalism. J. Chem. Phys., 121(8) 3393-3404, 2004. 

We consider an ensemble of systems each containing n atoms. Thus, q = (<71, , < 3n), P = (pi, , P3n), and dpdq = Ilf" (dp dqi). We assume that all interactions are known. As time evolves, each point will trace out a trajectory that will be independent of the trajectories of the other systems, since they represent isolated systems with no coupling between them. Since the Hamilton equations of motion, Eq. (4.63), determine the trajectory of each system point in phase space, they must also determine the density p(p, q, t) at any time t if the dependence of p on p and q is known at some initial time to. This trajectory is given by the Liouville equation of motion that is derived below. 

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... 

Section III is devoted to illustrating the first theoretical tool under discussion in this review, the GME derived from the Liouville equation, classical or quantum, through the contraction over the irrelevant degrees of freedom. In Section III.A we illustrate Zwanzig s projection method. Then, in Section III.B, we show how to use this method to derive a GME from Anderson s tight binding Hamiltonian The second-order approximation yields the Pauli master equation. This proves that the adoption of GME derived from a Hamiltonian picture requires, in principle, an infinite-order treatment. The case of a vanishing diffusion coefficient must be considered as a case of anomalous diffusion, and the second-order treatment is compatible only with the condition of ordinary... 

The approach proposed by Zwanzig [15,16] for a formal derivation of the master equation proceeds as follows. We assume the whole Universe, namely, the system of interest and its environment, to obey the Liouville equation... 

A second problem with the GME derived from the contraction over a Liouville equation, either classical or quantum, has to do with the correct evaluation of the memory kernel. Within the density perspective this memory kernel can be expressed in terms of correlation functions. If the linear response assumption is made, the two-time correlation function affords an exhaustive representation of the statistical process under study. In Section III.B we shall see with a simple quantum mechanical example, based on the Anderson localization, that the second-order approximation might lead to results conflicting with quantum mechanical coherence. 

The rigorous approach to a kinetic-theory derivation of the fluid-dynamical conservation equations, which begins with the Liouville equation and involves a number of subtle assumptions, will be omitted here because of its complexity. The same result will be obtained in a simpler manner from a physical derivation of the Boltzmann equation, followed by the identification of the hydrodynamic variables and the development of the equations of change. For additional details the reader may consult [1] and [2]. 

In the absence of the external potential V, Eqs. (52) can be given a rigorous derivation from a microscopic Liouville equation (see Chapter I). We make the naive assumption that when an external potential driving the reaction coordinate is present, the two contributions (the deterministic motion resulting from the external potential and the fluctuation-dissipation process described by the standard generalized Langevin equation) can simply be added to each other. 

Equation [43] was first derived in Ref. 15, where we represented V (r) by a generic Jacobian function/(F), and it represents a correct generalization of the Liouville equation to account for the nonvanishing compressibility of phase space. Equation [43] can be derived in many ways. A general approach starts with a statement of continuity valid for a space with any metric (see Appendix 2). One can examine the transformation from one set of phase space coordinates r to another F. The metric determinant transforms according to... 

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

We may easily carry out a linear response theory derivation of transport properties based on the quantum-classical Liouville equation that parallels the... 

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