Liouville equation representations

The research was greatly facilitated by two important elements. The (formal, perturbative) solution of the Liouville equation is greatly simplified by a Fourier representation (see Appendix). The latter allows one to easily identify the various types of statistical correlations between the particles. The traditional dynamics thus becomes a dynamics of correlations. The latter is completed by... 

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. 

In the next section we describe how the QCL equation may be expressed in any basis that spans the subsystem Hilbert space. Here we observe that the subsystem may also be Wigner transformed to obtain a phase-space-like representation of the subsystem variables as well as those of the environment. Taking the Wigner transform of Eq. (8) over the subsystem, we obtain the quantum-classical Wigner-Liouville equation [24],... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

Given this correspondence between the matrix elements of a partially Wigner transformed operator in the subsystem and mapping bases, we can express the quantum-classical Liouville equation in the continuous mapping coordinates [53]. The first step in this calculation is to introduce an n-dimensional coordinate space representation of the mapping basis,... 

Carrying out the this change of representation on the quantum-classical Liouville equation and using the product rule formula for the Wigner transform... 

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... 

The next steps are the following Step 1 Passage to the entropy representation and specification of the dissipative thermodynamic forces and the dissipative potential E. Step 2 Specification of the thermodynamic potential o. Step 3 Recasting of the equation governing the time evolution of the np-particle distribution function/ p into a Liouville equation corresponding to the time evolution of np particles (or p quasi-particles, Up > iip —see the point 4 below) that then represent the governing equations of direct molecular simulations. 

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 current transfer described here is a coherent phenomenon, sensitive to dephasing. To investigate environmental dephasing effects, we incorporate additional relaxation of coherences in the site representation of the Liouville equation for the system s density matrix (DM) p ... 

Vm fn n. In the same representation the density operator is p = 12 ZZ Show that in this representation the Liouville equation is... 

Using this reduction operation we may obtain interesting relationships by taking traces over bath states of the equations of motion (10.15) and (10.21). Consider for example the Liouville equation in the Schrodinger representation, Eq. (10.15). (Note below, an operator A in the interaction representation is denoted Ai while in the Schrodinger representation it carries no label. Labels S and 5 denote system and bath.)... 

Practical solutions of dynamical problems are almost always perturbative. We are interested in the effect of the thennal environment on the dynamical behavior of a given system, so a natural viewpoint is to assume that the dynamics of the system alone is known and to take the system-bath coupling as the perturbation. We have seen (Section 2.7.3) that time dependent perturbation theory in Hilbert space is most easily discussed in the framework of the interaction representation. Following this route" we start from the Liouville equation in this representation (cf. Eq. (10.21))... 

The algebra of the Wigner-Weyl representation has been extensively reviewed51 and we only call attention to the form that Eqs. (3.5) and (3.6) take in this representation. The Liouville equation becomes... 

The time evolution in the N body reacting fluid is, in general, given by the Liouville operator introduced earlier. If, however, we make the additional assumption that the strongly repulsive solute-solvent and solvent-solvent forces can be approximated by effective hard-sphere interactions, the theory can be formulated in a way that greatly simplifies the calculation. This can be accomplished by the use of the pseudo-Liouville representation for the dynamics in a hard-sphere system. In a hard-sphere system, the time evolution of a dynamic variable is given by the pseudo-Liouville equation... 

We now conclude with a derivation of the basic transport equations starting from the Boltzmann equation rather than from the Fokker-Planck equation. We already noted that both the Fokker-Planck and the Boltzmann equations are related to the Liouville equation and that our goal is to obtain equations for the charge distribution and the current density (Eqs. [55] and [54]) using an appropriate representation of the collisional term in the left-hand side of Eq. [60]). The method described here is the well-known method of moments. It consists of multiplying the Boltzmann equation by a power of the velocity, and by integrating over the velocity. For the moment of order zero... 

Finally, before leaving this section, we note another important aspect of the Liouville equation regarding transformation of phase space variables. We noted in Chap. 1 that Hamilton s equations of motion retain their form only for so-called canonical transformations. Consequently, the form of the Liouville equation given above is also invariant to only canonical transformations. Furthermore, it can be shown that the Jacobian for canonical transformations is unity, i.e., there is no expansion or contraction of a phase space volume element in going from one set of phase space coordinates to another. A simple example of a single particle in three dimensions can be used to effectively illustrate this point.l Considering, for example, two representations, viz., cartesian and spherical coordinates and their associated conjugate momenta, we have... 

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 subscript W refers to this partial Wigner transform, N is the eoordinate space dimension of the bath and X = R, P). In this partial Wigner representation, the Hamiltonian of the system takes the form Hw R,P) = P /2M + y-/2m+ V q,R). If the subsystem DOF are represented using the states of an adiabatic basis, a P), which are the solutions of hw R) I R)=Ea R) I where hw K)=p /2m+ V q,R) is the Hamiltonian for the subsystem with fixed eoordinates R of the bath, the density matrix elements are p i -, 0 = ( I Pw( 01 )- From the solution of the quantum Liouville equation given some initial state of the entire quantum system, the reduced density matrix elements of the quantum subsystem of interest can be obtained by integrating over the bath variables, p f t) = dX p X,t), in order to find the populations and off-diagonal elements (coherences) of the density matrix. 

Using the interaction representation, p (t) will evolve under the new Liouville equation ... 

The density matrix representation of spin and orbital angular momentum is capable of expressing a static state of matter and its time-dependent response to an external perturbation. Our application necessitates that we follow the response of the orbital and spin momenta subject to full or partial excitations, and the density matrix provides a direct solution to the stochastic Liouville equation. But the density matrix representation in a rotating operator is algebraically ambiguious, and we must also clarify the algebraic description of selective excitation of multiquantum systems. 

Matrix representations of the quantum Liouville equation and Heisenberg s equation of motion can be obtained by sandwiching both sides of Eqs. (225) and (236) or Eqs. (228) and (237) between the vectors 4>j and 4>k), where ( y) and k) are members of the orthonormal basis ( ). Tins procedure yields... 

Although the above matrix representations of the quantum Liouville equation and Heisenberg s equation of motion are formally correct, the solution of time evolution problems for quantum systems can be more readily accomplished by working in a representation called the superstate representation. The basis vectors of this representation are the superstates A y/t). These states are associated with the operators Njk= j) 4 k formed from the basis vectors < y) used in the formulation of Eqs. (239) and (240). The matrix element A(j,k) = (pj A (l)k) of the operator A is given by A j, k) = Njk A), which can be thought to represent a component of the vector A) in the superstate representation. We define the inner product A B) of A and 5) by A B) —Tr The matrix elements C(Jk, Im)... 

With the superstate representation at our disposal, we can rewrite the quantum Liouville equation and Heisenberg s equation of motion as vector equations of motion that are identieal in form to the vector equations of motion given by Eqs. (215) and (216) for elassieal systems. The only differenee between the quantum and elassieal veetor equations of motion is the manner in which the matrix elements of and the eomponents of 1) and ) are determined. Nonetheless, the expressions for the average of a qrrantrrm dynamieal variable differ from the eorrespond-ing expressions for elassieal systems. [See Eqs. (219a)-(219c).] For the qrrantttm ease, we have... 

Most detailed studies of spin effects in the literature are based on the Stochastic Liouville equation (SLE), which treats the system as an ensemble and requires the use of the density matrix Pij(r, t) [7-9], The density matrix (i) contains all the information about the ensemble physical observables of the system (ii) describes the distribution of spin states for an ensemble of particles and (iii) is constructed from the vector representation of the spin function (c ) relative to some predefined basis, such that pij(r, t) = c Cj). 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

Another important hierarchy of equations is obtained by applying the (MCM) to the matrix representation of the Liouville-von Neumann Equation LVNE) [12,13]. In this way the p-order Contracted Liouville-von Neumann Elquation (p-CLVNE) is obtained [4]. It will be shown here that The structure of a particular p-CSE, that involves the higher order CSE s for a given state, can be replaced by an equivalent set of equations, 1-CSE and 1-CLVNE, but for the whole spectrum, i. e. involving all the states. 

In this paragraph, we consider the equation obtained by applying the same MCM to both sides of the matrix representation of the Liouville-von Neumann... 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

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