Liouville formulation

Tuckennan et al. [38] showed how to systematically derive time-reversible, areapreserving MD algorithms from the Liouville formulation of classical mechanics. Here, we briefly introduce the Liouville approach to the MTS method. The Liouville operator for a system of N degrees of freedom in Cartesian coordinates is defined as... 

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

The previous discussion shows that the relaxation processes emerge from the quantum dynamics under appropriate circumstances leading to the formation of time-dependent quasiclassical parts in the observable quantities. Let us add that quasiclassical and semiclassical methods have been recently applied to the optical response of quantum systems in several works [65, 66] where the relation to the Liouville formulation of quantum mechanics has been discussed, without however pointing out the existence of Liouvillian resonances as we discussed here above. The connection between the property of chaos and n-time correlation functions or the nth-order response of a system in multiple-pulse experiments has also been discussed [67, 68]. 

In the Liouville formulation one obtains the well-known connection between x and the imaginary part T of the complex resonance eigenvalue... 

As we have frequently pointed out that the present analytic formulation carries over straightforwardly to the Liouville formulation. For some technical discussions see Refs. [28,102,122]. In principle we need to replace the Schrodinger equation with the Liouville equation, i.e.,... 

It is important to note that, in general, the pa(p, q) are complex functions. Additional conditions on acceptable distributions must therefore be imposed, as discussed subsequently, to recover traditional classical mechanics. What is clear at this point is that several of the features normally attributed to quantum mechanics, specifically all of the presumed quantum properties numbered (1)-(5) in the previously given list are, in fact, characteristics of classical mechanics in the Liouville formulation. Further identification of... 

IV. LIOUVILLE FORMULATION OF EQUATIONS OF MOTION—MULTIPLE TIME STEP ALGORITHMS... 

In Sections II.A and II.B we presented the most popular algorithms for integrating Newton s equations of motion, some of which are not reversible in time. Recently, Tuckerman et al. [15] and Martyna et al. [16] have shown how one can systematically derive time reversible MD algorithms from the Liouville formulation of classical mechanics. 

We will first demonstrate the subtleties involved in the derivations of the proper dynamical equations. Although the presentation below can, without problem, be extended to a Liouville formulation we will, for simplicity review the time reversible case of the Schrbdinger equation based on the self-adjoint Hamiltonian H. Thus we write the following causal expressions h = 2n) of the time-independent and time-dependent Schrbdinger Equation assuming the existence of an absolutely continuous spectrum (Tac... 

Szymanski S, Gryff-Keller A M and Binsch G A 1986 Liouville space formulation of Wangsness-Bloch-Redfield theory of nuclear spin irelaxation suitable for machine computation. I. Fundamental aspects J. Magn. Reson. 68 399-432... 

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. 

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

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. 

The similar appearance of the quantum and classical Liouville equations has motivated several workers to construct a mixed quantum-classical Liouville (QCL) description [27 4]. Hereby a partial classical limit is performed for the heavy-particle dynamics, while a quantum-mechanical formulation is retained for the light particles. The quantities p(f) and H in the mixed QC formulation are then operators with respect to the electronic degrees of freedom, described by some basis states 4> ), and classical functions with respect to the nuclear degrees of freedom with coordinates x = x, and momenta p = pj — for example. 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

The stochastic Liouville equation, in the form relevant for the ESR line shape calculation, can be written in a form reminiscent of the Redfield equation in the superoperator formulation, Eq. (19) (70-73) ... 

The density operator is the key quantity in quantum statistical physics and as the approach from the quantum world to the classical domain can conveniently be formulated by the Liouville equation... 

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]. 

With symmetric boundary conditions at the chosen time t = 0, the microscopic formulation conforms to time reversible laws as expected. The same conclusion follows from an analogous examination of the Liouville equation. In this setting, the initial data at time, t = 0, is a statistical density distribution or density matrix. Although there are celebrated discussions on the problem of the approach to equilibrium, we nevertheless observe that without course graining or any other simplifying approximations the exact subdynamics would submit to the same physical laws as above, i.e., time reversibility and therefore constant entropy. 

These results were obtained by using the time-dependent quantum mechanical evolution of a state vector. We have generalized these to non-equilibrium situations [16] with the given initial state in a thermodynamic equilibrium state. This theory employs the density matrix which obeys the von Neumann equation. To incorporate the thermodynamic initial condition along with the von Neumann equation, it is advantageous to go to Liouville (L) space instead of the Hilbert (H) space in which DFT is formulated. This L-space quantum theory was developed by Umezawa over the last 25 years. We have adopted this theory to set up a new action principle which leads to the von Neumann equation. Appropriate variants of the theorems above are deduced in this framework. 

We may take the Wigner transform of the quantum Liouville equation to obtain an alternate formulation of the equation of motion. The Wigner transforms of the density matrix and an operator A are defined, respectively,... 

In the Hamiltonian formulation the Liouville equation can be seen as a continuity or advection equation for the probability distribution function. This theorem is fundamental to statistical mechanics and requires further attention. 

The classical phase space is formally defined in terms of generalized coordinates and momenta because it is in terms of these variables that Liouville s theorem holds. However, in Cartesian coordinates as used in the present section it is usually stiU true that pi = mci under the particular system conditions specified considering the kinetic theory of dilute gases, hence phase space can therefore be defined in terms of the coordinate and velocity variables in this particular case. Nevertheless, in the general case, for example in the presence of a magnetic field, the relation between pi and Cj is more complicated and the classical formulation is required [83]. 

The model derivation given above using the Liouville theorem is in many ways equivalent to the Lagrangian balance formulation [83]. Of course, a consistent Eulerian balance formulation would give the same result, but includes some more manipulations of the terms in the number balance. However, the Eulerian formulation is of special interest as we have adopted this framework in the preceding discussion of the governing equations of classical fluid dynamics, chap 1. 

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