Operators Liouville

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

NVT, and in die course of the simulation the volume V of the simulation box is allowed to vary, according to the new equations of motion. A usefid variant allows the simulation box to change shape as well as size [89, 90], It is also possible to extend the Liouville operator-splitting approach to generate algoritlnns for MD in these ensembles examples of explicit, reversible, integrators are given by Martyna et al [91],... 

In order to apply the techniques discussed above to the MD simulation o biomolecules, one takes the Liouville operator for a macromolecule in vacuo containing N atoms to be... 

In addition, the non-bonded forces can be divided into several regions according to pair distances. The near region is normally more important than the distant region because the non-bonded forces decay with distance. Since most of the CPU time in a MD simulation is spent in the calculation of these non-bonded interactions, the separation in pair distance results in valuable speedups. Using a 3-fold distance split, the non-bonded forces are separated in 3 regions near, medium, and fax distance zones. Thus, the Liouville operator can be express as a sum of five terms... 

Liouville operator LnZi = using the common Poisson brackets... 

The exponential exp(lLf) in Equation (7.31) involves the so-called Liouville operator, L, whic in the case of a molecular system containing N atoms (and so 3N coordinates) can b expressed ... 

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... 

The displaced position on the left-hand side reflects the free streaming between collisions generated by the free streaming Liouville operator,... 

This is Liouville s equation, with the Liouville operator... 

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 ... 

The formal similarity between Eq. (10) and the time-dependent Schrodinger equation is striking, and we shall indeed develop methods which are very reminiscent of quantum mechanics. In particular, we may calculate the eigenfunctions and eigenvalues of the unperturbed Liouville operator L0. We look for solutions of ... 

Here La is the Liouville operator of ion a plus the fluid it corresponds to the Hamiltonian ... 

In Eq. (2), LN is the Liouville operator, which is decomposed into two terms ... 

Let us also introduce the eigenfunctions of the unperturbed Liouville operator (4)... 

For the numerical implementation of the QCL equation, the Liouville operator S is decomposed into a zero-order part Sq which is easy to evaluate and a nonadiabatic transition part if whose evaluation is difficult. The splitting suggests that we (a) employ a short-time expansion of the full exponential by use of a first-order Trotter formula... 

Thus, p(f + 8f) is a sum of contributions with the phase factors (( >ab t)< b t)), the weight factors w t)Wb t)), and phase-space distributions Pb t) which appear with a probability Tab t). Associating M(f) with the Liouville operator 1 — the scheme becomes somewhat more involved. In addition to... 

Here, t is the Liouville operator and the operator Q projects onto the orthogonal complement ofs.There are arguments" that suggest that it is a good approximation to calculate (0) by clamping the reaction coordinate at the... 

Here we will summarize some known results for the simplest example of random frequency modulation as defined by Eq. (2). Let us assume that the process 2( ) in Eq. (2) is a projection of a Markovian process characterized by the evolution operator T. This is possible in principle, because the dynamical motion of the environment can be described in terms of a Liouville operator. The set of variables defining the Markovian process is designated by X. If the variable fi itself is Markovian, X consists only of 2, but in general it has to be supplemented by additional variables to complete the set. Let the function W(x, X, t) be the probability or the probability density for finding the random variables x and X at the respective values at the time t. Then a systematic method of treating the problem, Eq. (2), is to rewrite it in the form... 

These results lead to the following physical interpretation of the two terms in (4.7). The first term alone is the Liouville operator belonging to the deterministic equation... 

Where T is the initial phase point of the system, L is the Liouville operator, y(tf)(F) is the canonical distribution function, and Bk(T) and k(T) are the values of the classical properties Bk and iLk when the system is in the classical state T. Much work has been done to determine how the quantum-mechanical functions approach the corresponding classical functions. 

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 ... 

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