Green time-dependent

Green M S 1954 Markov random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids J. Chem. Phys. 22 398... 

In a subsequent treatment from the time-dependent response point of view, connection with the Greens function... 

The time-dependent approach is thus just one teclmique for evaluating the action of the Green s fiinction on the initial wavepacket. 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

The name Propagator, also known as a Greens function, arises from a time-dependent evolution of a given quantity. For two time-dependent operators P(t) and Q(f), a propagator may be defined as... 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

From both the time-dependent plot and the time-independent projection, it is clear that the transition path crosses the space-fixed dividing surface qu = 0 several times. These crossings are indicated by thick green dots. As expected, therefore, the fixed surface is not free of recrossings and thus does not satisfy the fundamental requirement for an exact TST dividing surface. The moving surface, by contrast, is crossed only once, at the reaction time head = 8.936 that is marked by the blue cut. The solid blue line in this cut shows the instantaneous position of the dividing surface dotted lines indicate coordinate axes. 

We first find the Green function Go for H0 and then obtain pertur-batively the wave functional for the total Hamiltonian. In fact, each mode of the quadratic part Ho can be solved exactly in terms the time-dependent creation and annihilation operators (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003)... 

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. 

The decomposition of D2 in Eq. (25b) is sometimes called the Levy-Lieb partition of the 2-RDM [57,58]. Formulas essentially equivalent to Eqs. (25a)-(25e) were known long ago, in the context of time-dependent Green s functions [59-61], but this formalism was rediscovered in the present context by Mazziotti [33]. 

The exponential term can be thought of as a Green function, with the time dependence always implicit. Thus an excitation at x causes a response at x whose phase is delayed by the distance between them multiplied by the real part of kp (this corresponds approximately to 2 /Ar), and whose amplitude is decreased exponentially by the distance between them multiplied by the imaginary part of kp (this corresponds to the decay associated with the propagation of the leaky Rayleigh wave). The magnitude x — x is used because... 

The diffusion equation analysis is discussed in Sect. 2. It has been used very much more frequently in studies of diffusion-limited reactions rates than the analysis based on molecular pair behaviour, which is discussed in Sect. 3. This is probably because the diffusion equation approach is rather more direct, clear and versatile than the molecular pair analysis (furthermore, time-dependent Green s functions are required for the molecular pair approach). Besides, the probability that a molecular pair will reencounter one another is often derived from a diffusion equation analysis in any case and under these circumstances the two approaches are identical. 

All that remains, then, is to substitute y and y2 from eqns. (326) into eqn. (330), multiply by ePu = exp rc/r (which ensures that Green s function is symmetric to interchange of r and r0) and invert the expression for the Green function to get the time-dependent form. 

The average value of any operator O can be written as (O) = (t Os t) in the Schrodinger representation or (O) = (0 Off(t) 0) in the Heisenberg representation, where 0) is some initial state. This initial state is in principle arbitrary, but in many-particle problems it is convenient to take this state as an equilibrium state, consequently without time-dependent perturbation we obtain usual equilibrium Green functions. 

We now turn to approaches that begin with an arbitrary initial function and can, in principle, be iterated to an exact or accurate solution of the SE. The earliest approach is Green s function Monte Carlo (GFMC) in which the time-independent Schrodinger equation is employed [24] DMC was developed later and follows from the time-dependent SE (TDSE) in imaginary time. 

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