Phase-space point

F after transients have decayed. This final set of phase-space points is tire attractor, and tire set of all initial conditions tliat eventually reaches tire attractor is called its basin of attraction. 

The point (phase space point or configuration) is accepted or rejected according to the criterion... 

The resonant timesteps of order n m (where n and m axe integers), correspond to a sampling of n phase-space points in m revolutions. This condition implies that... 

Which implies that (x) is determined solely by the set of numbers representing the frequencies of two-site blocks appearing in the phase space point a . 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

As mentioned, xA(r) is half of the total adiabatic change in the subsystem macrostate associated with the current phase space point T. The factor of is used to compensate for double counting of the past and future changes. In the steady state, the subsystem most likely does not change macrostate, and hence this change has to be compensated by the change in the reservoir, Axr = xA(r). 

This tells the odds of the current phase space point compared to its conjugate. The quantity xA(T) Xr may be called the odd work. 

Now let us examine the phase space of an intermediate 7 defined by (6.73). Consider a phase space point /. If /) is outside of / q but inside / (i.e., exp —6/0 (/)) is large but exp[—/ E/i (.T )] is small), we can expect that the value of U0 will be small (negative) or moderate and that of C will be very large (positive). Consequently, the right-hand side of (6.73) will be small, which indicates that t/7 is a large positive number. Therefore, this specific I, is unlikely to be part of / , the important phase... 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

The Jarzynski identity can be used to calculate the free energy difference between two states 0 and 1 with Hamiltonians J%(z) and -A (z). To do that we consider a Hamiltonian -AA iz, A) depending on the phase-space point z and the control parameter A. This Hamiltonian is defined in such a way that A0 corresponds to the Hamiltonian of the initial state, Af(z, A0) = Atfo (z), and Ai to the Hamiltonian of the final state, Ai) = Aif z). By changing A continuously from A0 to Ai the Hamiltonian of the initial state is transformed into that of the final state. The free energy difference ... 

As before, we imagine that we can define the stable regions si and 38 with the help of an order parameter (z). A phase-space point z is in region sZ if the order parameter is within a certain range, < (z) < " ax, and the order parameters... 

Note also that the choice of what the move X- X from one phase space point to the next means microscopically depends on the type of problem that one wishes to study e.g., for a simulation of surface difiusion in the framework of the lattice gas model (see section 4.2), this move may mean a hop of a randomly chosen adatom to a randomly chosen nearest neighbor site (and W = 0 it this latter site is already taken). 

The prefactor for the HK propagator contains the derivatives 3q/3Fo, 0Ft/3po, 3pt/3Fo, and 3pt/3po. These derivative are evaluated for hopping trajectories, as discussed above, such that the changes in the hopping points, accompanying changes in the initial phase space point for the trajectory, always occur in a direction... 

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For 5(n) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. 

The phase space of eqn. (73) is the space of vectors c. Its points are specified by the coordinates cx,. . . , cn. The set of phase space points is the set of all possible states of the system. Phase space can be not only the whole vector space but also a certain part. Thus in chemical kinetic equations, variables are either concentrations or quantities of substances in the system. Their values cannot be negative. It is therefore natural to restrict ourselves to the set of those c all the components of which are not negative, i.e. Ci > 0. In what follows we shall refer to these d values as non-negative. Hence positive are those c values all the components of which are positive, i.e. Cj > 0. 

The quantum-classical Liouville equation in the force basis has been solved for low-dimensional systems using the multithreads algorithm [42,43]. Assuming that the density matrix is localized within a small volume of the classical phase space, it is written as linear combination of matrices located at L discrete phase space points as... 

Each phase-space point has a weight Pyfi(to), which reflects the particular quantum mechanical state of the parent molecule in the electronic ground state. 

This cloud of system points is very dense, since we consider a large number of systems, and we can therefore define a number density p(p, q, t) such that the number of systems in the ensemble whose phase-space points are in the volume element dp dq about (p, q) at time t is p(p,q,t)dpdq. Clearly, we must have that... 

We also need a function that shows whether a given trajectory, starting on the reactant side of the dividing surface, ends up at the product side at time t —> oo and thereby contributes to the formation of products. The Heaviside step function in Eq. (5.50) may also be used to specify whether the phase-space point of a system is at the dividing surface (S(q(t)) - 0), on the product side (say S(q(t)) > 0), or on the reactant side (S(q(t)) < 0). We then define the function P(p,q) according to... 

To calculate the normalization constant Na in feintra Zhao and Rice proceeded as follows. Assuming that the system is prepared in a state with all the phase-space points inside the intramolecular bottleneck dividing surface, then the density of these phase-space points can be written as... 

One can also assume that after the intramolecular vibrational relaxation process is completed the phase-space points are uniformly distributed inside the system... 

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