Many Coupled Degrees of Freedom

Consider the set of points in phase space at energy located at the barrier top q-i = q with pj = 0. All such points will, in addition to being located at the barrier top, have H, = [, and H2,..n = = E - Ei, where b is the barrier 

The above set of phase-space points forms a surface of dimension 2N - 3. This surface can be looked upon as a constant-energy surface for the bath modes and has the topology of a hypersphere This hypersphere is 

The surface t v-1 is a particular example of what Wiggins has referred to as a hyperbolic manifold and what De Leon and Ling have termed a normally invariant hyperbolic manifoldd Hyperbolic manifolds are unstable and constitute the formal multidimensional generalization of unstable periodic orbits. Hyperbolic manifolds, like PODS, can be either repulsive or attractive. - If motion near a hyperbolic manifold falls away without recrossing it in configuration space, the hyperbolic manifold is said to be repulsive. On the other hand, it is often the case that motion near a hyperbolic manifold will cross it several times in configuration space as it falls away, and in this case it is said to be attractive. 

Assume that is repulsive. Since Tv-i is an invariant surface, motion precisely on Xy-i can never fall away. However, speaking somewhat loosely for the moment, a slight push along q will cause motion initially on tv, to roll away from the barrier top. The set of motions that will roll away most slowly are motions that are asymptotic to the repulsive manifold and the surface formed by these motions constitutes the multidimensional version of a sep-aratrix. As motion asymptotic to Tn,. i falls away, it will generate a surface embedded in the full phase space whose geometry is the direct product of the sphere and the real line, x that is, a hypercylinder. The dimension of this hypercylinder is thus IN - As we will see, this (2N - 2)-dimen- 

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In order to resolve the outstanding issues of the quantum-classical transition, and study the control of entanglement and decoherence without the foregoing restrictions, we must venture into the domain of Quantum Complex Systems (QUACS), either consisting of a large number of inseparable elements or having many coupled degrees of freedom. Modern statistical physics copes with... 

Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as in evaluating a multivariable statistical mechanics integral. That is, they can be used to obtain the expectation value for a macroscopic variable. A, for a system of IV particles in which [1] the Hamiltonian, U r), is known [2], the system is at some temperature,... 

Monte Carlo methods (or Monte Carlo experiments) are used to simulate the probability of failure for a slope. Because of many coupled degrees of freedom such as the soil physical characteristics, statistical parameters and mathematical models, the Monte Carlo methods are especially useful to solve such a complicated system. The computational algorithms of the Monte Carlo method seek numerical solution by repeating random sampling. Its formulation is given by... 

Blake N P and Metiu H 1995 Efficient adsorption line shape calculations for an electron coupled to many quantum degrees of freedom, applications to an electron solvated in dry sodalites and halo-sodalites J. Chem. Phys. 103 4455... 

For long (infinite) /am.v-polyacclylene chains, the treatment of quantum lattice fluctuations is very complicated, because many lattice degrees of freedom couple in a non-linear way to the lowest electronic transitions. We have recently shown that for chains of up to 70 CH units, the amount of relevant lattice degrees of freedom reduces to only one or two, which makes it possible to calculate the low-energy part of the absorption spectrum in an essentially exact way [681. It remains a challenge to study models in which both disorder and the lattice quantum dynamics are considered. 

Since A, B, and C are regions in the phase space of single closed system, the transitions between A and represent a unimolecular reaction or isomerization, rather than a general reaction in the sense of chemical kinetics. Unlike some unimolecular reactions, (e.g the decomposition of diatomic molecules) the molecular dynamics system of eq. 1 will be assumed to have sufficiently many well-coupled degrees of freedom that transitions between reactant and product regions occur spontaneously, without outside interference. 

The coherent dynamics are reversible since the system is isolated and there are no bath modes in the simplified one-dimensional model. Therefore, if the two electronic states are coherently excited by a Unearly polarized pulse, the dynamic behaviors are invariant with respect to change in the direction of polarization, = 1 (e+) or = —1 (e ). Time-dependent behavior of PcuiW with e+ (e ) is the same as Pbo,w t) with = — 1 ( = 1). In real molecules with many vibrational degrees of freedom, the invariance is broken, and dephasing time of the vibrational coherence in lower excited state b is shorter than that in higher state c because multimode effects induced by potential couplings and/or anharmonicity play a much more dominant role in lower state b than in higher state c. 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

Of course, in a real system with many atoms, the coupling of the property to the individual degrees of freedom is more complicated, and there is no guarantee that A A will be an odd function. Nevertheless, the assumption that Eq. (9.51) is equal to zero is often sufficiently accurate for everyday computational predictions. 

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