Ehrenfest model

The classical-path approximation introduced above is common to most MQC formulations and describes the reaction of the quantum DoF to the dynamics of the classical DoF. The back-reaction of the quantum DoF onto the dynamics of the classical DoF, on the other hand, may be described in different ways. In the mean-field trajectory (MFT) method (which is sometimes also called Ehrenfest model, self-consistent classical-path method, or semiclassical time-dependent self-consistent-field method) considered in this section, the classical force F = pj acting on the nuclear DoF xj is given as an average over the quantum DoF... 

Exercise. Consider a one-step process with natural boundaries at n = 0 and at n = N for example, the continuous-time version of the Ehrenfest model (IV.5.4) ... 

Example 2.35 is the Ehrenfest diffusion model [6, p.21] for a simple random walk with reflecting barrier presented in example 2.18. The model assumes two containers A and B containing Z molecules. The containers are separated by a permeable membrane so that the molecules may move freely back and forth between the containers. It is assumed that at each instant of time t, one of the Z molecules chosen at random, is moving from one container to the other. The system are molecules in container A and the state Sj of the system is the number of molecules in container A which equals j -1. Thus, the following states are assumed Si = 0, S2 = 1, S3 = 2, S4 = 3,. .., Sz+i = Z molecules. In the Ehrenfest model, if A has j molecules, i.e., it is in state Sj+i, it can on the next step move to Sj or to Sj+2 with probabilities... 

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. 

Muller and Stock [227] used the vibronic coupling model Hamiltonian, Section III.D, to compare surface hopping and Ehrenfest dynamics with exact calculations for a number of model cases. The results again show that the semiclassical methods are able to provide a qualitative, if not quantitative, description of the dynamics. A large-scale comparison of mixed method and quantum dynamics has been made in a study of the pyrazine absorption spectrum, including all 24 degrees of freedom [228]. Here a method related to Ehrenfest dynamics was used with reasonable success, showing that these methods are indeed able to reproduce the main features of the dynamics of non-adiabatic molecular systems. 

Ehrenfest dynamics with the MMVB method has also been applied to the study of intermolecular energy transfer in anthryl-naphthylalkanes [85]. These molecules have a naphthalene joined to a anthracene by a short alkyl —(CH)n— chain. After exciting the naphthalene moiety, if n = 1 emission is seen from both parts of the system, if n = 3 emission is exclusively from the anthracene. The mechanism of this energy exchange is still not clear. This system is at the limits of the MMVB method, and the number of configurations required means that only a small number of trajectories can be run. The method is also unable to model the zwitterionic states that may be involved. Even so, the calculations provide some mechanistic information, which supports a stepwise exchange of energy, rather than the conventional direct process. 

Figure 5. Comparison of prediction (4) with numerical data. Normal diffusion ( ). The ballistic motion ( ). Superdiffusion ID Ehrenfest gas channel (Li et al, 2005)(v) the rational triangle channel (Li et al, 2003) (empty box) the polygonal billiard channel with (i = (V > — 1)7t/4), and 2 = 7r/3 (Alonso et al, 2002)(A) the triangle-square channel gas(Li et al, 2005) (<>) / values are obtained from system size L e [192, 384] for all channels except Ehrenfest channel (Li et al, 2005). The FPU lattice model at high temperature regime (Li et al, 2005) ( ), and the single walled nanotubes at room temperature ( ). Subdiffusion model from Ref. (Alonso et al, 2002) (solid left triangle). The solid curve is f3 = 2 — 2/a.

Exercise. To illustrate the approach to equilibrium, Ehrenfest invented the following model. ) N balls, labelled 1, 2,..., N, are distributed over two urns. Every second a numeral is selected at random (equal probabilities) from the set 1,2,..., N and the ball with that numeral is transferred from its urn to the other. The state of the system is specified by the number n of balls in one of the urns. The process is a Markov chain with... 

Exercise. The following modifications of Ehrenfest s urn model is nonlinear.510 Two urns each contain a mixture of black and white balls. Every second I draw with one hand a ball from one urn and with the other a ball from the other urn, and transfer both. Write the difference equation for the probability pn(t) of having n white balls in the left urn. 

Exercise. Reformulate Ehrenfest s urn model (IV.5.4) by noticing that each ball has two states. ... 

Example 2.36 is the Bernoulli-Laplace model of diffusion [15, p.378], similar to the one suggested by Ehrenfest. It is a probabilistic analog to the flow of two incompressible liquids between two containers A and B. This time we have a total of 2Z particles among which Z are black and Z white. Since these particles are supposed to represent incompressible liquids, the densities must not change, and so the number Z of particles in each container remains constant. The system are particles in container A of a certain color and the state Si of the system, is the number of these particles in container A where... 

We have seen in previous sections that the two-dimensional Ising model yields a symmetrical heat capacity curve that is divergent, but with no discontinuity, and that the experimental heat capacity at the A--transition of helium is finite without a discontinuity. Thus, according to the Ehrenfest-Pippard criterion these transitions might be called third-order. 

We apply our second-order Ehrenfest method to a model system benzene radical cation. Ionization of the neutral from the degenerate HOMO/HOMO-1 leads to the Jahn-Teller [15] effect in the cation. There is a peaked conical intersection between the two lowest-energy eigenstates Dq and Dy at geometries with D h symmetry. Figure 1 represents the surrounding moat of the conical intersection seen from above. It contains several valence bond (VB) resonance structures three equivalent quinoid structures... 

Different pathways of vibrational relaxation of diatoms in thermal collisions with atoms are discussed in the framework of the Ehrenfest adiabatic principle and generalized Landau-Teller model. Since the efficiency of different energy-transfer channels depend very strongly on the value of the Ehrenfest exponent, it is possible to assign, for given collision partners and the heat-bath temperature, the vibrational energy transfer events to VT, VRT or VR processes. 

The process of vibrational excitation and deexcitation of a diatom in a collision with an atom represents a simplest example from the host of processes which are relevant to gas-phase chemical kinetics. Experimental techniques available now allow one to measure directly state-to-state energy transfer rate coefficients. Theoretically, it is possible to accomplish completely ab initio calculation of these coefficients. One can therefore, regard the existing models of the vibrational relaxation from a new standpoint as a means for helping to understand more clearly the dynamics of the energy transfer provided that all the models are related to a single fundamental principle. This is the Ehrenfest adiabatic principle as formulated by Landau and Teller in the application to the collisional vibrational transitions of diatomic molecules. 

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