Ehrenfest approach

The methods outlined above can be contrasted with the Ehrenfest approach in which the bath follows a single mean field trajectory satisfying classical-like equations of motion R t) = P t)/M and P t) = Fmf = — ip t) dhei/dR ip t)). The mean field force on the trajectory is determined by the time dependent quantum subsystem state Cc t) a) which... 

An alternative formulation that uses classical trajectories to model non-adiabatic behaviour is the Ehrenfest approach [79, 80]. In this, each trajectory point (p, q) is driven by a mean-field force... 

Shalashilin DV (2011) Multiconfigurational Ehrenfest approach to quantum coherent dynamics in large molecular systems. Faraday Discuss 153 105... 

The terms and in (8.28) are those responsible for collision-induced phonon excitation. These terms depend upon time through the time-dependence of the trajectory. If only the linear terms are included, the oscillators are called linearly forced. From extensive studies on the linearly forced (and also more general) atom-molecule interactions, we know that good agreement between the semiclassical description and the full quantum description of the excitation process is obtained through the so-called symmetrized Ehrenfest approach. This approach is based upon the following two conjectures ... 

The semiclassical values are obtained using the symmetrized Ehrenfest approach. The energies are in units of hco. The three systems correspond to various reduced atom oscillator masses. System no. 1 has a relatively large oscillator mass, whereas systems no. 6 and 8 have small oscillator masses. The atoms of the surface are often much heavier than the gas atoms. Thus this case corresponds mainly to system no. 1. The numbering of the system refers to that of the first paper where the problem... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

In his book States of Matter, [(1985), Prentice Hall, Dover] David L. Goodstein writes Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. ... 

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... 

Called Z-star by P. and T. Ehrenfest, in Encyklopadie der mathematischen Wissenschaf-ten, Band 4, Nr. 32 (Teubner, Leipzig 1912) translated by M.J. Moravcsik under the title Conceptual Foundations of the Statistical Approach in Mechanics (Cornell University Press, Ithaca 1959). 

The transitions between the bottom five phases of Fig. 2 may occur close to equilibrium and can be described as thermodynamic first order transitions (Ehrenfest definition 17)). The transitions to and from the glassy states are limited to the corresponding pairs of mobile and solid phases. In a given time frame, they approach a second order transition (no heat or entropy of transition, but a jump in heat capacity, see Fig. 1). 

Therefore, it is more appropriate to start from an approach which is known as Ehrenfest dynamics. In the present case it is based on the following time-dependent Schrodinger equation for the CC electronic wave function... 

Ehrenfest, P., Ehrenfest T. "The conceptual foundations of statistical approach in mechanics", The Cornell University Press, Ithaca, NY (1959). 

In this concise classic, Paul Ehrenfest, one of the 20th century s greatest physicists, reformulated the foundations of the statistical approach in mechanics. Originally published in 1912 as an article for the German Encyclopedia of Mathematical Sciences , it has lost little of its scientific and didactic value, and no serious student of statistical mechanics can afford to remain ignorant of this great work. 

The conceptual foundations of the statistical approach in mechanics / by Paul and Tatiana Ehrenfest translated by Michael J. Moravcsik. p. cm. 

In most of the more recent classical approaches [18], no allusion to Ehrenfest s (adiabatic) principle is employed, but rather the differential equations of motion from classical mechanics are solved, either exactly or approximately, subject to a set of initial conditions (masses, force constants, interaction potential, phase, and initial energies). The amount of energy, AE, transferred to the oscillator is obtained for these conditions. This quantity may then be averaged over all phases of the oscillating molecule. In approximate classical and semiclassical treatments, the interaction potential is expanded in a Taylor s series and only the first two terms are retained. 

In order to determine time-dependent molecular properties utilizing the MCSCF/MM approach it is necessary to consider the time evolution of the appropriate operators and this is done by applying the Ehrenfest s equation for the evolution of an expectation value of an operator, X... 

A different theory of local control has been derived from the viewpoint of global optimization, applied to finite time intervals [58-60]. This approach can also be applied within a classical context, and local control fields from classical dynamics have been used in quantum problems [61]. In parallel, Rabitz and coworkers developed a method termed tracking control, in which Ehrenfest s equations [26] for an observable is used to derive an explicit expression for the electric field that forces the system dynamics to reproduce a predefined temporal evolution of the control observable [62, 63]. In its original form, however, this method can lead to singularities in the fields, a problem circumvented by several extensions to this basic idea [64-68]. Within the context of ground-state vibration, a procedure similar to tracking control has been proposed in Ref. 69. In addition to the examples already mentioned, the different local control schemes have found many applications in molecular physics, like population control [55], wavepacket control [53, 54, 56], control within a dissipative environment [59, 70], and selective vibrational excitation or dissociation [64, 71]. Further examples include isomerization control [58, 60, 72], control of predissociation [73], or enantiomer control [74, 75]. 

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