Entangled trajectories

Assume that the trajectory of a reactant B is engangled (case (b) in Fig. 1). Assume also that the reactant B carries a charge. Reactants A need not carry any charge. If an external electric field E is applied on the system, the entangled trajectory of reactant B will become more or less straight along the electric field (case (c)). Consequently the effective volume swept per unit time should Increase, and thus the rate of reaction should increase. 

The conceptual framework for the - semiclassical simulation of ultrafast spectroscopic observables is provided by the Wigner representation of quantum mechanics [2, 3]. Specifically, for the ultrafast pump-probe spectroscopy using classical trajectories, methods based on the semiclassical limit of the Liouville-von Neumann equation for the time evolution of the vibronic density matrix have been developed [4-8]. Our approach [4,6-8] is related to the Liouville space theory of nonlinear spectroscopy developed by Mukamel et al. [9]. It is characterized by the ability to approximately describe quantum phenomena such as optical transitions by averaging over the ensemble of classical trajectories. Moreover, quantum corrections for the nuclear dynamics can be introduced in a systematic manner, e.g. in the framework of the entangled trajectory method [10,11]. Alternatively, these effects can be also accounted for in the framework of the multiple spawning method [12]. In general, trajectory-based methods require drastically less computational effort than full quantum mechanical calculations and provide physical insight in ultrafast processes. Additionally, they can be combined directly with quantum chemistry methods for the electronic structure calculations. 

Fig. 17 Molecular trajectory in the late stage of chain adsorption. The chain atoms on the thin crystalline substrate and the amorphous substrate are depicted in dark gray and in white, respectively. The chain entanglements are pushed out of the crystalline substrate...

The word W consisting of a sequence of letters corresponding to different entanglements (introduced in Sect. 2.21) plays a role of full topological invariant for the PCAO-model. It is closely connected with the concept of the primitive path obtained by means of roughing of the microscopic chain trajectory up to the scale of the lattice cell and by exclusion of all loop fragments not entangled with the obstacles (Fig. 5). 

G. A. Arteca, Biopolymers, 35, 393 (1995). Fluctuations in Molecular Size, Entanglement Complexity, and Anisometry Along Molecular Dynamics Trajectories of Short a-Helices. 

We can surmise a slowing down of the DPhPC Upid motion from the enhanced entanglements among the neighboring lipid chains. In this section, we will see how and which motion is actually decreased or not. One of the bluest advantages of the MD simulation is that the dynamical properties are straightforwardly evaluated from the MD trajectories using a variety of correlation frmctions [54]. 

Yet, some theoretical problems are left to be discussed to seek for the ultimate and idealistic features as a nonadiabatic-transition theory Although a trajectory thus hopping plural times converges to run on an adiabatic potential surface asymptotically, the off-diagonal density matrix element Pij t) does not vanish practically, as in the original SET. This is ascribed to an incomplete treatment of the nuclear-electronic entanglement. This issue, often referred to as the problem of decoherence, is originated from the nuclear wavepacket bifurcation due to different slopes of potential surfaces, which will be discussed more precisely below. 

Molecular Interpretation. Stresses in a polymer melt may lead to alignment of the polymer chains along the direction of the imposed stress vector. A reduction in entanglements accompanies this orientation, so that translational motion of the polymer is enhanced. The tip is thought to behave as a molecular comb which orients the polymer chains parallel to the tip trajectory. Successive visits of the tip to the same polymer chains result in greater net orientation of the chains and hence enhanced translational mobility. 

On the other hand, when a reactant B Is frequently scattered by the solvent, its trajectory is entangled (case (b)) and It sweeps the same place repeatedly. Consequently the effective volume swept per unit time becomes smaller than ra2v. This is the reason the rate of reaction or the rate constant decreases with decreasing mean free path of reactant. Case (a) corresponds to low-pressure gas phase reactions, while case (b) corresponds to (narrowly defined) diffusion-controlled reactions. 

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