Bath modes

A reactive species in liquid solution is subject to pemianent random collisions with solvent molecules that lead to statistical fluctuations of position, momentum and internal energy of the solute. The situation can be described by a reaction coordinate X coupled to a huge number of solvent bath modes. If there is a reaction... 

Static bath mode. Feed enters at one end of the dmm and the floats exit from the other end. The sink product is removed continuously from the rotating dmm through the use of lifters attached to the dmm which empty into a launder as they move to the top. A modification of the simple dmm separator is the two-compartment dmm separator which allows a two-stage separation. In the cone-type separator (up to 6.1 m in dia and 450 t/h) feed is introduced at the top. The medium in the cone is kept in suspension by gentle agitation. The sink product is removed from the bottom of the cone either directly or by airlift in the center of the cone. The maximum particle size that can be separated is limited to 10 cm. Other separators include the Drewboy bath and the Norwaltbath (2). 

Figure 1. Phase-space portrait of the dynamics described by the linearized Hamiltonian in Eq. (5), projected onto (a) the reactive mode and (b) a bath mode.

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

A convenient quantitative characterization of the stable and unstable manifolds themselves as well as of reactive and nonreactive trajectories can be obtained by noting that the special form of the Hamiltonian in Eq. (5) allows one to separate the total energy into a sum of the energy of the reactive mode and the energies of the bath modes. All these partial energies are conserved. The value of the energy... 

Assuming that the pj (t) and Qj (t) can be interpreted as a TS trajectory, which is discussed later, we can conclude as before that ci = ci = 0 if the exponential instability of the reactive mode is to be suppressed. Coordinate and momentum of the TS trajectory in the reactive mode, if they exist, are therefore unique. For the bath modes, however, difficulties arise. The exponentials in Eq. (35b) remain bounded for all times, so that their coefficients q and q cannot be determined from the condition that we impose on the TS trajectory. Consequently, the TS trajectory cannot be unique. The physical cause of the nonuniqueness is the presence of undamped oscillations, which cannot be avoided in a Hamiltonian setting. In a dissipative system, by contrast, all oscillations are typically damped, and the TS trajectory will be unique. 

In Ref. 40 it was assumed that the driving forces vanish asymptotically for t —> oo. Under this condition it can easily be derived from the explicit representation (21) of the 5-functional that in the reactive mode P (t) and Q (t) tend to zero for t —> oo and that the bath modes approach the autonomous dynamics... 

In particular, the TS trajectory remains bounded for all times, which satisfies the general definition. The constants c and c in Eq. (39) depend on the specific choice of the TS trajectory. Because the saddle point of the autonomous system becomes a fixed point for large positive and negative times, one might envision an ideal choice to be one that allows the TS trajectory to come to rest at the saddle point both in the distant future and in the remote past. However, this is impossible in general because the driving force will transfer energy into or out of the bath modes in such a way that... 

Figure 3. Phase portrait of the noiseless dynamics (43) corresponding to the linear Langevin equation (15) (a) in the unstable reactive degree of freedom, (b) in a stable oscillating bath mode, and (c) in an overdamped bath mode. (From Ref. 37.)...

Because in an autonomous system many of the invariant manifolds that are found in the linear approximation do not remain intact in the presence of nonlinearities, one should expect the same in the time-dependent case. In particular, the separation of the bath modes will not persist but will give way to irregular dynamics within the center manifold. At the same time, one can hope to separate the reactive mode from the bath modes and in this way to find the recrossing-free dividing surfaces and the separatrices that are of importance to TST. As was shown in Ref. 40, this separation can indeed be achieved through a generalization of the normal form procedure that was used earlier to treat autonomous systems [34]. 

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

This choice still allows us to achieve the separation of the reactive degree of freedom from the bath mode that is required for TST because we can eliminate all monomials with j / kt. Thus if we carry out the normal form transformation up to an arbitrarily high order M, we find normal form coordinates... 

For time-dependent Hamiltonian systems we chose in Section IVB to use a normal form that decouples the reactive mode from the bath modes, but does not attempt a decoupling of the bath modes. This procedure is always safe, but in many cases it will be overly cautious. If it is relaxed, the dynamics within the center manifold is also transformed into a (suitably defined) normal form. This opens the possibility to study the dynamics within the TS itself, as has been done in the autonomous case, for example in Ref. 107. One can then try to identify structures in the TS that promote or inhibit the transport from the reactant to the product side. 

Eq. (II.4) play different roles in vibrational relaxation and dephasing and in spectral shifts. The above treatment for vibrational relaxation can only be applied to the case where the system oscillator frequency 00 is not too much larger than the bath mode frequencies coFor the case where 00 00 multi-... 

In the secular approximation [89], we can eliminate the coherence terms [e.g., pr, (x)(u / u")] in Eq. (III.9) such that the only diagonal terms contribute to the vibrational transitions through which the vibrational populations in various states are coupled. By applying the ladder model [89] to the interaction between the vibrational and heat-bath modes, the vibrational population decay constant is expressed as... 

Previously, stochastic Schrodinger equations for a quantum Brownian motion have been derived only for the particle component through approximated equations, such as the master equation obtained by the Markovian approximation [18]. In contrast, our stochastic Schrodinger equation is exact. Moreover, our stochastic equation includes both the particle and the field components, so it does not rely on integrating out the field bath modes. 

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

For analytic purposes, it is usefiil to define a spectral density of the bath modes coupled to the reaction coordinate in a given frequency range ... 

Because the generalized coordinate f is a linear combination of all bath modes and the potential is quadratic in the bath variables one can express the potential of mean force w[f] in terms of a single quadrature over the system coordinate... 

In many cases, when the damping is weak there is hardly any difference between the unstable mode and the system coordinate, while in the moderate damping limit, the depopulation factor rapidly approaches imity. Therefore, if the memory time in the friction is not too long, one can replace the more complicated (but more accurate) PGH perturbation theory, with a simpler theory in which the small parameter is taken to be for each of the bath modes. In such a theory, the average energy loss has the much simpler form ... 

This semiclassical turnover theory differs significantly from the semiclassical turnover theory suggested by Mel nikov, who considered the motion along the system coordinate, and quantized the original bath modes and did not consider the bath of stable normal modes. In addition, Mel nikov considered only Ohmic friction. The turnover theory was tested by Topaler and Makri, who compared it to exact quantum mechanical computations for a double well potential. Remarkably, the results of the semiclassical turnover theory were in quantitative agreement with the quantum mechanical results. 

We, however, are interested in the extremely non-Markovian timescales, much shorter than on which all bath modes excitations oscillate in unison and the... 

The relaxation rate R t) described by Eqs. (4.49)-(4.51) embodies our universal recipe for dynamically controlled relaxation [10, 21], which has the following merits (i) it holds for any bath and any type of interventions, that is, coherent modulations and incoherent interruptions/measurements alike (ii) it shows that in order to suppress relaxation, we need to minimize the spectral overlap of G( ), given to us by nature, and Ffo)), which we may design to some extent (iii) most importantly, it shows that in the short-time domain, only broad (coarse-grained) spectral features of G( ) and Ffa>) are important. The latter implies that, in contrast to the claim that correlations of the system with each individual bath mode must be accounted for, if we are to preserve coherence in the system, we actually only need to characterize and suppress (by means of Ffco)) the broad spectral features of G( ), the bath response function. The universality of Eqs. (4.49)-(4.51) will be elucidated in what follows, by focusing on several limits. 

The bath response function is usually associated with a characteristic correlation or memory time, t, which separates the non-Markovian (t t ) and the Markovian (t t ) temporal regimes, for example, 0(t) a Within the bath memory time, the bath modes oscillate coherently and in unison, and maintain memory of their interaction with the system, whereas after the correlation time has passed, the modes lose their coherent oscillations and forget their prior interactions [94]. 

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