Bath interaction

From the research on electrocodeposition to date, a number of variables appear to be influential in the process, which include hydrodynamics, current density, particle characteristics, bath composition, and the particle-bath interaction. The influence that a particular variable has on the process is typically assessed by the change in the amount of particle incorporation obtained when that variable is adjusted. Although the effect of each of these process variables has been reported in the literature, the results are often contradictory. The effects of the process variables, of which many are interrelated, can also vary for different particle-electrolyte systems and electrodeposition cell configurations used. This review will summarize these effects and the contradictions in the literature on electrocodeposition. 

Another concern regards the initial conditions. Here we have assumed a factorized initial state of the system and bath. This prevents us from taking into account system-bath interactions that may have occurred prior to that time. In particular, if the system is in equilibrium with the bath, their states are entangled or correlated [24, 94]. 

We use now the results of the foregoing section to discuss the electronic transport properties of our model in some limiting cases for which analytic expressions can be derived. We will discuss the mean-field approximation and the weak-coupling regime in the electron-bath interaction as well as to elaborate on the strong-coupling limit. Furthermore, the cases of ohmic (s = 1) and superohmic (s = 3) spectral densities are treated. 

Employing V — b)F bQ exp(— J2j xj9j) b as the system-heat bath interaction, where qj denotes the vibrational coordinate of theyth heat bath mode, the vibrational population decay rate constant is obtained as... 

To obtain the oscillator-bath interaction term, we argue that the solute s instantaneous size depends linearly on the breathing coordinate q multiplied by a dimensionless coefficient a. The latter is treated as the single adjustable parameter in the theory, which should on physical grounds be less than but on the order of unity. This leads to (2)... 

To define the oscillator-bath interaction term, we write the full solute-... 

Thus we again assume a Lennard-Jones form, where now the well depth and range parameters depend on the solute s internal vibrational coordinates. Without loss of generality we can define these coordinates so that q = Q = 0 corresponds to the minimum in the intramolecular potential. The solute-solvent potential in Hb above is actually then

(r, 0, 0), where clearly e = e(0, 0) and a = cr(0, 0). The oscillator-bath interaction term is... 

GX(i e5(ti-tj) represents the time-development matrix element of the intermolecular coherence between (i X, and e 8 from time to time t in the presence of the molecule-heat bath interaction. This matrix element is defined by... 

The initial temperature of the molecule is very low and certainly is much lower than hcajk of totally symmetric modes. We may divide the modes of the molecule into those which are optically active (predominantly totally symmetric or relevant R) and those which are not excited directly by the laser (bath modes B). This division of the system-bath interactions accounts for dephasing and energy relaxation by T2 and Ti time constants as discussed elsewhere. ... 

Momentum representation of the solute/bath interaction in the dynamic theory of chemical processes in condensed phase Bosanac S.D. 

Starting from the gap Hamiltonian (33) and the interactions with the reservoirs (34) and (35), eliminating the reservoir degrees of freedom within the standard Bom-Markov approach we derive a master equation for the reduced density operator of the atomic ensemble. The calculation is lengthy but straight forward. Disregarding level shifts caused by the bath interaction we find for the populations in states cj ) the following density matrix equations in the interaction picture ... 

The Hamiltonian in question is the sum of the system (S), reservoir bath (B) and system-bath interaction (7) terms,... 

Such correlation functions are often encountered in treatments of systems coupled to their thennal environment, where the mode 1 for the system-bath interaction is taken as a product of A or B with a system variable. In such treatments the coefficients Cj reflect the distribution of the system-bath coupling among the different modes. In classical mechanics these functions can be easily evaluated explicitly from the definition (6.6) by using the general solution of the harmonic oscillator equations of motion... 

The system-bath interaction term in (8.48) is xf, where f = cjqjis, the force exerted by the thermal environment on the system. The random force 7 (t), Eq. (8.56) is seen to have a similar form. 

Comparing Eqs (J.liy-ij.19 ) we see that Z t) is essentially the Fourier transform of the spectral density associated with the system-bath interaction. The differences are only semantic, originating from the fact that in Eqs (7.77)-(7.79) we used mass renormalized coordinates while here we have associated a mass nij with each harmonic bath mode j. 

The first two tenns on the right describe the system and the bath , respectively, and the last tenn is the system-bath interaction. This interaction consists of terms that annihilate a phonon in one subsystem and simultaneously create a phonon in the other. The creation and annihilation operators in Eq. (9.44) satisfy the commutation relations ... 

We note in passing that had we included the bath-average interaction (10.105) as part of the system s Hamiltonian, then the first term on the right of (10.113) would not appear. This is indeed the recommended practice for system-thermal bath interactions, however, we keep this term explicitly in (10.113) and below because, as will be seen, an equivalent time-dependent term plays an important role in describing the interaction of such system with an external electromagnetic field. 

What did we achieve so far We have an equation, (10.133) or (10.134), for the time evolution of the system s density operator. All terms in this equation are strictly defined in the system sub-space the effect of the bath enters through correlation functions of bath operators that appear in the system-bath interaction. These correlation functions are properties of the unperturbed equilibrium bath. Another manifestation of the reduced nature of this equation is the appearance of... 

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