Evolution of coupled systems

If we consider the change of affinity with time at constant temperature and pressure, we have 

It is possible to split the dCs into two parts dCs = deCs + c/,Cs, which describe the part resulting from the exchange with the surrounding and the part due to a chemical reaction. The rate of the second part is the reaction velocity d1Cs/dt = vsJI. With these relations, Eq. (9.171) yields 

Therefore, affinity changes at the rate of exchanged matter and chemical reaction velocity. Depending on the rate of exchanged matter, the first term in Eq. (9.172) may counterbalance the reaction velocity, and the affinity may become a constant. This represents as system where one of the forces is fixed, and may lead to a specific behavior in the evolution of the whole system. 

These equations suggest that the degrees of coupling besides the other parameters control the evolution and stability of the system. Comparing Eq. (9.174) with a simple rate expression 

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Concluding this section, one should mention also the method of molecular dynamics (MD) in which one employs again a bead-spring model [33,70,71] of a polymer chain where each monomer is coupled to a heat bath. Monomers which are connected along the backbone of a chain interact via Eq. (8) whereas non-bonded monomers are assumed usually to exert Lennard-Jones forces on each other. Then the time evolution of the system is obtained by integrating numerically the equation of motion for each monomer i... 

The present analysis shows that when a thermodynamic gradient is first applied to a system, there is a transient regime in which dynamic order is induced and in which the dynamic order increases over time. The driving force for this is the dissipation of first entropy (i.e., reduction in the gradient), and what opposes it is the cost of the dynamic order. The second entropy provides a quantitative expression for these processes. In the nonlinear regime, the fluxes couple to the static structure, and structural order can be induced as well. The nature of this combined order is to dissipate first entropy, and in the transient regime the rate of dissipation increases with the evolution of the system over time. 

In order to affect the system-bath coupling and control, or modulate, the decoherence due to this coupling, one must dynamically modulate the system faster than the correlation time. Slower modulation will have no effect on the loss of coherence and will thus not be able to control it. Modulating the system faster than the correlation time can effectively reset the clock. Applying a modulation sequence repeatedly can thus drastically change the decoherence and impose a continued coherent evolution of the system-bath coupling [46, 94]. 

Figure 17.7 Splitting of energy curves for a reduction reaction. In (a) the strong coupling leads to a considerable splitting and to a continuous curve connecting the reactant and the products configurations, favoring the evolution of the system toward the product state. In (h) the small splitting favors the tendency of the system to remain in the reactant curve (O + e).

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

As a first approach to describing the fast time evolution of the system quantitatively (71), a description using a harmonic basis set with anhar-monic coupling of the high-frequency CO stretch and the other modes can... 

Simulation packages such as GAMMA take advantage of the fact that evolution of the density matrix under the Liouville-von Neumann equation is well approximated by a small number of easily applied transformations of the density matrix, namely free evolution can be represented by a simple unitary transformation and application of ideal RF pulses can be represented by a simple rotation. Real RF pulses can be effectively modelled as a succession of ideal RF pulses. The beauty of this method is that fairly complex, realistic effects, such as evolution of coupled spin systems through complex pulses, can be modelled by a straightforward combination of these simple building blocks. 

If the characteristic time of changing Hik is fixed, the evolution of the system proceeds almost adiabatically beyond the coupling region (the region outside the circle in Figure 5.1). In this region the N matrix can be calculated using a theory of almost adiabatic perturbations. If, on the other hand, H12... 

The last equations prove that the Markov chains [4.6] are able to predict the evolution of a system with only the data of the current state (without taking into account the system history). In this case, where the system presents perfect mixing cells, probabilities p and p j are described with the same equations as those applied to describe a unique perfectly stirred cell. Here, the exponential function of the residence time distribution (p in this case, see Section 3.3) defines the probability of exit from this cell. In addition, the computation of this probability is coupled with the knowledge of the flows conveyed between the cells. For the time interval At and for i= 1,2,3,. ..N and j = 1,2,3,..N - 1 we can write ... 

In a time-dependent picture, one can imagine an electron being annihilated from a specific valence bond orbital at time t=0. This is obviously not a stationary state of the ion and hence one must follow the time dependent evolution of the system from this initial condition. This requires solving the time-dependent Schrodinger equation which may be written as the following set of coupled equations for the time dependent probabihty amplitudes, aj (t) ... 

Abstract Interaction between a quantum system and its surroundings - be it another similar quantum system, a thermal reservoir, or a measurement device - breaks down the standard unitary evolution of the system alone and introduces open quantum system behaviour. Coupling to a fast-relaxing thermal reservoir is known to lead to an exponential decay of the quantum state, a process described by a Lindblad-type master equation. In modern quantum physics, however, near isolation of individual quantum objects, such as qubits, atoms, or ions, sometimes allow them only to interact with a slowly-relaxing near-environment, and the consequent decay of the atomic quantum state may become nonexponential and possibly even nonmonotonic. Here we consider different descriptions of non-Markovian evolutions and also hazards associated with them, as well as some physical situations in which the environment of a quantum system induces non-Markovian phenomena. 

We consider a coupled quantum system consisting of two subsystems, one of which is referred to as dynamic part and the other one as ancilla. The evolution of the system is governed by the Schrodinger equation... 

The first tenn, iPCPpU describes the time evolution that would be observed if the system was uncoupled from the bath throughout the process (i.e. if QCP = PCQ = 0). The second term is the additional contribution to the time evolution of the system that results from its coupling to the bath. This contribution appears as a memory term that depends on the past history, / p(r), of the system. Consider the integrand in this term, written in the form ... 

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