Weakly open systems

Let us consider open systems at low vin in two stages. First let us assume that both uin and vOM are low and time-dependent, but are such that the gas pressure in the reactor is in the range Pmax P Pmm or, equivalently, 6max 2 cf bmm 0, where bmm = PmJRT and 6max = Pmax/RT. This agrees well with reality, i.e. even if we want to, we cannot obtain a pressure in the reactor which would be either equal to zero or higher than some very high Pmax. 

Kinetic equations in weakly open systems will take the form  

Using the condition of smallness of vm S.c, in and vout CjjOUt we obtain from eqn. (123) 

In each reaction polyhedron, the region specified by the inequality 

These regions in all reaction polyhedra can be described by the same inequality. For this purpose let us recall (Sect. 2) that we constructed G(c) using an arbitrary PDE not necessarily lying in the examined reaction polyhedron and showed that this function is a Lyapunov function for any reaction polyhedron. Now let us introduce one more Lyapunov function which differs from the previous one in every reaction polyhedron by a constant, depending, nevertheless, on this polyhedron. Let us prescribe a function c (c) whose value is PDE accounting for the initial conditions c (lying in the same reaction polyhedron). Let us determine 

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Fig. 2. Stochastic accretion models for an open system. The infalling gas is assumed to be extragalactic material with standard Big Bang nucleosynthetic abundances (Xo = 0.758, Yo = 0.242, 2D=6.5xlCP5, SBBN) and zero metals, (a) Star formation rate vs. time for the thin disk. From the top to the bottom the curves refer to 44%, 10%, 5%, 1% and no mass added, (b) Metallicity vs. time for the thin disk. From the top to the bottom the curves refer to standard case (no mass added), 1%, 5%, 10%, 44% of mass added. The metallicity evolution curve illustrates the relatively weak dilution effects that are offset by continuing star formation. Details for the Deuterium abundances are shown in Fig. 3...

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

It turns out that this analysis applies only to systems with a bounded phase space. It is possible that topological restrictions on the accessible phase space - and not only the form of the particular Hamiltonian -play a crucial role in determining when the weak form of the QCT actually applies. For example, this might explain why the open-system quantum delta-kicked rotor is a counter-example to naive expectations regarding the QCT (S. Habib et.al., 2002). 

We turn now to the finite open and closed chain and compare the pair correlations obtained in the different systems. First, we note that in the m —> °o limit all the sites become identical in the weak sense, i.e., there is only one intrinsic binding constant, but different pair (and higher-order) correlations as shown in Eq. (7.4.28). It should be noted, however, that owing to the translational invariance of the infinite system there is only one nn pair correlation, only one second nn pair correlation, etc. In other words, it does not matter where in the chain we choose the pair of nn neighbors, or the second nn neighbors, etc. This translational invariance is lost in the finite open system. 

Ammonium sulfate decomposes upon heating at 100°C in an open system, forming ammonium bisulfate, NH4HSO4. As a salt of a strong acid and weak base, its solution is acidic pH of 0. IM solution is 5.5. 

We now connect the analysis given above with the equation of motion displayed in Eq. (5.5). That equation of motion follows from subdivision of a system into an open subsystem S and a complementary reservoir R. When the coupling between S and R is weak, the evolution of the open system 5, due to the internal dynamics of 5 and the interaction with the reservoir R, can be described in density matrix form by Eq. (5.5). Now writing... 

Open systems with PCB. An efficient means to establish whether this point exists is to check the equality (169) M — l = S and a weak reversibility (these are sufficient but, generally speaking, not necessary conditions). 

The response of the acids is surprising in that significant uptake is observed despite the fact that only a small proportion of the compound exists in the neutral form. This could be attributed to the fact that this is an open system and the compound is taken away by the circulation as soon as it crosses the membrane. An ion trap effect may be of more significance. For example, in the case of a weak acid (piifa = 3), when there is a pH differential across a membrane and only the neutral species moves across that membrane, under equilibrium conditions, it can be seen that ions are trapped in the compartment where the pH favors the production of the anion (i.e., where the pH > p fa)- Since the pH of the circulating... 

Decoherence phenomena are most easily observed for small quantum systems that are well isolated from the rest of the world so that the latter (called the environment ) interacts only weakly with the system under consideration. Examples of such weakly coupled open systems are found in experiments on particles isolated in cavities under extreme vacua, where simple systems (pairs of entangled atoms) have been shown to stay internally coherent for milliseconds, but it has also been observed that the decoherence rate increases strongly with the size of the system, such as in the experiments by Brune et al. [1] on photons enclosed in a cavity. 

In the case of open-system dynamics, assuming weak coupling of the conical intersection with an environment, the time evolution of the system is determined by the Redfield Eq. (7) for the reduced density matrix. In this case, the time-dependent population probabilities of diabatic and adiabatic states are given by... 

In classical conceptions weakly distorted systems travel via relaxation towards an equilibrium state [5, 6, 8, 9]. To get the same logistics we have to postulate that in dissipative open systems optimised stationary states operate as temporary states of reference. If sufficiently fast processes are available there is no objection against the hypothesis that relaxation towards such stationary states should exhibit analogous features as in classical models [9]. If the external conditions vary steadily it should also be possible that relaxation is running off while the state of reference passes slowly through a logical sequence of temporary stationary patterns. 

Before we introduce the concept of an open system, it is useful to discuss the specific heat of the subsystem itself. The Hamiltonian "Xg in Equation 11.7 does qualify for describing the prototype thermodynamics of a smah quantum system, like a harmonic oscillator. What we have to do is to imagine Jig to be weakly coupled to a classical heat bath, with which the system undergoes exchange of energy. The consequent energy fluctuations provide the temperature of the system. All this can be put into statistical mechanical perspective in terms of the Gibbsian partition function... 

Some systems converge poorly, particularly those with multiple bonds or weak interactions between open-shell systems. HyperChem includes two convergence accelerators. One is the default con verge rice accelerator, effective in speed in g up ri orm ally... 

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