Decay master equation

The Montroll-Shuler equation can also predict how fast a molecule which is created in a highly excited vibrational state will decay to the equilibrium state. This is of interest in connection with chemiluminescence phenomena. In certain cases one finds experimentally that this relaxation is much faster than what one would expect from the master equation of Montroll and Shuler and improved versions of this equation. One possible mechanism for this fast relaxation is that although most of the collisions in which the diatomic molecule participates are between the diatomic molecule and an inert gas atom, there will also be some collisions between diatomic molecules. In the latter case we have the situation where two diatomic molecules in quantum state n collide producing, with fairly high probability, molecules in quantum states n I and n + 1, respectively. The number of such collisions is, of course quite small compared to the number of collisions of the first kind, but since they are so extremely efficient they may still be of importance. This mechanism, we believe, was first suggested in connection with chemiluminescence by Norrish in a Faraday Society discussion.5 The equations describing this relaxation had, however, been discussed several years earlier by Shuler6 and Osipov.7... 

As an example we treat the decay process of IV.6 in terms of the master equation. The decay probability y per unit time is a property of the radioactive nucleus or the excited atom, and can, in principle, be computed by solving the Schrodinger equation for that system. To find the long-time evolution of a collection of emitters write P(n, t) for the probability that there are n surviving emitters at time t. The transition probability for a... 

Exercise. Find the stationary distribution for the radioactive decay process described by the master equation (V.1.7). 

The reason why the boundaries in physical problems are often natural becomes obvious by looking at the simple example of radioactive decay in IV.6. The probability for an emission to take place is proportional to the number n of radioactive nuclei, and therefore automatically vanishes at n = 0. The same consideration applies when n is the number of molecules of a certain species in a chemical reaction, or the number of individuals in a population. Whenever by its nature n cannot be negative any reasonable master equation should have r(0) = 0. However, this does not exclude the possibility that something special happens at low n by which the analytic character of r(n) is broken, as in the example of diffusion-controlled reactions. A boundary that is not natural will be called artificial in section 7. 

Exercise. A long-lived radioactive substance A decays into B through two shortlived intermediaries. A -> X - Y - B. When the amount of A is supposed constant, the joint distribution of the numbers n, m of nuclei X, Y obeys the bivariate master equation... 

Exercise. In the radioactive decay process the state n = 0 is an absorbing state. Show that the equations for the remaining pn (n = 1,2,...) constitute a master equation with absorbing boundary. The state n = 0 functions as limbo state. 

Exercise. For the same decay process show that for any N 1 the states n N obey a master equation with absorbing boundary. Its limbo state consists of all states with n[Pg.156]

In the more difficult case of asymmetry in both mobilities (DA = 0, >b > 0) and initial concentrations, Sn > 0, a kind of master equation was derived and solved numerically, along with the Monte Carlo simulations [32], As it is seen in Fig. 6.26, for equal particle concentrations the decay asymptotics, shown by dashed lines, is reached very fast for both symmetrical and asymmetrical reactant mobilities (curves 1). It is no longer tme, however, for asymmetrical concentrations shown in curves 2. These latter demonstrate clearly an existence of the two different classes of universality for reactions with unequal concentrations one class corresponds to the case when both (A and B) reactants are mobile or those which are in majority whereas... 

Fig. 6.26. Time decay of the particle concentration, tia( ), for cases of equal (curves 1) and unequal (curves 2) concentrations involved into the A+B - 0 reaction for d = 1 [32]. Curves 1 riA(O) = riB(O) =0.1 curves 2 n =0.1, ub(0) = 0.2. Symbols A, B, AB correspond to types of mobile species (one type or both types respectively). Full lines show results of the Monte Carlo simulations whereas dashed lines - solution of a master equation.

As earlier in the case of diffusion-controlled concentration decay, one-species reactions, A +A -> A and A + A —> 0, in one-dimension, could be used as a proving ground for the testing theory of particle accumulation, when random particle creation is added. For the former reaction the master equation could be easily derived in the form [95]... 

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. 

Keywords Irreversible time evolution, Master Equation, non-exponential decay... 

These results leave several basic questions open How to derive a non-Markovian master equation (ME) for arbitrary time-dependent driving and modulation of a thermally relaxing two-level system Would the two-level system (TLS) model hold at all for modulation rates, that are comparable to the TLS transition frequency u)a (between its states e) and g)) which may invalidate the standard rotating-wave approximation (RWA), [to hen-Tannoudji 1992] Would temperature effects, which are known to incur upward g) —> e) transitions, [Lifshitz 1980], further complicate the dynamics and perhaps hinder the suppression of decay How to control decay in an efficient, optimal fashion We address these questions by outlining the derivation of a ME of a TLS that is coupled to an arbitrary bath and is driven by an arbitrary time-dependent field. 

T+ being the time-ordering operator. In the derivation of Eq. (168) we assumed that B(t)) = 0. It needs to be stressed that Eq. (168) generalizes previously known master equations to arbitrary time-dependent hamiltonians, Hs t) for the system and Hi(t) for system-bath coupling, [Cohen-Tannoudji 1992], Henceforth, we explicitly consider a driven TLS undergoing decay, whose resonant frequency and dipolar coupling to the reservoir are dynamically modulated, so that... 

This equation can also be used for obtaining the master equation in the Markovian limit. In that case, the exponential decays faster than the state p evolves p(s) may therefore be replaced with pit) and taken out of the integral. Assuming still Re(f n) < 0, the integral may be performed explicitly and the master equation is... 

The Pauli Master equation approach to calculating RET rates is particularly useful for simulating time-resolved anisotropy decay that results from RET within aggregates of molecules. In that case the orientation of the aggregate in the laboratory frame is also randomly selected at each Monte Carlo iteration in order to account for the rotational averaging properly. 

Ti = r2 the transformation (66) diagonalizes the dispersive part of the master equation. Furthermore, if Ti2 = /T I 2, then Taa = Tsa = Tas = 0 regardless of the ratio between Ti and r2. In this case the antisymmetric superposition does not decay. This implies that spontaneous emission can be controlled and even suppressed by appropriately engineering the cross-damping rate ri2 arising from the dissipative interaction between the systems. 

Much attention has been given to the relaxation of various properties toward their equilibrium values, especially in the glasses literature. Kohlrausch first proposed a stretched exponential form as a description of viscoelasticity, while Williams and Watts suggested the same form for dielectric relaxation exp[-(t/r) ], where 0 < 9 < 1 and 0 = 1 corresponds to the Debye limit. The master equation solution, Eq. (1.41), has a decaying multiexponential form that could lead to a wide variety of behavior depending upon the system. 

An unambiguous way to define the relaxation time, x, of a given property is to normalize its decay profile so that it starts at 1 and decays to 0, and then evaluate the total area under the curve [37]. For pure Debye relaxation, exp(—Xr), one simply obtains the inverse of the decay constant Xr = 1/. For the multiexponential decay given by the master equation, the relaxation... 

The analytic solution of the master equation decomposes the flow of probability into a series of exponentially decaying modes, each of which has a characteristic decay constant. It is instructive to look at how the contributions from these modes vary across the spectrum of time scales. From Eq. (1.41) mode j makes an important contribution to the probability evolution of minimum / if is large in magnitude. The mode... 

Relaxation times for the total energy were calculated for each of the three samples as a function of temperature by integrating the scaled, shifted energy decay curve as described in Section V.C.2. The master equation was solved analytically by matrix diagonalization (Section III.D) using two different starting distributions. The results in Figure 1.37a were obtained... 

We have already mentioned in Section II.G that the Greens function approach predicts nonexponential decay when the coupling matrix element is allowed to vary with energy. Lin used a very different approach to investigate the limitations of the master equation description. He showed, using a density matrix formalism, that memory effects appear if perturbation theory is extended to fourth order in the coupling between two vibrational levels. 

El < Eo) non-dissociative states (dissociative states are rapidly depopulated by the fast intramolecular dissociation process). As is well known, the time evolution of the populations [A(i)] is given by a series of exponentially decaying terms which ctHTiespond to an initial rovibrational relaxation, a subsequent incubation period with overlap of vibrational rriaxation of upper levels and dissociation, and the final dissociation period with steady-state of all populations [A(i)]. Explicit solutions of the master equation for the dissociation of diatomic molecules have been extensively reviewed by H. O. Pritchard in Volume 1 of this series. Such... 

Next we present the ease for three-level system in the Vee configuration as shown in Fig. 6. The master equation for the field-atom system has the same form as Eq. (11). Different from the Lambda system, the spontaneous decay in the Vee system occurs frxim two excited states 1 , 2 ) to a single ground state 3 ), the damping term in the master equation reads as L p = 2 3/P -... 

By tracing out the atomic variables, i.e., Pc=TratomP we can obtain the information of the cavity fields. Following the standard techniques for the mixing interactions [4,84] we treat the cavity fields b, bf) linearly. Assuming that the atoms decay much more rapidly than the cavity fields, we can eliminate atomic variables adiabatically and derive the master equation of cavity modes i,2. For the case of Qi/Qomaster equation for Pc has the same form as Eq. (35), and the parameters are the same as in Eq. (36). Also we have taken 713=723=/ and Ki=K2=k for simplicity. The only difference lies in the dressed populations at steady state in the absence of the cavity fields. They are calculated as... 

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