Time evolution equations master equation

Writing down the time evolution equation for the Wigner function, we find from the Master Equation (26)... 

Fortunately a substantial amount of relevant physics can be extracted by considering the low-order tenns in this expansion. The lowest order is the mean potential approximation (10.107). The next order is obtained by neglecting the last term on the right-hand side of (10.110) and inserting the remaining expression for Qp into Eq. (10.104). The resulting approximate time evolution equation for the system density operator d is what is usually referred to as the quantum master equation. 

Kinetic studies such as these use the master equation to follow the flow of probability between the states of the model. This equation is a basic loss-gain equation that describes the time evolution of the probability pi(t) for finding the system in state i [24]. The basic form of this equation is... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

The relaxation of a thermodynamic system to an equilibrium configuration can be conveniently described by a master equation [47]. The probability of finding a system in a specific state increases by the incoming jump from adjacent states, and decreases by the outgoing jump from this state to the others. From now on we shall be specific for the lattice-gas model of crystal growth, described in the previous section. At the time t the system will be found in the state. S/ with a probability density t), and its evolution... 

For a macroscopic variable A Sj), the time evolution of the expectation value, Eq. (11), is obtained by the master equation explicitly as... 

Mean Field Approximation as a first order approximation, we will ignore all correlations between values at different sites and parameterize configurations purely in terms of the average density at time t p. The time evolution of p under an arbitrary rule [Pg.73]

The chemist s view of a reaction is phenomenological. One assumes the existence of reactants, labeled a and products labeled b. The time evolution of normalized reactant (na) and product (nt) populations, na(t) + nb(t) = 1, is described by the coupled set of master equations ... 

In a realistic simulation, one initiates trajectories from the reactant well, which are thermally distributed and follows the evolution in time of the population. If the phenomenological master equations are correct, then one may readily extract the rate constants from this time evolution. This procedure has been implemented successfully for example, in Refs. 93,94. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref 95. 

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... 

Equation (5.1) described the vibrational response of a single particle to an applied forceF(t). In a (crystalline) system of many mobile particles (ensemble), the problem is analogous but the question now is how the whole system responds to an external force or perturbation Let us define the system s state (a) as a particular configuration of its particles and the probability of this state as pa. In a thermodynamic system, transitions from an a to a p configuration occur as thermally activated events. If the transition frequency a- /5 is copa and depends only on a and / (Markovian), the time evolution of the system is given by a master equation which links atomic and macroscopic parameters (dynamics and kinetics)... 

Be aware of the fact that we have to consider the non-Markovian version of the quantum master equation to stay at a level of description where the emission rate, Eq. (39), can be deduced. Moreover, to be ready for a translation to a mixed quantum classical description a variant has been presented where the time evolution operators might be defined by an explicitly time-dependent CC Hamiltonian, i.e. exp(—iHcc[t — / M) has been replaced by the more general expression Ucc(t,F). 

We remark that the simulation scheme for master equation dynamics has a number of attractive features when compared to quantum-classical Liouville dynamics. The solution of the master equation consists of two numerically simple parts. The first is the computation of the memory function which involves adiabatic evolution along mean surfaces. Once the transition rates are known as a function of the subsystem coordinates, the sequential short-time propagation algorithm may be used to evolve the observable or density. Since the dynamics is restricted to single adiabatic surfaces, no phase factors... 

Eq. (2.25) is often referred to as generalized master equation (GME). It should be noted that Eq. (2.20) or (2.25) describes the time evolution of an isolated system. 

Each site is occupied by adspecies of one kind (jjyj — A,yy)). Let I = jp, yj- denote the total set of the y j values of all the sites of the lattice and P I, t) denote the probability of finding the system at the time instant t in a certain state I. The time evolution of the system is described by the master equation [82-84] ... 

The time evolution of the probability / ,( ) is given by the following master equation ... 

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

This is the Markovian memory-less approximation to the Master Equation. In this approximation, the effective time evolution operator becomes independent of t and the integral may be extended to infinity. It is also consistent to assume that the system lost memory of the initial state of the reservoir, whatever this was. In the limit when Uq is calculated in perturbation theory and pq(0) = 0, we obtain the conventional Born-Markov time evolution which has a long and successful history. 

The problem is, however, that we need to fix the exact conditions of validity of this approximation, this was attempted already in Ref. [Fano 1954], In particular, it has turned out that introduction of the memory effect is a very sensitive issue [Barnett 2001], Highly reasonable but unprecise approximations may lead to non-physical time evolution. An additional problem is that the procedure does not necessarily lead to Master Equations of the Lindblad type, see above. If this is not its form, we may find well known complications, which have to be avoided if we want to escape unphysical results. 

Figure 3 Time evolution for two different initial conditions for over-damped situation with 7 = 3 and 0 = 1. Now w(t) i=- 0 for all finite t and the memory-less Master Equation has a unique solution.

Another problem is that memory kernels seem to be delicate entities. Erroneous kernels can destroy the physical sense of the time evolution of an initially acceptable density matrix. We do not have a general criterion to help us judge from the Master Equation with memory if the evolution is acceptable. In the Markovian limit, we know that the Lindblad form is certain to preserve the physical interpretation. It is a challenge for the theory of irreversibility in quantum systems to find such a criterion when memory effects are important. 

Abstract We consider the time evolution of a dynamic quantum system coupled to a repeatedly measured ancilla. Given the time lapse At between two subsequent measurements, the combined system may be described using a difference master equation whereas, in the Zeno-limit At — 0, the evolution of the dynamic system is unitary and defined by the state of the ancilla. For an arbitrary At, we also formulate a master equation that interpolates smoothly the exact evolution given by the difference equation. 

In the special case of a nondemolition interaction Hamiltonian, the master equation of the total system reduces to uncoupled systems of first order differential equations, whose dimensions are the same as the dimension Na of the ancilla Hilbert space. After having traced over the ancilla state, the master equation of the dynamic system can be expressed either as N,fh order differential equations in time or, equivalently, as Zwanzig equations with an explicit memory over the system evolution. 

In order to develop the mathematical framework for the master equation, we consider three descriptions of the system time evolution. 

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