Periodic boundary conditions temporal evolution

Thus, in practice, the potential distribution within the electrolyte is obtained by solving Laplace s equation subject to a time-dependent, Dirichlet-type boundary condition at the end of the double layer of the WE, a given value of (j> at the end of the double layer of the CE and zero-flux or periodic boundary conditions at all other domain boundaries. Knowing the potential distribution, the electric field at the WE can be calculated, and the temporal evolution of the double layer potential is obtained by integrating Eq. (11) in time, which results in changed boundary conditions (b.c.) at the WE. 

Althouh QBs are characterized by various methods [82, 102], here our main focus will be on the temporal evolution of the number of quanta, as it is possible to find out the critical time of redistribution that is proportional to QB s lifetime in femtoseconds, which might be useful for quantum computation application. QBs have been studied for dimer and trimer cases, and that also by (mainly) periodic boundary condition approach. However, a real material consists of many subunits, i.e., thousands of domains make ferroelectrics, each acting as sites and phonons act here as bosons or quanta. Again, how the increase of number of sites and bosons affects a system can also be regarded as an interesting topic. Hence, it drives us to a study that considers more number of sites and quanta. This is also the main aim of this chapter. 

Quantum localization behavior in K-G lattice has been studied by many researchers in terms of four atom lattice with periodic function, notably by Proville [106], delocalization and spreading behavior of wave-packets by Flach et al. [38], dimer case for targeted energy transfer by Aubry et al. [43], Here, we present a generalized method for any number of sites and quanta without periodic boundary condition to show the QB states. In K-G lattice, it is important to calculate the critical time of redistribution of quanta under various physical conditions. It is the time when the temporal evolution of the number of quanta first meets or tends to meet. 

FIGURE 5 Temporal evolution of 6 quanta on 3 sites under non-periodic boundary condition approach in hthium tantalate ferroelectrics with a low coupling value of 0.9 and a nonhnearity value of 421. At the point on the time-axis where three quanta meet each other, i.e., the critical time of redistribution that is proportional to quantum breathers lifetime in femtosecond can be derived. 

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