Random environment

Due to difficulties and uncertainties in the experimental separation of the porous media [93], and the inevitability of approximations in the analytical treatment [87,89], the nature of the chain movement in a random environment is still far from being well understood, and theoretical predictions are controversial [87,89]. Thus, on the ground of replica calculations within a variational approach, one predicts three regimes [87] in which the chain gyration radius Rg scales with the number of repeatable units N as rI (X for low, R x N for medium, and R x for high... 

A natural extension of the studies in Sec. VIIA would be the investigation of the drift of a polymer chain in random environment when a constant external field B is applied in one direction [21,22]. 

Hughes, BD, Random Walks and Random Environments Volume I Random Walks Clarendon Press Oxford, 1995. 

Let us consider that the solvent has nonoverlapping domains that have distinct static and dynamic characteristics. The diffusion in such a system can be considered as a random walk in a random environment formed by the independently relaxing domain [95], Hence the diffusions in three dimensions is the weighted average of the diffusion in all these different domains [84], The average diffusion D can then be written as... 

The collective behavior of condensed modulated structures like charge or spin density waves (CDWs/SDWs) [23, 22, 4], flux line lattices [2, 36] and Wigner crystals [4] in random environments has been the subject of detailed investigations since the early 1970s. These were motivated by the drastic influence... 

Ebert U, Baumgartner A, Schafer L (1996) Universal short-time motion of a polymer in a random environment Analytical calculations, a blob picture, and Monte Carlo results. 

This fractured surface may be modelled as a random surface passing through the cutting bonds, so that the surface has the least total energy. Obviously, the surface can not have any overhang the height z x y) can not be multi-valued at any point (x,y). This random surface problem is therefore analogous to that of a directed polymer in a random environment, in two dimensions, as mentioned before. 

B. D. Hughes, Random Walks and Random Environments, Vol. 1 Random Walks, Oxford University Press, Oxford, 1995. Note that Hughes coins the term leapers for Levy flights. 

Many solids do dissolve in liquids by endothermic processes, however. The reason such processes can occur is that the endothermicity can be outweighed by a large increase in disorder of the solute during the dissolution process. The solute particles are highly ordered in a solid crystal, but are free to move about randomly in liquid solutions. Likewise, the degree of disorder in the solvent increases as the solution is formed, because solvent molecules are then in a more random environment. They are surrounded by a mixture of solvent and solute particles. 

In diluted polymer solution macromolecules are in random environment, influencing on the process of either aggregation or degradation (more rarely on both of them) that results in the stochastic equation instead of the Eq. (6) [13] ... 

In this section we consider a random walk in random time. In this case the particle position X depends on the random time T(t), X(T t)), rather than on the conventional time t. An insight into this model can be obtained by considering a particle moving in a nonstationary random environment for which the intensity of jumps is random. We use the standard formula for the particle position as the sum of ED random jumps Z,, ... 

In paper [102] the theoretical treatment of a cluster-cluster aggregation accounting for the existence of coalescing of particles or of clusters (monomers or macromolecular coils) to aggregate in real polymerization processes, and their disconnection (destruction) was reported. Macromolecules are in a random environment, influencing the processes of aggregation or destmction in dilnte polymeric solution that is described by the stochastic equation [101] ... 

The conformational statistics of a polymer chain will change if the chain is placed in a random environment. We can model the effect of the random environment by introducing an interaction energy... 

In this chapter we have demonstrated the rich behavior of polymer chains embedded in a quenched random environment. As a starting point, we considered the problem of a Gaussian chain free to move in a random potential with short-ranged correlations. We derived the equilibrium conformation of the chain using a replica variational ansatz, and highlighted the crucial role of the system s volume. A mapping was established to that of a quantum particle in a random potential, and the phenomenon of localization was explained in terms of the dominance of localized tail states of the Schrodinger equation. 

Models of population growth in random environment have been constructed by inserting white noise fluctuations in the deterministic growth equations (May, 1972 Capocielli Ricciardi, 1974 Ricciardi, 1977 Nobile Ricciardi, 1984a, b). The structure of these models is similar to those described in Section 5.8 ( external fluctuations ) and we shall not discuss them here. 

Capocielli, R. M. Ricciardi, L. M. (1974). Growth with regulation in random environment, Kybernetik, 15, 145-57. 

In recent years, more complex types of transport processes have been investigated and, from the point of view of solid state science, considerable interest is attached to the study of transport in disordered materials. In glasses, for example, a distribution of jump distances and activation energies are expected for ionic transport. In crystalline materials, the best ionic conductors are those that exhibit considerable disorder of the mobile ion sublattice. At interfaces, minority carrier diffusion and discharge (for example electrons and holes) will take place in a random environment of mobile ions. In polycrystalline materials the lattice structure and transport processes are expected to be strongly perturbed near a grain boundary. 

Fourth, in Section 7.4 we examine the closure problem arising in aggregating systems as well as in random environments and some approximations that have been used to obtain closure. 

Various authors have developed methods based on Raman spectroscopy for determining crystallinity. In some polymers, specific vibrational bands have been attributed to the crystalline phase. In others, the ratio of the intensities of trans conformation vibrations to gauche conformation vibrations has been used. Still others have used the width of a particular vibrational band to indicate crystallinity. The last method is based on the effect of the local environment on the frequency of the band. In a random environment, many different local environments exist and the band is broadened by these variations. In a crystal, the local environment is the same for all of the chain segments. Then all of these segments have (nearly) the same frequency and the band is narrow. For a band that is sensitive to the local environment, a narrow band indicates high crystallinity and a broad band indicates low crystallinity. 

Kenzin, M., Frostig, E. (2009). M out of n inspected systems subject to shocks in random environment. Reliability Engineering and System Safety 94 1322-1330. 

Fig. 1 The emergence of a conducting ""open ) path across the disordered system. The corresponding percolation current in the random environment of an amorphous organic semiconductor is distributed quite inhomogeneous...

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