Nonlinear transport

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

The antisymmetric part of nonlinear transport matrix is not uniquely defined (due to the nonuniqueness of fh). However, the most likely terminal position is given uniquely by any that satisfies Eq. (122). 

The nonlinear transport matrix satisfies the reciprocal relation... 

Robert A. Brown is Warren K. Lewis Professor of Chemical Engineering and Provost at the Massachusetts Institute of Technology. He received his B.S. (1973) and M.S. (1975) from the University of Texas, Austin, and his Ph.D. from the University of Minnesota in 1979. His research area is chemical engineering with specialization in fluid mechanics and transport phenomena, crystal growth from the melt, microdefect formation in semiconductors and viscoelastic fluids, bifurcation theory applied to transitions in flow problems, and finite element methods for nonlinear transport problems. He is a member of the National Academy of Engineering, the National Academy of Sciences, and the American Academy of Arts and Sciences. 

In an important paper (TNC.l), they offered for the first time an extension of nonequilibrium thermodynamics to nonlinear transport laws. As could be expected, the situation was by no means as simple as in the linear domain. The authors were hoping to find a variational principle generalizing the principle of minimum entropy production. It soon became obvious that such a principle cannot exist in the nonlinear domain. They succeeded, however, to derive a half-principle They decomposed the differential of the entropy production (1) as follows ... 

Yuen JD, Menon R, Coates NE, Namdas EB, Cho S, Hannahs ST, Moses D, Heeger AJ (2009) Nonlinear transport in semiconducting polymers at high carrier densities. Nat Mater 8 572... 

The fully ionic solids (region I) afforded band insulators, 1 1 Mott insulators with ground states of antiferromagnets (E b(21) and F b(22) in Fig. 1) or spin-Peierls systems, ferroelectrics, ferromagnets, spin-ladders, and nonlinear transport materials (switching and memory). 

The Porath et al. experiment [14], reviewed in Sect. 2, reports nonlinear transport measurements on 10.4-nm-long poly(dG)-poly(dC) DNA, corresponding to 30 consecutive GC base pairs, suspended between platinum leads (GC-device). DFT calculations indicated that the poly(dG)-poly(dC) DNA molecule has typical electronic features of a periodic chain [58]. Thus, in both models (assuming dephasing or r-stack hybridization) the poly(dG)-poly(dC) DNA molecule is grained into a spinless linear TB chain. A generalization of the dephasing model to spin-transport has been proposed by Zwolak et al. [123]. 

This book treats a selection of topics in electro-diffusion—a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context [1]—[3]. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors [4]. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to ion separation and semiconductor device technology. Some aspects of electrodiffusion are relevant for electrophysiology. 

The plot in Figure 6.4 illustrates the differences in passive (linear) versus carrier-mediated (nonlinear) transport. At relatively low concentrations of drug, carrier-mediated processes may appear to be first order since the protein carriers are not saturated. However, at higher concentrations, zero-order behavior becomes evident. It is in plots such as this that the terms linear (first order) and nonlinear (zero order) come into existence. 

CDW can bear an electric current while the system is insulating below Tp in the sense of the single particle transport. The current is carried by a CDW sliding in the lattice with no restoring force at T=0 if 2kp is incommensurate with the underlying reciprocal lattice. In real materials impurities or lattice defects interact with the CDW leading to various phenomena such as the nonlinear transport, a type of mode-locking etc. [63] However, we will leave these problems out of the scope of this article. 

A. Wacker Semiconductor superlattices A model system for nonlinear transport, Phys. Rep. 357, 1 (2002). 

For this generic setup, we focus on the following topics (1) Resolving transport mechanisms. We study the role of harmonic/anharmonic internal molecular interactions on the heat current characteristics, as well as the effect of the molecular structure, system size, dimensionality, and the interaction strength with the solids. (2) Proposing molecular level mechanical device. We discuss the operation principle of various devices, e.g., a thermal rectifier and a heat pump. Such systems are of interest both fundamentally, manifesting nonlinear transport characteristics, and for practical applications. 

We have presented here several theoretical tools for exploring mechanisms of heat flow in molecular junctions a generalized Langevin equation approach, the master equation formalism, and classical MD simulations. Using these techniques, we have inferred that the thermal conductance of nanosystems results from an intricate interplay between the molecular structure, the contact properties, and the reservoir s spectral functions. Our minimal single-mode junction model clearly exposed the role of anharmonic interactions in bringing out nonlinear transport characteristics. This has lead us to propose unique thermal devices, a thermal rectifier and a heat pump. 

Of course the results concerning nonlinear transport phenomena must be transformed correspondingly. 

In this book, we use Truesdell s conceptually most simple idea of mixture [10-12] and we confine ourselves to a classical task important in applications we study the mixture of chemically reacting fluids (mechanically non-polar, cf. Secf.4.3 and Rem. 17 in Chap. 3), with the same temperature of all constituents and with linear transport properties (like diffusion, heat conduction, viscosity generalization on nonlinear transport, see [60, 71, 72, 104]). This model, called shortly the linear fluid mixture, contains as special cases non-reacting fluid mixtures and some further ones (see Sect. 4.8). 

Nonlinear diffusion. The voltammetric behavior related to linear (e.g. at short times) and spherical (e.g. at large times or small electrodes) diffusion has been discussed in Sect. 2.1.2. Of course, there are intermediate situations, in which mixed behavior is observed, which may be regarded as a distortion of either of the extreme types of transport. In particular, use of conventionally sized electrodes at slow scan rates causes the increase of peak currents (normahzed to with decreasing v since additional nonlinear transport of material across the edge of the electrode occurs ( edge diffusion or edge effect [47]). Consequently, too slow scan rates should be avoided. 

Other kinds of Fokker-Planck equations can be also derived. The continuous state-space stochastic model of a chemical reaction, which considers the reaction as a diffusion process , neglects the essential discreteness of the mesoscopic events. However, some shortcomings of (5.65) have been eliminated by using a direct Fokker-Planck equation obtained by means of nonlinear transport theory (Grabert et al., 1983). 

B. H. Hamadani and D. Natelson. Gated nonlinear transport in organic polymer field effect transistors. J. Appl. Phys. 95(3), 1227-1232 (2004). 

Rohrlich, B. Parisi, J. Peinke, J. Rossler, O. E. 1986. A Simple Morphogenetic Reaction-Diffusion Model Describing Nonlinear Transport Phenomena in Semiconductors, Z. Phys. B Cond. Matter 65, 259-266. 

A numerical model was developed to simulate MeBr movement in soil and volatilization into the atmosphere. The model simultaneously solves partial differential equations for nonlinear transport of water, heat, and solute in a variably saturated porous medium. Henry s Law is used to express partitioning between the liquid and gas phases and both liquid and vapor diffusion are included in the simulation. Soil degradation is simulated using a first-order decay reaction and the rate coefficients may differ in each of the three phases (i.e., liquid, solid, or gaseous). 

The result we obtain in (60) is that the linearization procedure always takes the evolution equation to a form of the type (53) that will provide an exponential decay of the small perturbation excess density. A general feature of small perturbation methods is that aU the nonlinear transport-conservation equations become linear. 

Fifth, at elevated applied fields one enters a nonlinear transport regime in which jx is not independent of . On entering this regime, one also crosses over from an exponential to a weak power-law dependence of pu on P. The transition location is a complex joint function of E, probe size, and matrix concentration. The nonlinear transport regime, and the probe size regime P > Pc described above, appear to be the same. Qualitatively, the transition to nonlinear behavior seems to be correlated with the total force applied to the solution by a probe. Nonlinear polymer dynamic behaviors are often difficult to study in a well-controlled way, suggesting that this transition may be of general interest. 

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