It has been known for many years that the Boltzmann equation provides a good description of the transport coefficients for a dilute gas of particles interacting with short-range forces. While this equation for the phase space (or Wigner) distribution function f(r, p, t) is closed, the rigorous mechanical evolution equation for f is not according to the investigation of Bogoliubov, Born and Green, Kirkwood, Yvon and others.D, 2 l Recently many efforts have been made to derive and generalize the former equation from the latter by introducing approximations.ll,3J Unfortunately. most of the approximations previously proposed by various authors seem to be motivated by mathematical tractability rather than by physical reasoning. To this category of approximations belong e. g. Bogoliubov's ffunctional dependence assumption on the many-body distribution functions,ll the factorizability of the initial many-body density into one-body densities, 3 l, <11fJ) I,where p(t) is a many-particle density operator, .Q the volume; I11CJl > and <11

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With the help of quantum-scattering-theory methods and the approximation of stationary phase, a one-dimensional coherent random-walk model which describes both tunneling and scattering above the potential diffusion of particles in a periodic one-dimensional lattice is proposed. The walk describes for each lattice cell, the time evolution of modulating amplitudes of two opposite-moving Gaussian wave packets as they are scattered by the potential barriers. Since we have a coherent process, interference contributions in the probabilities bring about strong departures from classical results. In the near-equilibrium limit, Fick’s law and its associated Landauer diffusion coefficient are obtained as the incoherent contribution to the quantum current density along with a novel coherent contribution which depends on the lattice properties as [(1−R)/R]1/2.

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The electrical conductivity of carbon nanotubes varies, depending on the temperature and the radius and pitch of the sample. In majority cases, the resistance decreases with increasing temperature, suggesting a thermally activated process. The standard band theory based on the Wigner–Seitz cell model predicts a gapless semiconductor, which does not account for the thermal activation. A new band model in which an “electron” (“hole”) has a carbon hexagon size for graphene is proposed. The normal charge carriers in graphene transport are electrons and holes. The electrons (holes) wavepackets extend over the carbon hexagon and carry the charges −e(+e). Electrons or holes thermally activated are shown to generate the observed temperature behavior of the conductivity in the nanotubes.

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For the study of the dynamics of a polymer molecule in dilute solution a continuous wire model is proposed and investigated. This model is a refinement of the wormlike chain model introduced by Kratky and Porod such that the configurational energy depends not only on the curvature κ but also on the torsion τ, both of which may be considered as functions of the arc lengths measured from one end. The physical basis of considering such energy dependence is given from a general differential–geometrical viewpoint, and also by the explicit calculation of the elastic energy of an ideally thin wire, which is found to be ∫ 0lds18πR4[E(κ − κ0)2 + 2μ(τ − τ0)2], where E and μ are Young's modulus and the modulus of rigidity of the wire material; R is the radius of the circular cross section of the wire; κ0(s) and τ0(s) are the curvature and torsion of the space curve characterizing the wire of minimum energy. The dynamics of the model is formulated with the aid of Hamilton's principle of least action. In particular the wave propagation along the axis of a helical coil is investigated in detail. It is shown that the measurement of the propagation speed of a longitudinal wave along a polymer chain can lead to the quantitative estimate of the bending and twisting characteristics of the chain.