Transport properties of the Lorentz gas: Fourier's law

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of , i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT ₁ andT ₂ respectively,T ₁ <T ₂. Letting be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit,,R 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in theL ¹-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T ₂-T ₁)/L for largeL, whereL is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.Research supported in part by NSF Grant no. Phy 77-22302.On leave of absence from the Fachbereich Physik der Universität, München. Work supported by a DFG fellowship.     

On the statistical independence of various column contributions to band broadening. Part 1: Second moment contributions from statistically independent,longitudinal diffusion in both phases

The total length-based second moment contribution from longitudinal sample diffusion in both phases on a column, σD , is derived by adding individual partial differential contributions to a partial differential equation accounting for the longitudinal diffusion processes only. Although each diffusion-dispersed sample part is equilibrated between two phases, the resulting σ,D (= 2D _mt _m + 2D _st _s) can be interpreted as the sum of two independent contributions in accordance with the variance addition rule. (D _m and D _s are the mean diffusion coefficients and t _mand t _s the mean residence times of the sample in the mobile and stationary phases, respectively.) The same σD expression is derived from the random walk model of Giddings by treating the diffusional process in each phase as statistically independent of the other processes. Under these conditions the broadening contribution from longitudinal diffusion in the mobile phase is shown to be independent of the velocity profile.     

Improving the lower detection limit of potentiometric sensors by covalently binding the ionophore to a polymer backbone

The lower detection limit of polymeric membrane ion-selective electrodes (ISEs) is impaired by zero-current ion fluxes through the organic phase. This adverse effect is largely eliminated by covalently attaching the ionophore to a polymer backbone. To this purpose, the Pb²⁺-selective ligand, 4-tert-butylcalix[4] arene-tetrakis(N,N′-dimethylthioacetamide) is substituted on its upper rim by a diol derivative which is subsequently copolymerized with poly(tetrahydrofuran)diol and 2,2,4-trimethylhexamethylene diisocyanate to the corresponding polyurethane. By measurements on sandwich membranes, it is shown that through binding the ionophore to the polymer, the mobility of Pb²⁺ in the ISE membrane is strongly reduced. As a consequence, the response range of such an ISE is extended by several orders of magnitude. This is the case even when using an internal electrolyte that with an ISE based on a mobile ionophore leads to strong deviations from the linear response because of ion uptake from the sample into the membrane or ion release from the membrane into the sample. With a conventional inner filling solution of 10⁻¹ M Pb(NO₃)₂, a lower detection limit of 1.7×10⁻⁹ M Pb²⁺ has been achieved in the presence of 10⁻⁴ M Na⁺.     

Derivation of the transport equation for electrons moving through random impurities

A single (nonrelativistic, spinless) electron subject to a constant external electric field interacts with impurities located on an infinitely extended lattice by a potential of random strength. The random strength is given by a field of Gaussian random variables. We show the existence of the averaged dynamics and prove that in the weak coupling limit, 0, ₂ t= fixed, one obtains the usual transport equation for the velocity distribution.Work supported by a Max Kade Foundation fellowship.On leave of absence of the Fachbereich Physik der Universität München.     

A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

The spherical limit forn-vector correlations

We investigate then limit of then-vector model single-spin and pairspin correlation functions. In this limit we show that the correlation functions become those of the corresponding spherical model.     

Conducting polymer membranes for low activity potentiometric ion sensing

Conducting polymer (CP) films, used as ion-sensing membranes under open circuit potentiometric conditions, are usually characterised with rather high detection limit, in the range of 10⁻⁴-10⁻⁵ mol dm⁻³. This effect is unfavourable, not only from the point of view of CP applications in potentiometry as ion sensitive membranes, but also when these materials are used as ion-to-electron transducers (solid contacts) for ion-selective electrodes. The theoretical considerations presented underline the crucial role of spontaneous processes of polymer charging/discharging—the source of observed high detection limit of sensors comprising CP layer under zero current conditions. Although the mechanism of occurring process is different from that observed for plastic, solvent polymeric based ion-selective electrodes, the ultimate result—alteration of activity of electrolyte at the membrane/solution interface leading to elevation of the detection limit—is the same.The method of estimation of parameters characterising spontaneous charge transfer processes is presented. The values obtained can be used to calculate the resulting polymer/solution interface activity of electrolyte ions, thus the detection limit of CP membrane can be theoretically predicted.A method of lowering of the detection limit of conducting polymer membranes, applying galvanostatic polarisation to compensate the spontaneous process of polymer charging/discharging, is presented.The experimental results obtained for poly(pyrrole), poly(N-methylpyrrole) and poly(3,4-ethylenedioxythiophene) are in good accordance with predictions of the presented model.     

Random walks on inhomogeneous lattices

For lattices with two kinds of points (black and white), distributed according to a translation-invariant joint probability distribution, we study statistical properties of the sequence of consecutive colors encountered by a random walker moving through the lattice. The probability distribution for the single steps of the walk is considered to be independent of the colors of the points. Several exact results are presented which are valid in any number of dimensions and for arbitrary probability distributions for the coloring of the points and the steps of the walk. They are used to derive a few general properties of random walks on lattices containing traps.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.     

Random walk properties of lattices and correlation factors for diffusion via the vacancy mechanism in crystals

Random walk properties and correlation factors for diffusion via the vacancy mechanism are calculated and compared for various three-dimensional lattices. By applying the theory of random walks on an imperfect lattice, the correlation factor for impurity diffusion is calculated rigorously for the five jump frequency model in the fee lattice.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.     

Novel random poly(propylene isophthalate/adipate) copolyesters: Synthesis and characterization

Poly(propylene adipate) (PPA) and poly(propylene isophthalate/adipate) (PPI-PPA) random copolymers of various compositions were synthesized in bulk and characterized in terms of chemical structure and molecular weight. Furthermore, the thermal behavior was examined by thermogravimetric analysis and differential scanning calorimetry. All the polymers showed a good thermal stability. At room temperature they appeared as semicrystalline materials, except the copolymers containing 20 and 30 mol% of PI units: the main effect of copolymerization was a lowering in the amount of crystallinity and a decrease of melting temperature with respect to homopolymers. The crystalline phase of PPI and PPA was evidenced at high content of propylene isophthalate or propylene adipate units, respectively. Amorphous samples were obtained after melt quenching and an increment of T_g as the content of PI units is increased was observed. This behavior was explained as due to the stiff phenylene groups in the polymeric chain. The Wood equation was found to describe well T_g-composition data. Lastly, the presence of a rigid-amorphous phase was evidenced in the copolymers, differently from PPA homopolymer.