Stability and exponential convergence for the Boltzmann equation

We prove existence, uniqueness and stability for solutions of the nonlinear Boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function. The results are given for a range of spaces, including L ¹, and extend previous results in L for the non-homogeneous equation, as well as the more developed L ^p -theory for the spatially homogeneous Boltzmann equation.We also give new L -estimates for the spatially homogeneous equation in the case of Maxwellian interactions.     

On Stability and Strong Convergence for the Spatially Homogeneous Boltzmann Equation for Fermi-Dirac Particles

 The paper considers the stability and strong convergence to equilibrium of solutions to the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Under a cutoff condition on the collision kernel, we prove a strong stability in L ¹ topology at any finite time interval, and, for hard and Maxwellian potentials, we prove that the solutions converge strongly in L ¹ to equilibrium under a high temperature condition. The basic tools used are moment-production estimates and the strong compactness of the collision gain term. (Accepted 25, October 2002) Published online March 14, 2003 Communicated by P.-L. Lions     

Peroxy radical isomerization in the oxidation of isoprene

We report experimental evidence for the formation of C(5)-hydroperoxyaldehydes (HPALDs) from 1,6-H-shift isomerizations in peroxy radicals formed from the hydroxyl radical (OH) oxidation of 2-methyl-1,3-butadiene (isoprene). At 295 K, the isomerization rate of isoprene peroxy radicals (ISO2?) relative to the rate of reaction of ISO2? + HO2 is k(isom)(295)/(k(ISO2?+HO2)(295)) = (1.2 ± 0.6) x 10(8) mol cm(-3), or k(isom)(295) ? 0.002 s(-1). The temperature dependence of this rate was determined through experiments conducted at 295, 310 and 318 K and is well described by k(isom)(T)/(k(ISO2?+HO2)(T)) = 2.0 x 10(21) exp(-9000/T) mol cm(-3). The overall uncertainty in the isomerization rate (relative to k(ISO2?+HO2)) is estimated to be 50%. Peroxy radicals from the oxidation of the fully deuterated isoprene analog isomerize at a rate ～15 times slower than non-deuterated isoprene. The fraction of isoprene peroxy radicals reacting by 1,6-H-shift isomerization is estimated to be 8-11% globally, with values up to 20% in tropical regions.     

OH-stretch vibrational spectroscopy of hydroxymethyl hydroperoxide

We report measurement and analysis of the photodissociation spectrum of hydroxymethyl hydroperoxide (HOCH(2)OOH) and its partially deuterated analogue, HOCD(2)OOH, in the OH-stretching region. Spectra are obtained by Fourier transform infrared spectroscopy in the 1nu(OH) and 2nu(OH) regions, and by laser induced fluorescence detection of the OH fragment produced from dissociation of HOCH(2)OOH initiated by excitation of the 4nu(OH) and 5nu(OH) overtone regions (action spectroscopy). A one-dimensional local-mode model of each OH chromophore is used with ab initio calculated OH-stretching potential energy and dipole moment curves at the coupled-cluster level of theory. Major features in the observed absorption and photodissociation spectra are explained by our local-mode model. In the 4nu(OH) region, explanation of the photodissocation spectrum requires a nonuniform quantum yield, which is estimated by assuming statistical energy distribution in the excited state. Based on the estimated dissociation threshold, overtone photodissociation is not expected to significantly influence the atmospheric lifetime of hydroxymethyl hydroperoxide.     

Brownian Approximation and Monte Carlo Simulation of the Non-Cutoff Kac Equation

The non-cutoff Boltzmann equation can be simulated using the DSMC method, by a truncation of the collision term. However, even for computing stationary solutions this may be very time consuming, in particular in situations far from equilibrium. By adding an appropriate diffusion, to the DSMC-method, the rate of convergence when the truncation is removed, may be greatly improved. We illustrate the technique on a toy model, the Kac equation, as well as on the full Boltzmann equation in a special case.     

On the Distribution of Free Path Lengths¶for the Periodic Lorentz Gas

Consider the domain

Vibrational overtone initiated unimolecular dissociation of HOCH2OOH and HOCD2OOH: evidence for mode selective behavior

The vibrational overtone induced unimolecular dissociation of HMHP (HOCH(2)OOH) and HMHP-d(2) (HOCD(2)OOH) into OH and HOCH(2)O (HOCD(2)O) fragments is investigated in the region of the 4nu(OH) and 5nu(OH) bands. The unimolecular dissociation rates in the threshold region, corresponding to the 4nu(OH) band, exhibit measurable differences associated with excitation of the OH stretch of the alcohol versus the peroxide functional group, with the higher energy alcohol OH stretching state exhibiting a slower dissociation rate compared to the lower energy peroxide OH stretch in both HMHP and HMHP-d(2). Predictions using the Rice-Ramsperger-Kassel-Marcus theory give rates that are in reasonably good agreement with the measured dissociation rate for the alcohol OH stretch but considerably differ from the measured rates for the peroxide OH stretch in both isotopomers. The present results are interpreted as suggesting that the extent of intramolecular vibrational energy redistribution (IVR) is different for the two OH stretching states associated with the two functional groups in HMHP, with IVR being substantially less complete for the peroxide OH stretch. Analysis of the OH fragment product state distributions in conjunction with phase-space theory simulation gives a D(0) value of 38+/-0.7 kcal/mole for breaking the peroxide bond in HMHP.     

Atmospheric fate of methacrolein. 1. Peroxy radical isomerization following addition of OH and O2

Peroxy radicals formed by addition of OH and O(2) to the olefinic carbon atoms in methacrolein react with NO to form methacrolein hydroxy nitrate and hydroxyacetone. We observe that the ratio of these two compounds, however, unexpectedly decreases as the lifetime of the peroxy radical increases. We propose that this results from an isomerization involving the 1,4-H-shift of the aldehydic hydrogen atom to the peroxy group. The inferred rate (0.5 ± 0.3 s(-1) at T = 296 K) is consistent with estimates obtained from the potential energy surface determined by high level quantum calculations. The product, a hydroxy hydroperoxy carbonyl radical, decomposes rapidly, producing hydroxyacetone and re-forming OH. Simulations using a global chemical transport model suggest that most of the methacrolein hydroxy peroxy radicals formed in the atmosphere undergo isomerization and decomposition.     

Atmospheric fate of methacrolein. 2. Formation of lactone and implications for organic aerosol production

We investigate the oxidation of methacryloylperoxy nitrate (MPAN) and methacrylicperoxy acid (MPAA) by the hydroxyl radical (OH) theoretically, using both density functional theory [B3LYP] and explicitly correlated coupled cluster theory [CCSD(T)-F12]. These two compounds are produced following the abstraction of a hydrogen atom from methacrolein (MACR) by the OH radical. We use a RRKM master equation analysis to estimate that the oxidation of MPAN leads to formation of hydroxymethyl-methyl-α-lactone (HMML) in high yield. HMML production follows a low potential energy path from both MPAN and MPAA following addition of OH (via elimination of the NO(3) and OH from MPAN and MPAA, respectively). We suggest that the subsequent heterogeneous phase chemistry of HMML may be the route to formation of 2-methylglyceric acid, a common component of organic aerosol produced in the oxidation of methacrolein. Oxidation of acrolein, a photo-oxidation product from 1,3-butadiene, is found to follow a similar route generating hydroxymethyl-α-lactone (HML).     

Free Path Lengths in Quasi Crystals

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter d, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann–Grad limit, a Poisson distribution of scatterers leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the free path length distribution asymptotically behaves as a power law.