B.S., The Cooper Union, 1957

Ph.D., Applied Physics, Harvard University, 1964

#### Research Interests

**Chemical kinetics of non-arrhenius chemical reactions. **A new theory of chemical kinetics is being developed that accounts for deviations from the Arrhenius equation and predicts nonlinear dependencies of the rate constants for certain systems of chemical reactions.

The bimolecular rate constant of a chemically reacting gas is usually given by the Arrhenius equation k = A exp(-E_{a}/RT) where E_{a} is the activation energy of the reaction and is independent of concentration and temperature but dependent on the two reacting species. However, when the reacting system is not near equilibrium, the kinetic energy distribution may be significantly different from the Maxwellian and the kinetics becomes considerably more complex. For example, if hydrogen atoms are introduced into a reacting system with translational energies that are much greater than thermal energies, the translational energy distribution may be greatly displaced from equilibrium, and the rate constants may no longer be given by the Arrhenius expression. Asymptotic expressions for the rate constants have been derived, and explicit expressions have been obtained for rate constants that do not have the Arrhenius form. For certain conditions it was shown that when energetic atoms are present, the measured values of the activation energies may depend on the concentrations of the species and on the temperature.

In complex systems where the energetic species can engage in several competitive chemical reactions, the expressions for the rate constants can be used to predict how the system will behave when some of the parameters are changed. For example, in a system in which the energetic or hot species is introduced at a constant rate, the ratio of rate constants will give the relative reaction rates as a function of temperature. For Arrhenius kinetics, the ratio of the number of reactions of hydrogen atoms with species ri to the number of reactions of hydrogen atoms with species rj is approximately proportional to exp((E_{j}-E_{i})/k_{b}T), where E_{i} and E_{j} are the energies of activation of the two reactions. When energetic or hot atoms are present, this ratio can be changed significantly. A comprehensive theory is being developed for predicting reaction rates of energetic or excited chemical species which are not near equilibrium. For certain reacting systems that are far from equilibrium we have derived explicit expressions showing that reaction rates and reaction yields can be much higher than those for systems near equilibrium. This work is being continued and used to study the principles and applications of highly non-equilibrium chemical reactions.

**Theory of rate constants. **Significant advances have been made in the literature in calculating the energy of interaction between molecules. Although these calculations are of scientific interest, they do not give us what is of real importance, namely, the rate constants as a function of temperature. In a series of published articles, we have developed a theory for obtaining the rate constants from the energies of interaction. This research is being continued so that the essential features of chemical reactions can be better understood and rate constants can be obtained as a function of temperature and other parameters.

**Applications of quantum theory. **A wide range of problems involving quantum theory and its applications has been studied. For example, new derivations of quantum equations using stochastic theory have been advanced. It has been shown how friction can be incorporated into quantum theory. New partial differential equations for the temperature-dependent probability density of quantum statistical mechanics have been derived and published. New algorithms for solving the time-dependent Schroedinger equation and time-dependent Dirac equation have been derived and published. The algorithms are unconditionally stable and explicit and allow all values to be updated simultaneously. This means that they are suitable for new types of massively parallel computers and have significant advantages over other algorithms. Work on extending these new algorithms to solve other equations of quantum theory is continuing. New partial differential equations for the entropy density, energy density, and number density of quantum statistical mechanics have been derived. The application of quantum theory to problems of engineering interest, such as the prediction of rate constants, is also being investigated.

- Kostin, M. D. (1994). "Differential-Equations for the Entropy Density, Energy Density, and Number Density of Statistical-Mechanics." Journal of Mathematical Physics
**35**(5): 2371-2377. - Kostin, M. D. (1995). "Number Density, Entropy Density, and Energy Density of Statistical-Mechanics for D-Dimensional Grand-Canonical Ensembles." Journal of Mathematical Physics
**36**(2): 750-755. - Kostin, M. D. (1995). "Thermal Schrodinger-Equation." Physical Review E
**52**(1): 229-233. - Kostin, M. D. (1996). "Non-negativity condition for cellular automata and other massively parallel computer algorithms." Journal of Physics a-Mathematical and General
**29**(24): 8173-8175. - Kostin, M. D. (1997). "Transition state theory and grand canonical partition functions." Journal of Chemical Physics
**107**(15): 5811-5814.