Kramers solutions

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The objective is to find the steady-state escape rate k out of the potential well. Before presenting the Kramers solution it is important to note that for such a (quasi) steady state to be established, a clear separation of time scales has to exist, whereupon the escape occurs on a time scale much longer than all time scales associated with the motion inside the well. In particular this implies that the well should be deep enough (see below). 

Leis, J.R., and Kramer, M.A., "The Simultaneous Solution and Sensitivity Analysis of Systems Described by Ordinary Differential Equations", ACM transactions on Mathematical Software, 14,45-60 (1988). 

V. I. Mel nikov and S. V. Meshkov, Theory of activated rate processes exact solution of the Kramers problem, J. Chem. Phys. 85, 1018 (1986). 

Kramer, F., Henig, Y. S., Garin, T. A., and Vogel, G. J., Selective adsorption from solutions such as coffee extracts, UK Pat. Appl. 2,027,576, 1980. (CA93 69060g)... 

We apply Kramer s rule to find C2 just as we had with C,. The solution has the same characteristic polynomial in (10-22). The transfer functions ... 

EVOLUTION TIMES OF PROBABILITY DISTRIBUTIONS AND AVERAGES—EXACT SOLUTIONS OF THE KRAMERS PROBLEM... 

Kramer, R. (1994). Functional principles of solute transport systems concepts and perspectives, Biochim. Biophys. Acta, 1185, 1-34. 

In general, it is computationally expensive to find the solution to problem (11.12). An alternative approach based on a surrogate objective function was developed by Johnston and Kramer (1998). This approach, for the unconstrained and linearly constrained cases, has an analytical solution thus simplifying the calculations. The complete procedure can be found in the aforementioned publication. 

Sigg, L., W. Stumm, and B. Zinder (1984), "Chemical Processes at the Particle/Water Interface Implications Concerning the Form of Occurrence of Solute and Adsorbed Species", in C. J. H. Kramer and J. Duinker, Eds., Complexation of Trace Metals in Natural Waters, Dr. W. Junk Publishers, The Hague, Netherlands, 251-266. 

Mechanical processing (e.g., abrasion) of metallic surfaces causes the emission of electrons this is known as the Kramer effect (Kramer 1950). The effect has been shown by the measurement of selfgenerated voltages between two metallic surfaces under boundary lubrication (Anderson et al. 1969, Adams and Foley 1975). The exoelectrons have a kinetic energy from 1 to 4 eV (Kobzev 1962) and they may initiate some chemical reactions. For instance, if the metal (whose surface has been worked) is placed in an aqueous solution of acrylonitrile, the latter forms an abundant amount of an insoluble... 

Exchange studies were carried out by solvolysing a series of butyl chlorides in 2m solutions of antimony pentafluoride in fluoro-sulfonic acid at —50° and —78°. The acid contained tracer levels of TjO and small amounts of water to provide sufficient nucleophiles to catalyse proton exchange reactions with some of the intermediates formed in the butyl system (Kramer, 1970, 1973). 

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