Short rate probability distribution

This equation relates the bimolecular rate constant to the state-to-state rate constant ka(ij l) and ultimately to vap(ij, v l). Note that the rate constant is simply the average value of vcrR(ij,v l). Thus, in a short-hand notation we have ka = (vap(ij,v l)). The average is taken over all the microscopic states including the appropriate probability distributions, which are the velocity distributions /a va) and /b( )(vb) in the experiment and the given distributions over the internal quantum states of the reactants. 

Fig. 24.8. Computational simulation analysis of conformational dynamics in T4 lysozyme enzymatic reaction, (a) Histograms of fopen calculated from a simulated single-molecule conformational change trajectory, assuming a multiple consecutive Poisson rate processes representing multiple ramdom walk steps, (b) Two-dimensional joint probability distributions <5 (tj, Tj+i) of adjacent pair fopen times. The distribution <5(ri, Ti+i) shows clearly a characteristic diagonal feature of memory effect in the topen, reflecting that a long topen time tends to be followed by a long one and a short fopen time tends to be followed by a short one...

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

The short rate follows a stochastic process, or probability distribution. So, although the rate itself can assume a range of possible future values, the process by which it changes from value to value can be modeled. A one-factor model of interest rates specifies the stochastic process that describes the movement of the short rate. 

In Vasicek s model, the short rate r is normally distributed. It therefore has a positive probability of being negative. Model-generated negative rates are an extreme possibility. Their occurrence depends on the initial interest rate and the parameters chosen for the model. They have been generated, for instance, when the initial rate was very low, like those seen in Japan for some time, and volatility was set at market levels. This possibility, which other interest rate models also allow, is inconsistent with a no-arbitrage market as Black (1995) states, investors will hold cash rather than invest at a negative interest rate. For most applications, however, the model is robust, and its tractability makes it popular with practitioners. 

The laser intensities required to exploit gaseous nonllnearltles in a short time period dictate employment of pulsed lasers. These are typically frequency-doubled neodymiumrYAG lasers at 532 nm which are ideally spectrally situated for CARS work from both a dye laser pumping and optical detection standpoint. These lasers operate at repetition rates in the 20-50 Hz range. The combustion medium cannot be followed in real time, but is statistically sampled by an ensemble of single shot measurements which form a probability distribution function (pdf). From the pdf, the parameter time average can be ascertained as well as the magnitude of the turbulent fluctuations. 

Although the general circulation patterns are fairly well known, it is difficult to quantify the rates of the various flows. Abyssal circulation is generally quite slow and variable on short time scales. The calculation of the rate of formation of abyssal water is also fraught with uncertainty. Probably the most promising means of assigning the time dimension to oceanic processes is through the study of the distribution of radioactive chemical tracers. Difficulties associated with the interpretation of radioactive tracer distributions lie both in the models used, nonconservative interactions, and the difference between the time scale of the physical transport phenomenon and the mean life of the tracer. 

Yet unless very detailed information is available to describe the initial distribution of separations, p(r, 0), it will not be possible to use measured time-dependent survival probabilities to probe details of dynamic liquid structure. Currently, experimental uncertainties at 30% are so large that such a probe is not possible, since the effects of the short-range caging region are only 30%, at the most, of the rate coefficient or escape probability. 

The net-event KMC (NE-KMQ or lumping approach has been introduced by our group. The essence of the technique is that fast reversible events are lumped into an event with a rate equal to the net, i.e., the difference between forward and backward transition probabilities per unit time (Vlachos, 1998). The NE-KMC technique has recently been extended to spatially distributed systems (Snyder et al., 2005), and it was shown that savings are proportional to the separation of time scales between slow and fast events. The method is applicable to complex systems, and is robust and easy to implement. Furthermore, the method is self-adjusted, i.e., it behaves like a conventional KMC when there is no separation of time scales or at short times, and gradually switches to using the net-event construct, resulting in acceleration, only as PE is approached. A disadvantage of the method is that the noise is reduced. 

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