Stochastic trajectories

Here, we will explain the most effective of them, the so-called shooting algorithm. For simplicity we will focus on how to do it for deterministic trajectories such as those obtained from a molecular dynamics simulation. We note, however, that very similar algorithms can be applied to stochastic trajectories [5, 7, 8]. 

More informative are the stochastic trajectory simulations run by Muhl-hausen et al. (M WT), on empirical interaction potential surfaces for scattering and desorption Although the major thrust was to understand the direct beam scattering results of NO/Ag(l 11), extension of these calculations allows for comparison to the desorption of NO from Pt(lll) Important insights derived from the NO/Ag(lll) calculations were ... 

To give an impression of the virtues and shortcomings of the QCL approach and to study the performance of the method when applied to nonadiabatic dynamics, in the following we briefly introduce the QCL working equation in the adiabatic representation, describe a recently proposed stochastic trajectory implementation of the resulting QCL equation [42], and apply this numerical scheme to Model 1 and Model IVa. 

U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett. 95, 040602 (2005). 

Simulation of the random walks on a site lattice is presented in Figs 2.13 and 2.14 they show that stochastic trajectories deviate systematically from the stationary solution [16]. Alongside those which correspond to the damping oscillations, above mentioned catastrophes are also observed and characterized by A b = 0, and Aa - oo. These results demonstrate indirectly... 

Typical Lagrangian approaches include the deterministic trajectory method and the stochastic trajectory method. The deterministic trajectory method neglects all the turbulent transport processes of the particle phase, while the stochastic trajectory method takes into account the effect of gas turbulence on the particle motion by considering the instantaneous gas velocity in the formulation of the equation of motion of particles. To obtain the statistical... 

Two basic trajectory models, i.e., the deterministic trajectory model [Crowe etal., 1977] and the stochastic trajectory model [Crowe, 1991], are introduced in this section. The deterministic approach, which neglects the turbulent fluctuation of particles, specifically, the turbulent diffusion of the mass, momentum, and energy of particles, is considered the most... 

Thus, by coupling Eqs. (5.247), (5.248), and (5.249) with the governing equations for the gas phase, the instantaneous particle velocities can be obtained. Moreover, the stochastic trajectories of the particles can also be obtained by... 

In computation using the stochastic trajectory model, the Monte Carlo approach is commonly employed. It is necessary to calculate several thousands, or even tens of thousands, of trajectories to simulate the particle flow field. The central issue in developing the stochastic trajectory model is how to model the instantaneous turbulent gas flow field. The method... 

In this section we analyze experimental data and make comparisons with theory. Data were obtained for 100 CdSe-ZnS nanocrystals at room temperature.1 We first performed data analysis (similar to standard approach) based on the distribution of on and off times and found that a+= 0.735 0.167 and v = 0.770 0.106,2 for the total duration time T = T = 3600 s (bin size 10 ms, threshold was taken as 0.16 max I(t) for each trajectory). Within error of measurement, a+ a k 0.75. The value of a 0.75 implies that the simple diffusion model with a = 0.5 is not valid in this case. An important issue is whether the exponents vary from one NC to another. In Fig. 13 (top) we show the distribution of a obtained from data analysis of power spectra. The power spectmm method [26] yields a single exponent apSd for each stochastic trajectory (which is in our case a+ a apSd). Figure 13 illustrates that the spread of a in the interval 0 < a < 1 is not large. Numerical simulation of 100 trajectories switching between 1 and 0, with /+ (x) = / (x) and a = 0.8, and with the same number of bins as the experimental trajectories, was performed and the... 

It is usually not possible (and never easy ) to solve equations such as Equation (11.27) analytically. So computational simulation of the stochastic trajectories are necessary. The numerical method to obtain stochastic trajectories by Monte Carlo sampling, which we shall discuss in Section 11.4.4, is known as the Gillespie algorithm [68], However, it happens that the steady state of Equation (11.27) can be obtained in closed form. This is because in steady state, the probability of leaving state 0, v0po has to exactly balance the probability of entering state 0 from state 1, wopi. Similarly, since v0po = vjqp, we have v p = w p2, and so on ... 

To use a computer to simulate a stochastic trajectory of the chemical master equation such as described in Figure 11.4, one must establish the rules of how to move the system from one grid point to its neighboring points. The essential idea is to draw random moves from the appropriate distribution and to assign random times (also drawn from the appropriate distribution) to each move. Thus each simulation step in the simulation involves two random numbers, one to determine the associated time step and one to determine the grid move. 

Figure 11.7 A schematic illustration of the Monte Carlo simulation method for computing the stochastic trajectories of a chemical reaction system following the CME. Two random numbers, r and r2, are sampled from a uniform distribution to simulate each stochastic step r determines when to move, and r2 determines where to move. For a given state of a master equation graph shown in the upper panel, there are four outward reactions, labeled 1-4, each with their corresponding rate constants q, (i = 1, 2, , 4). The upper and lower panels illustrate, respectively, the calculation of the random time T associated with a stochastic move, and the probability pm of moving to state m.

Figure 11.8 An example of the stochastic trajectory from Monte Carlo simulation according to the Gillespie algorithm for reaction system given in Equation (11.19) and corresponding master equation graph given in Figure 11.4. Here we set Ns = 100 and Nes = 0 at time zero and total enzyme number Ne = 10. (A) The fluctuating numbers of S and ES molecules as functions of time. (B) The stochastic trajectory in the phase space of (m, n).

After completing a step (a jump in state) the corresponding outward rate constants for the new state are now all different. To continue the simulation, we draw another two random numbers, make another move, and so on. A stochastic trajectory is thus obtained. One notes that the trajectory has randomly variable time steps, a feature indicative of the Gillespie algorithm. 

Perform a computer simulation of the CME for the system of Equation (11.25), using the parameters given in the legend of Figure 11.6. (Assume that A, B, and C are held at fixed numbers.) From the simulated stochastic trajectory, can you reproduce the steady state probability distribution shown in Figure 11.6 ... 

It can easily be observed that the considered case (4.149) corresponds to the situation for T = 0 only one marked particle evolves through a stochastic trajectory (a type 1 displacement vith Vj speed). This example corresponds to a Dirac type input and the model output response or the sum Pi(Ze,t) + P2(0, t), represents the distribution function of the residence time during the trajectory (see also application 4.3.1). 

The functional above was used already by Gauss [12] to study classical trajectories (which explains our choice of the action symbol). Onsager and Machlup used path integral formulation to study stochastic trajectories [13]. The origin of their trajectories is different from what we discussed so far, which are mechanical trajectories. However, the functional they derive for the most probable trajectories, O [X (t)] is similar to the equation above ... 

Problem 7.1. Consider the observation of the number of cars on a road segment as a function of time discussed above. In order to obtain the distribution Pin, Z) for this process one needs to obtain a representative set of stochastic trajectories. Using the timescales scenario discussed above for this system, suggest a procedure for obtaining an approximation for Pin, Z). 

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