Single joint probabilities

Second Order Stationarity. With only a single realization of the random function it would be impossible to make any meaningful inferences about the random function if we did not make some assumptions about its stationarity. A random function is said to be strictly stationary if the joint probability density function for k arbitrary points is invariant under simultaneous translation of all... 

The simplest of these models which permits a detailed discussion of the decay of correlations is a random walk model in which a set of random walkers whose positions are initially correlated is allowed to diffuse the motion of any single random walker being independent of any other member of the set. Let us assume that there are r particles in the set and motion occurs on a discrete lattice. The state of the system is, therefore, completely specified by the probabilities Pr(nlf n2,..., nr /), (tij = — 1, 0, 1, 2,. ..) in which Pr(n t) is the joint probability that particle 1 is at n1( particle 2 is at n2, etc., at time l. We will also use the notation Nj(t) for the random variable that is the position of random walker j at time t. Reduced probability distributions can be defined in terms of the Pr(n t) by summation. We will use the notation P nh, rth,..., ntj I) to denote the distribution of random walkers iu i2,..., i at time t. We define... 

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...

FIGURE 8.7 The top graph shows the separate probabilities over the entire range 0 to 1 for two isomorphous heavy atom derivatives I and II. Both are bimodal and each separately predicts the two most likely phase angles for the native structure factor. At the bottom the joint probability distribution strongly predicts a single most probable phase. 

Figure 10. Simulation of the EPR state preparation in an optical lattice with 25 sites, at three consecutive times. First row shows the joint probability distribution in x representation, the second one in p representation, (ol) and (a2) initially (t = 0), the atoms are cooled down to the external harmonic potential ground state, whereas the LIDDI is off. (61) and (62) at t = 1.4 x 10-4 s LIDDI and the repulsive linear potential (with the slope 0.04 Erec per lattice site) are on, whereas the harmonic potential is off. The diatoms are moving through the lattice very slowly in comparison to the single atoms, (cl) and (c2) at t = 2.16 x 10 4 s single atoms are ejected out of the lattice and discarded and the diatoms are separated out.

Our understanding of diffusion and reaction in single-file systems is impaired by the lack of a comprehensive analytical theory. The traditional way of analytically treating the evolution of particle distributions by differential equations is prevented by the correlation of the movement of distant particles. One may respond to this restriction by considering joint probabilities covering the occupancy and further suitable quantities with respect to each individual site. These joint probabilities may be shown to be subject to master equations. 

The benefit of the analytical treatment presented thus far for the calculation of the characteristic functions of the single-file system is only limited by the increasing complexity of the joint probabilities and the related master equations. This treatment, however, has suggested a most informative access to the treatment of systems subjected to particle exchange with the surroundings and to internal transport and reaction mechanisms [74,75]. Summing over all values (Ji = 0 and 1 and, subsequently, over all sites i, Eq. 31 may be transferred to the relation Eq. 34... 

A final important point is that all of the above has only been for isothermal conditions in single-phase systems. Extension to other cases requires the introduction of interactions with the second phase and/or heat exchange walls, and so on, and the age-distribution and micromixing functions depend nnuch more on the details of the system. A formal treatment would use joint probability distribution functions, but this rapidly gets extremely complex. The population balance models can give some insight into the two-phase situation, and will be discussed below. 

The mathematical model for a discrete stochastic process is a sequence of random variables for n = 0,1,2,..., where is the state at time n. Here the superscript (n) indicates time point n. All knowledge about a single random variable is in its probability distribution. Similarly, all knowledge about the probabilistic law for a stochastic process is contained in the joint probability distribution of every... 

The uncertainty of a parameter can be characterised by the upper and lower limits of the parameter or by the expected value and the variance of the parameter. Such descriptions of individual parameter uncertainty can, for example, be obtained from the data evaluation sources introduced in Chap. 3. The joint probability density function (pdf) of parameters gives the most complete information about the uncertainty of a parameter set. Methods of uncertainty analysis provide information about the uncertainty of the results of a model knowing the uncertainty of its input parameters. If such a lack of knowledge of model inputs is propagated through the model system then a model output becomes a distribution rather than a single value. Measures such as output variance can then be used to represent output uncertainty. 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

If there is no interaction between similar reactants (traps) B, they are distributed according to the Poisson relation, Ab (r, t) = 1. Besides, since the reaction kinetics is linear in donor concentrations, the only quantity of interest is the survival probability of a single particle A migrating through traps B and therefore the correlation function XA(r,t) does not affect the kinetics under study. Hence the description of the fluctuation spectrum of a system through the joint densities A (r, ), which was so important for understanding the A4-B — 0 reaction kinetics, appears now to be incomplete. The fluctuation effects we are interested in are weaker here, thus affecting the critical exponent but not the exponential kinetics itself. It will be shown below that adequate treatment of these weak fluctuation effects requires a careful analysis of many-particle correlations. 

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