Asymptotic probability density

Assuming that after their previous swap the two walks were sufficiently long to be in the asymptotic regime, this means that transient behavior has elapsed and the system has relaxed to equilibrium for the respective parameters. Then, the joint configurational probability density just before the current swap is simply... 

The tails prevent [19] convergence to the Gaussian distribution for N -= oo, but not the existence of a limiting distribution. These distributions as we have seen are called stable distributions. If the concept of a Levy distribution is applied to an assembly of temporal random variables such as the x, of the present chapter, then w(x) is a long-tailed probability density function with long-time asymptotic behavior [7,37],... 

Setting E = m being the mass of the emitted particles, makes q E) E / => p v) tP. Going back to Eq. (60) and considering the long-time regime t >>> t, the classical particle velocity can be approximated by V = r/t, so p r/t) = p v), which implies, as expected, that at large t the main contribution to the position probability density is from slow particles. If we consider the same dependence as in the quantum case, p v) v, Eq. (60) implies an asymptotic behavior Pc r,t) i.e., the classical model leads to... 

First we fix ko and observe the density as a function of t for different x values. Figure 9.7 shows the density at the transition versus tp. The curve depends parametrically on x, which increases from the bottom right comer upward. From this figure we can infer that the probability density at the transition point increases, as the observation point is moved away from the source, improving the possibility of observing the transition. A basic reason for this is the asymptotic growth of I with x, whereas the pole term is basically shifted by Sx/lkoR for a shift Sx in the observation coordinate in other words, the exponential behavior is delayed by increasing x. 

Statistics of the Spectral Estimates, The exact results for the statistics of the AR spectral estimator are not known. For large samples of stationary processes, the spectral estimates have approximately a Gaussian probability density function and are asymptotically unbiased and consistent estimates of the power spectral density. TTie variance of the estimate is given by... 

This is the usual behavior of probability density functions, i.e., asymptotically the probability of finding molecules at distances or momenta approaching infinity should decay rapidly to zero. 

The MC technique is a stochastic simulation method designed to generate a long sequence, or Markov chain of configurations that asymptotically sample the probability density of an equilibrium ensemble of statistical mechanics [105, 116]. For example, a MC simulation in the canonical (NVT) ensemble, carried out under the macroscopic constraints of a prescribed number of molecules N, total volume V and temperature T, samples configurations rp with probability proportional to, with, k being the Boltzmann constant and T the... 

For At -> O the probability density defined in (17) can be asymptotically approximated for the discrete case as... 

This leaking out of the wavefunction as it asymptotically approaches zero can be seen quantitatively by computing the probability density of the particle beyond the classical turning points . This is done specifically... 

The state is described by a Gaussian probability density with zero mean and a variance, a (t), that increases with time, eventually approaching the large time asymptote. 

With this simple acceptance criterion, the Metropolis Monte Carlo method generates a Markov chain of states or conformations that asymptotically sample the XTT probability density function. It is a Markov chain because the acceptance of each new state depends only on the previous state. Importantly, with transition probabilities defined by Eqs. 15.23 and 15.24, the transition matrix has the limiting, equihbrium distribution as the eigenvector corresponding to the largest eigenvalue of 1. 

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

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