Von Neumann

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

Contrary to what appears at a first sight, the integral relations in Eqs. (9) and (10) are not based on causality. However, they can be related to another principle [39]. This approach of expressing a general principle by mathematical formulas can be traced to von Neumann [242] and leads in the present instance to an equation of restriction, to be derived below. According to von Neumann complete description of physical systems must contain ... 

D. Marx and J. Hiitter, in Modern Methods and Algorithms of Quantum Chemistry, J. GrotendorsC, ed., John von Neumann Institute for Computing, Julich, Germany, 2000, pp. 301-449. See http //www.fz-juelich,de/nic-series. 

D.L. Hicks, von Neumann Stability of the Wondy Wavecode for Thermodynamic Equations of State, SAND77-0934, Sandia National Laboratories, Albuquerque, NM, 1977. 

Note that, since the von Neumann equation for the evolution of the density matrix, 8 j8t = — ih H, / ], differs from the equation for a only by a sign, similar equations can be written out for p in the basis of the Pauli matrices, p = a Px + (tyPy -t- a p -t- il- In the incoherent regime this leads to the master equation [Zwanzig 1964 Blum 1981]. For this reason the following analysis can be easily reformulated in terms of the density matrix. 

In view of the success of von Neumann s machine-based hydrodynamics in 1944, and at about the time when the fission bomb was ready, some scientists at Los Alamos were already thinking hard about the possible design of a fusion bomb. Von Neumann invited two of them, Nicholas Metropolis and Stanley Frankel, to try to model the immensely complicated issue of how jets from a fission device might initiate thermonuclear reactions in an adjacent body of deuterium. Metropolis linked... 

An improvement on the CJ model is the ZND (Zeldovich, von Neumann, and Doring) model, which takes the reaction rate into account (Nettleton 1987, Classman 1996, Lewis and von Elbe 1987). The ZND model describes the detonation wave as a shock wave, immediately fol-... 

A slightly more realistic concept is the Zel dovich-Von Neumann-Dohring (ZND) model. In this model, the fuel-air mixture does not react on shock compression beyond autoignition conditions before a certain induction period has elapsed (Figure 3.4). 

The formal study of CA really began not with the simpler one-dimensional systems discussed in the previous section but with von Neumann s work in the 1940 s with self-reproducing two-dimensional CA [vonN66]. Such systems also gained considerable publicity (as well as notoriety ) in the 1970 s with John Conway s introduction of his Life rule and its subsequent popularization by Martin Gardner in his Scientific American Mathematical Games department [gardner83] (see section 3.4-4). 

The description and overview of (mostly generic) two-dimensional systems given in this section focuses mainly on T- and OT- type rules. For the von-Neumann... 

Fig. 3.51 A few Snapshots of two-dimensional von-Neumann neighborhood totalistic rule X = l,2,3,4-> 1.

X — V, V, M, or M if M is the von Neumann, von Neumann with center-site excluded, Moore or Moore with center-site excluded neighborhoods, respectively. 

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