Space dimensionality

W.F. Noh, CEL A Time-Dependent, Two-Space-Dimensional, Coupled Eulerian-Lagrangian Code, in Methods in Computational Physics, Volume 3 (edited by B. Alder, S. Fernbach and M. Rotenberg), Academic Press, New York, 1964. 

It is noteworthy that the critical value given by Eq. (2.5.11) is exactly half as large as for two quadrupole orientations on a square lattice (cf. Eq. (2.5.7)). This is a vindication of the inference that a halved critical temperature results from a corresponding change in the orientation space dimensionality. Thus, one might with good reason anticipate that the critical temperature for a triangular lattice of quadruples with arbitrary planar orientations should also be approximately half as... 

Despite the conceptual elegance of partitioning in low-dimensional descriptor spaces, dimensional reduction is not essential for effective partitioning, as has been shown, for example, by application of statistical partitioning methods (4). 

This shows that with an increase in the space dimensionality, there is an extension of the time interval where the conventional expression of particle survival probability (5.2.41) is valid. 

NAN model [15-20]. This result could be easily generalized for the arbitrary space dimensionality d a = d/2. 

For d 4 singular term (d — l)(d — 3)/(4 j2) does not allow to find the solution at r/ = 0. It has simple interpretation in systems with so large space dimensionalities no variable rj = r/fo exists there. Similar to d = 3 in the linear approximation, for d 4 we can find the stationary solutions, Y (r, oo) = y0(r). For them the reaction rate K(oo) = Kq — const and the classical asymptotics n(t) oc Ya, ao = 1 hold. Therefore, for a set of kinetic equations derived in the superposition approximation the critical space dimension could be established for the diffusion-controlled reactions. 

The approach presented here allows us to calculate such a physically important observable quantity as the gyration radius of a closed chain unentangled with obstacles. Omitting the nonuniversal numerical coefficients depending on the space dimensionality and coordinational number of the lattice of obstacles, the following scaling relation obtained in Refs. [18,22] is fulfilled ... 

Furthermore, an increase in the space dimensionality results in restricting the time interval where an intermediate asymptotic limit (2.1.106) is true. This happens at the expense of a shift in its lower boundaty towards larger values of r, while the upper limit of the interval with J 1 does not depend on the space dimensionality. Indeed, with d I, w d /4 and the inequality (5.2.43) it takes the form... 

A -h B 0 reaction (provided equal reactant concentrations). Peculiarity of the diffusion-controlled regime of reaction is the existence of the marginal space dimensionality do-... 

As yet, these methods have not been applied in very large-scale simulations, but Henkelman and Jonsson (1999) have shown that the method is relatively insensitive to increasing the phase-space dimensionality of the system, so long as the system is large enough to allow all collective relaxations to take place. 

Repulsive chains in d-dimensional space dimensionality of a chain... 

The three polyad quantum numbers serve to block diagonalize the total vibrational H into individual, scaling-related polyads, Heff ([Astretch, AreKOnance],total) The coordinate space or state space dimensionality is reduced from 7 to (7 — 3) =... 

The Fock space dimensionality is finite when one deals with finite model systems. Thus, it appears that we can solve the model Hamiltonians exactly for finite systems and from the finite system properties, infer the behaviour of the system in the thermodynamic limit by suitable scaling techniques. However, the dimensionality increases as (25 4-1) for a spin-S chain and as 4 for fermions, where N is the number of sites. Thus, it is very difficult to carry out brute force numerical computations on a large system and the exact diag-onalization studies are primarily restricted to quasi-one-dimensional systems with very few sites per unit cell. 

In this approach, one begins by subdividing the total system into several blocks An and proceeds to iteratively build effective blocks so that at each iteration, each effective block represents two or more blocks of the previous iteration, without increasing the Fock space dimensionality of the blocks from what existed at the previous iteration. Usually, one starts with each An consisting of a single site. Since the Hilbert space grows exponentially with the increase in system size, one truncates the number of states kept at each iteration. The quantum RG procedure proceeds as follows ... 

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