Zero Temperature Limit

Let us first of all consider the deterministic Life rule, or zero temperature limit of our more general stochastic rule. Using the density p to represent our state of knowledge of the system at time t, our problem is then to estimate the time-evolution of p for T = 0. 

The simplest approximation to make is simply that the initial distribution of live sites is completely random and that any site-site correlations are negligible i.e. we first take a conventional Mean-Field approach (see section 7.4). In this case, the equilibrium density can be written down almost by inspection. The probability of a site having value 1 (= p) is equal to the probability that it had value 1 on the previous time step multiplied by the probability that it stays equal to 1 (i.e. the probability that a site has either 2 or 3 live neighboring sites) plus the probability that the site was previously equal to 0 multiplied by the probability that it become 1 (i.e. that it is surrounded by exactly 3 live sites). Letting p and p represent the density at times t and t + 1, respectively, simple counting yields  

Setting p = p = Pf, at equilibrium, we find that the only real stable nonzero solution for 0 pe 1 is Pe 0.370. This uncorrelated approximation actually describes the infinite temperature limit (effectively, T 1) rather well, since as the temperature increases, local correlations of the basic Life rule steadily decrease. 

In order to improve upon the mean-field approximation given in equation 7.112, we must somehow account for possible site-site correlations. Let us go back to the deterministic version of the basic Life rule (equation 7.110). We could take a formal expectation of this equation but we first need a way to compute expectation values of Kronecker delta functions. Schulman and Seiden [schul78] provide a simple means to do precisely that. We state their result without proof 

Schulman and Seiden first note that a (5-function can be written as a sum of products of cr s. Let L — and K be any subset of the set L of cardinality 

In spite of considering two-photon processes, we still find the energy levels (8.17) to be equidistant with respect to the photon occupation numbers n, -I- i. This suggests that it is permissible to neglect the effect of the electron-photon interaction on statistics and to occupy the electron states and the photon states according to Fermi statistics and to Bose statistics. We shall check this question in Section 8.4 by studying the creation and annihilation operators of the resulting quasi-electrons and quasi-photons. 

It is obvious from Eq. (8.17) that the effect of the electrons is basically covered by the quantities 

r) replaces the susceptibilities introduced in the preceding sections. In contrast to the quantities used before, it explicitly contains the position of the electron orbitals involved, i.e. X q, r) covers the spatial variation of the photon field as well. 

To obtain the dispersion energy between two particles 1 and 2, we split the total electronic susceptibility X q, r) into those of the electrons localized at particle 1 and particle 2, 

again interchanging the notation of photons q and r and of emission and absorption we find 

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In the case of the threshold rules defined in this section, we must consider sequential iterations of deterministic rules. Also, the choice of spins that may change state is not random but is fixed by some random permutation of the sites on the lattice. Such rules may be shown to correspond to spin glasses in the zero-temperature limit. 

The form of the stochastic transfer function p x) is shown in figure 10.7. Notice that the steepness of the function near a - 0 depends entirely on T. Notice also that this form approaches that of a simple threshold function as T —> 0, so that the deterministic Hopfield net may be recovered by taking the zero temperature limit of the stochastic system. While there are a variety of different forms for p x) satisfying this desired limiting property, any of which could also have been chosen, this sigmoid function is convenient because it allows us to analyze the system with tools borrowed from statistical mechanics. 

In some situations we have performed finite temperature molecular dynamics simulations [50, 51] using the aforementioned model systems. On a simplistic level, molecular dynamics can be viewed as the simulation of the finite temperature motion of a system at the atomic level. This contrasts with the conventional static quantum mechanical simulations which map out the potential energy surface at the zero temperature limit. Although static calculations are extremely important in quantifying the potential energy surface of a reaction, its application can be tedious. We have used ah initio molecular dynamics simulations at elevated temperatures (between 300 K and 800 K) to more efficiently explore the potential energy surface. 

More insight into these processes is obtained by studying the particle number dependent properties of density functionals. This of course requires a suitable definition of these density functionals for fractional particle number. The most natural one is to consider an ensemble of states with different particle number (such an ensemble is for instance obtained by taking a zero temperature limit of temperature dependent density functional theory [84]). We consider a system of N + co electrons where N is an integer and 0 < m < 1. For the corresponding electron density we then have... 

As verified by comparison with Eq. (80) and its zero temperature limit, this Fourier analysis is in agreement with the asymmetric Franck-Condon progression at zero temperature of the Marechal and Witkowski model [18]. 

Table P.l.b. Two half-spaces across a planar slab, separation /, zero-temperature limit...

By substituting the matrix S(E) from 5 into Landauer s formula 3 at zero-temperature limit, the tunneling conductance is exactly represented as... 

This expression can be interpreted as a decay rate of level g, 0) into the manifold x, v ) only if this manifold is (1) continuous or at least dense enough, and (2) satisfies other requirements specified in Section 9.1. Nevertheless, Eq. (12.59) can be used as a lineshape expression even when that manifold is sparse, leading to the zero temperature limit of (12.57)... 

The literature exhibits a strong theoretical bias toward the existence of a QLRO state. But our numerical studies are carried out in the zero temperature limit. We shall find that, that if the system is started in a random configuration, in the long-time limit, the system usually exhibits SRO with Imry-Ma-like features. 

Recoil-free fraction measurements for soUd krypton between 5 K and 85 K are also available [19,20], but in this case the / values are lower than expected. Some of the first calculations [20] omitted the effects of anharmonicity upon the phonon frequency spectrum. Inclusion of an harmonic effect [21] gives better agreement between experiment and theory, but the zero-temperature limit of / is anomalously low and has still not been explained. 

So the highest occupied Kohn-Sham orbital has a fractional occupation number to. The fact that is uniquely defined by follows directly from the Hohenberg-Kohn theorem applied to the non-interacting system. The proof of the Hohenberg-Kohn theorem for systems with noninteger number of electrons proceeds along the same lines as for systems with integer particle number (alternatively it can be obtained from the zero temperature limit of temperature dependent density functional theory [82]). We further split up v, as... 

The thermodynamic potential of the hardness representation, in the zero temperature limit, T = 0 K, is the system energy under constraints implied by the resolution in question ... 

Asymptotic decay. If one uses the zero-temperature limit of the grand canonical ensemble to interpolate between integer numbers of electrons, then... 

While the gradient expansion approach (Eq. 38) does not give results for the one-sided Fukui functions, / (r) and f r) may be obtained from the variational approach by using the one-sided hardness kernels, in Eqs. (40) and (42) [47]. If we use the onesided hardness kernels obtained from the zero-temperature limit of the grand canonical ensemble [6], then = rj = 0 hence, the one-sided hardness kernels have no inverse. 

Well below T, in the zero-temperature limit, the excitations out of the antifer-romagnetically ordered ground state of eq. (2) are spin waves. Neglecting interlayer coupling, conventional spin-wave theory in the classical (large-5) limit predicts a dynamic susceptibility (as measured ly inelastic neutron scattering) of the form... 

Fig. 8. Inverse magnetic correlation length for antiferromagnetic spin fluctuations in lightly doped La2, Sr,Cu04. The lines are based on a simple model in which the zero-temperature limit of K is governed by the doping, and the temperature dependence is derived from the renormalized classical expression (eq. 5). From Keimer et al. (1992).

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