Linear scaling with particle number

A model which is most often used in qualitative discussions of linear scaling with particle number is an array of well-separated identical subsystems. In particular, the model which is most often employed in such discussions is that of a linear array of well-separated, that is, non-interacting, helium atoms. In Section 3.1.1, we consider the exact Schrbdinger equation for this model system. In Section 3.1.2, we turn our attention to independent particle models. We make some general remarks about many-body methods in Section 3.1.3. 

We shall provide an overview of the applications that have been made over the period being review which demonstrate the many-body Brillouin-Wigner approach for each of these methods. By using Brillouin-Wigner methods, any problems associated with intruder states can be avoided. A posteriori corrections can be introduced to remove terms which scale in a non linear fashion with particle number. We shall not, for example, consider in any detail hybrid methods such as the widely used ccsd(t) which employs ccsd theory together with a perturbative estimate of the triple excitation component of the correlation energy. 

The key property of many-body methods in general and many-body perturbation theory in particular is recognized as their extensivity or linear scaling with the number of particles in the studied system. Rayleigh-Schrodinger perturbation theory contains terms in each order of the series which scale non-linearly with particle... 

Note again the characteristic linear quantum scaling with the number of particles r/pi,. We often refer to the variance of the coherent light state as the shot noise level. 

It must be based either directly or indirectly on the linked diagram theorem of many-body perturbation theory so as to ensure that the calculated energies and other expectation values scale linearly with particle number... 

Brillouin-Wigner theory has been largely neglected in the quantum chemistry and molecular physics literature, because it is found to lack extensivity. Brillouin-Wigner perturbation theory contains terms in each order which do not scale linearly with particle number and furthermore, these terms are not cancelled in a given order. This lack of extensivity is associated with the presence of the exact energy in the denominator factors. 

The theory of cumulants allows us to partition an RDM into contributions that scale differently with the number N of particles. Because aU of the particles are connected by interactions, the cumulant RDMs scale linearly with the number N of particles. The unconnected terms in the p-RDM reconstruction formulas scale between N and W according to the number of connected RDMs in the wedge product. For example, the term scales as NP since all p particles are statistically independent of each other. By examining the scaling of terms with N in the contraction of higher reconstruction functionals, we may derive an important set of relations for the connected RDMs. 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

The muon is about two hundred times heavier than the electron and its orbit lies 200 times closer to the nucleus. The nuclear structure effects scale with the mass of the orbiting particle as m3R2 (for the Lamb shift It is a characteristic value of the nuclear size) and as m R2 (for the hyperfine structure), while the linewidth is linear in m. That means, that from a purely atomic point of view the muonic atoms offer a way to measure the nuclear contribution with a higher accuracy than normal atoms. However, there are a number of problems with formation and thermalization of these atoms and with their collisions with the buffer gas. 

In geological literature dealing with particle size distribution [ 130,131 ] a very advantageous transformation of particle size is commonly used. This transformation replaces scale numbers based on linear millimeter values by the logarithms of these values. Because it is a logarithmic transformation it simplifies computation. 

The main advantage of the far formula (16) is that it allows for a product decomposition into the particle components and can be evaluated in linear time with respect to the number of particles with constant cutoffs. To draw advantage from this, some additional bookkeeping is necessary, see [17]. For MMM2D error estimates exist, which allow the tuning of the method this is important since the evaluation is split up into the near and far formula, leading to a highly non-uniform error distribution (see Fig. 1). With optimized parameters, the method can be tuned to scale like... 

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