We develop an approach of calculating the many-body path integral based on the linked cluster expansion method. First, we derive a linked cluster expansion and we give the diagrammatic rules for calculating the free-energy and the pair distribution function $g(r)$ as a systematic power series expansion in the particle density. We also generalize the hypernetted-chain (HNC) equation for $g(r)$, known from its application to classical statistical mechanics, to a set of quantum HNC equations (QHNC) for the quantum case. The calculated $g(r)$ for distinguishable particles interacting with a Lennard-Jones potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path integral Monte Carlo simulation even for densities as high as the equilibrium density of the strongly correlated liquid $^4$He. Our method is applicable to a wide range of problems of current general interest and may be extended to the case of identical particles and, in particular, to the case of the many-fermion problem.
Tertiary structure formation underlies the folding mechanics of many classes of polymers. A simplified model of helical polymers is a useful system in which to begin studying the formation and properties of compact conformations known from biomolecules. Hyper-phase diagrams and structural transitions are presented for polymers of length 40 and 50 over an array of model parameters.
The static structure factor of amorphous silicon (a-Si) models, containing 400,000 atoms with a density of 2.25 g.cm, has been studied by generating atomistic models using classical molecular-dynamics simulations. The behavior of the structure factor, S(Q), in the limit Q -> 0, is examined to determine the degree of hyperuniformity in a-Si and is compared with the results with those from earlier simulations and small-angle X-ray scattering experiments. The study suggests that the computed value of the relative variance of the number of atoms at large distances, and hence S(Q -> 0), lies in the range from 0.00736 to 0.00758, which is very close to the experimental value of 0.0076 +/- 0.0005, obtained from an extrapolation of transmission X-ray scattering data in the small-angle region. The non-zero value of the structure factor S(0) in a-Si can be attributed to density fluctuations on a very large length scale, which is a characteristic property of the structural and topological ordering of silicon atoms in the amorphous state.
The thermally driven differential mutation algorithm is an evolutionary algorithm dealing with the structural optimization of large amorphous systems represented by empirical potentials. It is a hybrid algorithm that combines a differential mutation evolutionary algorithm with a metropolis selection criterion and a cooling schedule inspired by simulated annealing. In this manuscript, the influence of the cooling rate on the quality of obtained amorphous graphene structures is discussed.
We find a lower bound on the computational complexity of Grover’s quantum search algorithms in low-dimensional networks using the renormalization group (RG). It highlights the competition between Grover’s abstract algorithm, i.e., a rotation in Hilbert space, and quantum transport in an actual geometry. It can be characterized in terms of the quantum walk dimension dwQ and the spatial (fractal) dimension df or, alternatively, the spectral dimension of the network, ds, even when translational invariance is broken.
