Local Sum Rules Research Articles

Noether\u2019s theorem in statistical mechanics

Noether’s calculus of invariant variations yields exact identities from functional symmetries. The standard application to an action integral allows to identify conservation laws. Here we rather consider generating functionals, such as the free energy and the power functional, for equilibrium and driven many-body systems. Translational and rotational symmetry operations yield mechanical laws. These global identities express vanishing of total internal and total external forces and torques. We show that functional differentiation then leads to hierarchies of local sum rules that interrelate density correlators as well as static and time direct correlation functions, including memory. For anisotropic particles, orbital and spin motion become systematically coupled. The theory allows us to shed new light on the spatio-temporal coupling of correlations in complex systems. As applications we consider active Brownian particles, where the theory clarifies the role of interfacial forces in motility-induced phase separation. For active sedimentation, the center-of-mass motion is constrained by an internal Noether sum rule.

TWO-POINT RESISTANCES IN SAILBOAT FRACTAL NETWORKS

The two-point resistance of fractal network has been studied extensively by mathematicians and physicists. In this paper, for a class of self-similar networks named sailboat networks, we obtain a recursive algorithm for computing resistance between any two nodes, using elimination principle, substitution principle and local sum rules on effective resistance.

Resistance Distances in Vertex-Face Graphs

Abstract The computation of two-point resistances in networks is a classical problem in electric circuit theory and graph theory. Let G be a triangulation graph with n vertices embedded on an orientable surface. Define K(G) to be the graph obtained from G by inserting a new vertex vϕ to each face ϕ of G and adding three new edges (u, vϕ ), (v, vϕ ) and (w, vϕ ), where u, v and w are three vertices on the boundary of ϕ. In this paper, using star-triangle transformation and resistance local-sum rules, explicit relations between resistance distances in K(G) and those in G are obtained. These relations enable us to compute resistance distance between any two points of Kk (G) recursively. As explanation examples, some resistances in several networks are computed, including the modified Apollonian network and networks constructed from tetrahedron, octahedron and icosahedron, respectively.

Random walks and the effective resistance sum rules

In this paper, using the intimate relations between random walks and electrical networks, we first prove the following effective resistance local sum rules: c i Ω i j + ∑ k ∈ Γ ( i ) c i k ( Ω i k − Ω j k ) = 2 , where Ω i j is the effective resistance between vertices i and j , c i k is the conductance of the edge, Γ ( i ) is the neighbor set of i , and c i = ∑ k ∈ Γ ( i ) c i k . Then we show that from the above rules we can deduce many other local sum rules, including the well-known Foster’s k -th formula. Finally, using the above local sum rules, for several kinds of electrical networks, we give the explicit expressions for the effective resistance between two arbitrary vertices.

Inhomogeneous spectral moment sum rules for the retarded Green function and self-energy of strongly correlated electrons or ultracold fermionic atoms in optical lattices

Spectral moment sum rules are presented for the inhomogeneous many-body problem described by the fermionic Falicov-Kimball or Hubbard models. These local sum rules allow for arbitrary hoppings, site energies, and interactions. They can be employed to quantify the accuracy of numerical solutions to the inhomogeneous many-body problem like strongly correlated multilayered devices, ultracold atoms in an optical lattice with a trap potential, strongly correlated systems that are disordered, or systems with nontrivial spatial ordering like a charge density wave or a spin density wave. We also show how the spectral moment sum rules determine the asymptotic behavior of the Green function, self-energy, and dynamical mean field, when applied to the dynamical mean-field theory solution of the many body problem. In particular, we illustrate in detail how one can dramatically reduce the number of Matsubara frequencies needed to solve the Falicov-Kimball model, while still retaining high precision, and we sketch how one can incorporate these results into Hirsch-Fye quantum Monte Carlo solvers for the Hubbard (or more complicated) models. Since the solution of inhomogeneous problems is significantly more time consuming than periodic systems, efficient use of these sum rules can provide a dramatic speed up in the computational time required to solve the many-body problem. We also discuss how these sum rules behave in nonequilibrium situations as well, where the Hamiltonian has explicit time dependence due to a driving field or due to the time-dependent change of a parameter like the interaction strength or the origin of the trap potential.

Resistance distance local rules

In [D.J. Klein, Croat. Chem. Acta. 75(2), 633 (2002)] Klein established a number of sum rules to compute the resistance distance of an arbitrary graph, especially he gave a specific set of local sum rules that determined all resistance distances of a graph (saying the set of local sum rules is complete). Inspired by this result, we give another complete set of local rules, which is simple and also efficient, especially for distance-regular graphs. Finally some applications to chemical graphs (for example the Platonic solids as well as their vertex truncations, which include the graph of Buckminsterfullerene and the graph of boron nitride hetero-fullerenoid B 12 N 12) are made to illustrate our approach.

Vibrational circular dichroism in amino acids and peptides. 7. Amide stretching vibrations in polypeptides.

AbstractVibrational circular dichroism (VCD) spectra for the principal amide stretching vibrations, amide A (NH stretch) and amide I (predominantly CO stretch), are presented and analyzed for a variety of polypeptides dissolved in chloroform, as well as for two examples in D2O. Our results for poly(γ‐benzyl‐L‐glutamate) confirm the first and only previous report of VCD in polypeptides carried out by Singh and Keiderling [(1981) Biopolymers 20, 237–240]. Collectively, our spectra show that the sense of the bisignate VCD in these two regions depends on the sense of α‐helicity and not on the absolute configuration of the constituent amino acids. This conclusion is established by obtaining VCD for the two polypeptides, poly(β‐benzyl‐L‐asparate) and poly(im‐benzyl‐L‐histidine), that form left‐handed as opposed to right‐handed α‐helices. A new amide band having significant VCD intensity owing to its Fermi resonance interaction with the NH stretching mode has been identified as a weak shoulder on the low‐frequency side of the amide A band near 3200 cm−1 and is assigned as a combination band of the amide I and amide II vibrations. VCD spectra of polypeptides in D2O solution, although weak, have been successfully measured in the amide I region, where spectra appear to be more complicated due to the presence of solvated and internally hydrogen‐bonded amide groups. Strong monosignate contributions to the VCD in the amide A and amide I regions for some of the polypeptides indicate coupling of an electronic nature between these two regions and is deduced by an application of the concept of local sum rules of rotational strength. It appears that a detailed understanding of the VCD obtained for polypeptides will not only be diagnostic of secondary structure, but also of more subtle structural and vibrational effects that give rise to local, intrinsic chirality in the amide vibrations.

Sum rules in classical scattering

This paper derives sum rules associated with the classical scattering of two particles. These sum rules are the analogs of Levinson’s theorem in quantum mechanics which provides a relationship between the number of bound-state wavefunctions and the energy integral of the time delay of the scattering process. The associated classical relation is an identity involving classical time delay and an integral over the classical bound-state density. We show that equalities between the Nth-order energy moment of the classical time delay and the Nth-order energy moment of the classical bound-state density hold in both a local and a global form. Local sum rules involve the time delay defined on a finite but otherwise arbitrary coordinate space volume Σ and the bound-state density associated with this same region. Global sum rules are those that obtain when Σ is the whole coordinate space. Both the local and global sum rules are derived for potentials of arbitrary shape and for scattering in any space dimension. Finally the set of classical sum rules, together with the known quantum mechanical analogs, are shown to provide a unified method of obtaining the high-temperature expansion of the classical, respectively the quantum-mechanical, virial coefficients.