Table of contents

Systems consisting of two interacting magnetic moments, or spins, in an applied magnetic field play a role in the study of quantum chaos where they are used as a classical analogue of a quantum spin system. The equations of motion for such systems are particularly amenable to solution via a geometric integrator constructed by splitting the Hamiltonian into integrable pieces. Such methods have the advantage of restricting numerical solutions to the correct manifold and respecting various geometric properties of the system. Some of these properties, such as approximate conservation of total energy, are important when producing Poincaré surfaces of section (PSSs) which are used to study the onset of chaos in the classical system. The choice of coordinate system for the equations of motion is also important with some choices leading to coordinate singularities. We compare PSSs obtained via a generalized 'leapfrog' integrator with those from a classical 'black-box' Runge–Kutta integrator. The effects of the coordinate singularity on the accuracy and the energy of the solution is also investigated.

Using the decay of the out-of-equilibrium spin–spin correlation function we compute the equilibrium Edward–Anderson order parameter in the three-dimensional binary Ising spin glass in the spin glass phase. We have checked that the Edward–Anderson order parameter computed from out-of-equilibrium numerical simulations follows with good precision the critical law as determined in experiments and in numerical studies at equilibrium (which allow us to estimate the β critical exponent). Finally, we present a large time study of the off-equilibrium fluctuation–dissipation relations and find strong discrepancies (in the low-temperature region) between the numerical data and the droplet theory predictions and agreement with the predictions of the replica symmetry breaking theory.

We present a formulation of the determination of the impedance between any two nodes in an impedance network. An impedance network is described by its Laplacian matrix L which has generally complex matrix elements. We show that by solving the equation Lu_α = λ_αu*_α with orthonormal vectors u_a, the effective impedance between nodes p and q of the network is Z_pq = ∑_α(u_αp − u_αq)²/λ_α, where the summation is over all λ_α not identically equal to zero and u_αp is the pth component of u_α. For networks consisting of inductances L and capacitances C, the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of λ_α. This curious result suggests the possibility of practical applications to resonant circuits. Our formulation is illustrated by explicit examples.

The existence of a dynamical potential with both local and global meanings in general nonequilibrium processes has been controversial. Following an earlier heuristic argument in a letter by one of us, in the present paper we show rigorously its existence for a generic class of situations in physical and biological sciences. The local dynamical meaning of this potential function is demonstrated via a special stochastic differential equation and its global steady-state meaning via a novel and explicit form of the Fokker–Planck equation. We also give a procedure to obtain the special stochastic differential equation for any given Fokker–Planck equation. No detailed balance condition is required in our demonstration. For the first time we obtain here a formula to describe the noise-induced shift in drift force compared to the steady-state distribution, a phenomenon extensively observed in numerical studies.

The properties of the mean first passage time in a system characterized by multiple periodic attractors are studied. Using a transformation from a high-dimensional space to 1D, the problem is reduced to a stochastic process along the path from the fixed point attractor to a saddle point located between two neighbouring attractors. It is found that the time to switch between attractors depends on the effective size of the attractors, τ, the noise, , and the potential difference between the attractor and an adjacent saddle point as ; the ratio between the sizes of the two attractors affects . The result is obtained analytically for small τ and confirmed by numerical simulations. Possible implications that may arise from the model and results are discussed.

Spherical harmonic expansions of the screened Coulomb interaction kernel involving the complementary error function are required in various problems in atomic, molecular and solid state physics, like for the evaluation of Ewald-type lattice sums or for range-separated hybrid density functionals. A general analytical expression is derived for the kernel, which is non-separable in the radial variables. With the help of series expansions a separable approximate form is proposed, which is in close analogy with the conventional multipole expansion of the Coulomb kernel in spherical harmonics. The convergence behaviour of these expansions is studied and illustrated by the electrostatic potential of an elementary charge distribution formed by products of Slater-type atomic orbitals.

We consider a μ-deformation of the Segal–Bargmann transform, which is a unitary map from a μ-deformed ground state representation onto a μ-deformed Segal–Bargmann space. We study the μ-deformed Segal–Bargmann transform as an operator between L^p spaces and then we obtain sufficient conditions on the Lebesgue indices for this operator to be bounded. A family of Hirschman inequalities involving the Shannon entropies of a function and of its μ-deformed Segal–Bargmann transform are proved. We also prove a parametrized family of log-Sobolev inequalities, in which a new quantity that we call 'dilation energy' appears. This quantity generalizes the 'energy term' that has appeared in a previous work.

We show how the radial position–momentum uncertainty product can be obtained analytically for the Dirac hydrogen-like atoms. Some interesting features for this system are found. First, for the same principal quantum number n, as the azimuthal quantum number l increases, the uncertainties Δp_r and ΔrΔp_r decrease. However, the uncertainty Δr does not always decrease, which is different from the non-relativistic hydrogen-like atoms case, where it always decreases. Second, for the same l both Δr and ΔrΔp_r increase as n increases. Third, all uncertainties for the same n and l = n − 1 are smallest in comparison with those for the same n but . Fourth, the uncertainty ΔrΔp_r is independent of the value of charge Z in the non-relativistic case, while it is related with the value of charge Z in the relativistic case. Fifth, the relativistic corrections to the non-relativistic values of uncertainties are very small when the values of charge Z are not too big, while the relativistic corrections to them will appear explicitly for a large value of charge Z.

We consider the problem of a self-interacting scalar field nonminimally coupled to the three-dimensional BTZ metric such that its energy–momentum tensor evaluated on the BTZ metric vanishes. We prove that this system is equivalent to a self-dual system composed by a set of two first-order equations. The self-dual point is achieved by fixing one of the coupling constant of the potential in terms of the nonminimal coupling parameter. At the self-dual point and up to some boundary terms, the matter action evaluated on the BTZ metric is bounded below and above. These two bounds are saturated simultaneously yielding to a vanishing action for configurations satisfying the set of self-dual first-order equations.

We discuss the crossover between the small and large field cutoff (denoted as x_max) limits of the perturbative coefficients for a simple integral and anharmonic oscillators. We show that in the limit where the order k of the perturbative coefficient a_k(x_max) becomes large and for x_max in the crossover region, . The order-independent constant A and the function x₀(k) are determined empirically and compared with exact (for the integral) and approximate (for the anharmonic oscillator) calculations. Numerically, our best estimates are A ≃ 1.5, 1.3 and 1.2 for a λx⁴, λx⁶ and λx⁸ perturbation respectively.

It has recently been determined that, within the framework of the exact renormalization group, continuum computations can be performed to any loop order in SU(N) Yang–Mills theory without fixing the gauge or specifying the details of the regularization scheme. In this paper, we summarize and refine the powerful diagrammatic techniques which facilitate this procedure and illustrate their application in the context of a calculation of the two-loop β function.

Relativistic quantum plasma dispersion functions (RQPDFs) are defined and the longitudinal and transverse response functions for an electron (plus positron) gas are written in terms of them. The dispersion is separated into Landau-damping, pair-creation and dissipationless regimes. Explicit forms are given for the RQPDFs in the cases of a completely degenerate distribution and a nondegenerate thermal (Jüttner) distribution. Particular emphasis is placed on the relation between dissipation and dispersion, with the dissipation treated in terms of the imaginary parts of RQPDFs. Comparing the dissipation calculated in this way with the existing treatments leads to the identification of errors in the literature, which we correct. We also comment on a controversy as to whether the dispersion curves in a superdense plasma pass through the region where pair creation is allowed.

We have derived rigorously the perimeter generating functions for the mean-squared radius of gyration of rectangular, Ferrers and pyramid polygons. These functions were found by Jensen recently. His nonrigorous results are based on the analysis of the long series expansions.

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