System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.     

Momentum autocorrelation function of a classical oscillator chain with alternating masses

A classical harmonic oscillator chain with alternating masses is studied using the recurrence relations method. The momentum autocorrelation function changes from combination of cosines to Bessel functions when the number of oscillators increases from finite to infinite bringing about irreversibility. Optic and acoustic branches of the momentum autocorrelation function are expanded in terms of even-order Bessel functions and are shown to be finite and well behaved. Irreversibility and ergodicity are discussed.     

Fourier’s Law for a Microscopic Model of Heat Conduction

We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove that the stationary state, in the limit as N→ ∞, satisfies Fourier’s law and the linear profile for the energy average     

Classical and quantum oscillators of sextic and octic anharmonicities

Classical oscillators of sextic and octic anharmonicities are solved analytically up to the linear power of λ (anharmonic constant) by using Taylor series method. These solutions exhibit the presence of secular terms which are summed up for all orders. The frequency shifts of the oscillators for small anharmonic constants are obtained. It is found that the calculated shifts agree nicely with the available results to-date. The solutions for classical anharmonic oscillators are used to obtain the solutions corresponding to quantum anharmonic oscillators by imposing fundamental commutation relations between position and momentum operators.     

Study of thermal conductivity and thermal rectification in exponential mass graded lattices

Concept of exponential mass variation of oscillators along the chain length of N oscillators is proposed in the present Letter. The temperature profile and thermal conductivity of one-dimensional (1D) exponential mass graded harmonic and anharmonic lattices are studied on the basis of Fermi-Pasta-Ulam (FPU) β model. Present findings conclude that the exponential mass graded chain provide higher conductivity than that of linear mass graded chain. The exponential mass graded anharmonic chain generates the thermal rectification of 70-75% which is better than linear mass graded materials, so far. Thus instead of using linear mass graded material, the use of exponential mass graded material will be a better and genuine choice for controlling the heat flow at nano-scale.     

The Waring formula and fusion rings

The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for su(N)_K is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for sp(N)_K from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers. In the case of su(N)_K the generating function for the generalized Fibonacci numbers obtained are in agreement with Lascoux using the theory of symmetric functions. For sp(N)_K, however, the generalized Fibonacci numbers are obtained from new sequences.     

Onsager-Zerlegung und Entropieproduktion für ein Brownsches Teilchen in der harmonischen Kette

The momentum autocorrelation function of a heavy particle in an infinite harmonic chain in thermal equilibrium is shown to be an example of “Onsager separation”. This separability ensures that thestatistical entropy production associated with the momentum relaxation of the particle contains aphenomenological part which proves to be always positive.     

On the validity of stochastic rate equations in finite systems with finite-strength interactions

Starting with the Hamiltonian for a linear harmonic chain of 2N particles of massm and one of massM, we have carried out numerical calculations for the momentum autocorrelation function of the mass defect particle for chains with finite numberN of mass points and for nonzero values of the mass ratio=m/M. These results have been compared with the well-known exponential relaxation of the momentum autocorrelation function which is found to be the rigorous result when passing to the thermodynamic and weak-coupling limit. In these limits, the dynamics of the mass defect particle is exactly described by a Fokker-Planck equation, i.e., a stochastic equation of motion. We have shown that, to an excellent approximation, an exponential relaxation of the momentum autocorrelation function is obtained for mass ratios as high as=0.1 and for chains with only 50 particles. Thus, for the harmonic chain considered here, the stochastic equations of motion can be applied to a very good approximation far outside the usually imposed thermodynamic and weak-coupling limits.Supported in part by the Advanced Research Projects Agency of the Department of Defense as monitored by the U.S. Office of Naval Research under Contract N00014-69-A-0200-6018 and by the National Science Foundation under Grant GP28257X.     

Fibonacci oscillators

We discuss the properties of oscillators whose spectrum is given by a generalized Fibonacci sequence. The properties include: invariance under the unitary quantum group, generalized angular momentum, coherent states and difference calculus, relativistic interpretation.     

Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust ‘slow switching’ oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate the implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking).For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S₃-transcritical homoclinic bifurcation, (b) a saddle-node/heteroclinic bifurcation and (c) a Z₃-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry.For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions.     

N-soliton quantum scattering in the sine-Gordon theory

The quantum treatment of soliton scattering in the sine-Gordon model, using the path integral collective coordinate method is generalized to N solitons. The solitions. The first quantum correction to the phase shift of N-soliton scattering is equal to the zero-point energy of an effective multi-soliton Hamiltonian. The energies of the oscillators of this Hamiltonian are shown to be equal to the stability angles of a complete set of solutions of the Schrödinger equation for small fluctuations around a classical N-soliton. Consequently, calculating the fluctuations and their stability angles by the inverse scattering method, we obtain the energies of the oscillators. The first quantum correction to the phase shift (the O(1) part in a development in powers of γ) is evaluated by summing the stability angles. This result is in agreement with the “exact” scattering amplitude conjectured by Faddeev, Kulish and Korepin.     

Transverse elastic waves in Fibonacci superlattices

We study the transverse elastic waves propagating in 6-mm class hexagonal crystals forming Fibonacci superlattices. These are formed by repetitions of CdS and ZnO slabs in A and B constituent blocks following the Fibonacci sequence. We study the periodic superlattices formed by the infinite repetition of a given Fibonacci generation together with the finite Fibonacci generations having stress-free surfaces, in order to compare the effects introduced by the different boundary conditions. We have also considered the effects of piezoelectricity when all the interfaces are metallized and kept at a fixed potential. We use the surface Green function matching method forNnonequivalent interfaces to obtain the dispersion relations and the density of states of these systems. We have studied the influence of the increasing order of the Fibonacci generations on the dispersion relation of the transverse elastic modes. The Fibonacci spectrum is clearly seen even for low-order Fibonacci generations and is almost not modified by the piezoelectric coupling when the interfaces are metallized.     

A strong-coupling expansion for fermion field theories and the large-N limit of the gross-neveu model

An expansion in the fermion propagator is formulated for the N-species Gross-Neveu model in the large-N limit. Different regularisation schemes may be adopted and we compare two. We find that a continuum momentum cut-off is easiest to work with and automatically avoids spurious fermionic states which afflict a naive lattice formulation. Chiral symmetry is broken at zeroth order and the resulting expansion is inverse powers of g²N simplifies considerably for large N. In this limit the strong-coupling expansion may be summed to all orders. Extrapolation techniques, like Padé approximants, are not needed. Using a momentum cut-off we recover all the exact results previously derived by summing weak-coupling expansions.     

Collective coupling of randomly dispersed oscillators with cavities filled with zero-index metamaterials

In cavity quantum electrodynamics, it is hard to enhance the coupling strength between quantum dot (QD) and cavity, owing to the limited choice of QDs and the positional uncertainty brought by the inhomogeneous cavity fields. In this paper, we randomly distribute N oscillators with oscillating strength G = G ₀ into a cavity filled with a zero-index metamaterial (ZIM). Because of the enhanced uniform fields, each oscillator couples to the field maximum and the N oscillators are equivalent to one oscillator with effective N G ₀. This provides a way to enhance the coupling strength just by adding the number of QDs. Both simulation and experiment demonstrate the adjustable coupling strength in ZIM-filled cavities.     

Fourier analysis of spherically averaged momentum densities for some gaseous molecules

The spherically averaged autocorrelation function,B(r), of the position-space wavefunction, ψ($\text{r}$), is calculated by numerical Fourier transformation from spherically averaged momentum densities, ?(p), obtained from either theoretical wavefunctions or (e,2e) electron-impact ionization experiments. Inspection of B(r) for the π molecular orbitals of C₄ H₆ establishes that autocorrelation function differences, ΔB(r), can be qualitatively related to bond lengths and numbers of bonding interactions. Differences between B(r) functions obtained from different approximate wavefunctions for a given orbital can be qualitatively understood in terms of wavefunction difference,Δψ($\text{r}$), maps for these orbitals. Comparison of the B(r) function for the 1a_u orbital of C₄H₆ obtained from (e,2e) momentum densities with that obtained from an ab initio SCF MO wavefunction shows differences consistent with expected correlation effects. Thus, B(r) appears to be a useful quantity for relating spherically averaged momentum distributions to position-space wavefunction differences.     

Variation of average charged particle multiplicity inp-nucleus interactions with energy and the two component description of particle production at high energies

Experimental data on average shower particle multiplicity (〈N _s〉) accumulated onp-nucleus interactions in the wide momentum region of 7.1–8000 GeV/c is investigated. It is observed that 〈N _s〉 is represented exceedingly well as a function of (v _vS). There are two physical processes which represent the experimental data reasonably well in the two momentum regionsviz 7.1–67.9 GeV/c and 67.9–8000 GeV/c. 〈N _s〉=a( _v ^S)/a+b fits the data in the low momentum region, whereas 〈N _s〉=a +b ln (v _vS) fits the experimental data in the high momentum region. The two physical processes are unified and represented by a single equation which is shown to be the consequence of two component theory and collective models.     

An impact parameter description of single and double diffraction dissociation

An s-channel impact parameter model of inelastic diffraction dissociation is constructed which describes quantitatively the momentum transfer distributions of both pion and nucleon induced reactions. It is stressed that the impact parameter profile is peripheral and is approximately the same for all helicity amplitudes. From the fits to the data, the concept of a universal limiting strength (ULS) for each amplitude is deduced and its consequence for other diffractive processes, and the approximate TCHC hypothesis are discussed. Finally, detailed predictions are presented the momentum transfer distributions in the double diffraction process NN → (Nπ)(Nπ), and compared with a simple t-channel picture of diffraction dissociation.     

Ghost vertices for the bosonic string using the group-theoretic approach to string theory

The N-string tree-level scattering vertices for the bosonic string are extended to include anticommuting (ghost) oscillators. These vertices behave correctly under the action of the BRST charge Q and reproduce the known results for the scattering of physical states. This work is an application of the group-theoretic approach to string theory.     

Singularrenormalization group transformations and first order phase transitions (I)

It is argued that standard position and momentum space renormalization group (RG) transformations are singular (i.e. lead to a singular hamiltonian after a finite number of RG steps) in large regions of the coupling constant space. It is shown in the d = 3, φ⁶ O(N)_N→∞ model that the momentum space RG transformation is singular in all those points of the coupling constant space, where metastable states exist. This region includes the full first order phase transition surface and its neighbourhood. Several other examples are discussed to illustrate that this phenomenon is generic and not a specific large N effect. Some earlier and recent anomalous Monte Carlo renormalization group results are consistent with this conclusion.