Collective motions of a quantum gas confined in a harmonic trap

Single-component quantum gas confined in a harmonic potential, but otherwise isolated, is considered. From the invariance of the system of the gas under a displacement-type transformation, it is shown that the center of mass oscillates along a classical trajectory of a harmonic oscillator. It is also shown that this harmonic motion of the center has, in fact, been implied by Kohn's theorem. If there is no interaction between the atoms of the gas, the system in a time-independent isotropic potential of frequency $\nu_c$ is invariant under a squeeze-type unitary transformation, which gives collective {\it radial} breathing motion of frequency $2\nu_c$ to the gas. The amplitudes of the oscillating and breathing motions from the {\it exact} invariances could be arbitrarily large. For a Fermi system, appearance of $2\nu_c$ mode of the large breathing motion indicates that there is no interaction between the atoms, except for a possible long-range interaction through the inverse-square-type potential.Comment: Typos in the printed verions are correcte

Coevolutionary dynamics on scale-free networks

We investigate Bak-Sneppen coevolution models on scale-free networks with various degree exponents $\gamma$ including random networks. For $\gamma >3$, the critical fitness value $f_c$ approaches to a nonzero finite value in the limit $N \to \infty$, whereas $f_c$ approaches to zero as $2<\gamma \le 3$. These results are explained by showing analytically $f_c(N) \simeq A/_N$ on the networks with size $N$. The avalanche size distribution $P(s)$ shows the normal power-law behavior for $\gamma >3$. In contrast, $P(s)$ for $2 <\gamma \le 3$ has two power-law regimes. One is a short regime for small $s$ with a large exponent $\tau_1$ and the other is a long regime for large $s$ with a small exponent $\tau_2$ ($\tau_1 > \tau_2$). The origin of the two power-regimes is explained by the dynamics on an artificially-made star-linked network.Comment: 5 pages, 5 figure

Coherent States and Geometric Phases in Calogero-Sutherland Model

Exact coherent states in the Calogero-Sutherland models (of time-dependent parameters) which describe identical harmonic oscillators interacting through inverse-square potentials are constructed, in terms of the classical solutions of a harmonic oscillator. For quasi-periodic coherent states of the time-periodic systems, geometric phases are evaluated. For the $A_{N-1}$ Calogero-Sutherland model, the phase is calculated for a general coherent state. The phases for other models are also considered.Comment: To appear in the Int. J. Mod. Phys.

Diffusive Capture Process on Complex Networks

We study the dynamical properties of a diffusing lamb captured by a diffusing lion on the complex networks with various sizes of $N$. We find that the life time $of a lamb scales as \sim N$ and the survival probability $S(N\to \infty,t)$ becomes finite on scale-free networks with degree exponent $\gamma>3$. However, $S(N,t)$ for $\gamma<3$ has a long-living tail on tree-structured scale-free networks and decays exponentially on looped scale-free networks. It suggests that the second moment of degree distribution $is the relevant factor for the dynamical properties in diffusive capture process. We numerically find that the normalized number of capture events at a node with degree$k$,$n(k)$, decreases as$n(k)\sim k^{-\sigma}$. When$\gamma<3$,$n(k)$still increases anomalously for$k\approx k_{max}$. We analytically show that$n(k)$satisfies the relation$n(k)\sim k^2P(k)$and the total number of capture events$N_{tot}$is proportional to$, which causes the $\gamma$ dependent behavior of $S(N,t)$ and $.Comment: 9 pages, 6 figure

`Operational' Energy Conditions

I show that a quantized Klein-Gordon field in Minkowski space obeys an `operational' weak energy condition: the energy of an isolated device constructed to measure or trap the energy in a region, plus the energy it measures or traps, cannot be negative. There are good reasons for thinking that similar results hold locally for linear quantum fields in curved space-times. A thought experiment to measure energy density is analyzed in some detail, and the operational positivity is clearly manifested. If operational energy conditions do hold for quantum fields, then the negative energy densities predicted by theory have a will-o'-the-wisp character: any local attempt to verify a total negative energy density will be self-defeating on account of quantum measurement difficulties. Similarly, attempts to drive exotic effects (wormholes, violations of the second law, etc.) by such densities may be defeated by quantum measurement problems. As an example, I show that certain attempts to violate the Cosmic Censorship principle by negative energy densities are defeated. These quantum measurement limitations are investigated in some detail, and are shown to indicate that space-time cannot be adequately modeled classically in negative energy density regimes.Comment: 18 pages, plain Tex, IOP macros. Expanded treatment of measurement problems for space-time, with implications for Cosmic Censorship as an example. Accepted by Classical and Quantum Gravit

Dynamical surface structures in multi-particle-correlated surface growths

We investigate the scaling properties of the interface fluctuation width for the $Q$-mer and $Q$-particle-correlated deposition-evaporation models. These models are constrained with a global conservation law that the particle number at each height is conserved modulo $Q$. In equilibrium, the stationary roughness is anomalous but universal with roughness exponent $\alpha=1/3$, while the early time evolution shows nonuniversal behavior with growth exponent $\beta$ varying with models and $Q$. Nonequilibrium surfaces display diverse growing/stationary behavior. The $Q$-mer model shows a faceted structure, while the $Q$-particle-correlated model a macroscopically grooved structure.Comment: 16 pages, 10 figures, revte

S matrix of collective field theory

By applying the Lehmann-Symanzik-Zimmermann (LSZ) reduction formalism, we study the S matrix of collective field theory in which fermi energy is larger than the height of potential. We consider the spatially symmetric and antisymmetric boundary conditions. The difference is that S matrices are proportional to momenta of external particles in antisymmetric boundary condition, while they are proportional to energies in symmetric boundary condition. To the order of $g_{st}^2$, we find simple formulas for the S matrix of general potential. As an application, we calculate the S matrix of a case which has been conjectured to describe a "naked singularity".Comment: 19 page, LaTe

Geometric Phase, Hannay's Angle, and an Exact Action Variable

Canonical structure of a generalized time-periodic harmonic oscillator is studied by finding the exact action variable (invariant). Hannay's angle is defined if closed curves of constant action variables return to the same curves in phase space after a time evolution. The condition for the existence of Hannay's angle turns out to be identical to that for the existence of a complete set of (quasi)periodic wave functions. Hannay's angle is calculated, and it is shown that Berry's relation of semiclassical origin on geometric phase and Hannay's angle is exact for the cases considered.Comment: Submitted to Phys. Rev. Lett. (revised version

Restrictions on negative energy density in a curved spacetime

Recently a restriction ("quantum inequality-type relation") on the (renormalized) energy density measured by a static observer in a "globally static" (ultrastatic) spacetime has been formulated by Pfenning and Ford for the minimally coupled scalar field, in the extension of quantum inequality-type relation on flat spacetime of Ford and Roman. They found negative lower bounds for the line integrals of energy density multiplied by a sampling (weighting) function, and explicitly evaluate them for some specific spacetimes. In this paper, we study the lower bound on spacetimes whose spacelike hypersurfaces are compact and without boundary. In the short "sampling time" limit, the bound has asymptotic expansion. Although the expansion can not be represented by locally invariant quantities in general due to the nonlocal nature of the integral, we explicitly evaluate the dominant terms in the limit in terms of the invariant quantities. We also make an estimate for the bound in the long sampling time limit.Comment: LaTex, 23 Page