We consider the Euclidean traveling salesman problem for $N$ cities randomly distributed in the unit $d$-dimensional hypercube, and investigate the finite size scaling of the mean optimal tour length $L_E$. With toroidal boundary conditions we find, motivated by a remarkable universality in the $k$th nearest neighbor distribution, that $L_E(d=2\mathrm{})=(0.7120\pm 0.0002)N^{1/2}[1+O(1/N\left)\right]$ and $L_E(d=3\mathrm{})=(0.6979\pm 0.0002)N^{2/3}[1+O(1/N\left)\right]$. We then consider a mean-field approach in the limit $N\to \infty$ which we find to be a good approximation (the error being less than 2.1% at $d=1,2\mathrm{}$, and 3), and which suggests that $L_E\left(d\right){=N}^{1-1/d}\sqrt{d/2\pi e}\left(\pi d{)}^{1/2d}\right[1+O\left(1/d\right)]$ at large $d$.

We propose an extended quantum mechanical formalism that is based on a wave operator $\hat{\varrho }$, which is related to the ordinary density matrix via $\rho =\hat{\varrho }{\hat{\varrho }}^{†}$. This formalism allows a (generalized) unitary evolution between pure and mixed states. It also preserves much of the connection between symmetries and conservation laws. The new formalism is illustrated for the case of a two-level system.

We find the exact finite size and lattice corrections, to the partition function $Z$, of a continuous spin Gaussian model on a toroidal triangular lattice, with periods $L_0$ and $L_1$. The spins carry a representation of the fundamental group of the torus labeled by $u_0$ and $u_1$ and a mass $m$. Summing $Z^{-1/2\mathrm{}}$ over $u_i$ gives the corresponding result for the Ising model. The limits $m\to 0$ and $u_i\to 0$ do not commute. With $m=0\mathrm{}$ and $u_i$ nonzero the model exhibits a vortex critical phase. In the continuum limit, for arbitrary $m$, the finite size corrections to $-\ln Z$ are modular invariant. In the limit $L_1\to \infty$ the “cylinder charge” $c\left(u_0{,m}^2{L}_0^2\right)$ ranges nonmonotonically from $2\left[{1+6u}_0\right(u_0-1\mathrm{}\left)\right]$ for $m=0\mathrm{}$ to zero for $m\to \infty$.

In a recent study of Thomas precession of electron spin in the context of the Dirac equation, Shankar and Mathur identify the non-Abelian Berry potential (arising from Kramers degeneracy) with the meron. We point out that there is a global mathematical subtlety which prevents such an identification. We go on to clarify the physical context in which merons do arise as Berry potentials.

Two separated observers, by applying local operations to a supply of not-too-impure entangled states (e.g., singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity (e.g., near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield $D\left(M\right)$ of pure singlets $\left(|{\Psi }^{-}〉\right)$ distillable from mixed states $M$, showing $D\left(M\right)>0\mathrm{}$ if $〈{\Psi }^{-}|M|{\Psi }^{-}〉>\frac{1}{2}$.

A model for a mesoscopic system where hyperspherical impurities are placed at random in the interior of hypercubical billiard is studied. The disorder is characterized by the elastic mean free path $l$ that ranges from the ballistic to the diffusive regime. The energy level correlator $K\left(s\right)$ and the form factor $\hat{K}\left(t\right)$ are calculated and characterized by analyzing the analytic structure of the dynamical zeta function of the corresponding classical system.

We derive quantum corrections to the semiclassical Thomas-Fermi distribution for systems obeying a fractional exclusion principle. We obtain a generalized Wigner Thomas-Fermi theory (GWTF) that describes, in a unified framework, boson, fermion, and fractional-statistics particles. In the high temperature limit it recovers the Wigner quantum corrections for the Gibbs-Boltzmann probability density. We apply this method to study the Calogero-Sutherland model with an external anharmonic potential. We show how quantum effects become more and more important as we increase the number of particles allowed to occupy a single quantum state.

We experimentally detect noise enhanced stability in an unstable physical system. The average escape time from a metastable, periodically driven, system is measured in the stable and unstable regimes in a noisy environment. In the unstable regime, we measure that the average escape time has a maximum for a finite value of the noise intensity. The scaling properties of the average escape time and of the variance of escape times are compared with the predictions obtained for a system in a marginal state.

A quantum generalization of the semiclassical theory of Gutzwiller is given. The new formulation leads to systematic orbit-by-orbit inclusion of higher $ħ$ contributions to the spectral determinant. We apply the theory to billiard systems, and compare the periodic orbit quantization including the first $ħ$ contribution to the exact quantum mechanical results.

The paper deals with the dynamics of a semiclassical system under the influence of the environment. The effect of the environment is shown to convert the quantum dynamics of the system into the classical one. The condition of correspondence between quantum and classical dynamics is obtained and checked numerically.

We describe a numerical procedure that clearly indicates whether or not a given statistical mechanical system is solvable (in the sense of being expressible in terms of $D$-finite functions). If the system is not solvable in this sense, any solution that exists must be expressible in terms of functions that possess a natural boundary. We provide compelling evidence that the susceptibility of the two-dimensional Ising model, the generating function of square lattice self-avoiding walks and polygons and of hexagonal lattice polygons, and directed animals are in the “unsolvable” class.

We introduce a self-organized critical model of punctuated equilibrium with many internal degrees of freedom ( $M$) per site. We find exact solutions for $M\to \infty$ of cascade equations describing avalanche dynamics in the steady state. This proves the existence of simple power laws with critical exponents that verify general scaling relations for nonequilibrium phenomena. Punctuated equilibrium is described by a devil's staircase with a characteristic exponent ${\tau }_{\mathrm{first}}=2-d/4$ where $d$ is the spatial dimension.

We study the interference of two Bose-Einstein condensates within an elementary model. The detection of the atoms is modeled by adapting the standard theory of photon detection. Even though the condensates are taken to be in number states with no phases whatsoever, our stochastic simulations of atom detection produce interference patterns as would also be predicted on the basis of the phases of the macroscopic wave functions describing the condensates. In statistical mechanics terms, we have devised a method to analyze spontaneous symmetry breaking for an arbitrary (not necessarily larger) number of particles.

Using the Thomas-Fermi approximation, we show that an interacting two dimensional electron gas may be described in terms of fractional exclusion statistics at zero and finite temperatures when the interaction has a short-range component. We argue that a likely physical situation for this phenomenon to occur may exist in two dimensional quantum dots.

We have evaluated numerically the zeros of the partition function of the $q$-state Potts model on the square lattice with reduced interactions $K$. On the basis of our numerical results, we conjecture that, both for finite planar self-dual lattices and for lattices with free or periodic boundary conditions in the thermodynamic limit, the zeros in the $\mathrm{Re}\left(x\right)>0\mathrm{}$ region of the complex $x=(e^K-1\mathrm{})/\sqrt{q}$ plane are located on the unit circle $|x|=1\mathrm{}$.

We consider a directed compact site lattice animal problem on the $d$-dimensional hypercubic lattice, and establish its equivalence with (i) the infinite-state Potts model and (ii) the enumeration of $\left(d-1\mathrm{}\right)$-dimensional restricted partitions of an integer. The directed compact lattice animal problem is solved exactly in $d=2,3\mathrm{}$ using known solutions of the enumeration problem. The maximum number of lattice animals of size $n$ grows as $\exp \left({\mathrm{cn}}^{\left(d-1\mathrm{}\right)/d}\right)$. Also, the infinite-state Potts model solution leads to a conjectured limiting form for the generating function of restricted partitions for $d>\mathrm{}3\mathrm{}$, the latter an unsolved problem in number theory.

A Comment on the Letter by D. P. Snowden-Ifft, E. S. Freeman, and P. B. Price, Phys. Rev. Lett. 74, 4133, (1995). The authors of the Letter offer a Reply.

A Reply to the Comment by Juan I. Collar.

We calculate the binary decision costs of different strategies for measurement of a given ensemble of $N$ independent and identically prepared polarized spin 1/2 particles. The results obtained prove that, for arbitrarily given values of the prior probabilities and an arbitrary number of constituent particles, the cost of a combined measurement is the same as that for sequential measurements of the individual particles.

We study the prospects for observing superfluidity in a spin-polarized atomic gas of ${}^6\mathrm{Li}$ atoms, using state-of-the-art interatomic potentials. We determine the spinodal line and show that a BCS transition to the superfluid state can indeed occur in the (meta)stable region of the phase diagram if the densities are sufficiently low. We also discuss the stability of the gas due to exchange and dipolar relaxation and conclude that the prospects for observing superfluidity in a magnetically trapped atomic ${}^6\mathrm{Li}$ gas are particularly promising for magnetic bias fields larger than 10 T.