Statistical mechanics of the discrete nonlinear Schrödinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of $T=\infty$, we identify a phase transition through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breatherlike localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D $\mathrm{XY}$ model. We demonstrate that this function describes the fluctuations of global quantities in other correlated equilibrium and nonequilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sandpiles, avalanches, and granular media in a self-organized critical state. We discuss the relationship with both Gaussian and extremal statistics.

A Comment on the Letter by Qi-Kun Xue, et al., Phys. Rev. Lett. 82, 3074 (1999). The authors of the Letter offer a Reply.

A Reply to the Comment by V. Ramachandran, et al.

Nonlocal twist operators are introduced for the $\mathrm{O}\left(n\right)$ and $Q$-state Potts models in two dimensions which count the numbers of self-avoiding loops (respectively, percolation clusters) surrounding a given point. Their scaling dimensions are computed exactly. This yields many results: for example, the number of percolation clusters which must be crossed to connect a given point to an infinitely distant boundary. Its mean behaves as $\left(1/3\sqrt{3}\pi \right)|\ln (p_c-p)|$ as $p\to p_c-$. As an application we compute the exact value $\sqrt{3}/2$ for the conductivity at the spin Hall transition, as well as the shape dependence of the mean conductance in an arbitrary simply connected geometry with two extended edge contacts.

We present a classification scheme for phase transitions in finite systems like atomic and molecular clusters based on the Lee-Yang zeros in the complex temperature plane. In the limit of infinite particle numbers the scheme reduces to the Ehrenfest definition of phase transitions and gives the right critical indices. We apply this classification scheme to Bose-Einstein condensates in a harmonic trap as an example of a higher order phase transition in a finite system and to small Ar clusters.

Excited states and excitation energies of weakly bound systems, e.g., atomic few-body systems and clusters, are difficult to study experimentally. For this purpose we propose a new and very general atom-optical method which is based on inelastic diffraction from transmission gratings. The technique is applicable to the recently found helium trimer molecule ${}^4{\mathrm{He}}_3$, allowing for the first time an investigation of the possible existence of an excited trimer state and determination of its excitation energy. This would be of fundamental importance for the famous Efimov effect.

Calculated Hugoniots and 293-K isotherms at pressures up to 1 TPa (10 Mbar) for the five reference metals Al, Cu, Ta, Mo, and W are reported using the classical mean-field approach where both the cold and the thermal parts of the Helmholtz free energy are derived entirely from the 0-K total energies and electronic density of states calculated with the full-potential linearized augmented plane wave method within the generalized gradient approximation to exchange correlational functional. Our approach permits efficient computation and invokes no empirical parameters. Both the experimental Hugoniots and 293-K isotherms are reproduced excellently.

We present one-dimensional numerical simulations reporting the temperature evolution of a pure fluid subjected to heating near its liquid-vapor critical point under weightlessness. In this model, thermal boundary conditions are imposed at the outer edges of the solids in contact with the fluid. Our investigations concern the thermal conditions at the edges of the fluid and their consequences on the fluid's global response. The results for piston effect heating are shown to be significantly affected by the simulation of the solid boundaries. Concerning critical speeding up, it is even found that taking conductive solids into account can make the bulk fluid temperature change in a way opposed to that predicted in their absence.

In the limit of infinite yield time for stresses, the hydrodynamic equations for viscoelastic, non-Newtonian liquids such as polymer melts must reduce to that for solids. This piece of information suffices to uniquely determine the nonlinear convective derivative, an ongoing point of contention in the rheology literature.

We demonstrate a versatile new technique that provides a phase coherent link between optical frequencies and the radio frequency domain. The regularly spaced comb of modes of a mode-locked femtosecond laser is used as a precise ruler to measure a large frequency gap between two different multiples (harmonics or subharmonics) of a laser frequency. In this way, we have determined a new value of the hydrogen $1S-2S$ two-photon resonance, $f_{1S-2S}=2\mathrm{}466061413187.29\left(37\right)\mathrm{kHz}$, representing now the most accurate measurement of an optical frequency.

We find exact localized time-periodic solutions with frequencies inside the linearized spectrum [intraband discrete breathers (IDBs)] in random nonlinear models using a new self-consistent method. The IDB frequencies belong to intervals between forbidden gaps generated by resonances with the linear modes, becoming fat Cantor sets in infinite systems. When localized IDBs are continued versus frequency, they delocalize and become multisite IDBs (not predicted by existing theorems), which can propagate energy. Some implications for energy relaxation in glasses are discussed.

We experimentally investigate the escape from a metastable state over a fluctuating barrier of a physical system. The system is switching between two states under electronic control of a dichotomous noise. We measure the escape time and its probability density function as a function of the correlation rate of the dichotomous noise in a frequency interval spanning more than four frequency decades. We observe resonant activation, namely a minimum of the average escape time as a function of the correlation rate. We detect two regimes in the study of the shape of the escape time probability distribution: (i) a regime of exponential and (ii) a regime of nonexponential probability distribution.

The scaling behavior of cyclical surface growth (e.g., deposition/desorption), with the number of cycles, $n$, is investigated. The roughness of surfaces grown by two linear primary processes follows a scaling behavior with asymptotic exponents inherited from the dominant process while the effective amplitudes are determined by both. Relevant nonlinear effects in the primary processes may remain so or be rendered irrelevant. Numerical simulations for several pairs of generic primary processes confirm these conclusions. Experimental results for the surface roughness during cyclical electrodeposition/dissolution of silver show a power-law dependence on $n$, consistent with the scaling description.

We show that quantum mechanics predicts a contradiction with local hidden variable theories for photon number measurements which have limited resolving power, to the point of imposing an uncertainty in the photon number result which is macroscopic in absolute terms. We show how this can be interpreted as a failure of a new premise, macroscopic local realism.

The nonextensivity parameter $q$ occurring in some of the applications of Tsallis statistics (known also as index of the corresponding Lévy distribution) is shown to be given, in the $q>\mathrm{}1\mathrm{}$ case, entirely by the fluctuations of the parameters of the usual exponential distribution.

We report upon the numerical computation of the Euler characteristic $\chi$ (a topologic invariant) of the equipotential hypersurfaces ${\Sigma }_v$ of the configuration space of the two-dimensional lattice ${\varphi }^4$ model. The pattern $\chi \left({\Sigma }_v\right)$ versus $v$ (potential energy) reveals that a major topology change in the family $\{{\Sigma }_v{\}}_{v\in R}$ is at the origin of the phase transition in the model considered. The direct evidence given here—of the relevance of topology for phase transitions—is obtained through a general method that can be applied to any other model.

We discuss Bose-Einstein condensation (BEC) in quasi-2D trapped gases and find that well below the transition temperature $T_c$ the equilibrium state is a true condensate, whereas at intermediate temperatures $T<\mathrm{}{T}_c$ one has a quasicondensate (condensate with fluctuating phase). The mean-field interaction in a quasi-2D gas is sensitive to the frequency ${\omega }_0$ of the (tight) confinement in the “frozen” direction, and one can switch the sign of the interaction by changing ${\omega }_0$. Variation of ${\omega }_0$ can also reduce the rates of inelastic processes. This offers promising prospects for tunable BEC in trapped quasi-2D gases.

Fluctuational escape from a multiwell potential is shown to display new features, as compared to the conventional single-well case. The flux $J$ may depend on friction $\Gamma$ exponentially strongly, over an exponentially long period; for small enough temperatures, $J\left(\Gamma \right)$ undergoes marked oscillations in the range of small $\Gamma$, and the time evolution of $J$ changes drastically as $\Gamma$ exceeds a critical value.

We demonstrate, by construction, that simple renormalizable matrix potentials with ${\mathrm{S}}_N$, as opposed to $\mathrm{O}\left(N\right)$, symmetry can exhibit an exponentially large number of inequivalent deep local minima.