The mean escape time $\overline{T}$ for thermal diffusion in a fluctuating metastable potential is investigated. A new type of “resonant activation” (minimum of $\overline{T}$ as a function of the correlation time $\tau$ of the potential fluctuations) is predicted analytically, explained in simple terms, and confirmed numerically. In its pure form it generically occurs for potentials without fluctuations near the barrier and the well. The effect is dominated by a minimum in the exponentially leading Arrhenius factor at a correlation time $\tau$ that stays finite for asymptotically small thermal and potential fluctuations.

We derive the universal geometrical exponents of contour loops on equilibrium rough surfaces, using analytical scaling arguments (confirmed numerically): the fractal dimension $D_f$, the distribution of contour lengths, and the probability that two points are connected by a contour. This is sufficient to calculate exact critical exponents in certain nontrivial two-dimensional spin models that can be mapped to interface models. The novel scaling relation between $D_f$ and the roughness exponent that we find can be used to analyze scanning tunneling microscopy images of rough metal surfaces.

Zero temperature phase transitions in the quantum rotor model of disordered, frustrated, and unfrustrated $n$-component short-range-coupled spin systems are investigated in the spherical limit, $n\to \infty$. We find a transition to a spin glass ordered phase in three dimensions but not in two. We map out the phase diagram as a function of frustration, finding that the ferromagnetic and spin glass critical lines meet at a bicritical point which also terminates a first order line separating the spin glass and ferromagnetic ordered phases. Finite size scaling of numerically obtained exact solutions on finite lattices is used to estimate critical exponents.

Using a liquid-state approach based on Ornstein-Zernike equations, we study the behavior of a fluid inside a porous disordered matrix near the liquid-gas critical point. The results obtained within various standard approximation schemes such as lowest-order $\gamma$ ordering and the mean-spherical approximation suggest that the critical behavior is closely related to that of the random-field Ising model.

A tomographical scheme is proposed to infer the quantum states of finite–dimensional systems from experiments. For this a new discrete Wigner formalism is developed.

It is shown that a series of repeated weak quantum nondemolition measurements performed on a single quantum system gives no information about the wave function of the system. The physical explanation, based on the quantum Brownian motion and the continuous collapse of the wave function which originate in the projection postulate, is discussed in two specific examples.

The quasiclassical approximation to the scattering amplitude is expressed, for the first time, as a ratio of absolutely convergent series. We exploit the Fredholm theory of integral equations and the result is shown to be a generalization and simplification of techniques used in resumming the Gutzwiller trace formula. A numerical example is given which displays rapid convergence of the Fredholm series for the scattering amplitude. The method is not restricted to hard chaos or to cases where the curvature expansion is good, but it naturally exploits these simplifications.

Simulations of the two dimensional versions of a ballistic coalescence (or aggregation) model show significant deviations from the predictions of a mean-field scaling analysis. The growth exponents vary with density, due to multiple coalescence events. Kinetic theory combined with a self-similarity assumption leads to a hyperscaling relation between the exponents associated with the mass and the kinetic energy.

We study random-matrix ensembles with a non-Gaussian probability distribution $P\left(H\right)\sim \exp [-N\mathrm{tr}V(H\left)\right]$, where $N$ is the dimension of the matrix $H$ and $V\left(H\right)$ is independent of $N$. Using Efetov's supersymmetry formalism, we show that in the limit $N\to \infty$ both energy level correlation functions and correlation functions of $S$-matrix elements are independent of $P\left(H\right)$ and hence universal on the scale of the local mean level spacing. This statement applies to each of the three generic ensembles (unitary, orthogonal, and symplectic). Universality is also found for correlation functions depending on some external parameter. Our results generalize previous work by Brezin and Zee [Nucl. Phys. B402, 613 (1993)].

We present analytical results for the average density of levels $\rho \left(\lambda \right)$ in an ensemble of random matrices, which is a generalization of the standard Gaussian unitary ensemble. The generalized ensemble contains a parameter $\mu \left(N\right)$ which is allowed to scale, in an arbitrary way, with the matrix size $N$. The results crucially depend on the behavior of $\mu \left(N\right)$. For a sublinear dependence of $\mu$ on $N$ a modified Wigner semicircle is obtained, in the large $N$ limit. For a superlinear dependence the ensemble approaches the Poissonian limit of uncorrelated levels, with a Gaussian shape for $\rho \left(\lambda \right)$. For a strictly linear dependence, i.e., when $\underset{N\to \infty }{limop}\left(\mu /N\right)=\mathrm{const}$, an intermediate situation occurs.

We study relaxation properties of the fractal time random walk model in which the waiting time distribution is given by the power law type $t^{-1-a}$. By means of theoretical analyses as well as of Monte Carlo simulations of this model, we find that the relaxation becomes anomalous in the case of $a<\mathrm{}1\mathrm{}$ where the complex susceptibility is described by the Cole-Cole form. On the other hand, the normal Debye type relaxation is observed for $a>\mathrm{}1\mathrm{}$. We also find the scaling laws both for the relaxation function and for the particle density.

Time-dependent solutions of the planar, Chern-Simons–gauged nonlinear Schrödinger equation are proved to collapse in a finite time for certain initial data including, e.g., a negative energy. Radially symmetric collapsing structures are found to blow up with a twice-logarithmic contraction rate and to self-focus with a central core whose extension is bounded in space. Finally, interaction of Chern-Simons vortices with an external magnetic field is shown to shorten the collapse time.

We use the Thomas-Fermi method to examine the thermodynamics of particles obeying Haldane exclusion statistics. Specifically, we study Calogero-Sutherland particles placed in a given external potential in one dimension. For the case of a simple harmonic potential (constant density of states), we obtain the exact one-particle spatial density and a closed form for the equation of state at finite temperature, which are both new results. We then solve the problem of particles in a $x^{2/3}$ potential (linear density of states) and show that Bose-Einstein condensation does not occur for any statistics other than bosons.

We consider the classical $\mathrm{XY}$ model [O(2) nonlinear $\sigma$ model] in a class of lattices with the (fractal) dimensions $1<D<\mathrm{}2\mathrm{}$. Berezinskii's harmonic approximation suggests that the model undergoes a phase transition in which the low temperature phase is characterized by stretched exponential decay of correlations. We prove an exponentially decaying upper bound for the two-point correlation functions at nonzero temperatures, thus excluding the possibility of such a phase transition.

Non-negative probabilities that obey the sum rules may be assigned to a much wider family of sets of histories than decohering histories. The resulting linearly positive histories avoid the highly restrictive decoherence conditions and yet give the same probabilities when those conditions apply. Thus linearly positive histories are a broad extension of decohering histories. Moreover, the resulting theory is manifestly time-reversal invariant.

Many-dimensional quantum calculations of energy flow are presented, revealing dynamical scaling in agreement with theoretical predictions. A model system of coupled anharmonic oscillators which undergoes a localization transition as a function of coupling strength was studied. For strong coupling, action diffusion is clearly seen. For near critical coupling, the average survival probability varies inversely with time but shows no simple relationship to the average extent of the spread of the wave function in action space.

We discuss periodic Schrödinger operators for a particle on a rectangular lattice of sides ${\ell }_1,{\ell }_2$. In addition to the standard ( $\delta$-type) coupling with continuous wave functions at lattice nodes, we introduce two other boundary conditions which generalize naturally the one-dimensional ${\delta }^{\prime }$ interaction and its symmetrized version; both of them can be used as models for geometric scatterers. We show that the band spectrum of these models depends on number-theoretic properties of the parameters. In particular, the $\delta$ lattice has no gaps above the threshold if ${\ell }_2/{\ell }_1$ is badly approximable by rationals and the coupling constant is small enough.

It is shown that a gas of interacting bosons with weakly nonlinear commutation relations behaves as a Fermi gas rather than a Bose gas if there is a gap in the frequency spectrum. Despite the quasifermionic behavior of such a gas (absence of Bose condensation, Fermi-like distributions, etc.) superfluidity is predicted. Some systems with such quasifermions are discussed.

We describe a new test for possible variations of the fine structure constant $\alpha$ by comparisons of rates between clocks based on hyperfine transitions in alkali atoms with different atomic number $Z$. H-maser, Cs, and Hg ${}^{+}$ clocks have a different dependence on $\alpha$ via relativistic contributions of order $(Z\alpha {)}^2$. Recent H-maser vs ${\mathrm{Hg}}^{+}$ clock comparison data improve laboratory limits on a time variation by 100-fold to give $\dot{\alpha }/\alpha \le 3.7\times 10^{-14\mathrm{}}/$yr. Future laser cooled clocks (Be ${}^{+}$, Rb, Cs, Hg ${}^{+}$, etc.), when compared, will yield the most sensitive of all tests for $\dot{\alpha }/\alpha$.

A Comment on the Letter by Boris Levitan, Phys. Rev. Lett. 72, 4057 (1994).