Consider a system rendered unstable by both quantum tunneling and thermodynamic fluctuation. The tunneling rate $\Gamma$, at temperature ${\beta }^{-1\mathrm{}}$, is related to the free energy $F$ by $\Gamma =\left(\frac{2}{\hslash }\right)\mathrm{Im}F$. However, the classical escape rate is $\Gamma =\left(\frac{\omega \beta }{\pi }\right)\mathrm{Im}F$, $-{\omega }^2$ being the negative eigenvalue at the saddle point. A general theory of metastability is constructed in which these formulas are true for temperatures, respectively, below and above $\frac{\omega \hslash }{2\pi }$ with a narrow transition region of $O\left({\hslash }^{\frac{3}{2}}\right)$.

A quantum system which can tunnel, at $T=0\mathrm{}$, out of a metastable state and whose interaction with its environment is adequately described in the classically accessible region by a phenomenological friction coefficient $\eta$, is considered. By only assuming that the environment response is linear, it is found that dissipation multiplies the tunneling probability by the factor $\exp [-\frac{A\eta {\left(\Delta q\right)}^2}{\hslash }]$, where $\Delta q$ is the "distance under the barrier" and $A$ is a numerical factor which is generally of order unity.

Starting from a genuine discrete version of the Feynman path-integral representation for the partition function, calculations have been made of the energy, specific heat, and the static density-density correlation functions for a one-dimensional lattice model at nonzero temperatures. A Monte Carlo technique has been used to calculate the temperature-dependent properties. The results are compared with exact calculations for short chains, the Hartree-Fock approximation, and a classical model.

It is shown (by means of a perturbation series) that for a class of potentials $V\left(x\right)\mathrm{}$ the stationary distribution of the solution $x\left(t\right)\mathrm{}$ of the quantum Langevin equation approaches in the weak-coupling limit ($f\to 0$) the quantum mechanical canonical distribution of the displacement of the oscillator, subject to the potential $V\left(x\right)\mathrm{}$, if and only if $E\left(t\right)\mathrm{}$ is the operator version of the purely random Gaussian process so that, in particular, higher symmetrized averages ${〈E\left(t_1\right)\cdots E\left(t_n\right)〉}_s$ are expressible in terms of pair correlations, in the usual way.

The results of a test of general relativity with use of a hydrogen-maser frequency standard in a spacecraft launched nearly vertically upward to 10 000 km are reported. The agreement of the observed relativistic frequency shift with prediction is at the 70×${10}^{-6\mathrm{}}$ level.

A new class of sum rules in nonrelativistic scattering theory is obtained. By extending the concept of the Fredholm determinant to a local form defined at each space point $x$, a function with analytic properties analogous to the Fredholm determinant, but containing more information, is found. These rules assume the form of the power-moment problem and are valid in both quantum and classical mechanics.

The ray description in the WKB (or eikonal) limit is extended to all orders in the expansion in powers of wavelength/characteristic length. The equations for the ray trajectories are given by a Hamiltonian system, and the intensity is given by a conservation law with flow along the rays, just as in the eikonal limit. The generalization requires that the Hamiltonian depends weakly on the solution, in a way which can be solved iteratively.

An argument is presented for the validity of the quadrupole formula for gravitational radiation energy loss in the far field of nearly Newtonian (e.g., binary stellar) systems. This agrument differs from earlier ones in that it determines beforehand the formal accuracy of approximation required to describe gravitationally self-interacting systems, uses the corresponding approximate equation of motion explicitly, and evaluates the appropriate asymptotic quantities by matching along the correct space-time light cones.

A new derivation of the quadrupole formulas and balance equations of general relativity for gravitationally bound systems is given which overcomes objections raised to previous derivations.

Equilibrium polymerization can be described by the $n\to 0$ limit of the $n$-vector model of magnetism in a small magnetic field. Nonclassical critical effects are predicted. The earlier theory of Tobolsky and Eisenberg is a mean-field approximation to the present theory. An application is made to the polymerization in sulfur.

The decay of an unstable state by the stochastic time of intersection of a given threshold is described. This stochastic parameter is characterized by closed moment equations exactly soluble, whereas previous approaches, in terms of stochastic amplitudes sampled at a fixed time, led to an open hierarchy of equations requiring approximate solutions. The relevance of this new description is shown by the fit with the experimental transient fluctuations of an unstable system.

The static and dynamic critical properties of a nonionic amphiphile aqueous solution showing a lower consolution point are investigated by laser-light scattering. The osmotic isothermal compressibility and the correlation length of the concentration fluctuations diverge with mean-field-theory critical exponents. The behavior of the mass diffusion constant in the hydrodynamic region is in good agreement with the predictions of the mode-mode-coupling theory.

The Raychaudhuri equation is used to analyze the effect of the Hawking radiation back reaction upon a black-hole event horizon. It is found that if the effective stress-energy tensor of the Hawking radiation has negative energy density as expected, then an evaporating black hole initially a solar mass in size must disappear in less than a second. This implies that either the evaporation process, if it occurs at all, must be quite different from what is commonly supposed, or else black-hole event horizons—and hence black holes—do not exist.

It is shown that ${\mathcal{L}}^1$-clustering equilibrium states of classical systems of point particles are translationally invariant.

Renormalization-group techniques are applied to Ising-model spins placed on the sites of several self-similar fractal lattices. The resulting critical properties are shown to vary with the (noninteger) fractal dimensionality $D$, but also with several topological factors: ramification, connectivity, lacunarity, etc. For any $D>~1$, there exist systems with both $T_c=0\mathrm{}$, and $T_c>0\mathrm{}$; hence a lower critical dimensionality is not defined. The nonvanishing values of $T_c$ and the critical exponents depend on all these factors.

The two-point function of the two-dimensional Ising model at arbitrary temperature is expressed in terms of the solution of a nonlinear partial difference equation. From this difference equation the known results for the two-point function of the Ising field theory may be regained as a special case.

It is argued that the leading corrections to scaling for planar Ising models, which occur as analytic factors, arise from the quadratic terms of the nonlinear thermal and ordering fields (rather than from irrelevant variables). This yields ${{\alpha }_{\mathrm{eff}}}^{\prime }+2\mathrm{}{\beta }_{\mathrm{eff}}+{{\gamma }_{\mathrm{eff}}}^{\prime }=2\mathrm{}$ for the effective critical exponents, with $\mathrm{no}$ leading corrections, and is confirmed by exact square-lattice results for arbitrary anisotropy $\frac{J_2}{J_1}$. For isotropic lattices the ratios of correction amplitudes and quadratic nonlinear-field terms appear universal.

The critical behavior of the $O\left(n\right)\mathrm{}$ classical Heisenberg model in $d$ dimensions close to $n=d=2\mathrm{}$ is discussed, assuming analyticity in $n$ and $d$ of the renormalization-group equations. In the ($d$,$n$) plane there is a line passing through (2,2) across which the critical exponents are nonanalytic. For $d=2\mathrm{} \mathrm{and} -2\mathrm{}<~n<~2$, a conjecture is presented for the exact form of the leading and subdominant thermal eigenvalues.

Very-low-angle Brillouin light-scattering experiments have been performed in liquid water subjected to a temperature gradient. The line intensities are unequal, and both the sign and the order of magnitude of this effect agree with recent theoretical calculations for nonequilibrium systems.

An exact coherent-state representation is given for fermion systems which avoids the Gaussian overlap approximation in the reduction of the Hill-Wheeler-Griffin integral equation to a differential equation. An application shows that the random-phase-approximation ground-state correlation energy, derived with use of the Gaussian overlap approximation, is a factor of 2 too large.