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A new class of exactly solvable Frenkel-Kontorova models is studied. For any fixed irrational winding number w, we find examples of nonrecurrent minimum-energy configurations, the existence of which has hitherto been in doubt. Such incommensurate defects nucleate a «devil's-staircase» type of ground-state phase transitions, corresponding to discontinuous transformations of cantori in the associated area-preserving maps. These minimizing cantori may have several independent orbits of gaps and are accompanied by an infinity of metastable cantori with the same w

The probabilistic element of quantum theory can be combined with the unitary time evolution of Schr\"odinger's equation in a natural and consistent way using the idea of a quantum trajectory, the quantum analog of the trajectory traced out in phase space as a function of time by a point representing the state of a closed classical system. A family of quantum trajectories can be defined using bases for the quantum Hilbert space at different times chosen so that an appropriate noninterference condition, related to the Gell-Mann and Hartle notion of medium decoherence, is satisfied. The result is a generalization of the consistent histories approach to quantum mechanics.

It is shown that certain types of position-space, or "cell," renormalization-group transformations from an Ising-model object Hamiltonian onto another Ising-model image Hamiltonian are well defined and smooth in the thermodynamic limit, provided the object Hamiltonian has a large enough magnetic field. In certain other cases there is evidence (though not a rigorous proof) that no thermodynamic limit exists, or, if it exists, the transformation is not smooth.

A new method, involving the solution of a nonlinear eigenvalue equation, is proposed for finding the ground-state configurations and energies of the Frenkel-Kontorova and similar onedimensional models which can show complicated transitions between commensurate and incommensurate phases. It is used to show that a distinctive phase diagram can arise if a cosine potential is perturbed by the addition of suitable harmonics.

The two-fluid critical mixing point in ${\mathrm{He}}^{3}$ - ${\mathrm{He}}^{4}$ differs from ordinary critical points in that it occurs at the intersection of three lines of critical points, in a suitable variable space. A free-energy function is proposed which removes certain discrepancies between classical (Landau) theory and experimental thermodynamic measurements. Certain solid-state transitions (e.g., the metamagnetic-antiferromagnetic transition in Fe${\mathrm{Cl}}_{2}$) are thermodynamic analogs of critical mixing in ${\mathrm{He}}^{3}$ - ${\mathrm{He}}^{4}$.

It is shown that in a class of randomly diluted Ising ferromagnets the magnetization fails to be an analytic function of the field $H$ at $H=0$ for a range of temperatures above that at which spontaneous magnetization first appears.