The Marriage Problem and the Fate of Bachelors

In the marriage problem, a variant of the bi-parted matching problem, each member has a `wish-list' expressing his/her preference for all possible partners; this list consists of random, positive real numbers drawn from a certain distribution. One searches the lowest cost for the society, at the risk of breaking up pairs in the course of time. Minimization of a global cost function (Hamiltonian) is performed with statistical mechanics techniques at a finite fictitious temperature. The problem is generalized to include bachelors, needed in particular when the groups have different size, and polygamy. Exact solutions are found for the optimal solution (T=0). The entropy is found to vanish quadratically in $T$. Also other evidence is found that the replica symmetric solution is exact, implying at most a polynomial degeneracy of the optimal solution. Whether bachelors occur or not, depends not only on their intrinsic qualities, or lack thereof, but also on global aspects of the chance for pair formation in society.Comment: 14 pages revtex, submitted to Physica

Thermodynamic picture of the glassy state gained from exactly solvable models

A picture for thermodynamics of the glassy state was introduced recently by us (Phys. Rev. Lett. {\bf 79} (1997) 1317; {\bf 80} (1998) 5580). It starts by assuming that one extra parameter, the effective temperature, is needed to describe the glassy state. This approach connects responses of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized, non-equilibrium way. Similar universal relations do not hold between energy fluctuations and the specific heat. In the present paper the underlying arguments are discussed in greater length. The main part of the paper involves details of the exact dynamical solution of two simple models introduced recently: uncoupled harmonic oscillators subject to parallel Monte Carlo dynamics, and independent spherical spins in a random field with such dynamics. At low temperature the relaxation time of both models diverges as an Arrhenius law, which causes glassy behavior in typical situations. In the glassy regime we are able to verify the above mentioned relations for the thermodynamics of the glassy state. In the course of the analysis it is argued that stretched exponential behavior is not a fundamental property of the glassy state, though it may be useful for fitting in a limited parameter regime.Comment: revised version, 38 pages, 9 figure

Exact Schwarzschild-de Sitter black holes in a family of massive gravity models

The Schwarzschild-de Sitter and Reissner-Nordstr\"om-de Sitter black hole metrics appear as exact solutions in the recently formulated massive gravity of de Rham, Gabadadze and Tolley (dRGT), where the mass term sets the curvature scale. They occur within a two-parameter family of dGRT mass terms. They show no trace of a cloud of scalar graviton modes, and in the limit of vanishing graviton mass they go smoothly to the Schwarzschild and Reissner-Nordstr\"om metrics.Comment: 4 page

Thermodynamics of the glassy state

A picture for thermodynamics of the glassy state is introduced. It assumes that one extra parameter, the effective temperature, is needed to describe the glassy state. This explains the classical paradoxes concerning the Ehrenfest relations and the Prigogine-Defay ratio. As a second part, the approach connects the response of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized non-equilibrium way.Comment: 12 pages, including 2 figures. To appear in: 8th Tohwa University Int'l Symposium on Slow Dynamics in Complex System

A puzzle on fluctuations of weights in spin glasses

In certain mean field models for spin glasses there occurs a one step replica symmetry breaking pattern. As an example of general $1/N$-corrections in such systems, the fluctuations in the internal energy are calculated. For this specific quantity the outcome is known from the specific heat. It is shown that the correct result can be derived by assuming that the both the height and the location of the breakpoint fluctuate. This effect enlarges the commonly considered space of allowed fluctuations of the order parameter in loop calculations for short range spin glasses of this type. The phenomenon does not occur in spin glasses with infinite order replica symmetry breaking.Comment: 7 pages RevTex Revised version, to appear in: Journal de Physique (January 96

Classical Phase Space Density for the Relativistic Hydrogen Atom

Quantum mechanics is considered to arise from an underlying classical structure (``hidden variable theory'', ``sub-quantum mechanics''), where quantum fluctuations follow from a physical noise mechanism. The stability of the hydrogen ground state can then arise from a balance between Lorentz damping and energy absorption from the noise. Since the damping is weak, the ground state phase space density should predominantly be a function of the conserved quantities, energy and angular momentum. A candidate for this phase space density is constructed for ground state of the relativistic hydrogen problem of a spinless particle. The first excited states and their spherical harmonics are also considered in this framework. The analytic expression of the ground state energy can be reproduced, provided averages of certain products are replaced by products of averages. This analysis puts forward that quantum mechanics may arise from an underlying classical level as a slow variable theory, where each new quantum operator relates to a new, well separated time interval.Comment: 15pages AIP tex with 1 Figur

Ginzburg-Landau theory of the cluster glass phase

On the basis of a recent field theory for site-disordered spin glasses a Ginzburg-Landau free energy is proposed to describe the low temperatures glassy phase(s) of site-disordered magnets. The prefactors of the cubic and dominant quartic terms change gradually along the transition line in the concentration-temperature phase diagram. Either of them may vanish at certain points $(c_*, T_*)$, where new transition lines originate. The new phases are classifiedComment: 6 pages Revtex, 5 figures. To appear in J. Phys. A. Let