Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points

For the mean-field version of an important lattice-spin model due to Blume and Capel, we prove unexpected connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg-Landau polynomials. The model depends on the parameters n, beta, and K, which represent, respectively, the number of spins, the inverse temperature, and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m(beta_n,K_n) for appropriate sequences (beta_n,K_n) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as (beta_n,K_n) converges to one of these points (beta,K), m(beta_n,K_n) ~ c |beta - beta_n|^gamma --> 0. In this formula gamma is a positive constant, and c is the unique positive, global minimum point of a certain polynomial g that we call the Ginzburg-Landau polynomial. This polynomial arises as a limit of appropriately scaled free-energy functionals, the global minimum points of which define the phase-transition structure of the model. For each sequence (beta_n,K_n) under study, the structure of the global minimum points of the associated Ginzburg-Landau polynomial mirrors the structure of the global minimum points of the free-energy functional in the region through which (beta_n,K_n) passes and thus reflects the phase-transition structure of the model in that region. The properties of the Ginzburg-Landau polynomials make rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena, and the asymptotic formula for m(beta_n,K_n) makes rigorous the heuristic scaling theory of the tricritical point.Comment: 70 pages, 8 figure

Quickest Sequence Phase Detection

A phase detection sequence is a length-$n$ cyclic sequence, such that the location of any length-$k$ contiguous subsequence can be determined from a noisy observation of that subsequence. In this paper, we derive bounds on the minimal possible $k$ in the limit of $n\to\infty$, and describe some sequence constructions. We further consider multiple phase detection sequences, where the location of any length-$k$ contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences. We study the optimal trade-offs between the lengths of the sequences, and describe some sequence constructions. We compare these phase detection problems to their natural channel coding counterparts, and show a strict separation between the fundamental limits in the multiple sequence case. Both adversarial and probabilistic noise models are addressed.Comment: To appear in the IEEE Transactions on Information Theor

Unified perspective on proteins: A physics approach

We study a physical system which, while devoid of the complexity one usually associates with proteins, nevertheless displays a remarkable array of protein-like properties. The constructive hypothesis that this striking resemblance is not accidental leads not only to a unified framework for understanding protein folding, amyloid formation and protein interactions but also has implications for natural selection.Comment: 26 pages, 15 figures, to appear on Phys. Rev.

Temperature phase transition and an effective expansion parameter in the O(N)-model

The temperature phase transition in the N-component scalar field theory with spontaneous symmetry breaking is investigated in the perturbative approach. The second Legendre transform is used together with the consideration of the gap equations in the extrema of the free energy. Resummations are performed on the super daisy level and beyond. The phase transition turns out to be weakly of first order. The diagrams beyond the super daisy ones which are calculated correspond to next-to-next-to-leading order in 1/N. It is shown that these diagrams do not alter the phase transition qualitatively. In the limit N goes to infinity the phase transition becomes second order. A comparison with other approaches is done.Comment: 28 pages, 5 figures, corrected for some misprints, unnecessary section remove

Oscillatory convective modes in red giants: a possible explanation of the long secondary periods

We discuss properties of oscillatory convective modes in low-mass red giants, and compare them with observed properties of the long secondary periods (LSPs) of semi-regular red giant variables. Oscillatory convective modes are very nonadiabatic g$^{-}$ modes and they are present in luminous stars, such as red giants with \log L/{\rm L}_\odot \ga 3. Finite amplitudes for these modes are confined to the outermost nonadiabatic layers, where the radiative energy flux is more important than the convective energy flux. The periods of oscillatory convection modes increase with luminosity, and the growth times are comparable to the oscillation periods. The LSPs of red giants in the Large Magellanic Cloud (LMC) are observed to lie on a distinct period-luminosity sequence called sequence D. This sequence D period-luminosity relation is roughly consistent with the predictions for dipole oscillatory convective modes in AGB models if we adopt a mixing length of 1.2 pressure scale height ($\alpha = 1.2$). However, the effective temperature of the red-giant sequence of the LMC is consistent to models with $\alpha=1.9$, which predict periods too short by a factor of two.Comment: 7 pages, 6 figures, accepted for publication in MNRA

Differential Phase Estimation with the SeaMARC II Bathymetric Sidescan Sonar System

A maximum-likelihood estimator is used to extract differential phase measurements from noisy seafloor echoes received at pairs of transducers mounted on either side of the SeaMARC II bathymetricsidescan sonar system. Carrier frequencies for each side are about 1 kHz apart, and echoes from a transmitted pulse 2 ms long are analyzed. For each side, phase difference sequences are derived from the full complex data consisting of base-banded and digitized quadrature components of the received echoes. With less bias and a lower variance, this method is shown to be more efficient than a uniform mean estimator. It also does not exhibit the angular or time ambiguities commonly found in the histogram method used in the SeaMARC II system. A figure for the estimation uncertainty of the phasedifference is presented, and results are obtained for both real and simulated data. Based on this error estimate and an empirical verification derived through coherent ping stacking, a single filter length of 100 ms is chosen for data processing application