Decoherence produces coherent states: an explicit proof for harmonic chains

We study the behavior of infinite systems of coupled harmonic oscillators as t->infinity, and generalize the Central Limit Theorem (CLT) to show that their reduced Wigner distributions become Gaussian under quite general conditions. This shows that generalized coherent states tend to be produced naturally. A sufficient condition for this to happen is shown to be that the spectral function is analytic and nonlinear. For a rectangular lattice of coupled oscillators, the nonlinearity requirement means that waves must be dispersive, so that localized wave-packets become suppressed. Virtually all harmonic heat-bath models in the literature satisfy this constraint, and we have good reason to believe that coherent states and their generalizations are not merely a useful analytical tool, but that nature is indeed full of them. Standard proofs of the CLT rely heavily on the fact that probability densities are non-negative. Although the CLT generally fails if the probability densities are allowed to take negative values, we show that a CLT does indeed hold for a special class of such functions. We find that, intriguingly, nature has arranged things so that all Wigner functions belong to this class.Comment: Final published version. 17 pages, Plain TeX, no figures. Online at http://astro.berkeley.edu/~max/gaussians.html (faster from the US), from http://www.mpa-garching.mpg.de/~max/gaussians.html (faster from Europe) or from [email protected]

Relating the thermodynamic arrow of time to the causal arrow

Consider a Hamiltonian system that consists of a slow subsystem S and a fast subsystem F. The autonomous dynamics of S is driven by an effective Hamiltonian, but its thermodynamics is unexpected. We show that a well-defined thermodynamic arrow of time (second law) emerges for S whenever there is a well-defined causal arrow from S to F and the back-action is negligible. This is because the back-action of F on S is described by a non-globally Hamiltonian Born-Oppenheimer term that violates the Liouville theorem, and makes the second law inapplicable to S. If S and F are mixing, under the causal arrow condition they are described by microcanonic distributions P(S) and P(S|F). Their structure supports a causal inference principle proposed recently in machine learning.Comment: 10 page

Exploratory study of ultraviolet B (UVB) radiation and age of onset of bipolar disorder

Background: Sunlight contains ultraviolet B (UVB) radiation that triggers the production of vitamin D by skin. Vitamin D has widespread effects on brain function in both developing and adult brains. However, many people live at latitudes (about > 40 N or S) that do not receive enough UVB in winter to produce vitamin D. This exploratory study investigated the association between the age of onset of bipolar I disorder and the threshold for UVB sufficient for vitamin D production in a large global sample. Methods: Data for 6972 patients with bipolar I disorder were obtained at 75 collection sites in 41 countries in both hemispheres. The best model to assess the relation between the threshold for UVB sufficient for vitamin D production and age of onset included 1 or more months below the threshold, family history of mood disorders, and birth cohort. All coefficients estimated at P ≤ 0.001. Results: The 6972 patients had an onset in 582 locations in 70 countries, with a mean age of onset of 25.6 years. Of the onset locations, 34.0% had at least 1 month below the threshold for UVB sufficient for vitamin D production. The age of onset at locations with 1 or more months of less than or equal to the threshold for UVB was 1.66 years younger. Conclusion: UVB and vitamin D may have an important influence on the development of bipolar disorder. Study limitations included a lack of data on patient vitamin D levels, lifestyles, or supplement use. More study of the impacts of UVB and vitamin D in bipolar disorder is needed to evaluate this supposition

Variations in seasonal solar insolation are associated with a history of suicide attempts in bipolar I disorder

Background: Bipolar disorder is associated with circadian disruption and a high risk of suicidal behavior. In a previous exploratory study of patients with bipolar I disorder, we found that a history of suicide attempts was associated with differences between winter and summer levels of solar insolation. The purpose of this study was to confirm this finding using international data from 42% more collection sites and 25% more countries. Methods: Data analyzed were from 71 prior and new collection sites in 40 countries at a wide range of latitudes. The analysis included 4876 patients with bipolar I disorder, 45% more data than previously analyzed. Of the patients, 1496 (30.7%) had a history of suicide attempt. Solar insolation data, the amount of the sun’s electromagnetic energy striking the surface of the earth, was obtained for each onset location (479 locations in 64 countries). Results: This analysis confirmed the results of the exploratory study with the same best model and slightly better statistical significance. There was a significant inverse association between a history of suicide attempts and the ratio of mean winter insolation to mean summer insolation (mean winter insolation/mean summer insolation). This ratio is largest near the equator which has little change in solar insolation over the year, and smallest near the poles where the winter insolation is very small compared to the summer insolation. Other variables in the model associated with an increased risk of suicide attempts were a history of alcohol or substance abuse, female gender, and younger birth cohort. The winter/summer insolation ratio was also replaced with the ratio of minimum mean monthly insolation to the maximum mean monthly insolation to accommodate insolation patterns in the tropics, and nearly identical results were found. All estimated coefficients were significant at p < 0.01. Conclusion: A large change in solar insolation, both between winter and summer and between the minimum and maximum monthly values, may increase the risk of suicide attempts in bipolar I disorder. With frequent circadian rhythm dysfunction and suicidal behavior in bipolar disorder, greater understanding of the optimal roles of daylight and electric lighting in circadian entrainment is needed