Geometric phases in quantum control disturbed by classical stochastic processes

We describe the geometric (Berry) phases arising when some quantum systems are driven by control classical parameters but also by outer classical stochastic processes (as for example classical noises). The total geometric phase is then divided into an usual geometric phase associated with the control parameters and a second geometric phase associated with the stochastic processes. The geometric structure in which these geometric phases take place is a composite bundle (and not an usual principal bundle), which is explicitely built in this paper. We explain why the composite bundle structure is the more natural framework to study this problem. Finally we treat a very simple example of a two level atom driven by a phase modulated laser field with a phase instability described by a gaussian white noise. In particular we compute the average geometric phase issued from the noise

Exotic Smoothness and Physics

The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space-time models, especially the simplest one, ${\bf R^4}$, possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structures available for space-time models. These can be thought of as source of new global, but not properly topological, features. This paper reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory, recalling that diffeomorphisms are at the heart of the principle of general relativity. Some of the history of the discovery of exotic, i.e., non-standard, differentiable structures is reviewed. Some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories.Comment: 13 pages, LaTe

Natural extensions and entropy of α\alpha-continued fractions

We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy)^{-2}$. In particular, we show that, for all $0 < \alpha \le 1$, the product of the entropy with the measure of the domain equals $\pi^2/6$. As a key step, we give the explicit relationship between the $\alpha$-expansion of $\alpha-1$ and of $\alpha$

Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent

One dimensional intermittent maps with stretched exponential separation of nearby trajectories are considered. When time goes infinity the standard Lyapunov exponent is zero. We investigate the distribution of $\lambda_{\alpha}= \sum_{i=0}^{t-1} \ln \left| M'(x_i) \right|/t^{\alpha}$, where $\alpha$ is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of $\lambda_{\alpha}$ is determined by the infinite invariant density. Using semi analytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it obtain excellent agreement between numerical simulation and theory. We show that \alpha \left is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that \left and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.Comment: 12 pages, 10 figure

Pesin-type relation for subexponential instability

We address here the problem of extending the Pesin relation among positive Lyapunov exponents and the Kolmogorov-Sinai entropy to the case of dynamical systems exhibiting subexponential instabilities. By using a recent rigorous result due to Zweim\"uller, we show that the usual Pesin relation can be extended straightforwardly for weakly chaotic one-dimensional systems of the Pomeau-Manneville type, provided one introduces a convenient subexponential generalization of the Kolmogorov-Sinai entropy. We show, furthermore, that Zweim\"uller's result provides an efficient prescription for the evaluation of the algorithm complexity for such systems. Our results are confirmed by exhaustive numerical simulations. We also point out and correct a misleading extension of the Pesin relation based on the Krengel entropy that has appeared recently in the literature.Comment: 10 pages, 4 figures. Final version to appear in Journal of Statistical Mechanics (JSTAT

The entropy of alpha-continued fractions: numerical results

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.Comment: 33 pages, 14 figure

Identification of the Major Expressed S-Layer and Cell Surface-Layer-Related Proteins in the Model Methanogenic Archaea: Methanosarcina barkeri Fusaro and Methanosarcina acetivorans C2A

Many archaeal cell envelopes contain a protein coat or sheath composed of one or more surface exposed proteins. These surface layer (S-layer) proteins contribute structural integrity and protect the lipid membrane from environmental challenges. To explore the species diversity of these layers in the Methanosarcinaceae, the major S-layer protein in Methanosarcina barkeri strain Fusaro was identified using proteomics. The Mbar_A1758 gene product was present in multiple forms with apparent sizes of 130, 120, and 100 kDa, consistent with post-translational modifications including signal peptide excision and protein glycosylation. A protein with features related to the surface layer proteins found in Methanosarcina acetivorans C2A and Methanosarcina mazei Goel was identified in the M. barkeri genome. These data reveal a distinct conserved protein signature with features and implied cell surface architecture in the Methanosarcinaceae that is absent in other archaea. Paralogous gene expression patterns in two Methanosarcina species revealed abundant expression of a single S-layer paralog in each strain. Respective promoter elements were identified and shown to be conserved in mRNA coding and upstream untranslated regions. Prior M. acetivorans genome annotations assigned S-layer or surface layer associated roles of eighty genes: however, of 68 examined none was significantly expressed relative to the experimentally determined S-layer gene

Pyrophosphate-Dependent ATP Formation from Acetyl Coenzyme A in Syntrophus aciditrophicus, a New Twist on ATP Formation.

UnlabelledSyntrophus aciditrophicus is a model syntrophic bacterium that degrades key intermediates in anaerobic decomposition, such as benzoate, cyclohexane-1-carboxylate, and certain fatty acids, to acetate when grown with hydrogen-/formate-consuming microorganisms. ATP formation coupled to acetate production is the main source for energy conservation by S.&nbsp;aciditrophicus However, the absence of homologs for phosphate acetyltransferase and acetate kinase in the genome of S.&nbsp;aciditrophicus leaves it unclear as to how ATP is formed, as most fermentative bacteria rely on these two enzymes to synthesize ATP from acetyl coenzyme A (CoA) and phosphate. Here, we combine transcriptomic, proteomic, metabolite, and enzymatic approaches to show that S.&nbsp;aciditrophicus uses AMP-forming, acetyl-CoA synthetase (Acs1) for ATP synthesis from acetyl-CoA. acs1 mRNA and Acs1 were abundant in transcriptomes and proteomes, respectively, of S.&nbsp;aciditrophicus grown in pure culture and coculture. Cell extracts of S.&nbsp;aciditrophicus had low or undetectable acetate kinase and phosphate acetyltransferase activities but had high acetyl-CoA synthetase activity under all growth conditions tested. Both Acs1 purified from S.&nbsp;aciditrophicus and recombinantly produced Acs1 catalyzed ATP and acetate formation from acetyl-CoA, AMP, and pyrophosphate. High pyrophosphate levels and a high AMP-to-ATP ratio (5.9 ± 1.4) in S.&nbsp;aciditrophicus cells support the operation of Acs1 in the acetate-forming direction. Thus, S.&nbsp;aciditrophicus has a unique approach to conserve energy involving pyrophosphate, AMP, acetyl-CoA, and an AMP-forming, acetyl-CoA synthetase.ImportanceBacteria use two enzymes, phosphate acetyltransferase and acetate kinase, to make ATP from acetyl-CoA, while acetate-forming archaea use a single enzyme, an ADP-forming, acetyl-CoA synthetase, to synthesize ATP and acetate from acetyl-CoA. Syntrophus aciditrophicus apparently relies on a different approach to conserve energy during acetyl-CoA metabolism, as its genome does not have homologs to the genes for phosphate acetyltransferase and acetate kinase. Here, we show that S.&nbsp;aciditrophicus uses an alternative approach, an AMP-forming, acetyl-CoA synthetase, to make ATP from acetyl-CoA. AMP-forming, acetyl-CoA synthetases were previously thought to function only in the activation of acetate to acetyl-CoA

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n} f(T^{i_1}x,...,T^{i_d}x),\, n=1,2,...,$$ where $f$ (called the \emph{kernel}) is a function from $X^d$ to $\R$ and $C_1, C_2,...$ are appropriate normalizing constants. We observe that the above random variables are well defined and belong to $L_r(\mu)$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,\mathcal F_1, \mu_1) \otimes_{\pi}...\otimes_{\pi} L_p(X_d,\mathcal F_d, \mu_d)\subset L_p(\mu^d)$$ with $p=d\,r,\, r\ \in [1, \infty).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form $\sum_{m=1}^{\infty} \lambda_m\eta^2_m$ in independent standard Gaussian variables $\eta_1, \eta_2,...$. Our results on the distributional convergence use a $T$--\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations

Exotic Differentiable Structures and General Relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as $S^7$, \R and $S^3\times {\bf R^1}.$ Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.Comment: 11 pages, LaTe