Extension of Information Geometry to Non-statistical Systems: Some Examples

Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.Comment: 8 pages, to be published in the proceedings of GSI2015, Lecture Notes in Computer Science, Springe

Grand canonical ensemble in generalized thermostatistics

We study the grand-canonical ensemble with a fluctuating number of degrees of freedom in the context of generalized thermostatistics. Several choices of grand-canonical entropy functional are considered. The ideal gas is taken as an example.Comment: 14 pages, no figure

The q-exponential family in statistical physics

The notion of generalised exponential family is considered in the restricted context of nonextensive statistical physics. Examples are given of models belonging to this family. In particular, the q-Gaussians are discussed and it is shown that the configurational probability distributions of the microcanonical ensemble belong to the q-exponential family.Comment: 18 pages, 4 figures, proceedings of SigmaPhi 200

Covariance systems

We introduce new definitions of states and of representations of covariance systems. The GNS-construction is generalized to this context. It associates a representation with each state of the covariance system. Next, states are extended to states of an appropriate covariance algebra. Two applications are given. We describe a nonrelativistic quantum particle, and we give a simple description of the quantum spacetime model introduced by Doplicher et al.Comment: latex with ams-latex, 23 page

Data Set Models and Exponential Families in Statistical Physics and Beyond

The exponential family of models is defined in a general setting, not relying on probability theory. Some results of information geometry are shown to remain valid. Exponential families both of classical and of quantum mechanical statistical physics fit into the new formalism. Other less obvious applications are predicted. For instance, quantum states can be modeled as points in a classical phase space and the resulting model belongs to the exponential family

Methane Fluxes from a High Intensity Seep Area west of Crimea, Black Sea

Methane seepage is a wide-spread phenomenon in the Black Sea with an increase in density and intensity west of the Crimea in the Paleo Dnepr area between 70 and 250m water depth. Within the EU funded project CRIMEA we studied the impact of high intensity seeps on the methane distribution in the water column and its possible transport into the atmosphere. Here we present data which allow flux calculations of free methane from an area of 1 by 1.23 miles between 80 and 95m water depth. Our calculations are based on direct and hydroacoustic flux measurements of single seeps or small-scaled seep areas (several m2); the spatial extrapolation of these fluxes use the very strong correlation between the bubble seep occurrence and a high backscattering seafloor; the temporal variability of bubble release was detected via the lander-based hydroacoustic system GasQuant.More than 1000 bubbling seep sites were identified during two cruises in 2003 and 2004 by hydroacoustic water column surveys. The hydroacoustic detection of bubbles uses the strong backscattering of the free gas phase caused by the great impedance difference of bubbles in water (equivalent to the detection of fish and their swim bladder). In echograms, bubble streams or even single bubbles can be detected, traced and used for special analyses such as bubble rising speed, bubble size and shrinking rates. Because of the flare-like appearance of bubble streams in echograms we call these features ’flares’.Parallel multi beam mapping allowed the detection of the seafloor morphology together with the spatial backscatter intensity of the seafloor. The combination of flare occurrences with high backscatter areas provided a very good correlation. Normalized, the backscatter ranged from -12.5 to 7.1 dB for an area of 4.23 km2. All seep positions plott in areas with more than -2.7 dB, which is almost the entire area of investigation (95.8 %). However, 75% of the flares occur within only 20.1% of the area, half of the flares occur in only 9.2% and 25% even occur in only 3.8% of the area with backscattering values above 2.4 dB. This correlation allows to predict and extrapolate active bubble seeps even without direct or hydroacoustic observations.One reason for the high backscattering seafloor are patches of carbonate cemented seafloor (formed via AOM) which typically occurs just below bright white Beggiatoa mats. In addition, high resolution seismic studies with a 5kHz sub-bottom profiler clearly show a shallow gas front in normally 3m sediment depth. In those areas where strong gas front reflectors dome up and reach the seafloor surface the backscatter values and flare density are the highest. This clearly shows that the bubbles released are fed from shallow gas which also might have an impact on the physical properties of the seafloor and its backscatter behaviour. Seeps in lower or even very low backscatter areas possibly indicate a rather young or weak activity which did not (so far) cause a remarkable carbonate cementation detectable during multi beam surveys.However, the backscatter data are the base for our spatial flux calculations which use direct bubble trapping to distinguish the flux rate from one single seep hole and hydroacoustic methods for small seep areas of several m2. Direct bubble flux measurements were performed with the submersible JAGO by trapping the bubbles with a funnel. Fluxes vary between 0.55 and 1.44 ml/s (or 1.98 to 5.18 l/h at in situ volume; or 0.24 to 0.64 mmol/s). Subsequent GC-based gas analyses onboard confirmed that the gas phase consists exclusively of methane. Visual observations by JAGO and towed camera systems showed bubble diameters between 1 and 15mm with typical sizes between 3 and 7mm. Together with bubble rising speeds of typically 25cm/s both attributes are in very good agreement with detailed hydroacoustic measurements using a dual frequency scientific echo sounder EK500 (120 and 38kHz). Flux estimat

A generalized quantum microcanonical ensemble

We discuss a generalized quantum microcanonical ensemble. It describes isolated systems that are not necessarily in an eigenstate of the Hamilton operator. Statistical averages are obtained by a combination of a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.Comment: improved version, new titl

Ensemble averaged entanglement of two-particle states in Fock space

Recent results, extending the Schmidt decomposition theorem to wavefunctions of identical particles, are reviewed. They are used to give a definition of reduced density operators in the case of two identical particles. Next, a method is discussed to calculate time averaged entanglement. It is applied to a pair of identical electrons in an otherwise empty band of the Hubbard model, and to a pair of bosons in the the Bose-Hubbard model with infinite range hopping. The effect of degeneracy of the spectrum of the Hamiltonian on the average entanglement is emphasised.Comment: 19 pages Latex, changed title, references added in the conclusion

A one-dimensional model for theoretical analysis of single molecule experiments

In this paper we compare two polymer stretching experiments. The outcome of both experiments is a force-extension relation. We use a one-dimensional model to show that in general the two quantities are not equal. In certain limits, however, both force-extension relations coincide.Comment: 11 pages, 5 figure