On the Non-Uniqueness of Statistical Ensembles Defining a Density Operator and a Class of Mixed Quantum States with Integrable Wigner Distribution

It is standard to assume that the Wigner distribution of a mixed quantum state consisting of square-integrable functions is a quasi-probability distribution, i.e., that its integral is one and that the marginal properties are satisfied. However, this is generally not true. We introduced a class of quantum states for which this property is satisfied; these states are dubbed “Feichtinger states” because they are defined in terms of a class of functional spaces (modulation spaces) introduced in the 1980s by H. Feichtinger. The properties of these states were studied, giving us the opportunity to prove an extension to the general case of a result due to Jaynes on the non-uniqueness of the statistical ensemble, generating a density operator.

$$\mu PT$$ statistical ensemble: systems with fluctuating energy, particle number, and volume

Within the theory of statistical ensemble, the so-called $$\mu PT$$ μ P T ensemble describes equilibrium systems that exchange energy, particles, and volume with the surrounding. General, model-independent features of volume and particle number statistics are derived. Non-analytic points of the partition function are discussed in connection with divergent fluctuations and ensemble equivalence. Quantum and classical ideal gases, and a model of Bose gas with mean-field interactions are discussed as examples of the above considerations.

Beyond the relaxation time approximation

The relaxation time approximation (RTA) is a well known method of describing the time evolution of a statistical ensemble by linking distributions of the variables of interest at different stages of their temporal evolution. We show that if all the distributions occurring in the RTA have the same functional form of a quasi-power Tsallis distribution the time evolution of which depends on the time evolution of its control parameter, nonextensivity q(t), then it is more convenient to consider only the time evolution of this control parameter.

Designing an optimal flood forecasting chain using convective-scale ensembles: a sensitivity study.

<p>Flooding is New Zealand’s most frequent natural disaster with an average annual cost of approximately NZ$51 million. Accurately forecasting convective and orographically enhanced precipitation for hydrometeorological ensemble prediction systems is challenging in Aotearoa New Zealand’s complex topographic regions with fast-responding and mostly ungauged catchments. Globally, designing convection-permitting ensemble flood forecasting chains is still a work in progress, with errors in the forecast rainfall amount and the location or timing of storm events a significant contributor to uncertainties in river flow forecasts. Given operational, computational and model representation constraints, compromises are often required on ensemble size, frequency of forecast issue times, NWP model resolution, domain size and data assimilation strategies. This research aims to design an optimal operational forecasting chain for convective-scale flood forecasting in New Zealand.  In doing so, our goal is to improve uncertainty representation in hydrometeorological forecasts during flood events by understanding the impact of convective-scale ensemble strategies.</p><p>The NWP model used is a local implementation of the UK Met Office-developed Unified Model.  The New Zealand Convective-Scale Model (NZCSM) is NIWA’s 1.5km high-resolution operational forecast model, configured such that convective processes develop explicitly. The New Zealand Ensemble (NZENS) is configured with similar convection-permitting model physics but operates with a 4.5km horizontal resolution and features up to 18 members.  Flood forecasts were produced by coupling several weather ensemble configurations with the semi-distributed hydrological model TopNet and its built-in statistical ensemble generation tool. TopNet is based on TOPMODEL concepts of runoff generation controlled by sub-surface water storage.</p><p>In this study, we evaluated three ensemble strategies for flood forecasting. The experiment design allowed for the effect of model horizontal resolution (and thus the representation of orography) to be investigated using ensemble forecasts from consecutive initialization times (a “lagged ensemble”), and from the same initialisation time (a “dynamical ensemble”). The third forecasting chain is a “statistical ensemble” generated by perturbing the deterministic 1.5km NWP model and hydrological states. For recent flood events across multiple case study catchments, we evaluated the impact of each approach on flood forecast performance. Flood forecasts were most sensitive to convective-scale forecasts with consecutive issue time initialisations (lagged ensemble) over other hydrometeorological ensemble configurations considered. Given dynamical ensembles are computationally expensive, the study suggests an optimal strategy might be to produce a small ensemble pool of dynamical forecasts at more frequent issue times combined with statistically post-processed ensembles rather than a larger ensemble pool generated less frequently.</p>

Quantum Invariance

Quantum invariance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble after measurement.A set-theory corollary is the curious invariance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. It should be equated to a well-ordered set after measurement and thus requires the axiom of choice.Quantum invariance underlies quantum information and reveals it as the relation of an unordered quantum “much” (i.e. a coherent state) and a well-ordered “many” of the measured results (i.e. a statistical ensemble). It opens up to a new horizon, in which all physical processes and phenomena can be interpreted as quantum computations realizing relevant operations and algorithms on quantum information. All phenomena of entanglement can be described in terms of the so defined quantum information.Quantum invariance elucidates the link between general relativity and quantum mechanics and thus, the problem of quantum gravity.

Monthly Reservoir Inflow Forecasting for Dry Period Using Teleconnection Indices: A Statistical Ensemble Approach

Reliable long-range reservoir inflow forecast is essential to successfully manage water supply from reservoirs. This study aims to develop statistical reservoir inflow forecast models for a reservoir watershed, based on hydroclimatic teleconnection between monthly reservoir inflow and climatic variables. Predictability of such a direct relationship has not been assessed yet at the monthly time scale using the statistical ensemble approach that employs multiple data-driven models as an ensemble. For this purpose, three popular data-driven models, namely multiple linear regression (MLR), support vector machines (SVM) and artificial neural networks (ANN) were used to develop monthly reservoir inflow forecasting models. These models have been verified using leave-one-out cross-validation with expected error S as a measure of forecast skill. The S values of the MLR model ranged from 0.21 to 0.55, the ANN model ranged from 0.20 to 0.52 and the SVM from 0.21 to 0.56 for different months. When used as an ensemble, Bayesian model averaging was more accurate than simple model averaging and naïve forecast for four target years tested. These were considered to be decent prediction skills, indicating that teleconnection-based models have the potential to be used as a tool to make a decision for reservoir operation in preparing for droughts.

Cognition according to Quantum information: Three Epistemological Puzzles Solved

The cognition of quantum processes raises a series of questions about ordering and information connecting the states of one and the same system before and after measurement: Quantum measurement, quantum invariance and the nonlocality of quantum information are considered in the paper from an epistemological viewpoint.The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble after measurement. Quantum invariance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results.A set-theory corollary is the curious invariance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. However the above equivalence requires it to be equated to a well-ordered set after measurement and thus requires the axiom of choice for it to be able to be obtained.Quantum invariance underlies quantum information and reveals it as the relation of an unordered quantum “much” (i.e. a coherent state) and a well-ordered “many” of the measured results (i.e. a statistical ensemble). It opens up to a new horizon, in which all physical processes and phenomena can be interpreted as quantum computations realizing relevant operations and algorithms on quantum information. All phenomena of entanglement can be described in terms of the so defined quantum information. Quantum invariance elucidates the link between general relativity and quantum mechanics and thus, the problem of quantum gravity.The nonlocality of quantum information unifies the exact position of any space-time point of a smooth trajectory and the common possibility of all space-time points due to a quantum leap. This is deduced from quantum invariance.Epistemology involves the relation of ordering and thus a generalized kind of information, quantum one, to explain the special features of the cognition in quantum mechanics.,