Nonextensive thermodynamic functions in the Schr\"odinger-Gibbs ensemble

Schr\"odinger suggested that thermodynamical functions cannot be based on the gratuitous allegation that quantum-mechanical levels (typically the orthogonal eigenstates of the Hamiltonian operator) are the only allowed states for a quantum system [E. Schr\"odinger, Statistical Thermodynamics (Courier Dover, Mineola, 1967)]. Different authors have interpreted this statement by introducing density distributions on the space of quantum pure states with weights obtained as functions of the expectation value of the Hamiltonian of the system. In this work we focus on one of the best known of these distributions, and we prove that, when considered in composite quantum systems, it defines partition functions that do not factorize as products of partition functions of the noninteracting subsystems, even in the thermodynamical regime. This implies that it is not possible to define extensive thermodynamical magnitudes such as the free energy, the internal energy or the thermodynamic entropy by using these models. Therefore, we conclude that this distribution inspired by Schr\"odinger's idea can not be used to construct an appropriate quantum equilibrium thermodynamics.Comment: 32 pages, revtex 4.1 preprint style, 5 figures. Published version with several changes with respect to v2 in text and reference

Ehrenfest dynamics is purity non-preserving: a necessary ingredient for decoherence

We discuss the evolution of purity in mixed quantum/classical approaches to electronic nonadiabatic dynamics in the context of the Ehrenfest model. As it is impossible to exactly determine initial conditions for a realistic system, we choose to work in the statistical Ehrenfest formalism that we introduced in Ref. 1. From it, we develop a new framework to determine exactly the change in the purity of the quantum subsystem along the evolution of a statistical Ehrenfest system. In a simple case, we verify how and to which extent Ehrenfest statistical dynamics makes a system with more than one classical trajectory and an initial quantum pure state become a quantum mixed one. We prove this numerically showing how the evolution of purity depends on time, on the dimension of the quantum state space $D$, and on the number of classical trajectories $N$ of the initial distribution. The results in this work open new perspectives for studying decoherence with Ehrenfest dynamics.Comment: Revtex 4-1, 14 pages, 2 figures. Final published versio

A general framework for quantum splines

Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on u∗(n) which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.The work of L. Abrunheiro was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (“FCT–Fundaçao para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013. The work of M. Camarinha and P. Santos was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. The work of J. Clemente-Gallardo and J. C. Cuchí was partially covered by MICINN grants FIS2013-46159-C3-2-P and MTM2012-33575 and by DGA Grant 2016-24/1

Statistics and Nos\'e formalism for Ehrenfest dynamics

Quantum dynamics (i.e., the Schr\"odinger equation) and classical dynamics (i.e., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well defined Poisson bracket allows to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of a Ehrenfest system, it is straightforward to extend the formalism of Nos\'e (invented to do constant temperature Molecular Dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.Comment: 28 pages, 1 figure. Published version. arXiv admin note: substantial text overlap with arXiv:1010.149

Hydrogeological model from hydrochemical and geocryologic records on the Cape Lamb of Vega Island, northern Antarctic Peninsula

The rapid changes that are taking place in the climate of the Antarctic Peninsula are triggering hydrological processes which had been limited or inactive for relatively long periods of time. These processes are evident in ice-free areas on the northern edge of the Antarctic Peninsula, such as Cape Lamb, and lead to intense draining of the surface and groundwater system throughout the brief Antarctic summer. The result is a movement of large amounts of water, sediments and nutrients that would be immobilized in other scenarios. This study proposes a model of the operation of the surface and groundwater system that could be valid for several ice-free areas in the region, based on the interpretation of field observations in Cape Lamb, the most extensive ice-free area on Vega Island. The model proposed is further supported by the interpretation of 56 chemical analyses of samples representing groundwater, active layer water, glacier ice and snowfall. The hydrochemical interpretation is supported primarily by four indicators (pH, electrical conductivity, D/18O ratio and SO4/Cl ratio) which have proven to be the most appropriate hydrochemical variables to differentiate the origin and interaction of the waters in the various sections of the system

Extreme value distributions and Renormalization Group

In the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent, identically distributed random variables are studied. So far, only affine rescalings have been considered. We show, however, that more general rescalings are natural and lead to new limit distributions, apart from the Gumbel, Weibull, and Fr\'echet families. The problem is approached using the language of Renormalization Group transformations in the space of probability densities. The limit distributions are fixed points of the transformation and the study of the differential around them allows a local analysis of the domains of attraction and the computation of finite-size corrections.Comment: 16 pages, 5 figures. Final versio

Ehrenfest Statistical Dynamics in Chemistry: Study of Decoherence Effects

In previous works, we introduced a geometric route to define our Ehrenfest statistical dynamics (ESD) and we proved that, for a simple toy model, the resulting ESD does not preserve purity. We now take a step further: we investigate decoherence and pointer basis in the ESD model by considering some uncertainty in the degrees of freedom of a simple but realistic molecular model, consisting of two classical cores and one quantum electron. The Ehrenfest model is sometimes discarded as a valid approximation to nonadiabatic coupled quantum-classical dynamics because it does not describe the decoherence in the quantum subsystem. However, any rigorous statistical analysis of the Ehrenfest dynamics, such as the described ESD formalism, proves that decoherence exists. In this article, decoherence in ESD is studied by measuring the change in the quantum subsystem purity and by analyzing the appearance of the pointer basis to which the system decoheres, which for our example is composed of the eigenstates of the electronic Hamiltonian.We have received support by Research Grants E24/1 and E24/3 (DGA, Spain), MINECO MTM2015-64166-C2-1-P and FIS2017-82426-P, and MICINN FIS2013-46159-C3-2-P and FIS2014-55867-P. Support from Scholarships B100/13 (DGA) and FPU13/01587 (MECD) for J.A.J.-G. is also acknowledged

A Computational Channel Model for Magnetic Induction-Based Subsurface Applications

There are many underground applications based on magnetic fields generated by an oscillating magnetic source. For them, a magnetic dipole in a three-layered region with upper semi-infinite air layer can be a convenient idealization used for their planning, development, and operation. Solutions are in the form of the well-known Sommerfeld integral expressions that can be evaluated by numerical methods. A set of field expressions to be numerically evaluated by an efficient algorithm are not collected comprehensively yet, or at least in a directly usable form. In this paper, the explicit magnetic field solutions for the vertical magnetic dipole and the horizontal magnetic dipole for a general source-observer location are derived from the Hertz vector. They can be properly combined to model the problem of a tilted magnetic dipole source for horizontally or inclined stratified media. As a result, a complete set of integral equations of the Sommerfeld type valid from the near zone to the far zone are formulated. A method for numerical evaluation of the field expressions for high accurate computations is described. The numerical results are validated using the finite element method for all the possible source-receiver configurations and three well-spanned frequencies of typical subsurface applications. Both numerical solutions agree according to the normalized root-mean-square error-based fit metric. Numerical results for two cases of study are presented to see its usefulness for subsurface applications. A MATLAB implementation of the mathematical description outlined in this paper and the proposed evaluation method is freely available for download