An information theoretic approach to statistical dependence: copula information

We discuss the connection between information and copula theories by showing that a copula can be employed to decompose the information content of a multivariate distribution into marginal and dependence components, with the latter quantified by the mutual information. We define the information excess as a measure of deviation from a maximum entropy distribution. The idea of marginal invariant dependence measures is also discussed and used to show that empirical linear correlation underestimates the amplitude of the actual correlation in the case of non-Gaussian marginals. The mutual information is shown to provide an upper bound for the asymptotic empirical log-likelihood of a copula. An analytical expression for the information excess of T-copulas is provided, allowing for simple model identification within this family. We illustrate the framework in a financial data set.Comment: to appear in Europhysics Letter

Intersubband-induced spin-orbit interaction in quantum wells

Recently, we have found an additional spin-orbit (SO) interaction in quantum wells with two subbands [Phys. Rev. Lett. 99, 076603 (2007)]. This new SO term is non-zero even in symmetric geometries, as it arises from the intersubband coupling between confined states of distinct parities, and its strength is comparable to that of the ordinary Rashba. Starting from the $8 \times 8$ Kane model, here we present a detailed derivation of this new SO Hamiltonian and the corresponding SO coupling. In addition, within the self-consistent Hartree approximation, we calculate the strength of this new SO coupling for realistic symmetric modulation-doped wells with two subbands. We consider gated structures with either a constant areal electron density or a constant chemical potential. In the parameter range studied, both models give similar results. By considering the effects of an external applied bias, which breaks the structural inversion symmetry of the wells, we also calculate the strength of the resulting induced Rashba couplings within each subband. Interestingly, we find that for double wells the Rashba couplings for the first and second subbands interchange signs abruptly across the zero bias, while the intersubband SO coupling exhibits a resonant behavior near this symmetric configuration. For completeness we also determine the strength of the Dresselhaus couplings and find them essentially constant as function of the applied bias.Comment: 16 pages, 12 figure

Spin dynamics in semiconductors

This article reviews the current status of spin dynamics in semiconductors which has achieved a lot of progress in the past years due to the fast growing field of semiconductor spintronics. The primary focus is the theoretical and experimental developments of spin relaxation and dephasing in both spin precession in time domain and spin diffusion and transport in spacial domain. A fully microscopic many-body investigation on spin dynamics based on the kinetic spin Bloch equation approach is reviewed comprehensively.Comment: a review article with 193 pages and 1103 references. To be published in Physics Reports

An information theoretic approach to statistical dependence: copula information

We discuss the connection between information and copula theories by showing that a copula can be employed to decompose the information content of a multivariate distribution into marginal and dependence components, with the latter quantified by the mutual information. We define the information excess as a measure of deviation from a maximum entropy distribution. The idea of marginal invariant dependence measures is also discussed and used to show that empirical linear correlation underestimates the amplitude of the actual correlation in the case of non-Gaussian marginals. The mutual information is shown to provide an upper bound for the asymptotic empirical log-likelihood of a copula. An analytical expression for the information excess of T-copulas is provided, allowing for simple model identification within this family. We illustrate the framework in a financial data set.

Topics in statistical physics of complex systems: a statistical mechanical approach to two topics of interest in finance, economics and sociology

No presente trabalho, exploramos dois temas de interesse em finanças, economia e antropologia social, através da aplicação de técnicas da teoria da informação e da mecânica estatística. No primeiro tópico, estudamos a conexão entre teoria de dependência estatística, teoria de informação e teoria da cópulas. O conceito de distribuição cópula é revisto e aplicado em reformulação das definições de medida de dependência dadas por Rényi 1. Em seguida, mostramos que a informação mútua satisfaz todos os requisitos para ser uma boa medida de dependência. Obtemos uma identidade entre a informação mútua e a entropia da distribuição cópula e uma decomposição mais específica da informação mútua de uma distribuição elíptica nas suas partes linear e não-linear. Avaliamos o risco de usar quantidades ingênuas como medidas de dependência estatística, mostrando que a correlação linear pode subestimar grosseiramente a dependência. Esses resultados são utilizados para desenvolver um método de detectação de desvios de dependência gaussiana em pares de variáveis aleatórias e aplicá-lo a séries temporais financeiras. Finalmente, discutimos um método para ajustar t-cópulas a dados empíricos 2 através da medida da informação mútua e do tau de Kendall. No segundo tópico, desenvolvemos um modelo para o surgimento de autoridade em sociedades humanas pré-agrícolas. Discutimos motivações empíricas com raízes em neurociência, primatologia e antropologia para um modelo matemático capaz de explicar a ampla variabilidade de formas de organização social humana no eixo igualitário hierárquico. O modelo resulta da aplicação de teoria da informação a uma hipótese sobre os custos evolutivos envolvidos na vida social. O modelo gera um diagrama de fase rico, com diferentes regimes que podem ser interpretados como diferentes tipos de organização social. Os parâmetros de controler do modelo estão ligados à capacidade cognitiva da espécie em questão, ao tamanho do grupo e a pressões ecológicas e sociais.In the present work we explore two topics of interest in finance, economics and social anthropology through the application of techniques from information theory and statistical mechanics. In the first topic we study the connexion between statistical dependency theory, information theory and copula theory. The concept of copula distribution is reviewed and applied to the reformulation of the definition of dependency measures given by Rényi 3. It is then shown that the mutual information satisfy all the requirements to be a good dependency measure. We derive an identity between mutual information and the entropy of the copula distribution and a more specific decomposition of the mutual information of an elliptical distribution into its linear and non-linear parts. We evaluate the risk of using naive measures as statistical dependency measures by showing that linear correlation can grossly underestimate dependency. Those results are used to develop a method to detect deviation from gaussian dependence in pairs of random variables and apply it to financial time series. Finally, we discuss a method to adjust t-copulas to empirical data4 through the determination of the mutual information and Kendalls tau. In the second topic we develop a model for the emergence of authority in pre-agricultural human societies. We discuss empirical motivations with roots in neuroscience, primatology and anthropology for a mathematical model able to explain the ample variability of forms of human social organization in the egalitarian-hierarchical axis. The model results from the application of information theory on a hypothesis about the evolutive costs involved in social life. It generates a rich phase diagram, with different regimes which can be interpreted as different types of societal organization, from egalitarian to hierarchical. The control parameters of the model are connected to the cognitive capacity of the species in question, the size of the group and ecological and social pressures

Inter-subband spin-orbit coupling in semiconductor heterostructures

Neste trabalho apresentamos a determinação autoconsistente da constante de interação spin-órbita em heteroestruturas com duas sub-bandas. Como recentemente proposto, ao obter o hamiltoneano de um sistema com duas sub-bandas na aproximação de massa efetiva, constata-se a presença de um acoplamento inter-subbanda que não se anula mesmo em heteroestruturas simétricas. Apresentamos aqui as deduções teóricas que levaram à proposição desse novo acoplamento e mostramos o cálculo autoconsistente da intensidade do acoplamento e a comparamos com a intensidade do acoplamento Rashba, já amplamente estudado. Discutimos o método k.p e a Aproximação da Função Envelope e mostramos a obtenção do modelo de Kane 8x8 para semicondutores com estrutura zincblende. Aplicamos o método do \"folding down\'\' ao hamiltoneano de Kane isolando o setor correspondente à banda de condução. Escrevemos dessa forma um hamiltoneano efetivo para a banda de condução no contexto de um poço quântico com uma barreira. Através da projeção desse hamiltoneano nos dois primeiros estados da parte orbital verifica-se o surgimento de um acoplamento inter-subbanda. Finalmente escrevemos o hamiltoneano efetivo 4x4 que descreve as duas primeiras subbandas de um poço quântico e obtivemos seus autoestados e autoenergias. Finalmente fizemos o cálculo autoconsistente das funções de onda e energias de um gás de elétrons em poços quânticos simples e duplos através da aproximação de Hartree e a partir desses resultados determinamos o valor da constante de acoplamento Rashba e da nova constante inter-subbanda. Entre os resultados obtidos destacam-se o controle elétrico da constante de acoplamento inter-subbanda através de um eletrodo externo e um efeito de renormalização da massa efetiva que pode chegar até 5% em algumas estruturas.In this work we present the self-consistent determination of the spin-orbit coupling constant in heterostructure with two subbands.As recently proposed, the effective hamiltonian for the conduction band in the effective mass approximation contains an inter-subband spin-orbit coupling which is non-zero even for symmetric heterostructures. We present the theoretical derivation which leads to this proposal and show a selfconsistent determination of the coupling constant. We also compare the magnitude of the new coupling constant with the usual Rashba coupling. Starting with a discussion of the k.p method and the Envelope Function Approximation (EFA) we show the derivation of the 8x8 Kane model for semiconductors with zincblende structure. We then apply the \"folding down\'\' method, isolating the conduction band sector of the EFA hamiltonian. By projecting this hamiltonian in the first two states of the orbital part, we find an effective 4x4 hamiltonian that contains an inter-subband spin orbit coupling. The eingenvalues and eigenvectors of this hamiltonian are shown and, specializing the model for single and double quantum wells, we self-consistently determine the inter-subband and Rashba coupling constants in the Hartree approximation. The results indicate the possibility of electrical control of the coupling constant and show an effective mass renormalization effect that can be up to 5% in some cases