Information and correlation in 
statistical mechanical systems.

David Richard Wolf

May 1996

• Contents
• List of Figures
• Dedication, Abstract, Acknowledgements, Vita
• Copyright
• Dedication
• Committee
• Abstract
• Acknowledgements
• Vita
• Introduction
• Entropy
• Entropy
• Convexity properties of entropy
• Information correlation, cumulants, clusters, and partitions
• Cumulants
• Cumulants in one dimension
• Cumulants in many dimensions
• Independence property
• Ursell functions and the density hierarchy
• Summation property
• The Ursell functions, partitions, and graphs
• Partition theorem, cluster theorems
• Factorization property
• Clustering
• Function hierarchies
• Information correlation functions
• Interpretation of the information correlation functions.
• Another interpretation of the information correlations
• Information correlation functions - example
• Correlation functions
• Information flow
• Estimating the information functions
• Constraint induced correlations
• What is done here
• The bit string system and observed states
• Distribution of the observed states
• High order entropies of the observed states
• First and second order entropies
• Relationship of asymptotic entropies
• Relationship of entropies per degree of freedom.
• Correction terms in second order entropy.
• The Kirkwood phase transition
• General 
• Information correlation functions
• Moments, Correlations, Cumulants
• Appendix A - Combinatorial identities
• Classical equations of information flow
• Liouville equation
• Reduced density functions and the BBGKY hierarchy
• Geometry of the BBGKY equations - general flows
• Information flow, correlation flow
• Maxent outside implies no flow into subsystem
• Information and correlation in the Ising system
• Entropy in the Ising System
• Ising Model - Closed chain
• Probability of states
• Reduced probabilities
• Computing the partition functions
• Simplifying the transfer matrix
• Expression for the partition function
• High order entropies in the Ising system
• The full Ising entropy
• Thermodynamic limit of transfer matrix
• Thermodynamic limit of full Ising entropy
• Thermodynamic limit of high order entropies
• Eigenvalues and Eigenvectors
• Eigenvalues and Eigenvectors in zero external field
• High order entropies in zero external field
• High order reduced distributions of non-contiguous sites
• High order entropies of non-contiguous sites
• Information correlation functions in the Ising system
• Forms of the information correlation function
• Moments, correlations, and cumulants
• Quantum equations of information flow
• Equation of motion of pure states
• Density of states, quantum Liouville equation
• Quantum BBGKY equations
• Schrodinger representation and Heisenberg representation
• Time development of pure state measurements.
• Information and correlation in the quantum Heisenberg system
• Description of system, Hamiltonian
• Hamiltonian forms and known solutions
• Dynamics, comparison of classical and quantum systems.
• Symmetric spin Hamiltonian
• Energy eigenvectors of the symmetric spin Hamiltonian
• Spin correlations in the symmetric spin Hamiltonian equilibrium distribution
• Symmetric spin Hamiltonian with external field
• Entropy of the spin system
• Mutual information between measured and unmeasured variables
• Entropies
• Information correlation functions
• Moments, correlations, cumulants
• Mathematica for the symmetric spin Hamiltonian
• Estimating information and correlation from finite data
• Inferring unknown parameters from data
• Inference increases information about the true distribution on average
• Estimating functions of probability distributions from finite samples.
• Statement of the problem solved here
• Estimating from finite data is ubiquitous
• The distribution of data
• The Bayes' estimator
• The Bayes' estimator minimizes posterior mean square error
• Form of the prior
• Form of the integrals giving the estimator
• Integrating
• Estimators for the first and second moments of the entropy
• Entropy estimator comparison
• Estimators for moments, correlations, cumulants.
• Extended notation for more complicated functions of probability distributions
• More integration techniques
• Estimators for functions involving up-to-pairwise overlap integrals
• Multiple overlap integration
• Appendix A. Hypergeometric functions
• Appendix B. Hypergeometric function identities
• Appendix C. Transforms
• Appendix C.1. The T transform
• Appendix C.2. The Z transform
• Appendix C.3. The Laplace transform and inverse
• Appendix D. Commuting linear operators
• Appendix D.1. Commuting two integrals
• Appendix D.2. Commuting integrals and derivatives
• Appendix E. Analytic continuation: Expanding 's domain
• Appendix F. Existence conditions
• Appendix G. Derivatives of overlap convolutions: Poles
• Derivatives of system functions, fluctuations, and correlations
• Correlations and power spectra
• Relationship of equilibrium correlations to power spectrum of dynamics: Weiner-Khinchin theorem
• Derivatives and correlations in spin systems
• Derivatives of the reduced entropy
• Ordering properties of the reduced entropy derivatives
• Conclusion
• Clebsch-Gordon transformation of spin bases
• Energy eigenbasis.
• Measurement basis. Energy to measurement basis transformation.
• Mathematica for the Heisenberg spin system
• Generating the coupled spin basis
• Transforming to the measurement basis
• Finding the coupled spin basis probability distribution
• Distribution of measured states
• References
• About this document ...

Tue Mar 25 08:11:49 CST 1997