Electron Phonon Collisions, Fermi Dirac Distribution and Bloch's T5T^5 law

In this paper, we model exchange of energy between electrons in solids and the phonon bath as electron-phonon collisions. Phonons are modelled as packets which create a lattice deformation potential of which electrons scatter. We show how these collisions exchange energy between electrons and phonons, leading to Fermi-Dirac distribution for electrons. Using these collisions, we derive the temperature dependence of resistivity of metals and the Bloch's $T$ and $T^5$ law for high and low temperature regime respectively. Unlike standard derivations of $T$ dependence of high temperature resistivity, our derivation rests fundamentally on the temperature dependence of the scattering angle.Comment: 14 pages, 10 Figures. arXiv admin note: text overlap with arXiv:1710.0348

On Minimal Tree Realizations of Linear Codes

A tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree. A tree decomposition can be extended to a tree realization, i.e., a cycle-free realization of the code on the underlying tree, by specifying a state space at each edge of the tree, and a local constraint code at each vertex of the tree. The constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes. A measure of the complexity of maximum-likelihood decoding for a code is its treewidth, which is the least constraint complexity of any of its tree realizations. It is known that among all tree realizations of a code that extends a given tree decomposition, there exists a unique minimal realization that minimizes the state space dimension at each vertex of the underlying tree. In this paper, we give two new constructions of these minimal realizations. As a by-product of the first construction, a generalization of the state-merging procedure for trellis realizations, we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree. The second construction relies on certain code decomposition techniques that we develop. We further observe that the treewidth of a code is related to a measure of graph complexity, also called treewidth. We exploit this connection to resolve a conjecture of Forney's regarding the gap between the minimum trellis constraint complexity and the treewidth of a code. We present a family of codes for which this gap can be arbitrarily large.Comment: Submitted to IEEE Transactions on Information Theory; 29 pages, 11 figure

Constraint Complexity of Realizations of Linear Codes on Arbitrary Graphs

A graphical realization of a linear code C consists of an assignment of the coordinates of C to the vertices of a graph, along with a specification of linear state spaces and linear ``local constraint'' codes to be associated with the edges and vertices, respectively, of the graph. The \k-complexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes. \k-complexity is a reasonable measure of the computational complexity of a sum-product decoding algorithm specified by a graphical realization. The main focus of this paper is on the following problem: given a linear code C and a graph G, how small can the \k-complexity of a realization of C on G be? As useful tools for attacking this problem, we introduce the Vertex-Cut Bound, and the notion of ``vc-treewidth'' for a graph, which is closely related to the well-known graph-theoretic notion of treewidth. Using these tools, we derive tight lower bounds on the \k-complexity of any realization of C on G. Our bounds enable us to conclude that good error-correcting codes can have low-complexity realizations only on graphs with large vc-treewidth. Along the way, we also prove the interesting result that the ratio of the \k-complexity of the best conventional trellis realization of a length-n code C to the \k-complexity of the best cycle-free realization of C grows at most logarithmically with codelength n. Such a logarithmic growth rate is, in fact, achievable.Comment: Submitted to IEEE Transactions on Information Theor