Archive of posts with tag “mathematical physics”
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Eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition
Several features of the eigenfunctions of the Laplacian on an annulus with homogeneous Neumann boundary condition are discussed. The distribution of the eigenvalues is discussed in detail, making use of a phase angle function called . The limiting cases of a disk and a circle are discussed. -
The role of particle indistinguishability in statistical mechanics
Indistinguishability plays an important role in enumerative problems in combinatorics. This article explains the concept and significance of particle indistinguishability in statistical mechanics. -
The duality between two plane trajectories related by a conformal map
The conformal map transforms the trajectory with energy in potential into the trajectory with energy in potential . I will prove this beautiful result and show some implications of it. -
Rotational symmetry of plane lattices as a simple example of algebraic number theory
For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an -fold symmetry is compatible with translational symmetry is the same as whether . -
A measure-theoretic formulation of statistical ensembles (part 2)
For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms. -
A measure-theoretic formulation of statistical ensembles (part 1)
For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.
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