Related papers: Statistical Physics of Coding for the Integers
Machine learning has had a major impact on data compression over the last decade and inspired many new, exciting theoretical and applied questions. This paper describes one such direction -- relative entropy coding -- which focuses on…
Distribution matching transforms independent and Bernoulli(1/2) distributed input bits into a sequence of output symbols with a desired distribution. Fixed-to-fixed length, invertible, and low complexity encoders and decoders based on…
We construct an encoding of finite strings over a fixed finite alphabet as natural numbers, based on a block partition of the Fibonacci sequence. Each position in the string selects one Fibonacci number from a dedicated block, with unused…
We describe a method to compute thermodynamic quantities in the harmonic approximation for identical bosons and fermions in an external confining field. We use the canonical partition function where only energies and their degeneracies…
Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the $N-$body phase space with the given total energy. Due to Boltzmann's principle,…
A map $f{:}\,[0,1)\to [0,1)$ is a {\it piecewise contraction of $n$ intervals} ($n$-PC) if there exist $0<\lambda<1$ and a partition of $[0,1)$ into intervals $I_1,\ldots,I_n$ such that $f\vert_{I_i}$ is $\lambda$-Lipschitz for every $1\le…
We show that the particles in the Calogero-Sutherland Model obey fractional exclusion statistics as defined by Haldane. We construct anyon number densities and derive the energy distribution function. We show that the partition function…
We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. The codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix. The…
Using the quantum Hamiltonian for a gravitational system with boundary, we find the partition function and derive the resulting thermodynamics. The Hamiltonian is the boundary term required by functional differentiability of the action for…
Traditionally, phase transitions are defined in the thermodynamic limit only. We propose a new formulation of equilibrium thermo-dynamics that is based entirely on mechanics and reflects just the {\em geometry and topology} of the N-body…
Non-extensive systems do not allow to go to the thermodynamic limit. Therefore we have to reformulate statistical mechanics without invoking the thermodynamical limit. I.e. we have to go back to Pre-Gibbsian times. We show that Boltzmann's…
We clarify that an ideal gas obeying infinite statistics cannot undergo condensation. Then we derive the dynamic equation for an identical particle system obeying infinite statistics under external potential and inter-particle interaction.…
The hydrodynamic interpretation of quantum mechanics treats a system of particles in an effective manner. In this work, we investigate squeezed coherent states within the hydrodynamic interpretation. The Hamiltonian operator in question is…
The process of dynamic state estimation (filtering) based on point process observations is in general intractable. Numerical sampling techniques are often practically useful, but lead to limited conceptual insight about optimal…
We propose a simple algorithm for generating normally distributed pseudo random numbers. The algorithm simulates N molecules that exchange energy among themselves following a simple stochastic rule. We prove that the system is ergodic, and…
We study the fluctuation of the number of particles in ideal Bose-Einstein condensates, both within the canonical and the microcanonical ensemble. Employing the Mellin-Barnes transformation, we derive simple expressions that link the…
We derive a systematic approach to the thermodynamics of quantum systems based on the underlying symmetry groups. We show that the entropy of a system can be described in terms of group-theoretical quantities that are largely independent of…
We present a systematic study of quantum system compression for the evolution of generic many-body problems. The necessary numerical simulations of such systems are seriously hindered by the exponential growth of the Hilbert space dimension…
By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which…
In the Minimum Description Length (MDL) principle, learning from the data is equivalent to an optimal coding problem. We show that the codes that achieve optimal compression in MDL are critical in a very precise sense. First, when they are…