Diffusion and Superdiffusion from Hydrodynamic Projections

Hydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in many-body systems, such as the exact Drude weights. Focussing one one-dimensional systems, I show that this principle can be extended beyond the Euler scale, in particular to the diffusive and superdiffusive scales. By hydrodynamic reduction, Hilbert spaces of observables are constructed that generalise the standard space of conserved densities and describe the finer scales of hydrodynamics. The Green–Kubo formula for the Onsager matrix has a natural expression within the diffusive space. This space is associated with quadratically extensive charges, and projections onto any such charge give generic lower bounds for diffusion. In particular, bilinear expressions in linearly extensive charges lead to explicit diffusion lower bounds calculable from the thermodynamics, and applicable for instance to generic momentum-conserving one-dimensional systems. Bilinear charges are interpreted as covariant derivatives on the manifold of maximal entropy states, and represent the contribution to diffusion from scattering of ballistic waves. An analysis of fractionally extensive charges, combined with clustering properties from the superdiffusion phenomenology, gives lower bounds for superdiffusion exponents. These bounds reproduce the predictions of nonlinear fluctuating hydrodynamics, including the Kardar–Parisi–Zhang exponent 2/3 for sound-like modes, the Levy-distribution exponent 3/5 for heat-like modes, and the full Fibonacci sequence.

Large deviation principle for piecewise monotonic maps with density of periodic measures

We show that a piecewise monotonic map with positive topological entropy satisfies the level-2 large deviation principle with respect to the unique measure of maximal entropy under the conditions that the corresponding Markov diagram is irreducible and that the periodic measures of the map are dense in the set of ergodic measures. This result can apply to a broad class of piecewise monotonic maps, such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces.

Path integral based convolution and pooling for graph neural networks*

Graph neural networks (GNNs) extend the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of graph convolution and pooling is key to success. Borrowing ideas from physics, we propose path integral-based GNNs (PAN) for classification and regression tasks on graphs. Specifically, we consider a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It generalizes the graph Laplacian to a new transition matrix that we call the maximal entropy transition (MET) matrix derived from a path integral formalism. Importantly, the diagonal entries of the MET matrix are directly related to the subgraph centrality, thus leading to a natural and adaptive pooling mechanism. PAN provides a versatile framework that can be tailored for different graph data with varying sizes and structures. We can view most existing GNN architectures as special cases of PAN. Experimental results show that PAN achieves state-of-the-art performance on various graph classification/regression tasks, including a new benchmark dataset from statistical mechanics that we propose to boost applications of GNN in physical sciences.

Excursions to the cusps for geometrically finite hyperbolic orbifolds and equidistribution of closed geodesics in regular covers

We give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the orbifold. Using this criterion, we prove new results on the distribution of collections of closed geodesics on such an orbifold, and as a corollary, we prove the equidistribution of closed geodesics up to a certain length in amenable regular covers of geometrically finite orbifolds.

Designing a Block Cipher in Galois Extension Fields for IoT Security

This paper focuses on a block cipher adaptation of the Galois Extension Fields (GEF) combination technique for PRNGs and targets application in the Internet of Things (IoT) space, an area where the combination technique was concluded as a quality stream cipher. Electronic Codebook (ECB) and Cipher Feedback (CFB) variations of the cryptographic algorithm are discussed. Both modes offer computationally efficient, scalable cryptographic algorithms for use over a simple combination technique like XOR. The cryptographic algorithm relies on the use of quality PRNGs, but adds an additional layer of security while preserving maximal entropy and near-uniform distributions. The use of matrices with entries drawn from a Galois field extends this technique to block size chunks of plaintext, increasing diffusion, while only requiring linear operations that are quick to perform. The process of calculating the inverse differs only in using the modular inverse of the determinant, but this can be expedited by a look-up table. We validate this GEF block cipher with the NIST test suite. Additional statistical tests indicate the condensed plaintext results in a near-uniform distributed ciphertext across the entire field. The block cipher implemented on an MSP430 offers a faster, more power-efficient alternative to the Advanced Encryption Standard (AES) system. This cryptosystem is a secure, scalable option for IoT devices that must be mindful of time and power consumption.

Compacting the density matrix in quantum dynamics: Singular value decomposition of the surprisal and the dominant constraints for anharmonic systems

We introduce a practical method for compacting the time evolution of the quantum state of a closed physical system. The density matrix is specified as a function of a few time-independent observables where their coefficients are time-dependent. The key mathematical step is the vectorization of the surprisal, the logarithm of the density matrix, at each time point of interest. The time span used depends on the required spectral resolution. The entire course of the system evolution is represented as a matrix where each column is the vectorized surprisal at the given time point. Using singular value decomposition, SVD, of this matrix we generate realistic approximations for the time-independent observables and their respective time dependent coefficients. This allows a simplification of the algebraic procedure for determining the dominant constraints (the time-independent observables) in the sense of the maximal entropy approach. A nonstationary coherent initial state of a Morse oscillator is used to introduce the approach. We derive analytical exact expression for the surprisal as a function of time and this offers a benchmark for comparison with the accurate but approximate SVD results. We discuss two examples of a Morse potential of different anharmonicities, the H2 and I2 molecules. We further demonstrate the approach for a two coupled electronic states problem, the well studied non radiative decay of pyrazine from its bright state. Five constraints are found to be enough to capture the ultrafast electronic population exchange and to recover the dynamics of the wave packet in both electronic states.

Contracting and Involutive Negations of Probability Distributions

A dozen papers have considered the concept of negation of probability distributions (pd) introduced by Yager. Usually, such negations are generated point-by-point by functions defined on a set of probability values and called here negators. Recently the class of pd-independent linear negators has been introduced and characterized using Yager’s negator. The open problem was how to introduce involutive negators generating involutive negations of pd. To solve this problem, we extend the concepts of contracting and involutive negations studied in fuzzy logic on probability distributions. First, we prove that the sequence of multiple negations of pd generated by a linear negator converges to the uniform distribution with maximal entropy. Then, we show that any pd-independent negator is non-involutive, and any non-trivial linear negator is strictly contracting. Finally, we introduce an involutive negator in the class of pd-dependent negators. It generates an involutive negation of probability distributions.