Two extensions of exact non-equilibrium steady states of a boundary driven cellular automaton

Recently Prosen and Mej\'ia-Monasterio (J. Phys. A: Math. Theor. 49 (2016) 185003) obtained exact nonequilibrium steady states of an integrable and reversible cellular automaton driven by some stochastic boundary conditions. In this paper, we explore the possible extensions of their method by generalizing the boundary conditions. As the result, we find two cases where such an extension is possible. One is the case where a special condition is satisfied in a generalized boundary condition. The other is obtained by considering a conserved quantity as energy and boundaries as heat reservoirs. The latter includes the original solution as the special case. Properties of the both solutions are discussed.Comment: 27 pages, 13 figure

Quantum Gravity as a Dissipative Deterministic System

It is argued that the so-called holographic principle will obstruct attempts to produce physically realistic models for the unification of general relativity with quantum mechanics, unless determinism in the latter is restored. The notion of time in GR is so different from the usual one in elementary particle physics that we believe that certain versions of hidden variable theories can -- and must -- be revived. A completely natural procedure is proposed, in which the dissipation of information plays an essential role. Unlike earlier attempts, it allows us to use strictly continuous and differentiable classical field theories as a starting point (although discrete variables, leading to fermionic degrees of freedom, are also welcome), and we show how an effective Hilbert space of quantum states naturally emerges when one attempts to describe the solutions statistically. Our theory removes some of the mysteries of the holographic principle; apparently non-local features are to be expected when the quantum degrees of freedom of the world are projected onto a lower-dimensional black hole horizon. Various examples and models illustrate the points we wish to make, notably a model showing that massless, non interacting neutrinos are deterministic.Comment: 20 pages plain TeX, 2 figures PostScript. Added some further explanations, and the definitions of `beable' and `changeable'. A minor error correcte

Universal gauge-invariant cellular automata

Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justification for the fundamental forces. Usually, one starts from a non-interactive theory which governs `matter', and features a global symmetry. One then extends the theory so as make the global symmetry into a local one (a.k.a gauge-invariance). We formalise a discrete counterpart of this process, known as gauge extension, within the Computer Science framework of Cellular Automata (CA). We prove that the CA which admit a relative gauge extension are exactly the globally symmetric ones (a.k.a the colour-blind). We prove that any CA admits a non-relative gauge extension. Both constructions yield universal gauge-invariant CA, but the latter allows for a first example where the gauge extension mediates interactions within the initial CA

On two non-ergodic reversible cellular automata, one classical, the other quantum

We propose and discuss two variants of kinetic particle models - cellular automata in 1+1 dimensions, which have some appeal due to their simplicity and intriguing properties which could warrant further research and applications. The first model is a deterministic and reversible automaton describing two species of quasiparticles: stable massless matter particles moving with velocity $\pm 1$ and unstable, standing (zero velocity) field particles. We discuss two distinct continuity equations for three conserved charges of the model. While the first two charges and the corresponding currents have support three (3) lattice sites and represent a lattice analogue of conserved energy-momentum tensor, we find an additional conserved charge and current with support of nine (9) sites, implying non-ergodic behaviour and potentially signalling integrability of the model with a highly nested R-matrix structure. The second model represents a quantum (or stochastic) deformation of a recently introduced and studied charged hardpoint lattice gas, where particles of different binary charge ($\pm 1$) and binary velocity ($\pm 1$) can nontrivially mix upon elastic collisional scattering. We show that while the unitary evolution rule of this model does not satisfy the full Yang-Baxter equation, it still satisfies an intriguing related identity which gives birth to an infinite set of local conserved operators, the-so-called glider operators.Comment: 15 pages, 7 figures; submitted to Entropy for the special issue on the 80th birthday of Giulio Casat

Universal Gauge-Invariant Cellular Automata

Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justification for the fundamental forces. Usually, one starts from a non-interactive theory which governs "matter", and features a global symmetry. One then extends the theory so as make the global symmetry into a local one (a.k.a gauge-invariance). We formalise a discrete counterpart of this process, known as gauge extension, within the Computer Science framework of Cellular Automata (CA). We prove that the CA which admit a relative gauge extension are exactly the globally symmetric ones (a.k.a the colour-blind). We prove that any CA admits a non-relative gauge extension. Both constructions yield universal gauge-invariant CA, but the latter allows for a first example where the gauge extension mediates interactions within the initial CA