arXiv subjects

Jarmo Hietarinta

Publications and source records attributed to Jarmo Hietarinta.

39 records · Page 3Linked to original sources

arXiv1994

Labeling Schemes for Tetrahedron Equations and Dualities between Them

Zamolodchikov's tetrahedron equations, which were derived by considering the scattering of straight strings, can be written in three different labeling schemes: one can use as labels the states of the vacua between the strings, the states of the string segments, or the states of the particles at the intersections of the strings. We give a detailed derivation of the three corresponding tetrahedron equations and show also how the Frenkel-Moore equations fits in as a {\em nonlocal} string labeling. We discuss then how an analog of the Wu-Kadanoff duality can be defined between each pair of the above three labeling schemes. It turns out that there are two cases, for which one can simultaneously construct a duality between {\em all} three pairs of labelings.

hep-th↗

arXiv1993

The upper triangular solutions to the three-state constant quantum Yang-Baxter equation

In this article we present all nonsingular upper triangular solutions to the constant quantum Yang-Baxter equation $R_{j_1j_2}^{k_1k_2}R_{k_1j_3}^{l_1k_3}R_{k_2k_3}^{l_2l_3}= R_{j_2j_3}^{k_2k_3}R_{j_1k_3}^{k_1l_3}R_{k_1k_2}^{l_1l_2}$ in the three state case, i.e. all indices ranging from 1 to 3. The upper triangular ansatz implies 729 equations for 45 variables. Fortunately many of the equations turned out to be simple allowing us to start breaking the problem into smaller ones. In the end we had a total of 552 solutions, but many of them were either inherited from two-state solutions or subcases of others. The final list contains 35 nontrivial solutions, most of them new.

solv-int↗

arXiv1993

Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the 1-dimensional Newtonian gravitational three-body system (three masses on a line with $-1/r$ pairwise potential). We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change (with negative total energy). For two mass choices we have calculated 18000 full orbits (with initial states on a $100\times 180$ lattice on the Poincaré section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincaré maps for $10\times 18$ starting points. Our results show that the Poincaré section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: 1) There is a region of fast scattering, with a minimum of pairwise collisions and smooth dependence on initial values. 2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincaré section. For both 1) and 2) the initial and final states consists of a binary + single particle. 3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is the periodic Schubart orbit, whose stability turns out to correlate strongly with the global behavior.

chao-dyn↗