Stochastic interdependence of a probablility distribution on a product space is measured by its Kullback-Leibler distance from the exponential family of product distributions (called multi-information). Here we investigate low-dimensional exponential families that contain the maximizers of stochastic interdependence in their closure. Based on a detailed… (More)
The number-theoretical spin chain has exactly one phase transition , which is located at inverse temperature cr = 2. There the mag-netization jumps from one to zero. The energy density, being zero in the low temperature phase, grows at least linearly in cr ? .
We present numerical and analytical evidence for a rst-order phase transition of the ferromagnetic spin chain with partition function Z() = (? 1)==() at the inverse temperature cr = 2. In a recent paper 6] we established a link between analytic number theory and classical statistical mechanics by interpreting the quotient Z(s) = (s ? 1)==(s) of Riemann zeta… (More)
We analyze the Farey spin chain, a one dimensional spin system with effective interaction decaying like the squared inverse distance. Using a polymer model technique, we show that when the temperature is decreased below the (single) critical temperature T c = 1 2 , the magnetization jumps from zero to one.
The quotient (s ? 1)==(s) of Riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperature s.
We analyze the number-theoretical spin chain with partition function Z() = (? 1)==() using the polymer model technique. The nite (grand) canonical chains give bounds for the limit free energy and internal energy. The correlation functions for inverse temperature = ?1 are products of two-point functions. A combinatorial result for general interval graphs is… (More)
We consider motion in a periodic potential in a classical, quantum , and semiclassical context. Various results on the distribution of asymptotic velocities are proven.
We shortly review recent work interpreting the quotient (s ? 1)==(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive.
We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas. The comparison does not only support the general… (More)