Canonical and non-canonical equilibrium distribution

We address the problem of the dynamical foundation of non-canonical equilibrium. We consider, as a source of divergence from ordinary statistical mechanics, the breakdown of the condition of time scale separation between microscopic and macroscopic dynamics. We show that this breakdown has the effect of producing a significant deviation from the canonical prescription. We also show that, while the canonical equilibrium can be reached with no apparent dependence on dynamics, the specific form of non-canonical equilibrium is, in fact, determined by dynamics. We consider the special case where the thermal reservoir driving the system of interest to equilibrium is a generator of intermittent fluctuations. We assess the form of the non-canonical equilibrium reached by the system in this case. Using both theoretical and numerical arguments we demonstrate that Levy statistics are the best description of the dynamics and that the Levy distribution is the correct basin of attraction. We also show that the correct path to non-canonical equilibrium by means of strictly thermodynamic arguments has not yet been found, and that further research has to be done to establish a connection between dynamics and thermodynamics.Comment: 13 pages, 6 figure

Canonical varieties with no canonical axiomatisation

Accepted versio

Canonical DSR

For a certain example of a "doubly special relativity theory" the modified space-time Lorentz transformations are obtained from momentum space transformations by using canonical methods. In the sequel an energy-momentum dependent space-time metric is constructed, which is essentially invariant under the modified Lorentz transformations. By associating such a metric to every Planck cell in space and the energy-momentum contained in it, a solution of the problem of macroscopic bodies in doubly special relativity is suggested.Comment: 11 page

Canonical thermalization

For quantum systems that are weakly coupled to a much 'bigger' environment, thermalization of possibly far from equilibrium initial ensembles is demonstrated: for sufficiently large times, the ensemble is for all practical purposes indistinguishable from a canonical density operator under conditions that are satisfied under many, if not all, experimentally realistic conditions

Canonical Typicality

It is well known that a system, S, weakly coupled to a heat bath, B, is described by the canonical ensemble when the composite, S+B, is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true both for classical distributions on the phase space and for quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands and justifies remarks made by Schr\"odinger in 1952.Comment: 6 pages LaTeX, no figures; v2 minor improvements and addition

Canonical Truth

We introduce and study a notion of canonical set theoretical truth, which means truth in a `canonical model', i.e. a transitive class model that is uniquely characterized by some $\in$-formula. We show that this notion of truth is `informative', i.e. there are statements that hold in all canonical models but do not follow from ZFC, such as Reitz' ground model axiom or the nonexistence of measurable cardinals. We also show that ZF+$V=L[\mathbb{R}]$+AD has no canonical models. On the other hand, we show that there are canonical models for `every real has sharp'. Moreover, we consider `theory-canonical' statements that only fix a transitive class model of ZFC up to elementary equivalence and show that it is consistent relative to large cardinals that there are theory-canonical models with measurable cardinals and that theory-canonicity is still informative in the sense explained above