A partition Π of the set [n] = {1, 2, . . . , n} is a collection B1, B2, . . . , Bd of nonempty disjoint subsets of [n]. The elements of a partition are called blocks. We assume that B1, B2, . . . , Bd are listed in the increasing order of their minimum elements, that is minB1 < minB2 < · · · < minBd. The set of all partitions of [n] with d blocks is denoted by P (n, d). The cardinality of P (n...

where Z is the partition function Z( ) = R exp( x) dQx, and can := f j Z( ) < 1g is the canonical parameter space. We let sup = supf j 2 cang, and inf likewise. The elements of the exponential family are also parametrized by their mean value . We write for the mean value corresponding to the canonical parameter and for the canonical parameter corresponding to the mean value : For any x the maxi...

A coding partition of a set of words partitions this set into classes such that whenever a sequence, of minimal length, has two distinct factorizations, the words of these factorizations belong to the same class. The canonical coding partition is the finest coding partition that partitions the set of words in at most one unambiguous class and other classes that localize the ambiguities in the f...

A new model of non-equilibrium thermodynamic states has been investigated on the basis of the fact that all thermodynamic variables can be derived from partition functions. We have thus attempted to define partition functions for non-equilibrium conditions by introducing the concept of pseudo-temperature distributions. These pseudo-temperatures are configurational in origin and distinct from ki...

About the sign ambiguity in the evaluation of Grand Canonical traces for quasi-particle statistical density operators. Abstract A simple and general prescription for evaluating unambiguously the sign of the grand-canonical trace of quasi-particle statistical density operators (the so-called sign ambiguity in taking the square root of determinants) is given. Sign ambiguities of this kind appear ...

For ensemble of bosons trapped in a 1D harmonic potential well we have found an analytical formula for the canonical partition function and shown that, for 100 trapped atoms, the discrepancy between the grand canonical and the canonical predictions for the condensate fraction reaches 10% in the vicinity of the Bose-Einstein threshold. This discrepancy decreases only logarithmically as the numbe...

For ensemble of bosons trapped in a 1D harmonic potential well we have found an analytical formula for the canonical partition function and shown that, for 100 trapped atoms, the discrepancy between the grand canonical and the canonical predictions for the condensate fraction reaches 10% in the vicinity of the Bose-Einstein threshold. This discrepancy decreases only logarithmically as the numbe...