Bose-Einstein condensation and condensation of $q$-particles in equilibrium and non equilibrium thermodynamics: a new approach

In the setting of the principle of local equilibrium which asserts that the temperature is a function of the energy levels of the system, we exhibit plenty of steady states describing the condensation of free Bosons which are not in thermal equilibrium. The surprising facts are that the condensation can occur both in dimension less than 3 in configuration space, and even in excited energy levels. The investigation relative to non equilibrium suggests a new approach to the condensation, which allows an unified analysis involving also the condensation of $q$-particles, $-1leq qleq 1$, where $q=pm1$ corresponds to the Bose/Fermi alternative. For such $q$-particles, the condensation can occur only if $0<qleq1$, the case 1 corresponding to the standard Bose-Einstein condensation. In this more general approach, completely new and unexpected states exhibiting condensation phenomena naturally occur also in the usual situation of equilibrium thermodynamics. The new approach proposed in the present paper for the situation of $2^text{nd}$ quantisation of free particles, is naturally based on the theory of the Distributions, which might hopefully be extended to more general cases

Identification of the theory of multidimensional orthogonal polynomials with the theory of symmetric interacting Fock spaces with finite dimensional one particle space

The identification mentioned in the title allows a formulation of the multidi mensional Favard Lemma different from the ones currently used in the literature and which exactly parallels the original one dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of Hermitean matri ces of the same dimension. Moreover, in this identification, the multidimensional extension of the compatibility condition for the positive Jacobi sequence becomes the condition which guarantees the existence of the creator in an interacting Fock space. The above result opens the way to the program of a purely algebraic clas sification of probability measures on $mathbb{R}^d$ with finite moments of any order. In this classification the usual Boson Fock space over $mathbb{C}^d$ is characterized by the fact that the positive Jacobi sequence is made up of identity matrices and the real Jacobi sequences are identically zero. The quantum decomposition of classical real valued random variables with all moments is one of the main ingredients in the proof.

$C^*$-non-linear second quantization

Recently, we have constructed a non{linear (polynomial) extension of the 1-mode Heisenberg group and the corresponding Fock and Weyl representations. The transition from the 1-mode case to the current algebra level, in which the operators are indexed by elements of an appropriate test function space (second quantization), can be done at Lie algebra level. A way to bypass the difficulties of constructing a (non trivial) Hilbert space representation is to try and construct directly a $C^*$-algebra rep- resentation and then to look for its Hilbert space representations. In usual (linear) quantization, this corresponds to the construction of the Weyl $C^*$-algebra. In this paper, we produce such a construction for the above mentioned polynomial extension of the Weyl $C^*$-algebra. The result of this construction is a factorizable system of local alge- bras localized on bounded Borel subsets of $mathbb{R}$ and obtained as induc- tive limit of tensor products of finite sets of copies of the one mode $C^*$-algebra. The $C^*$-embeddings of the inductive system require some non{trivial re{scaling of the generators of the algebras involved. These re{scalings are responsible of a $C^*$-analogue of the no-go theorems, first met at the level of Fock second quantization, namely the proof that the family of Fock states defined on the inductive family of $C^*$-algebras is projective only in the linear case (i.e. the case of the usual Weyl algebra). Thus the solution of the representa- tion problem at $C^*$-level does not automatically imply its solution at Hilbert space level.

On Quantum Markov Chains on Cayley tree III: Ising model

In this paper, we consider the classical Ising model on the Cayley tree of order k and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the found critical temperature coincides with usual critical temperature.

Optimal Hamiltonian of Fermion Flows

After providing a general formulation of Fermion flows within the context of Hudson-Parthasarathy quantum stochastic calculus, we consider the problem of determining the noise coefficients of the Hamiltonian associated with a Fermion flow so as to minimize a naturally associated quadratic performance functional. This extends to Fermion flows results of the authors previously obtained for Boson flows .

Quadratic control of quantum processes

Within the framework of the Accardi-Fagnola-Quaegebeur (AFQ) representation free calculus of cite{b}, we consider the problem of controlling the size of a quantum stochastic flow generated by a unitary stochastic evolution affected by quantum noise. In the case when the evolution is driven by first order white noise (which includes quantum Brownian motion) the control is shown to be given in terms of the solution of an algebraic Riccati equation. This is done by first solving the problem of controlling (by minimizing an associated quadratic performance criterion) a stochastic process whose evolution is described by a stochastic differential equation of the type considerd in cite{b}. The solution is given as a feedback control law in terms of the solution of a stochastic Riccati equation.