QUANTUM MECHANICS IN AFC*-SYSTEMS

1996 ◽  
Vol 08 (06) ◽  
pp. 819-859 ◽  
Author(s):  
FUMIO HIAI ◽  
DÉNES PETZ
Keyword(s):  
Potential Theory ◽  
Chemical Potential ◽  
Lattice Systems ◽  
Quantum Random Walks ◽  
C Algebras ◽  
Localization Model ◽  
Dynamical Entropy ◽  
Invariant Interaction ◽  
Kms Condition ◽  
Shift Automorphism

Motivated from the chemical potential theory, we study quantum statistical thermodynamics in AF C*-systems generalizing usual one-dimensional quantum lattice systems. Our systems are C*-algebras [Formula: see text] which have a localization [Formula: see text] of finite-dimensional subalgebras indexed by finite intervals of Z and an automorphism γ acting as a right shift on the localization. Model examples are supplied from derived towers (string algebras) for type II1 factor-subfactor pairs. Given a (γ-invariant) interaction and a specific tracial state, we formulate the Gibbs conditions and the variational principle for (γ-invariant) states on [Formula: see text], and investigate the relationship among these conditions and the KMS condition for the time evolution generated by the interaction. Special attention is paid to C*-systems of gauge invariance (typical model in the chemical potential theory) and to C*-systems considered as quantum random walks on discrete groups. The CNT-dynamical entropy for the shift automorphism γ is also discussed.

Dynamical entropy for Bogoliubov actions of free abelian groups on the CAR-algebra

1997 ◽  
Vol 17 (4) ◽  
pp. 757-782 ◽  
Author(s):  
SERGEY I. BEZUGLYI ◽  
VALENTIN YA. GOLODETS
Keyword(s):  
Abelian Group ◽  
Infinite Number ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Car Algebra ◽  
C Algebras ◽  
Singular Spectrum ◽  
Number Of Generators ◽  
Dynamical Entropy ◽  
Free Abelian Groups

The notion of dynamical entropy for actions of a countable free abelian group $G$ by automorphisms of $C^*$-algebras is studied. These results are applied to Bogoliubov actions of $G$ on the CAR-algebra. It is shown that the dynamical entropy of Bogoliubov actions is computed by a formula analogous to that found by Størmer and Voiculescu in the case $G={\bf Z}$, and also it is proved that the part of the action corresponding to a singular spectrum gives zero contribution to the entropy. The case of an infinite number of generators has some essential differences and requires new arguments.

scholarly journals EQUILIBRIUM STATES AND THEIR ENTROPY DENSITIES IN GAUGE-INVARIANT C*-SYSTEMS

2005 ◽  
Vol 17 (04) ◽  
pp. 365-389 ◽  
Author(s):  
NOBUYUKI AKIHO ◽  
FUMIO HIAI ◽  
DÉNES PETZ
Keyword(s):  
Tensor Product ◽  
Chemical Potential ◽  
Entropy Density ◽  
Quantum Spin ◽  
Equilibrium States ◽  
Quantum Spin System ◽  
Translation Invariant ◽  
Gauge Invariant ◽  
Matrix Algebras ◽  
Kms Condition

A gauge-invariant C*-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In this paper, thermodynamics is studied in such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macroscopic uniformity.

Mass transfer based on chemical potential theory: ZnSO4/H2SO4/D2EHPA

AIChE Journal ◽  
1997 ◽  
Vol 43 (10) ◽  
pp. 2479-2487 ◽  
Author(s):  
Hartmut Klocker ◽  
Hans-Jörg Bart ◽  
Rolf Marr ◽  
Hansjourg MÜLler
Keyword(s):  
Mass Transfer ◽  
Potential Theory ◽  
Chemical Potential

MEAN-FIELD DYNAMICAL SEMIGROUPS ON C*-ALGEBRAS

1992 ◽  
Vol 04 (03) ◽  
pp. 383-424 ◽  
Author(s):  
N.G. DUFFIELD ◽  
R.F. WERNER
Keyword(s):  
State Space ◽  
Mean Field ◽  
Liapunov Function ◽  
The State ◽  
Mean Field Limit ◽  
Lattice Systems ◽  
C Algebra ◽  
C Algebras ◽  
Dynamical Semigroup ◽  
The Mean

We study a notion of the mean-field limit of a sequence of dynamical semigroups on the n-fold tensor products of a C*-algebra [Formula: see text] with itself. In analogy with the theory of semigroups on Banach spaces we give abstract conditions for the existence of these limits. These conditions are verified in the case of semigroups whose generators are determined by the successive resymmetrizations of a fixed operator, as well as generators which can be approximated by generators of this type. This includes the time evolutions of the mean-field versions of quantum lattice systems. In these cases the limiting dynamical semigroup is given by a continuous flow on the state space of [Formula: see text]. For a class of such flows we show stability by constructing a Liapunov function. We also give examples where the limiting evolution is given by a diffusion, rather than a flow on the state space of [Formula: see text].

scholarly journals REENTRANT BEHAVIOR IN LANDAU–FERMI LIQUIDS WITH SPIN-SPLIT POMERANCHUK INSTABILITIES

2012 ◽  
Vol 26 (19) ◽  
pp. 1250125 ◽  
Author(s):  
P. RODRÍGUEZ PONTE ◽  
D. C. CABRA ◽  
N. GRANDI
Keyword(s):  
Magnetic Field ◽  
Phase Space ◽  
Fermi Liquid ◽  
Chemical Potential ◽  
Square Lattice ◽  
Fermi Liquids ◽  
Lattice Systems ◽  
Fermi Surfaces ◽  
The Stability ◽  
The Magnetic Field

We study the effects of spin-antisymmetric interactions on the stability of a Landau–Fermi liquid on the square lattice, using the generalized Pomeranchuk method for two-dimensional lattice systems. In particular, we analyze interactions that could induce instabilities of the so-called spin-split type, that is when spin-up and spin-down Fermi surfaces are displaced with respect to each other. The phase space is studied as a function of the strength of the interaction V, the electron chemical potential μ and an external magnetic field h. We find that such interactions produce in general an enhancement of the instability region of the Landau–Fermi liquid. More interestingly, in certain regions of the V–μ phase space, we find a reentrant behavior as a function of the magnetic field h, similar to that found in recent experiments, e.g. in URu 2 Si 2 and Sr 3 Ru 2 O 7.

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