Gibbs states and equilibrium states for finitely presented dynamical systems

We studyfinitely presented dynamical systems (which generalize Axiom A systems) and show that the notions of equilibrium states and Gibbs states (for Hölder continuous functions) are equivalent. Our results extend those of Ruelle, Haydn, and others on Axiom A dynamical systems and statistical mechanics.     

Gibbs free energy and equilibrium states in the Si/Si oxide systems

In this paper, the equilibrium states in the Si/Si oxide systems formed as a result of the phase separation of nonstoichiometric silicon oxide films are studied. The expressions for the Gibbs free energy of Si oxide and Si/Si oxide systems are derived thermodynamically. The transformations of the Gibbs free energy in the amorphous Si/Si oxide and the crystalline Si/Si oxide systems with the change in the amount of separated silicon and the composition of the silicon oxide phase are analyzed. By minimizing the Gibbs free energy of these systems, the equilibrium stoichiometry indices of silicon oxide are calculated as functions of its initial stoichiometry and the temperature. The solubility limits of Si in SiO(2) in equilibrium with amorphous and crystalline Si are determined. The obtained results form the basis for the development of a complete thermodynamic theory of phase separation in nonstoichiometric silicon oxide films with the formation of Si nanoinclusions in the silicon oxide matrix.     

Equivalence of Gibbs ensembles for quantum systems obeying Maxwell-Boltzmann statistics

A generating functional of the canonical ensemble is introduced for quantum systems of particles obeying Maxwell-Boltzmann statistics. Power estimates for the quantum analogs of the particle distribution functions of the canonical and grand canonical ensembles are obtained for systems of particles with hard cores. It is shown on the basis of the power estimates that limiting generating functionals exist for the systems under consideration and satisfy the same Bogolyubov equation. The ensembles are equivalent in this sense.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–8, July, 1984.The authors thank É. A. Arinshtein for useful discussion.     

Complete Equivalence of the Gibbs Ensembles for One-Dimensional Markov Systems

We show the equivalence of the Gibbs ensembles at the level of measures for one-dimensional Markov-Systems with arbitrary boundary conditions. That is, the limit of the microcanonical Gibbs ensemble is a Gibbs measure with an interaction depending on the microcanonical constraint. In fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins (including the boundary spins) are fixed. When conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble. We show that the limit is a Gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures. For this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions. We establish analogous results for finite range potentials.     

Canonical and grand canonical Gibbs states for continuum systems

It is shown that for a large class of interactions any canonical Gibbs state satisfying a natural temperedness condition is a mixture of Gibbs states with appropriate activities, and vice versa. Some general results on Gibbs states and canonical Gibbs states are established. In particular, a differential characterization of Gibbs states is given.     

Equivalence Between Canonical Gibbs Measures and Stationary Measures for Stochastic Lattice-Gas Model

It is proved that, under appropriate conditions on the jump rate and potential, one- and two-dimensional stochastic lattice-gas models (exclusion process with speed change) have only canonical Gibbs measures as their stationary measures. This extends the previously known result, which treats only a special jump rate and potential.     

Purification and uniqueness of quantum Gibbs states

We give a new condition for uniqueness of Gibbs states of quantum spin models on lattices.Research fellow of Canon Foundation in Europe, on leave from Tokyo Metropolitan University, Hachioji, 192-03 Japan     

Uniqueness of one-dimensional continuum Gibbs states

We investigate one-dimensional continuum grandcanonical Gibbs states corresponding to finite range superstable many-body potentials. Absence of phase transitions in the sense of uniqueness of the tempered Gibbs state is proved for potentials without hard-core by first proving uniqueness of the Gibbs measures for related hard-core potentials and then taking an appropriate limit of those Gibbs measures.     

Resolvents,semigroups and Gibbs states for infinite coupling constant

The infinite coupling constant limit of the resolvent, the semigroup and the Gibbs state is obtained for a certain class of perturbations. As an example the infinite intrasite repulsion limit of the one-dimensional Hubbard model with nearest neighbour hopping terms is treated. This system exhibits a phase transition in the thermodynamic limit.     

Entropy,information, and complexity of the steady states of open systems not satisfying the principle of local equilibrium

The entropy of the steady state of an open electronic system is determined, and its relation to the entropy arriving at the environment is established. The notion of system ordering is defined, and quantitative parameters characterizing ordering are introduced. The information and conditional complexity of the system versus the state of thermodynamic chaos are determined. These values are numerically estimated for vacuum, magnetron, and Gunn diodes.     

Stability and equilibrium states

For an infinite dynamical quantum system idealized as aC*-algebra acted upon by time-translations automorphisms in an asymptotically abelian way, we propose to characterize equilibrium states by the three properties of stationarity, stability for local perturbations of the dynamics, and relative purity. We show that a state with these properties either gives rise to a one-sided energy spectrum or is a KMS- (i.e. essentially a limit Gibbs-) state.     

Symmetry and equilibrium states

Within the general framework ofC*-algebra approach to mathematical foundation of statistical mechanics, we prove a theorem which gives a natural explanation for the appearance of the chemical potential (as a thermodynamical parameter labelling equilibrium states) in the presence of a symmetry (under gauge transformations of the first kind). As a symmetry, we consider a compact abelian groupG acting as *-automorphisms of aC*-algebra (quasi-local field algebra) and commuting (elementwise) with the time translation automorphisms _t of . Under a technical assumption which is satisfied by examples of physical interest, we prove that the set of all extremal _t -KMS states (pure phases) ofG-fixed-point subalgebra (quasi-local observable algebra) of satisfying a certain faithfulness condition is in one-to-one correspondence with the set of all extremalG-invariant _t · _t -KMS states ^– of with varying over one-parameter subgroups ofG (the specification of being the specification of the chemical potential), where the correspondence is that the restriction of ^– to is .     

Uniqueness of continuum one-dimensional Gibbs states for slowly decaying interactions

We consider one-dimensional grand-canonical continuum Gibbs states corresponding to slowly decaying, superstable, many-body interactions. Absence of phase transitions, in the sense of uniqueness of the tempered Gibbs state, is proved for interactions with anNth body hardcore for arbitrarily largeN.     

Existence and uniqueness of Gibbs states for a statistical mechanical polyacetylene model

One-dimensional polyacetylene is studied as a model of statistical mechanics. In a semiclassical approximation the system is equivalent to a quantumXY model interacting with unbounded classical spins in one-dimensional lattice spaceZ. By establishing uniform estimates, an infinite-volume-limit Hilbert space, a strongly continuous time evolution group of unitary operators, and an invariant vector are constructed. Moreover, it is proven that any infinite-limit state satisfies Gibbs conditions. Finally, a modification of Araki's relative entropy method is used to establish the uniqueness of Gibbs states.     

Correlation inequalities and equilibrium states

For an infinite dynamical system, idealized as a von Neumann algebra acted upon by a time translation implemented by a HamiltonianH, we characterize equilibrium states (KMS) by stationarity, a Bogoliubov-type inequality and continuous spectrum ofH, except at zero.Aangesteld Navorser NFWO, Belgium     

Gibbs states of a one dimensional quantum lattice

A one dimensional infinite quantum spin lattice with a finite range interaction is studied. The Gibbs state in the infinite volume limit is shown to exist as a primary state of a UHF algebra. The expectation value of any local observables in the state as well as the mean free energy depend analytically on the potential, showing no phase transition. The Gibbs state is an extremal KMS state.On leave from Research Institute for Mathematical Sciences, Kyoto University Kyoto, Japan.     

Stability properties of equilibrium states

The significance of stability of an equilibrium state under local perturbations of the dynamics (as defined in [1]) and the different degree of stability with respect to extended perturbations of states at phase transition points are discussed. The general conclusions are tested and illustrated in the example of the free Bose gas. A more transparent proof of the relation between local stability and the Kubo-Martin-Schwinger relation is given.     

Ergodic properties of equilibrium states

Possible ergodic properties of Gibbs states are discussed by constructing a number of examples. In particular existence of Gibbs states which are mixing but not extremal is shown.