Statistical Mechanical Systems Research Articles

Ergodic properties of a semi-infinite one-dimensional system of statistical mechanics

We consider the dynamical system ( $$\mathfrak{X}$$ ,μ,T t ) where ( $$\mathfrak{X}$$ ,μ) is the Gibbs ensemble at some fixed temperature and density for a semi-infinite one-dimensional ideal gas of point particles. The first particle has massM, all the other particles massm<M. T t is the time evolution which describes free motion of the particles except for elastic collisions with each other and with the wall at the origin. We prove that ( $$\mathfrak{X}$$ ,μ,T t ) is aK-flow.

Distribution of the eigenvalues of the energy operator of a continuous system in quantum statistical mechanics

Distribution of the eigenvalues of the energy operator of a continuous system in quantum statistical mechanics

Baryons in lattice QCD at strong coupling

Lattice QCD with euclidean Susskind fermions and gauge group SU( N) is transformed into a statistical mechanical system of dimers and baryons for infinite coupling. For even N the baryons are bosons, all configurations have positive weights, and the system can be simulated in a straightforward way. The original quark determinant is fully and exactly taken into account. As a byproduct chiral symmetry breaking and stability against spontaneous breakdown of baryon number conservation is verified.

Improved Hamiltonian variational technique for lattice models.

In the context of the Hamiltonian variational approach we propose a systematic method for the improvement of any given trial wave function. The approach has many similarities with the Lanczos scheme and it may be used for quantum-mechanical problems as well as field-theory and statistical-mechanics systems. We apply the method to the Mathieu equation and to the Ising one-dimensional model in a finite lattice. The agreement between our results and the exact ones is excellent in the whole range of parameters. We also briefly discuss the application of these ideas to lattice gauge theories and the similarities with other recently proposed methods.

Being Discrete about Yang and Mills: Basic Techniques of Euclidean Lattice Gauge Theory

We discuss the formulation of gauge-invariant quantum field theories (without dynamical matter fields) as statistical mechanics systems on four-dimensional Euclidean lattices. Approximation methods including strong- and weak-coupling expansions, mean-field theory and Monte Carlo simulations are reviewed in detail, and Abelian duality transformations are derived. New models are discussed. An action is defined on 2 × 1 rectangular loops of links and its properties are investigated. It is found to result in phase transitions in 2, 3 and 4 dimensions with Z(2) and SU(2) gauge groups. A large class of models with Z(N) symmetry realised on plaquettes is investigated, and several phase diagrams are presented. A mixed model with interactions through both plaquettes and rectangles is found to have a line of phase transitions and a critical point associated with the crossover region in the Wilson SU(2) model.

A correlation decay theorem at high temperature

We derive a theorem of exponential decay of correlation functions at high temperature for a general statistical mechanical system following the approach introduced by L. Gross. The theorem is formulated so as to be useful for locality problems in lattice quantum gravity.

Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images

We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.

Path-integral Riemannian contributions to nuclear Schrödinger equation

Several studies in quantum mechanics and statistical mechanics have formally established that nonflat metrics induce a difference in the potential used to define the path-integral Lagrangian from that used to define the differential Schr\"odinger Hamiltonian. A recent study has described a statistical-mechanical biophysical system in which this effect is large enough to be measurable. This study demonstrates that the nucleon-nucleon velocity-dependent interaction derived from meson exchanges is a quantum-mechanical system in which this effect is also large enough to be measurable.

Microcanonical quantum gravity

Quantum gravity is equivalent to a statistical mechanical system of classical fields in 4 + 1 dimensions at temperatureħ. It is generically true that systems involving gravitation do not exist at fixed temperature. We therefore propose a fixed action description of quantum gravity. The formalism for a fixed action description of quantum field theory is developed. It is found to be equivalent to the canonical fixedħ formalism for renormalizable field theories. It is also found that, in precise analogy to the microcanonical formulation of statistical mechanics, it describes a wider class of theories than the canonical formalism.

Finite-size scaling and phenomenological renormalization (invited)

Research in recent years has shown that combining finite-size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two-dimensional statistical mechanical and one-dimensional quantum mechanical systems. We review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q-state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self-avoiding random walk, percolation, and Kosterlitz-Thouless transitions are briefly reviewed. The Kosterlitz-Thouless transitions and the critical fan in the antiferromagnetic 3-state Potts model are discussed at somewhat greater length.

The global Markov property for lattice systems

We prove the global Markov property for lattice systems of classical statistical mechanics, with bounded spins and finite range interactions. The method uses the one developed by two of us to prove the global Markov property of Euclidean generalized random fields. The result shows that the systems considered have a transition matrix, which together with a distribution on a hyperplane, describes completely the system.

Disorder variables for a non-abelian symmetry group

This paper is concerned with the construction of local disorder operators for two-dimensional statistical mechanical systems, and with the representation of these operators in transfer matrix language. Each internal global symmetry operation leads to a separate disorder variable. This idea is illustrated in the example of the Ashkin-Teller model, in which the symmetry operations form a non-abelian group. There are seven non-trivial disorder operators. Five of these are shown to be simply duals of various kinds of spin operators. Series analysis is used to describe the critical behavior of one of the remaining two operators. No essentially new scaling behavior is observed.

Nonequilibrium statistical mechanics. I. Mathematical formalism. Theorem of N. S. Krylov

Based on an algebraic formulation of quantum mechanics we introduce concepts playing a fundamental role in the constructionof the statistical mechanics of systems having direct classical analogs. We give the definition of a macrostate, mixing in the quantum version, and also demonstrate the existence of an upper bound to the relaxation time for an isolated system. It is shown that the theory constructed here contains both quantum and classical mechanics as limiting cases, but these two cases are not reducible to each other. The existence of a transition range not describable by the Schrodinger equation is noted.

Binding under a molecular ‘umbrella’ as a cooperative statistical mechanical system: tropomyosin-actin-myosin as an example

We Study models in which initial ligand molecules can bind on a linear array of sites only by lifting or moving an adjacent polymer molecule, over a range of a few binding sites, into a state of higher free energy. This inhibits the initial binding but subsequent ligand molecules can then bind more easily under the "umbrella" already lifted in the initial binding (a cooperative effect). Three subjects are considered: cooperative properties of a general model in which, besides the "umbrella" effect, the polymer is divided into m-site long segments, with interactions between the ends of these segments (section 3); application of this model to tropomyosin-troponin-myosin (section (a cooperative effect). Three subjects are considered: cooperative properties of a general model in which, besides the "umbrella" effect, the polymer is divided into m-site long segments, with interactions between the ends of these segments (section 3); application of this model to tropomyosin-troponin-myosin (section (a cooperative effect). Three subjects are considered: cooperative properties of a general model in which, besides the "umbrella" effect, the polymer is divided into m-site long segments, with interactions between the ends of these segments (section 3); application of this model to tropomyosin-troponin-myosin (section 2); and cooperative properties of simpler models in which the polymer is not divided into segments though the "umbrella" can have various ranges (section 4).

On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems

We develop a unified approach, based on Araki's relative entropy concept, to proving absence of spontaneous breaking of continuous, internal symmetries and translation invariance in two-dimensional statistical-mechanical systems. More precisely, we show that, under rather general assumptions on the interactions, all equilibrium states of a two-dimensional system have all the symmetries, compact internal and spatial, of the dynamics, except possibly rotation invariance. (Rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance.) We also prove that two-dimensional systems with a non-compact internal symmetry group, like anharmonic crystals, typically do not have Gibbs states.

Finite-size rescaling transformations

A theory of finite-size rescaling transformations (FSRT) is derived. It yields accurate estimates of the critical properties of statistical-mechanical systems of infinite extent. The arguments used are analogous to, but different from, conventional renormalization-group theory. The FSRT do not involve block spins and do not introduce any new couplings beyond those already in the problem. A technique which was suggested phenomenologically by Nightingale is derived as a special case of the FSRT theory. The optimization of convergence rates, and the relationship between FSRT and the finite-size scaling hypotheses of Fisher, are examined. Some new numerical results are presented.

A remark on Dobrushin's uniqueness theorem

Ten years ago, Dobrushin [1] proved a beautiful result showing that under suitable hypotheses, a statistical mechanical lattice system interaction has a unique equilibrium state. In particular, there is no long range order, etc. see [6,7] for related material, Israel [4] for analyticity results and Gross [3] for falloff of correlations. There does not appear to have been systematic attempts to obtain very good estimates on precisely when Dobrushin's hypotheses hold, except for certain spin models [6,4]. Our purpose here is to note that with one simple device one can obtain extremely good estimates which are fairly close to optimal. Let Ω be a fixed compact space (single spin configuration space), dμ0 a probability measure on Ω and for each α e Z, Ωα a copy of Ω. For X a finite subset of TD', let Ω — X Ωα. An interaction is an assignment of a continuous function, ΦpO, on Ω to each finite XcTL. While it is not necessary for Dobrushin's theorem, it is convient notationally to suppose Φ translation covariant in the obvious sense.

Limit Gibbs state for some classes of one-dimensional systems of quantum statistical mechanics

We prove the existence of the limit Gibbs state for one-dimensional continuous quantum fermion systems with non-hard-core, non-negative, rapidly decreasing pair interaction potentials. Existence of the limit Gibbs state is also established for one-dimensional continuous quantum boson systems with pair interaction potentials as above which, in addition, increase sufficiently fast at small distances.

Dynamics of the open BCS model

The dynamics of the strong coupling BCS model, considered as an open system interacting with a thermal bath, is solved rigorously and explicitly in the weak coupling limit and in the infinite-volume limit. The BCS system goes from the normal phase to the ordered phase by bifurcation. Fluctuations around trajectories of intensive observables are Gaussian and Markovian. Thermodynamic phases are global attractors in the physical domain. Structural stability is discussed. The model provides an example of a nonequilibrium statistical mechanical system with phase transition whose irreversible macroscopic dynamics can be calculated exactly from the underlying Hamiltonian quantum mechanics.

Equivalence of the canonical and grand canonical ensembles for one-dimensional systems of quantum statistical mechanics

Equivalence of the canonical and grand canonical ensembles for one-dimensional systems of quantum statistical mechanics