In one and two dimensions,every stationary measure for a stochastic Ising Model is a Gibbs state

It is shown that one and two dimensional (generalized) stochastic Ising models with finite range potentials have only Gibbs states as their stationary measures. This is true even if the stationary measure or the potential is not translation invariant. This extends previously known results which are restricted to translation invariant stationary measures and potentials. In particular if the potential has only one Gibbs state the stochastic Ising Model must be ergodic.Research supported in part by N.S.F. Grant MPS 74-18926Alfred P. Sloan Fellow     

Analogues of Non-Gibbsianness in Joint Measures of Disordered Mean Field Models

It is known that the joint measures on the product of spin-space and disorder space are very often non-Gibbsian measures, for lattice systems with quenched disorder, at low temperature. Are there reflections of this non-Gibbsianness in the corresponding mean-field models? We study the continuity properties of the conditional probabilities in finite volume of the following mean field models: (a) joint measures of random field Ising, (b) joint measures of dilute Ising, (c) decimation of ferromagnetic Ising. The conditional probabilities are functions of the empirical mean of the conditionings; so we look at the large volume behavior of these functions to discover non-trivial limiting objects. For (a) we find (1) discontinuous dependence for almost any realization and (2) dependence of the conditional probabilities on the phase. In contrast to that we see continuous behavior for (b) and (c), for almost any realization. This is in complete analogy to the behavior of the corresponding lattice models in high dimensions. It shows that non-Gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean-field models.     

Molecular motor that never steps backwards

We investigate the dynamics of a classical particle in a one-dimensional two-wave potential composed of two periodic potentials that are time independent and of the same amplitude and periodicity. One of the periodic potentials is externally driven and performs a translational motion with respect to the other. It is shown that, if one of the potentials is of the ratchet type, translation of the potential in a given direction leads to motion of the particle in the same direction, whereas translation in the opposite direction leaves the particle localized at its original location. Moreover, even if the translation is random, but still has a finite velocity, an efficient directed transport of the particle occurs.     

Quasipotentials for simple noisy maps with complicated dynamics

The theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems, leading to exponential estimates of invariant probabilities and mean first exit times. In the present article we develop the mathematical foundation of this theory for discrete-time systems, following and extending the work of Freidlin and Wentzell, and Kifer. We discuss strategies for calculating and estimating quasipotentials and show their application to one-dimensionalS-unimodal maps. The method proves to be especially suited for describing the noise scaling behavior of invariant probabilities, e.g., for the map occurring as the limit of the Feigenbaum period-doubling sequence. We show that the method allows statements about the scaling behavior in the case of localized noise, too, which does not originally lie within the scope of the quasipotential formalism.     

On a question of Bratteli and Robinson

We give a partial answer to a question raised by Bratteli and Robinson, by showing that the mean entropy equals the mean conditional entropy for a large class of translation invariant states. In the general case we show that equality holds if and only if the mean conditional entropy is upper semicontinuous in the w^*-topology.     

The localization properties of a random steady flow on a lattice

We consider a random stationary vector field on a multidimensional lattice and investigate flow-connected subsets of the lattice invariant under the action of the associated flow. The subsets of primary interest are cycles, and vortices each of which is the set of orbits terminating in the same cycle. We prove that with probability 1 each vortex only involves a finite number of sites of the lattice. Under the assumption of independence of the vector field in different sites, we find that with probability 1 the vortices exhaust all possible maximal flowconnected invariant subsets of the lattice if and only if the probability of existence of a cycle is positive. Thus, if cycles exist, a particle under the action of the flow only moves within a bounded region, i.e., it is completely localized.     

A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker-Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker-Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was offered that a Chapman-Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker-Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker-Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker-Planck pdes and Chapman-Kolmogorov equations for stochastic processes with finite memory.     

Weakly Gibbsian Measures and Quasilocality: A Long-Range Pair-Interaction Counterexample

We exhibit an example of a measure on a discrete and finite spin system whose conditional probabilities are given in terms of an almost everywhere absolutely summable potential but are discontinuous almost everywhere.     

Blocking and Dimer Processes on the Cayley Tree

We show that the equilibrium distribution for the dimer process on the finite Cayley tree tends to a translation invariant limit as the size of the tree tends to infinity. The same is true for the blocking process except when there is a phase transition, in which case there are two limits, each a one-step translation of the other. We also find correlations for occupation probabilities.     

Boltzmann Limit and Quasifreeness for a Homogenous Fermi Gas in a Weakly Disordered Random Medium

We discuss some basic aspects of the dynamics of a homogenous Fermi gas in a weak random potential, under negligence of the particle pair interactions. We derive the kinetic scaling limit for the momentum distribution function with a translation invariant initial state and prove that it is determined by a linear Boltzmann equation. Moreover, we prove that if the initial state is quasifree, then the time evolved state, averaged over the randomness, has a quasifree kinetic limit. We show that the momentum distributions determined by the Gibbs states of a free fermion field are stationary solutions of the linear Boltzmann equation; this includes the limit of zero temperature.     

Conditional Expectations Relative to a Product State and the Corresponding Standard Potentials

For a lattice system with a finite number of Fermions and spins on each lattice point, conditional expectations relative to an even product state (such as Fermion Fock vacuum) are introduced and the corresponding standard potential for any given dynamics, or more generally for any given time derivative (at time 0) of strictly local operators, is defined, with the case of the tracial state previously treated as a special case. The standard potentials of a given time derivative relative to different product states are necessarily different but they are shown to give the same set of equilibrium states, where one can compare states satisfying the variational principle (for translation invariant states) or the local thermodynamical stability or the Gibbs condition, all in terms of the standard potential relative to different even product states.     

Rationality in conformal field theory

We show that if the one-loop partition function of a modular invariant conformal field theory can be expressed as a finite sum of holomorphically factorized terms thenc and all values ofh are rational.     

Field algebras do not leave field domains invariant

It is proved that von Neumann algebras associated to Op*-algebra (P, D) cannot leave the domainD ofP invariant if they are type I or type III factors or finite direct sums of such factors. Hence it follows that in quantum field theory global and local von Neumann field algebras in typical cases do not leave invariant the definition domain of Wightman fields.     

On the inverse problem in statistical mechanics

For quantum spin systems it is known that for a suitable space of potentials the equilibrium states areW*-dense in the set of all translation invariant states. The problem discussed in this paper is how to recognize such equilibrium states and how to find the corresponding potential. A necessary and sufficient condition for a state to be an equilibrium state for some potential is given in Sect. 3.     

The classical mechanics of one-dimensional systems of infinitely many particles

We apply the existence theorem for solutions of the equations of motion for infinite systems to study the time evolution of measures on the set of locally finite configurations of particles. The set of allowed initial configurations and the time evolution mappings are shown to be measurable. It is shown that infinite volume limit states of thermodynamic ensembles at low activity or for positive potentials are concentrated on the set of allowed initial configurations and are invariant under the time evolution. The total entropy per unit volume is shown to be constant in time for a large class of states, if the potential satisfies a stability condition.On leave from: Department of Mathematics, University of California, Berkeley, California.     

Scalar quantum field theory on fractals

We investigate scale invariant measures over multiple variables for scalar field theories by imitating Wiener’s construction of the measure on the space of functions of one variable. We assign random fields values on the vertices of simple geometric shapes (triangles, squares, tetrahedra) which are subdivided into a finite number of similar shapes. We find several Gaussian measures with anomalous scaling associated with these field variables. A non-Gaussian fixed point arises from the Ising model on a fractal. In the continuum limit, we construct correlation functions that vary as a power of the distance. It is either a positive power (analogous to the Wiener process) or a negative power depending on the subdivision scheme used; however it is an irrational number for all the examples. This suggests that in the continuum limits it corresponds to quantum field theories (random fields) on spaces of fractional dimension.     

Time-delay in potential scattering theory

We show that a gauge field uniquely determines its potential if and only if its holonomy group coincides with the gauge group on every open set in spacetime, provided that the field is not degenerate as a 2-form over spacetime. In other words, there is no potential ambiguity whenever such a field is irreducible everywhere in spacetime. We then show that the ambiguous potentials for those gauge fields are partitioned into gauge-equivalence classes (modulo certain homotopy classes) as a consequence of the nontrivial connectivity of spacetime. These homotopy classes depend on the gauge group, on the holonomy group and on this last group's centralizer in the gauge group.To the Memory of Jorge André SwiecaResearch supported by C.N.Pq. and M.E.C. (Brazil)     

Quantum Hall Effect on the Hyperbolic Plane in the Presence of Disorder

We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in Comm. Math. Phys. 190 (1998), 629–673, to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.     

Translation invariant tensor product states in a finite lattice system

We show that the matrix (or more generally tensor) product states in a finite translation invariant system can be accurately constructed from a same set of local matrices (or tensors) that are determined from an infinite lattice system in one or higher dimensions. This provides an efficient approach for studying translation invariant tensor product states in finite lattice systems. Two methods are introduced to determine the size-independent local tensors.     

Abundance of translation invariant pure states on quantum spin chains

We construct a set of translation invariant pure states of a quantum spin chain, which is w -dense in the set of all translation invariant states of the chain. Each of the approximating states has exponential decay of correlations, and is the unique ground state of a finite range Hamiltonian with a spectral gap above the ground state energy.