One-sided versus two-sided stochastic descriptions

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.Comment: 13 pages, Contribution for "Statistical Mechanics of Classical and Disordered Systems

Higher Powers in Gravitation

We consider the Friedmann-Robertson-Walker cosmologies of theories of gravity that generalise the Einstein-Hilbert action by replacing the Ricci scalar, R, with some function, f(R). The general asymptotic behaviour of these cosmologies is found, at both early and late times, and the effects of adding higher and lower powers of R to the Einstein-Hilbert action is investigated. The assumption that the highest powers of R should dominate the Universe's early history, and that the lowest powers should dominate its future is found to be inaccurate. The behaviour of the general solution is complicated, and while it can be the case that single powers of R dominate the dynamics at late times, it can be either the higher or lower powers that do so. It is also shown that it is often the lowest powers of R that dominate at early times, when approach to a bounce or a Tolman solution are generic possibilities. Various examples are considered, and both vacuum and perfect fluid solutions investigated.Comment: 30 pages, 9 figure

The Analyticity of a Generalized Ruelle's Operator

In this work we propose a generalization of the concept of Ruelle operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle operator, that generalizes both the Ruelle operator proposed in [BCLMS] and the Perron Frobenius operator defined in [Bowen]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle operator and present some examples

Phonon distributions of a single bath mode coupled to a quantum dot

The properties of an unconventional, single mode phonon bath coupled to a quantum dot, are investigated within the rotating wave approximation. The electron current through the dot induces an out of equilibrium bath, with a phonon distribution qualitatively different from the thermal one. In selected transport regimes, such a distribution is characterized by a peculiar selective population of few phonon modes and can exhibit a sub-Poissonian behavior. It is shown that such a sub-Poissonian behavior is favored by a double occupancy of the dot. The crossover from a unequilibrated to a conventional thermal bath is explored, and the limitations of the rotating wave approximation are discussed.Comment: 21 Pages, 7 figures, to appear in New Journal of Physics - Focus on Quantum Dissipation in Unconventional Environment