Journal of Statistical Physics

For n + 1 particles moving independently on a straight line, we study the question of how long the leading position of one of them can last. Our focus is the asymptotics of the probability pT,n that the leader time will exceed T when n and T are large. It is assumed that the dynamics of particles are described by independent, either stationary or self-similar, Gaussian processes, not necessarily identically distributed. Roughly, the result for particles with stationary dynamics of unit variance is as follows: $$ L: = - \ln p_{T,n}/(T\ln n) = 1/d_{0} + o(1), $$ where d0/(2π) is the power of the zero frequency in the spectrum of the leading particle, and this value is the largest in the spectrum. Previously, in some particular models, the asymptotics of L was understood as a sequential limit first over T and then over n. For processes that do not necessarily have non-negative covariances, the limit over T may not exist. To overcome this difficulty, the growing parameters T and n are considered in the domain $$ c\;{\text{ln}}\;T < n \le CT $$ where c > 1. The Lamperti transform allows us to transfer the described result to self-similar processes by changing the $$ \ln p_{T,n} $$ normalization to the value $$ \ln T\ln n $$ .

The random variable number of cluster defined on the configurations of a ferromagnetic Ising model at zero field and inverse temperatureβ on a graphG is considered. The Gibbs average atβ=0 is proved to be greater than the one atβ>0 if the degree ofG is not greater than 3.

We consider the novel properties of the Stell–Hemmer core-softened potentials. First we explore how the theoretically predicted second critical point for these potentials is related to the occurrence of the experimentally observed solid–solid isostructural critical point. We then discuss how this class of potentials can generate anomalies analogous to those found experimentally in liquid water.

An interacting particle system (Glauber dynamics) which evolves on a finite subset in thed-dimensional integer lattice is considered. It is known that a mixing property of the Gibbs state in the sense of Dobrushin and Shlosman is equivalent to several very strong estimates in terms of the Glauber dynamics. We show that similar, but seemingly much milder estimates are again equivalent to the Dobrushin-Shlosman mixing condition, hence to the original ones found by Stroock and Zegarlinski. This may be understood as the absence of intermediate speed of convergence to equilibrium.

A proof of Anderson localization is obtained by ruling out any continuous spectrum on the basis of the space-time characteristic of its states.

We develop a strategy to establish a Wegner estimate and localization in random lattice Schrödinger operators on $\Bbb{Z}^{d}$ , which does not rely on the usual eigenvalue variation argument. Our assumption is that the potential V(ω) depends real analytically on ω and we use a distributional property of analytic functions in many variables. An application is given to models where V n is a self-adjoint matrix obtained by random unitary conjugation V n =U n AU * of a fixed matrix A.

We present the first example of an exponentially decaying interaction which gives rise to nonperiodic long-range order at positive temperatures.

Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.

A laser with intracavity saturable absorber showing optical bistability is investigated through the simultaneous injection of modulation and noise on the pumping parameter; stochastic resonance is exhibited in the measured signal-to-noise ratio of the laser intensity.

Rigorous correlation inequalities are presented for a class of even ferromagnets, which includes the spin-1/2 Ising model and scalar ϕ4 models. One of them leads to an extension of the Glimm and Jaffe uniform upper bound on the ϕ4 renormalized coupling constant into the nonsymmetric regime.