34 research outputs found
A Droplet within the Spherical Model
Various substances in the liquid state tend to form droplets. In this paper
the shape of such droplets is investigated within the spherical model of a
lattice gas. We show that in this case the droplet boundary is always
diffusive, as opposed to sharp, and find the corresponding density profiles
(droplet shapes). Translation-invariant versions of the spherical model do not
fix the spatial location of the droplet, hence lead to mixed phases. To obtain
pure macroscopic states (which describe localized droplets) we use generalized
quasi-averaging. Conventional quasi-averaging deforms droplets and, hence, can
not be used for this purpose. On the contrary, application of the generalized
method of quasi-averages yields droplet shapes which do not depend on the
magnitude of the applied external field.Comment: 22 pages, 2 figure
Random walks in random environment with Markov dependence on time
We consider a simple model of discrete-time random walk on Zν, ν = 1, 2, . . . in a random environment
independent in space and with Markov evolution in time. We focus on the application of methods based on
the properties of the transfer matrix and on spectral analysis. In section 2 we give a new simple proof of the
existence of invariant subspaces, with an explicit condition on the parameters. The remaining part is devoted
to a review of the results obtained so far for the quenched random walk and the environment from the point of
view of the random walk, with a brief discussion of the methods.Ми розглядаємо просту модель випадкового блукання з дискретним часом у Zν, ν = 1, 2, . . . у випадковому середовищi, що є незалежним у просторi i має маркiвську еволюцiю у часi. Ми зосереджуємось на застосуваннi методiв, що ґрунтуються на властивостях трансфер-матрицi i на спектральному аналiзi. У §2 ми подаємо просте доведення iснування iнварiантних пiдпросторiв, що використовує явну умову для параметрiв. Решта роботи присвячується огляду результатiв одержаних дотепер для замороженого випадкового блукання i оточення з точки зору випадкового блукання, а також короткому обговоренню методiв
Relaxation times for Hamiltonian systems
Usually, the relaxation times of a gas are estimated in the frame of the
Boltzmann equation. In this paper, instead, we deal with the relaxation problem
in the frame of the dynamical theory of Hamiltonian systems, in which the
definition itself of a relaxation time is an open question. We introduce a
lower bound for the relaxation time, and give a general theorem for estimating
it. Then we give an application to a concrete model of an interacting gas, in
which the lower bound turns out to be of the order of magnitude of the
relaxation times observed in dilute gases.Comment: 26 page
Short-time Gibbsianness for Infinite-dimensional Diffusions with Space-Time Interaction
We consider a class of infinite-dimensional diffusions where the interaction
between the components is both spatial and temporal. We start the system from a
Gibbs measure with finite-range uniformly bounded interaction. Under suitable
conditions on the drift, we prove that there exists such that the
distribution at time is a Gibbs measure with absolutely summable
interaction. The main tool is a cluster expansion of both the initial
interaction and certain time-reversed Girsanov factors coming from the
dynamics
Large Deviations in the Superstable Weakly Imperfect Bose Gas
The superstable Weakly Imperfect Bose Gas {(WIBG)} was originally derived to
solve the inconsistency of the Bogoliubov theory of superfluidity. Its
grand-canonical thermodynamics was recently solved but not at {point of} the
{(first order)} phase transition. This paper proposes to close this gap by
using the large deviations formalism and in particular the analysis of the Kac
distribution function. It turns out that, as a function of the chemical
potential, the discontinuity of the Bose condensate density at the phase
transition {point} disappears as a function of the particle density. Indeed,
the Bose condensate continuously starts at the first critical particle density
and progressively grows but the free-energy per particle stays constant until
the second critical density is reached. At higher particle densities, the Bose
condensate density as well as the free-energy per particle both increase
{monotonously}
A Contour Method on Cayley tree
We consider a finite range lattice models on Cayley tree with two basic
properties: the existence of only a finite number of ground states and with
Peierls type condition. We define notion of a contour for the model on the
Cayley tree. By a contour argument we show the existence of different
(where is the number of ground states) Gibbs measures.Comment: 12 page
The ^4He trimer as an Efimov system
We review the results obtained in the last four decades which demonstrate the
Efimov nature of the He three-atomic system.Comment: Review article for a special issue of the Few-Body Systems journal
devoted to Efimov physic
Directed polymers in Markov random media
We consider a model of directed polymers in discrete space and time assuming a Markov dependence of the environment in time. We extend results on the almost-sure validity of
the central limit theorem for small randomness in space dimension 3 which were previously obtained for an independent environment by relying on two main technical tools: the analysis of the spectrum of a kind of transfer matrix which allows one to treat the averaged model, and the
explicit construction of a multiplicative orthonormal basis in the appropriate L_2 space, together with cluster estimates of cumulants of the basis functions
Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive
We consider a general model of discrete-time random walk X-t on the lattice (nu), nu = 1,..., in a random environment xi={xi(t,x):(t,x)is an element of(nu+1)} with i.i.d. components xi(t,x). Previous results on the a.s. validity of the Central Limit Theorem for the quenched model required a small stochasticity condition. In this paper we show that the result holds provided only that an obvious non-degeneracy condition is met. The proof is based on the analysis of a suitable generating function, which allows to estimate L-2 norms by contour integrals
Interacting random walk of two particles in a dynamical random environment. Decay of correlations.
This paper continues the study of a family of models studied earlier by the authors. Two particles perform discrete-time symmetric random walks on the d-dimensional
integer lattice Z^d and interact locally with each other and with a random field (the “environment”) which is indexed by the lattice points. The environment evolves randomly in time its law of evolution is locally affected by the particles .
The whole system is Markovian, and all interactions are assumed to be sufficiently small.
It is shown that the correlations of the field at two fixed points decay in time as C t^{(−d/2)−1}. Under additional assumptions the constant C may be expressed to first
order as the sum of the corresponding constants for the one-particle model. The proofs are based on the analysis of the spectrum of the system’s transition operator. The
results may be extended to models containing a finite number of particles