29 research outputs found
A Phase Transition in a Quenched Amorphous Ferromagnet
Quenched thermodynamic states of an amorphous ferromagnet are studied. The
magnet is a countable collection of point particles chaotically distributed
over , . Each particle bears a real-valued spin with
symmetric a priori distribution; the spin-spin interaction is pair-wise and
attractive. Two spins are supposed to interact if they are neighbors in the
graph defined by a homogeneous Poisson point process. For this model, we prove
that with probability one: (a) quenched thermodynamic states exist; (b) they
are multiple if the particle density (i.e., the intensity of the underlying
point process) and the inverse temperature are big enough; (c) there exist
multiple quenched thermodynamic states which depend on the realizations of the
underlying point process in a measurable way
Stochastic dynamics of particle systems on unbounded degree graphs
We consider an infinite system of coupled stochastic differential equations
(SDE) describing dynamics of the following infinite particle system. Each
partricle is characterised by its position and internal
parameter (spin) . While the positions of particles
form a fixed ("quenched") locally-finite set (configuration)
, the spins and interact via a pair
potential whenever is a
fixed interaction radius. The number of particles interacting with a
particle in positionn is finite but unbounded in . The growth of
as creates a major technical problem for solving our SDE
system. To overcome this problem, we use a finite volume approximation combined
with a version of the Ovsjannikov method, and prove the existence and
uniqueness of the solution in a scale of Banach spaces of weighted sequences.
As an application example, we construct stochastic dynamics associated with
Gibbs states of our particle system
Stochastic differential equations in a scale of Hilbert spaces
A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in R
Gibbs states of continuum particle systems with unbounded spins : existence and uniqueness
We study an infinite system of particles chaotically distributed over a Euclidean space Rd. Particles are characterized by their positions x∈Rd and an internal parameter (spin) σx∈Rm and interact via position-position and (position dependent) spin-spin pair potentials. Equilibrium states of such system are described by Gibbs measures on a marked configuration space. Due to the presence of unbounded spins, the model does not fit the classical (super-) stability theory of Ruelle. The main result of the paper is the derivation of sufficient conditions of the existence and uniqueness of the corresponding Gibbs measures