Project 1: Cauchy-Born
rule at positive temperature
Participants
Stefan Adams, Jean-Dominique Deuschel, Roman Kotecký, Stephan
Luckhaus, Stefan Müller, Max
von Renesse
Summary
We consider gradient Gibbs fields in the context of both effective
interface models and lattice spring models of nonlinearly elastic
crystals.
In case of strictly convex interaction potential, Funaki and Spohn have
characterised the ergodic extremal gradient Gibbs states.
Also the microcanonical Gibbs distribution for fixed volume can be
derived via large deviation principle in this case.
In the context of lattice spring models a realistic interaction has to
be nonconvex in view of frame indifference.
Friesecke and Theil have shown for a model problem that despite the
lack of convexity the Cauchy-Born rule holds in a certain
parameter regime, i.e., the ground state for affine boundary conditions
is given by an affine deformation.
Their argument uses central ideas from the (continuum) calculus of
variations, in particular the existence on interesting null Lagrangians.
Our objective is to relax the convexity asumption of the interaction
potential and characterise the ergodic Gibbs states
for both the interface model (which corresponds to a scalar independent
variable) and lattice spring model
(which corresponds to a vector-valued independent variable).
Achievements of the research group:
C. Cotar, J.-D. Deuschel and S. Müller:
Strict convexity of the free energy for non-convex gradient models at moderate $\beta$
Comm. Math. Phys., to appear
preprint
C.Cotar and J.-D. Deuschel:
Decay of covariances, uniqueness of ergodic component and scaling limit for a class of $\nabla\phi$ systems with non-convex potential
preprint
F. Caravenna and J.-D. Deuschel:
Scaling limits of (1+1)-dimensional pinning models with Laplacian interaction
preprint
F. Caravenna and J.-D. Deuschel:
Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
Ann. Probab., to appear.
preprint