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Convexity of the surface tension for non-convex potentials
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Adams, Stefan (2008) Convexity of the surface tension for non-convex potentials. In: 8th German Open Conference on Probability and Statistics, Aachen, 4-7 Mar 2008
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Official URL: http://gocps2008.rwth-aachen.de/registration.php?s...
Abstract
Recently the study of gradient fields has attained a lot of attention because they are space-time analogy of Brownian motions, and are connected to the Schramm-Loewner evolution. The corresponding discrete versions arise in equilibrium statistical mechanics, e.g., as approximations of critical systems and as effective interface models. The latter models - seen as gradient fields - enables one to study effective descriptions of phase coexistence, a major mathematical challenge.
All models of gradient fields have a continuous symmetry and coexistence of different phases breaks this symmetry. In the probabilistic setting gradients field involve the study of strongly correlated random variables. One major problem has been open for several decades. This is the problem for non-convex interactions of the microscopic subsystems.
To understand phase transitions and apply this knowledge to physical complex systems it is of great importance to mathematically study the microscopic origin and to solve the mathematical problem of non-convex energy functions.
We present in the talk the first break through, an interdisciplinary cooperation between analysis and stochastics, using Gaussian measures and multi-scale formalism yielding an analysis in terms of dynamical systems. We outline also the connection to the Cauchy-Born rule which states that the deformation on the atomistic level is locally given by an affine deformation obtained from the deformation gradient in the continuum theory. The work has been done in cooperation with S. Mueller and R. Kotecky.
| Item Type: | Conference Item (Paper) | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science > Mathematics | ||||
| Official Date: | March 2008 | ||||
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| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| Conference Paper Type: | Paper | ||||
| Title of Event: | 8th German Open Conference on Probability and Statistics | ||||
| Type of Event: | Conference | ||||
| Location of Event: | Aachen | ||||
| Date(s) of Event: | 4-7 Mar 2008 |
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