Thermodynamics for Spatially Inhomogeneous Magnetization and Young-Gibbs Measures

Alessandro Montino,Nahuel Soprano-Loto,Dimitrios Tsagkarogiannis

doi:10.1007/s10955-016-1592-y

Alessandro Montino, Nahuel Soprano-Loto + Show 1 more

Open Access

https://doi.org/10.1007/s10955-016-1592-y

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Abstract

We derive thermodynamic functionals for spatially inhomogeneous magnetization on a torus in the context of an Ising spin lattice model. We calculate the corresponding free energy and pressure (by applying an appropriate external field using a quadratic Kac potential) and show that they are related via a modified Legendre transform. The local properties of the infinite volume Gibbs measure, related to whether a macroscopic configuration is realized as a homogeneous state or as a mixture of pure states, are also studied by constructing the corresponding Young-Gibbs measures.

Highlights

In order to describe the properties of a material, one studies a minimization problem of a given free energy functional with respect to an appropriate order parameter
Of particular interest is the case when we are in the regime of phase transition between pure states, which corresponds to a linear segment in the graph of the above functional with respect to the order parameter
We connect the two descriptions and derive a macroscopic continuum mechanics theory for scalar order parameter starting from statistical mechanics

Summary

Introduction

In order to describe the properties of a material, one studies a minimization problem of a given free energy functional with respect to an appropriate order parameter. By appropriately choosing ζ , δ and for γ small, the right hand side of (6.10) is bounded by 3e−C5(δ)| ε| for some constant C5(δ) > 0, and the result follows.

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