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Gaussian Potential

The potential is given by

\begin{equation}V(\eta) = \frac{c}{2}\eta^{2} c > 0.\end{equation}

For this choice of potential the partition function \(\widehat Z_\lambda\) introduced in \eqref{Cmeas} can be computed explicitly as a Gaussian integral

\[\widehat Z_\lambda =\quad \int_{\mathbb R} e^{\lambda \eta - \frac{c}{2} \eta^2} d\eta \, = \quad \sqrt{ \frac {2\pi}{c} } e^{\lambda^2/ 2c }.\]

Then by the definition of \(x(\lambda)\), taking logs and differentiating gives \(x(\lambda) = \lambda / c\), it follows that \(\lambda(x) = c x\) and hence

\begin{equation}f(x) = \frac{c}{2}x^2.\end{equation}

Superposition of Gaussian Potentials

A non convex example for which our method could be used is given by:

\begin{equation}\exp \big( - V(\eta) \big) = p \exp \left( - \frac{\kappa_1}{2} \eta^2 \right) + (1-p) \exp \left( - \frac{\kappa_2}{2}\eta^2 \right).\end{equation}

Here \(\kappa_1, \kappa_2>0\) are stiffness parameters and it is generally assumed that \(\kappa_1 \gg \kappa_2\). The measure can be reformulated so that with probability \(p\) (respectively \(1-p\)) the distribution is conditioned to behave according to the density \(\exp( - \frac{\kappa_1}{2} \eta^2 )\) ( respectively \(\exp( - \frac{\kappa_2}{2} \eta^2 )\)).

\begin{figure}[h]

\centering

\includegraphics[width=0.5\textwidth]{NonConvex.pdf}

\caption{Superposition of two Gaussian potentials.}

\label{SuperGauss}

\end{figure}

Using the same Gaussian integral evaluation we used for the potential \eqref{GaussianPotential}, we can easily compute the partition function \(\widehat Z_\lambda\) for the superposed Gaussian potential

\begin{equation}\hat {Z_{\lambda}} = \int_{\mathbb R} e^{\lambda \eta} \left ( p \exp \left( - \frac{\kappa_1}{2}\eta^2 \right) \end{equation}

of this lemma needs that in the limit the gradients across the bonds are i.i.d.\@ with distribution described by the Cram\'er measure.

Lemma

For potentials \(V : \mathbb R \rightarrow \mathbb R\) satisfying (\ref{Cond6})-(\ref{Cond2}), then given that a limit measure \(\mu^x\) exists, it is given by
\begin{equation}\mu^{x}(d\eta)=\prod_{b\in \mathcal B( \mathbb{Z})}\hat{\nu}_{\lambda(x)}(d\eta(b)), \qquad \eta \in \Omega \end{equation}

Proof

Remark

For condition \eqref{Cond1} on the moment generating function \(M\) to hold, we require that in the tail, \( { \lim_{\eta \rightarrow \pm \infty} \lambda \eta - V (\eta) = - \infty } \), so \(V\) needs to have superlinear growth for \(\eta \gg 1\), and for \(\eta \ll -1\), the decay must be slower than linear. If \(V\) is a symmetric polynomial satisfying \eqref{Cond1} then we note that \eqref{Cond2} holds since the derivative has lower order than \(V\).

In \cite{FS_97}, Funaki and Spohn provide a theory for nearest neighbour gradient models in dimensions \(d \geq 2\), when the potential satisfies the following conditions

\begin{equation}V \in C^{2}(\mathbb{R}),\end{equation}

\begin{equation}V(-\eta)=V(\eta) \hspace{5mm} \forall\eta\in\mathbb{R}\end{equation}

\begin{equation}c_- \leq V''(\eta) \leq c_+, \, \forall \eta \in \mathbb R, \, \text{for some } c_-, c_+ > 0.\end{equation}

In particular under these conditions, Funaki and Spohn use the Brascamp-Lieb inequality to prove tightness of the sequence \(\{\mu_N^x\}_{N \in \mathbb N}\), which ensures the existence of the limiting distribution \(\mu^x\) (where we understand this as the weak limit of a subsequence in \(\{\mu_N^x\}_{N \in \mathbb N}\)). Having estabished this they proceed to show the existence and convexity of the free energy, as well as describing results for \(\nabla f(x)\).

Question:
We take as our starting point the following question, asked in \cite{FS_97} p.5: what effect does relaxing the conditions (\ref{Cond5}-\ref{Cond4}) have on the shape of \(f\)? For instance, are there flat regions that correspond to a macroscopic phase transition, or could it have cusps which would correspond to a roughening transition?

Our aim is to first address this question for the special \(1\)-dimensional models under weaker conditions on the potential. In doing this we find a tractable expression for the free energy, which we apply to some concrete examples of potentials.

Introduction and Motivation

Our project focuses on the study of a gradient field model: in particular we consider a \(1\)-dimensional model on a lattice box \(\Lambda \subset \mathbb Z\) and consider the thermodynamic limit as \(\Lambda \rightarrow \mathbb Z\). The model is determined by a Gibbs distribution which weights possible configurations according to how nearby particles interact; the model is introduced formally in section \ref{theIntroduction}, but first we provide a motivation from material science.


Modelling Defects in Crystal Structures

The physical motivation for the work in this report is the following problem, in which we want to model a material which is made up of many interacting atoms. We would like to be able to study the effects that defects have on the atomic structure of the material, and whether microscopic defects can have macroscopic effect. Figure~\ref{DefectFoell} gives examples of the possible defects that may occur in such a material, and descriptions can be found in \cite{F_12}.


\begin{figure}[h]
\begin{center}
\includegraphics[height=60mm]{defekte_alte_uebersicht.pdf}
\end{center}
\caption{Possible defects that can occur in a crystal. Image courtesy of Helmut Föll, \cite{F_12}.}
\label{DefectFoell}
\end{figure}
Studying such materials in realistic dimensions (i.e. \(d = 2, 3\) ) is typically hard, so we turn to a toy model in \(1\)-dimension which we hope to capture some of the features of the material. To justify this approach we consider the model to be made up of many layers of particles, so that taking a cross section through the material results in a collection of \(1\)-dimensional systems stacked on top of one another, Figure~\ref{Defect1}. In our model we consider just one of these layers, and model it by considering the particles as the sites of a lattice segment which are then displaced by distorting forces.

\begin{figure}[h]
\begin{center}
\includegraphics[height=60mm]{Defect.pdf}
\end{center}
\caption{A force field is induced by a defect in the material, and is seen to effect the behaviour of surrounding layers of particles.}
\label{Defect1}
\end{figure}

The \(1\)-dimensional gradient Gibbs model takes into account the interactions between particles in one layer of the material which we model by configurations of particles in a box \(\Lambda \subset \mathbb Z\), where interactions between particles can have arbitrary distance (though we concentrate on the cases of nearest and next-nearest neighbour potentials); we are interested in describing the behaviour of the model as we take the thermodynamic limit and send \(\Lambda \rightarrow \mathbb Z\). We also assume that the material is under stress, which is expressed via a tilt boundary condition, as shown by the arrows in figure~\ref{Defect1}. When \(x>0\) this tilt can be interpreted as a stretch that is applied pulling the particles apart, whilst when \(x<0\) the tilt plays the role of a squeezing force.

For the most part of this project we concentrate on the case in which there is no defect in the material, though in section \ref{sec:defectSection} we take such extra forces into account. At the end of this section we describe how such forces can be encorporated into the model definition.

We assume that in one of the lower layers of particles there is a defect which creates a force field that impacts the layers around it as shown in figure~\ref{Defect1}. This force is modelled by adding another term to the Hamiltonian,

\begin{equation}
H^{f}_{N}(u):= H_{N}(u) + \sum_{i=0}^{N+1} f_{i}(u_{i})
\end{equation}



where the sequence of functions \(\{f_{i}\}_{i\in\Z}\) represents the strength of the force field on particles sitting at lattice sites \(i\in\Z\)and \(u\in\R^{N+1}\) is a configuration of particle displacements from their original lattice sites. A Taylor expansion of the functions \(f_{i}\) enables us to approximate the components of this new term as \(\tilde{f}_{i}\cdot u_{i}\) which significantly simplifies the Hamiltonian. We assume that interactions between particles in different layers of the material are incorporated into this force field. This is a very simplified form of the Frenkel-Kontorova model for vacancies and dislocations \cite{O_12}.

There are various forms that we can choose this force field to take. Initially we will consider the simplest case \(f\equiv0\) in order to establish a good modelling foundation on which to build up to cases such as,

  1. A homogeneous force field that exists only inside a finite region around the defect, ie. \(f_{i}=f\) for \(i\in\{-L,\dots,L\}\) and zero otherwise,

  2. An inhomogeneous force field that also exists only inside a finite region, ie. \(\{f_{i}\}_{i\in\{-L,\dots,L\}}\),

  3. An inhomogeneous force field that extends to the whole space but decays rapidly enough for certain quantities in the model to be finite, ie. \(\{f_{i}\}_{i\in\Z}\) such that \(f_{i}\rightarrow 0\) as \(i\rightarrow\pm\infty\).


Our overall purpose in this modelling will be to show existence of and analyse properties of the free energy function related to the above described system. The technicalities involved in the definition of this function are explained in section??

Another concept that is a key tool in such modelling, is coarse graining. In our setting we enclose the region around the defect by a finite box \(\Delta\subset\Z\) and freeze the positions of the particles \(u_{i}\) for \(i\in\Delta\), in effect making them parameters of the model. We then apply the same modelling as we did for the whole system before coarse graining (in terms of looking at free energy) to the regions to the right and left of \(\Delta\), as shown in figure??. In this setting we can again consider the force field \(\{f_{i}\}_{i\in\Z}\) in the cases mentioned above.

The Gradient Gibbs Distribution

Our project considers gradient models on the \(1\)-dimensional lattice \(\mathbb Z\), where configurations are weighted according to a Gibbs distribution; in particular we define the finite state space Gibbs distributions, and analyse the limiting behaviour by studying the specific free energy. In this section we introduce the basic technical notions and definitions which will be used throughout the report.

Let \(\Lambda = \Lambda_N = \{0, 1, \ldots, N\} \subset \mathbb Z\) denote a finite box. Our attention will focus on the case where we equip \(\Lambda\) with periodic boundary conditions, and often refer to it as the \(1\)-dimensional torus on \(N+1\) points, \(\Lambda = T_{N+1}\). The configuration space \(\Omega_N\) on \(\Lambda\) is taken to be the space of real-valued height fields \(\Omega_N = \mathbb R^ \Lambda\), and the weight of a configuration \(u \in \Omega_N\) is determined by a Hamiltonian, \(H(u) = H_N(u)\). In particular, we consider Hamiltonian functions which can be expressed as sums over `\(k\)-th neighbour' gradient potentials
\begin{equation}\label{Hamiltonian}
H_N(u) = \sum_{k=1}^K \sum_{\substack{i, j \in \Lambda \\ |i - j | = k }} V_k \left( \frac{u_i - u_j}{|i - j|} \right),
\end{equation}
where \(K \leq N\) is the greatest interaction distance, and we assume periodic boundary conditions. For the purposes of our essay we will consider only the cases of \(K = 1,2\), and in particular will focus initially on the pure cases of \(V_1 \not \equiv 0, \, V_2 \equiv 0\), and \(V_1 \equiv 0,\, V_2 \not \equiv 0\). Traditional statistical mechanics would then proceed by analysing the Gibbs distribution,
\[
\gamma_N^\beta ( u ) \propto \exp \big( - \beta H(u) \big), \qquad u \in \Omega_N,
\]
where \(\beta > 0\) is the inverse temperature. What makes the gradient model tractable (and gives it its name) is the reinterpretation of the model as a configuration of gradients over bonds. Let \(\mathcal B (\Lambda) = \{ b_i = (i,i+1) : i \in \Lambda \}\) denote the bond set of \(\Lambda_N\), again endowed with periodic boundary conditions so that \(b_N = (N,0) \in \mathcal B (\Lambda)\). Given a configuration \(u \in \Omega_N\) we obtain a bond configuration \(\eta = \eta_u \in \mathbb R^{\mathcal B(\Lambda)}\) by letting \(\eta(b_i) = u_{i+1} - u_i\), be the gradient across bond \(b_i\). Maintaining the interpretation that the value \(\eta(b_i)\) is the gradient across bond \(b_i\), the periodic boundary conditions on \(\Lambda\) enforce the requirement that
\begin{equation}\label{loopCondition}
\sum_{i=0}^N \eta(b_i) = 0,
\end{equation}
implying that the mean gradient must be \(0\): this condition is heuristically justified since the mean gradient of the bonds is \(|\Lambda|^{-1}(u_0 - u_0) = 0\). Henceforth we refer to (\ref{loopCondition}) as the \textit{loop condition}. Now given a configuration \(\eta\) of bonds satisfying (\ref{loopCondition}), if we fix the position of one site, \(u_0 = 0\) for instance, then \(\eta\) uniquely determines a site configuration in \(\Omega_N\). It follows that the models are equivalent up to translation invariance, and henceforth consider the gradient model to be described by a Gibbs distribution on the space of bond configurations satisfying the loop condition. In the case of pure nearest neighbour interactions, \(V_k \equiv 0\), \(k \geq 2\), this correspondence means we can replace the configuration space with
\begin{equation}\label{configurationSpace}
\Omega_N = \{ \eta_i = \eta(b_i) \in \mathbb R : b_i \in \mathcal B(\Lambda) \} = \mathbb{R}^{\mathcal B (\Lambda)}.
\end{equation}
This correspondence generalises to more general potentials, for instance when \(V_2 \not \equiv 0\) we add to \(\mathcal B(\Lambda)\) the set of edges joining next-to-nearest neighbours. Now given a bond configuration space \(\Omega_N\) and a suitably defined Hamiltonian \(H_N\), the gradient Gibbs distribution \(\mu_N \in \mathcal M^1(\Omega_N)\) is defined as
\begin{equation}\label{gibbsDistribution}
\mu_N(d \eta ) = \frac{1}{Z_N^\beta} e^{-\beta H_N( \eta ) } d \eta_N,
\end{equation}
where \(\beta > 0\), \(d \eta_N\) is the uniform measure on \(\Omega_N\), and \(Z_N^\beta\) is the partition function (or normalising constant).

To model physical systems, we endow the gradient model with a tilt, \(x \in \mathbb R\), which corresponds to stretching or compressing \(\Lambda\): if we consider the case that \(\Lambda\) has free boundary conditions as done by Blanc, Le Bris, Legoll and Patz \cite{BLLP_10}, this is implemented by restricting attention to configurations for which the mean bond gradient is \(x\), i.e. \(u_N - u_0 = Nx\). However, since we consider the model on the torus there is no way to add a tilt with this interpretation, since we already have the loop condition. Instead we add the tilt to each term in the potential, and see it as a stretching factor; that is we consider the tilted Hamiltonian in which we replace \(V_k( \cdot )\) with \(V_k( \, \cdot + x )\). Hence, in the case of pure nearest neighbour interactions, we consider the Gibbs distribution \(\mu_N^x \in \mathcal M^1(\Omega_N)\)
\begin{equation}\label{tiltedGibbsDistribution}
\mu_N^x(d\eta) = \frac{1}{Z_{N,x}^\beta} \exp \left( - \beta \sum_{b \in \mathcal B(\Lambda) }V \big( \eta(b) + x \big) \right) d\eta_N,
\end{equation}
and denote \(H_N^x\) for the tilted Hamiltonian. For the gradient model to be of interest it is important that we can show the existence of some form of limiting distribution on the space \(\Omega\) of bond configurations on \(\mathbb Z\); in \cite{FS_97} it is shown that for pure nearest neighbour interactions with convex potential \(V = V_1\) , there is a measure \(\mu^x\) to which a subsequence of \(\mu_N^x\) is converging weakly. In section \ref{TightnessSection} we confirm that convexity can be replaced by a weaker requirement.

The emphasis of our project is to analyse the behaviour of the specific free energy as \(|\Lambda| \rightarrow \infty\),
\begin{align}
& f \colon \mathbb R \rightarrow \mathbb R, \nonumber \\
\label{freeEnergy}
& f (x) = \lim_{N\rightarrow \infty} -\frac{1}{\beta |\Lambda|} \log Z^\beta_{N,x}.
\end{align}
In section \ref{sec:Tessa&Alex} we extend the results in \cite{FS_97, GPV_88} to show the existence of the free energy for a class of non-convex potentials, before applying the results to particular examples and obtaining bounds on \(f\). Section \ref{sec:Owen&Barney} concentrates on pure next nearest neighbour interactions, following the large deviations approach of Blanc et al.\@ we see that subject to the same conditions as in \cite{BLLP_10}, the free energy exists and can be expressed as the solution to a variational problem. Finally in section \ref{sec:maha}...

Introduction


Nearest neighbour models.


Pure Nearest Neighbour Interaction


Next-nearest neighbour models

Next Nearest Neighbour Interactions


Fourier analysis.

Quadratic Potentials