\section{Introduction}\label{sec_intro}
The motivation of this paper lies in the minimization of a differentiable function $F:\mathbb{R}^n\rightarrow \mathbb{R}$ with at least one minimizer. Inspired by Nesterov pioneering work \cite{nesterov1983method}, we study the following ordinary differential equation (ODE):
\begin{equation} \label{ODE}
\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\nabla F(x(t))=0,
\end{equation}
where
$\alpha>0$,
with $t_0>0$, $x(t_0)=x_0$ and $\dot x(t_0)=v_0$.
This ODE is associated to the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA)\cite{beck2009fast} or the Accelerated Gradient Method \cite{nesterov1983method} :
\begin{equation}
x_{n+1}=y_n-h \nabla F(y_n) \text{ and }
y_n=x_n+\frac{n}{n+\alpha}(x_n-x_{n-1}),
\end{equation}
with $h$ and $\alpha$ positive parameters.
This equation, including or not a perturbation term, has been widely studied in the literature \cite{attouch2000heavy,su2016differential,cabot2009long,balti2016asymptotic,may2015asymptotic}. This equation belongs to a set of similar equations with various viscosity terms. It is impossible to mention all works related to the heavy ball equation or other viscosity terms. We refer the reader to the following recent works \cite{Begout2015,jendoubi2015asymptotics,may2015asymptotic,cabot2007asymptotics,attouch2002dynamics,polyak2017lyapunov,attouch2017asymptotic} and the references therein.

Throughout the paper, we assume that, for any initial conditions $(x_0,v_0)\in \mathbb{R}^n\times\mathbb{R}^n$, the Cauchy problem associated with the differential equation \eqref{ODE}, has a unique global solution $x$ satisfying $(x(t_0),\dot x(t_0) )=(x_0,v_0)$. This is guaranteed for instance when the gradient function $\nabla F$ is Lipschitz  on bounded subsets of $\mathbb{R}^n$.

In this work we investigate the convergence rates of the values $F(x(t))-F^*$ for the trajectories of the ODE \eqref{ODE}. It was proved in \cite{attouch2018fast} that if $F$ is convex with Lipschitz gradient and if $\alpha>3$, the trajectory $F(x(t))$ converges  to the minimum $F^*$ of $F$. It is also known that for $\alpha\geqslant  3$ and $F$ convex we have:
\begin{equation}
F(x(t))-F^*=O \left(
t^{-2}
\right).
\end{equation}

Extending to the continuous setting the work of Chambolle-Dossal~\cite{chambolle2015convergence} of the convergence of iterates of FISTA, Attouch et al. \cite{attouch2018fast} proved that for
$\alpha>3$ the trajectory $x$ converges (weakly in infinite-dimensional Hilbert space) to a minimizer of $F$.
Su et al. \cite{su2016differential} proposed some new results, proving the integrability of $t\mapsto t(F(x(t))-F^*)$ when $\alpha>3$, and they gave more accurate
bounds on $F(x(t))-F^*$ in the case of strong convexity. Always in the case of the strong convexity of $F$, Attouch,   Chbani, Peypouquet and Redont proved in \cite{attouch2018fast} that the trajectory $x(t)$ satisfies $F(x(t))-F^*=O\left( t^{-\frac{2\alpha}{3}}\right)$ for any $\alpha>0$. More recently several studies including a perturbation term \cite{attouch2018fast,AujolDossal,aujol2015stability,vassilis2018differential} have been proposed.

In this work, we focus on the decay of $F(x(t))-F^*$ depending on more general geometries of $F$ around its set of minimizers than strong convexity.
Indeed, Attouch et al. in \cite{attouch2018fast} proved that if $F$ is convex then for any $\alpha>0$, $F(x(t))-F^*$ tends to $0$ when $t$ goes to infinity. Combined with the coercivity of $F$, this convergence implies that the distance
$d(x(t),X^*)$ between $x(t)$ and the set of minimizers $X^*$ tends to $0$. To analyse the asymptotic behavior of $F(x(t))-F^*$  we can thus only assume hypotheses on $F$ only on the neighborhood of $X^*$ and may avoid the tough question of the convergence of the the trajectory $x(t)$ to a point of $X^*$.

More precisely, we consider functions behaving like $\norm{x-x^*}^{\gamma}$ around their set of minimizers for any $\gamma\geqslant 1$. Our aim is to show the optimal convergence rates that can be obtained depending on this local geometry. In particular we prove that if $F$ is strongly convex with a Lipschitz continuous gradient, the decay is actually better than $O\left( t^{-\frac{2\alpha}{3}}\right)$. We also prove that the actual decay for quadratic functions is  $O\left(
t^{-\alpha}
\right)$. These results rely on two geometrical conditions: a first one ensuring that the function is sufficiently flat around the set of minimizers, and a second one ensuring that it is sufficiently sharp. In this paper, we will show that both conditions are important to get the expected convergence rates: the flatness assumption ensures that the function is not too sharp and may prevent from bad oscillations of the solution, while the sharpness condition ensures that the magnitude of the gradient of the function is not too low in the neighborhood of the minimizers.

The paper is organized as follows.
In Section~\ref{sec_geom}, we introduce the geometrical hypotheses we consider on the function $F$, and their relation with \L ojasiewicz property. We then recap the state of the art results on the ODE \eqref{ODE} in Section~\ref{sec_state}. We present the contributions of the paper in Section~\ref{sec_contrib}: depending on the geometry of the function $F$ and the value of the damping parameter $\alpha$, we give optimal rates of convergence. The proofs of the theorems are given in Section~\ref{sec_proofs}. Some technical proofs are postponed to Appendix~\ref{appendix}.

\section{Local geometry of convex functions}\label{sec_geom}
Throughout the paper we assume that the ODE \eqref{ODE} is defined in $\mathbb{R}^n$ equipped with the euclidean scalar product $\langle \cdot,\cdot\rangle$ and the associated norm $\|\cdot\|$. As usual $B(x^*,r)$ denotes the open euclidean ball with center $x^*$ and radius $r>0$ while $\bar B(x^*,r)$ denotes the closed euclidean ball with center $x^*$ and radius $r>0$.

In this section we introduce two notions describing the geometry of a convex function around its minimizers.
\begin{definition}
Let $F:\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex differentiable function, $X^*:=\textup{argmin}\, F\neq \emptyset$ and: $F^*:=\inf F$.
\begin{enumerate}
\item[(i)] Let $\gamma \geqslant 1$. The function $F$ satisfies the hypothesis $\textbf{H}_1(\gamma)$ if, for any minimizer $x^*\in X^*$, there exists $\eta>0$ such that:
$$\forall x\in B(x^*,\eta),\quad F(x) - F^* \leqslant \frac{1}{\gamma} \langle \nabla F(x),x-x^*\rangle.$$
\item[(ii)]
Let $r\geqslant 1$. The function $F$ satisfies the growth condition $\textbf{H}_2(r)$ if, for any minimizer $x^*\in X^*$, there exist $K>0$ and $\varepsilon>0$, such that:
\begin{equation*}
\forall x\in B(x^*,\varepsilon),\quad K d(x,X^*)^{r}\leqslant F(x)-F^*.
\end{equation*}
\end{enumerate}
\end{definition}

The hypothesis $\textbf{H}_1(\gamma)$ has already been used in \cite{cabot2009long} and later in \cite{su2016differential,AujolDossal}. This is a mild assumption, requesting slightly more than the convexity of $F$ in the neighborhood of its minimizers. Observe that any convex function automatically satisfies $\textbf{H}_1(1)$ and that any differentiable function $F$ for which $(F-F^*)^{\frac{1}{\gamma}}$ is convex for some $\gamma\geq 1$, satisfies $\textbf{H}_1(\gamma)$. Nevertheless having a better intuition of the geometry of convex functions satisfying $\textbf{H}_1(\gamma)$ for some $\gamma\geq 1$, requires a little more effort:
\begin{lemma}
Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a convex differentiable function with $X^*=\textup{argmin}\, F\neq \emptyset$, and $F^*=\inf F$. If $F$ satisfies $\textbf{H}_1(\gamma)$ for some $\gamma\geq 1$, then:
\begin{enumerate}
\item $F$ satisfies $\textbf{H}_1(\gamma')$ for all $\gamma'\in[1,\gamma]$.
\item For any minimizer $x^*\in X^*$, there exists $M>0$ and $\eta >0$ such that:
\begin{equation}
\forall x\in B(x^*,\eta),~F(x) -F^* \leqslant M \|x-x^*\|^\gamma.\label{hyp:H1}
\end{equation}
\end{enumerate} \label{lem:geometry}
\end{lemma}
\begin{proof}
The proof of the first point of Lemma \ref{lem:geometry} is straightforward. The second point relies on the following elementary result in dimension $1$: let $g:\mathbb{R}\rightarrow \mathbb{R}$ be a convex differentiable function such that $0\in \textup{argmin}\, g$, $g(0)=0$ and:
$$\forall t\in [0,1],~g(t) \leq \frac{t}{\gamma}g'(t),$$
for some $\gamma\geqslant 1$. Then the function $t\mapsto t^{-\gamma}g(t)$ is monotonically increasing on $[0,1]$ and:
\begin{equation}
\forall t\in [0,1],~g(t)\leqslant g(1)t^\gamma.\label{majo1D}
\end{equation}

Consider now any convex differentiable function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying the condition $\textbf{H}_1(\gamma)$, and $x^*\in X^*$. There then exists $\eta>0$ such that:
$$\forall x\in B(x^*,\eta),\quad 0\leqslant F(x) - F^* \leqslant \frac{1}{\gamma} \langle \nabla F(x),x-x^*\rangle.$$
Let $\eta'\in(0,\eta)$. For any $x\in\bar B(x^*,\eta')$ with $x \neq x^*$, we introduce the following univariate function:
$$g_x:t\in [0,1]\mapsto F\left(x^*+t\eta'\frac{x-x^*}{\|x-x^*\|}\right)-F^*.$$
First observe that, for all $x\in \bar B(x^*,\eta')$ with $x \neq x^*$ and for all $t\in [0,1]$, we have: $x^*+t\eta'\frac{x-x^*}{\|x-x^*\|}\in \bar B(x^*,\eta').$ Since $F$ is continuous on the compact set $\bar B(x^*,\eta')$, we deduce that:
\begin{equation}
\exists M>0,~\forall x\in \bar B(x^*,\eta')  \ \mbox{ with $x \neq x^*$},~ \forall t\in[0,1],~g_x(t)\leq M.\label{bounded}
\end{equation}
Note here that the constant $M$ only depends on the point $x^*$ and the real constant $\eta'$.

Then, by construction, $g_x$ is a convex differentiable function satisfying: $0\in \textup{argmin}\,(g_x)$, $g_{x}(0)=0$ and:
\begin{eqnarray*}
\forall t\in (0,1],~g_x'(t) &=& \left\langle \nabla F\left(x^*+t\eta'\frac{x-x^*}{\|x-x^*\|}\right),\eta'\frac{x-x^*}{\|x-x^*\|}\right\rangle\\%=\frac{1}{t}\langle \nabla F(x^*+t\frac{x-x^*}{\|x-x^*\|}),t\frac{x-x^*}{\|x-x^*\|}\rangle\\
&\geqslant & \frac{\gamma}{t}\left(F\left(x^*+t\eta\frac{x-x^*}{\|x-x^*\|}\right)-F^*\right) = \frac{\gamma}{t} g_x(t).
\end{eqnarray*}
Thus, using the one dimensional result \eqref{majo1D} and the uniform bound \eqref{bounded}, we get:
\begin{equation}
\forall x\in \bar B(x^*,\eta') \ \mbox{ with $x \neq x^*$},~\forall t\in [0,1],~g_{x}(t) \leqslant g_x(1)t^{\gamma}\leqslant Mt^{\gamma}.
\end{equation}
Finally by choosing $t=\frac{1}{\eta'}\|x-x^{\ast}\|$, we obtain the expected result.
\end{proof}

In other words, the hypothesis $\textbf{H}_1(\gamma)$ can be seen as a ``flatness'' condition on the function $F$ in the sense that it ensures that $F$ is sufficiently flat (at least as flat as $x\mapsto \|x\|^\gamma$) in the neighborhood of its minimizers.

The hypothesis $\textbf{H}_2(r)$, $r\geqslant 1$, is a growth condition on the function $F$ around any minimizer (any critical point in the non-convex case). It is sometimes also called $r$-conditioning \cite{garrigos2017convergence} or H\"olderian error bounds \cite{Bolte2017}. This assumption is motivated by the fact that, when $F$ is convex, $\textbf{H}_2(r)$ is equivalent to the famous \L ojasiewicz inequality \cite{Loja63,Loja93}, a key tool in the mathematical analysis of continuous (or discrete) subgradient dynamical systems, with exponent $\theta = 1-\frac{1}{r}$
\begin{definition}
\label{def_loja}
A differentiable function $F:\mathbb R^n \to \mathbb R$ is said to have the \L ojasiewicz property with exponent $\theta \in [0,1)$ if, for any critical point $x^*$, there exist $c> 0$ and $\varepsilon >0$ such that:
\begin{equation}
\forall x\in B(x^*,\varepsilon),~\|\nabla F(x)\| \geqslant c|F(x)-F(x^*)|^{\theta},\label{loja}
\end{equation}
where: $0^0=0$ when $\theta=0$ by convention.
\end{definition}

When the set $X^*$ of the minimizers is a connected compact set, the \L ojasiewicz inequality turns into a geometrical condition on $F$ around its set of minimizers $X^*$, usually referred to as H\"older metric subregularity \cite{kruger2015error}, and whose proof can be easily adapted from \cite[Lemma 1]{AttouchBolte2009}:
\begin{lemma}
Let $F:\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex differentiable function satisfying the growth condition $\textbf{H}_2(r)$ for some $r\geqslant 1$. Assume that the set $X^*=\textup{argmin}\, F$ is compact. Then there exist $K>0$ and $\varepsilon >0$ such that for all $x\in \mathbb{R}^n$:
$$d(x,X^*) \leqslant \varepsilon\Rightarrow K d(x,X^*)^{r}\leqslant F(x)-F^*.$$\label{lem:H2}
\end{lemma}
Typical examples of functions having the \L ojasiewicz property are real-analytic functions, $C^1$ subanalytic functions or semi-algebraic functions \cite{Loja63,Loja93}. Strongly convex functions satisfy a global \L ojasiewicz property with exponent $\theta=\frac{1}{2}$ \cite{AttouchBolte2009}, or equivalently a global version of the hypothesis $\textbf{H}_2(2)$, namely:
$$\forall x\in \mathbb{R}^n,  F(x)-F^*\geqslant \frac{\mu}{2}\|x-x^*\|^2,$$
where $\mu>0$ denotes the parameter of strong convexity and $x^*$ the unique minimizer of $F$. By extension, uniformly convex functions of order $p\geqslant 2$ satisfy the global version of the hypothesis $\textbf{H}_2(p)$ \cite{garrigos2017convergence}.

Let us now present two simple examples of convex differentiable functions to illustrate situations where the hypothesis $\textbf{H}_1$ and $\textbf{H}_2$ are satisfied. Let $\gamma > 1$ and consider the function defined by: $F:x\in \mathbb{R}\mapsto |x|^\gamma$.
We easily check that $F$ satisfies the hypothesis $\textbf{H}_1(\gamma')$ for some $\gamma'\geq 1$ if and only if $\gamma'\in [1,\gamma]$. By definition, $F$ also naturally satisfies $\textbf{H}_2(r)$ if and only if $r\geqslant \gamma$. Same conditions on $\gamma'$ and $r$ can be derived without uniqueness of the minimizer for functions of the form:
\begin{equation}
F(x) = \left\{\begin{array}{ll}
\max(|x|-a,0)^\gamma &\mbox{ if } |x| \geqslant a,\\
0 &\mbox{otherwise,}
\end{array}\right.\label{ex2}
\end{equation}
with $a>0$, and whose set of minimizers is: $X^*=[-a,a]$, since conditions $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(r)$ only make sense around the extremal points of $X^*$.

Let us now investigate the relation between the parameters $\gamma$ and $r$ in the general case: any  convex differentiable function $F$ satisfying both $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(r)$, has to be at least as flat as $x\mapsto \|x\|^\gamma$ and as sharp as $x\mapsto \|x\|^r$ in the neighborhood of its minimizers. Combining the flatness condition $\textbf{H}_1(\gamma)$ and the growth condition $\textbf{H}_2(r)$, we consistently deduce:
\begin{lemma}
If a convex differentiable function satisfies both $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(r)$ then necessarily $r\geqslant \gamma$. \label{lem:geometry2}
\end{lemma}

Finally, we conclude this section by showing that an additional assumption of the Lipschitz continuity of the gradient provides additional information on the local geometry of $F$: indeed, for convex functions, the Lipschitz continuity of the gradient is equivalent to a quadratic upper bound on $F$:
\begin{equation}\label{lipschitz}
\forall (x,y)\in \mathbb{R}^n\times \mathbb{R}^n, ~F(x)-F(y) \leqslant \langle \nabla F(y),x-y \rangle + \frac{L}{2}\|x-y\|^{2}.
\end{equation}
Applying \eqref{lipschitz} at $y=x^*$, we then deduce:
\begin{equation}\label{lipschitz2}
\forall x\in \mathbb{R}^n,~F(x)-F^* \leqslant \frac{L}{2}\|x-x^*\|^{2},
\end{equation}
which indicates that $F$ is at least as flat as $\|x-x^*\|^2$ around $X^*$. More precisely:

\begin{lemma}
Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a convex differentiable function
with a $L$-Lipschitz continuous gradient for some $L>0$.
Assume also that $F$ satisfies the growth condition $\textbf{H}_2(2)$ for some constant $K>0$. Then $F$
automatically satisfies $\textbf{H}_1(\gamma)$ with $\gamma=1+\frac{K}{2L}\in(1,2]$.\label{lem:Lipschitz}
\end{lemma}
\begin{proof}
Since $F$ is convex with a Lipschitz continuous gradient, we have:
$$\forall (x,y)\in \mathbb{R}^n, F(y)-F(x)-\langle \nabla F(x),y-x\rangle \geqslant \frac{1}{2L}\|\nabla F(y)-\nabla F(x)\|^2,$$
hence:
$$\forall x\in \mathbb{R}^n, F(x)-F^*\leqslant \langle\nabla F(x),x-x^*\rangle -\frac{1}{2L}\|\nabla F(x)\|^2.$$
Assume in addition that $F$ satisfies the growth condition $\textbf{H}_2(2)$ for some constant $K>0$. Then $F$ has the \L ojasiewicz property with exponent $\theta=\frac{1}{2}$ and constant $c=\sqrt{K}$. Thus:
$$\left(1+\frac{K}{2L}\right)(F(x)-F^*) \leqslant\langle\nabla F(x),x-x^*\rangle,$$
in the neighborhood of its minimizers, which means that $F$ satisfies $\textbf{H}_1(\gamma)$ with $\gamma=1+\frac{K}{2L}$.
\end{proof}

\begin{remark}
Observe that Lemma \ref{lem:Lipschitz} can be easily extended to the case of convex differentiable functions with a $\nu$-H\"older continuous gradient.
Indeed, let $F$ be a convex differentiable functions with a $\nu$-H\"older continuous gradient for some $\nu\geqslant 1$. If $F$ also satisfies the growth condition $\textbf{H}_2(1+\nu)$ (for some constant $K>0$), then $F$ automatically satisfies $\textbf{H}_1(\gamma)$ with $\gamma =1 + \frac{\alpha K}{(1+\nu)L^\frac{1}{\nu}}$. This result is based on a notion of generalized co-coercivity for functions having a H\"older continuous gradient.
\end{remark}

\section{Related results}\label{sec_state}
In this section, we recall some classical state of the art results on the convergence properties of the trajectories of the ODE \eqref{ODE}.

Let us first recall that as soon as $\alpha>0$,  $F(x(t))$ converges to $F^*$ \cite{AujolDossal,attouch2017rate}, but a larger value of $\alpha$ is required to show the convergence of the trajectory $x(t)$. More precisely, if $F$ is convex and $\alpha >3$, or if $F$ satisfies $\textbf{H}_1(\gamma)$ hypothesis and $\alpha>1+\frac{2}{\gamma}$ then:
\begin{equation*}
F(x(t))-F^*=o\left(\frac{1}{t^2}\right),
\end{equation*}
and the trajectory $x(t)$ converges (weakly in an infinite dimensional space)  to a minimizer $x^*$ of $F$ \cite{su2016differential,AujolDossal,may2015asymptotic}. This last point generalizes what is known on convex functions: thanks to the additional hypothesis $\textbf{H}_1(\gamma)$, the optimal decay $\frac{1}{t^2}$ can be achieved for a damping parameter $\alpha$ smaller that $3$.

In the sub-critical case (namely when $\alpha <3$), it has been proven in \cite{attouch2017rate,AujolDossal} that if $F$ is convex, the convergence rate is then given by:
\begin{equation}
F(x(t))-F^*=O\left(\frac{1}{t^\frac{2\alpha}{3}}\right),
\end{equation}
but we can no longer prove the convergence of the trajectory $x(t)$.

The purpose in this paper is to prove that by exploiting the geometry of the function $F$, better rates of convergence can be achieved for the values $F(x(t))-F^*$.

Consider first the case when $F$ is convex and $\alpha \leqslant 1+\frac{2}{\gamma}$. A first contribution  in this paper is to provide convergence rates for the values when $F$ only satisfies $\textbf{H}_1(\gamma)$. Although we can no longer prove the convergence of the trajectory $x(t)$, we still have the following convergence rate for $F(x(t))-F^*$:
\begin{equation}
F(x(t))-F^*=O\left(\frac{1}{t^{\frac{2\gamma\alpha}{2+\gamma}}}\right),
\end{equation}
and this decay is optimal and achieved for $F(x)=\vert x\vert^{\gamma}$ for any $\gamma\geqslant  1$. These results have been first stated and proved in the unpublished report \cite{AujolDossal} by Aujol and Dossal in 2017 for convex differentiable functions satisfying $(F-F^*)^\frac{1}{\gamma}$ convex. Observe that this decay is still valid for $\gamma=1$ i.e. with the sole assumption of convexity as shown in \cite{attouch2017rate}, and that the constant hidden in the big $O$ is explicit and available also for $\gamma<1$, that is for non-convex functions (for example for functions whose square is convex).

Consider now the case when $\alpha > 1+\frac{2}{\gamma}$. In that case, with the sole assumption $\textbf{H}_1(\gamma)$ on $F$ for some $\gamma \geqslant 1$, it is not possible to get a bound on the decay rate like $O(\frac{1}{t^\delta})$ with $\delta>2$. Indeed as shown in \cite[Example 2.12]{attouch2018fast}, for any $\eta>2$ and for a large friction parameter $\alpha$, the solution $x$ of the ODE associated to $F(x)=|x|^{\eta}$ satisfies:
$$F(x(t))-F^*=Kt^{-\frac{2\eta}{\eta-2}},$$
and the power $\frac{2\eta}{\eta-2}$ can be chosen arbitrary close to $2$. More conditions are thus needed to obtain a decay faster than $O\left(\frac{1}{t^2}\right)$, which is the uniform rate that can be achieved for $\alpha \geqslant 3$ for convex functions.

Our main contribution is to show that a flatness condition $\textbf{H}_1$ associated to classical sharpness conditions such as the \L ojasiewicz property provides new and better decay rates on the values $F(x(t))-F^*$, and to prove the optimality of these rates in the sense that they are achieved for instance for the function $F(x)=|x|^\gamma$, $x\in \mathbb{R}$, $\gamma\geqslant 1$.

We will then confront our results to well-known results in the literature. In particular we will focus on the case when $F$ is strongly convex or has a strong minimizer \cite{cabot2009long}. In that case, Attouch Chbani, Peypouquet and Redont in \cite{attouch2018fast} following Su, Boyd and Candes \cite{su2016differential} proved that for any $\alpha>0$ we have:
$$F(x(t))-F^*=O\left(t^{-\frac{2\alpha}{3}}\right),$$
(see also \cite{attouch2017rate} for more general viscosity term in that setting). In Section~\ref{sec_contrib}, we will prove the optimality of the power $\frac{2\alpha}{3}$ in \cite{attouch2016fast}, and that if $F$ has additionally  a Lipschitz gradient then the decay rate of $F(x(t))-F^*$ is always strictly better than $O\left(t^{-\frac{2\alpha}{3}}\right)$.

Eventually several results about the convergence rate of the solutions of ODE associated to the classical gradient descent :
\begin{equation}\label{EDOGrad}
\dot{x}(t)+\nabla F(x(t))=0,
\end{equation}
or the ODE associated to the heavy ball method
\begin{equation}\label{EDO}
\ddot{x}+\alpha\dot x(t)+\nabla F(x(t))=0
\end{equation}
under geometrical conditions such that the \L ojasiewicz property have been proposed, see for example Polyak-Shcherbakov~\cite{polyak2017lyapunov}. The authors prove that if the function $F$ satisfies $\textbf{H}_2(2)$ and some other conditions, the decay of $F(x(t))-F^*$ is exponential for the solutions of both previous equations. These rates are the continuous counterparts of the exponential decay rate of the classical gradient descent algorithm and the heavy ball method algorithm for strongly convex functions.

In the next section we will prove that this exponential rate is not true for solutions of \eqref{ODE} even for quadratic functions, and we will prove that from an optimization point of view, the classical Nesterov acceleration may be less efficient than the classical gradient descent.

\section{Contributions}\label{sec_contrib}
In this section, we state the optimal convergence rates that can be achieved when $F$ satisfies hypotheses such as  $\textbf{H}_1(\gamma)$ and/or $\textbf{H}_2(r)$.
The first result gives optimal control for functions whose geometry is sharp :
\begin{theorem}\label{Theo1}
Let $\gamma\geqslant  1$ and $\alpha >0$. If $F$ satisfies $\textbf{H}_1(\gamma)$ and if $\alpha\leqslant 1+\frac{2}{\gamma}$ then:
\begin{equation*}
F(x(t))-F^*=O\left(\frac{1}{t^{\frac{2\gamma\alpha}{\gamma+2}}}\right).
\end{equation*}
\end{theorem}
\begin{figure}[h]
\includegraphics[width=\textwidth]{RateTheo1.png}
\caption{Decay rate $r(\alpha,\gamma)=\frac{2\alpha\gamma}{\gamma+2}$ depending on $\alpha$ when $\alpha\leqslant 1+\frac{2}{\gamma}$ and when $F$ satisfies $\textbf{H}_1(\gamma)$ (as in Theorem \ref{Theo1}) for four values $\gamma$:
$\gamma_1=1.5$ dashed line, $\gamma_2=2$,  solid line, $\gamma_3=3$ dotted line and $\gamma_4=5$ dashed-dotted line.}
\end{figure}

Note that a proof of the Theorem \ref{Theo1} has been proposed in the unpublished report \cite{AujolDossal}. The obtained decay is proved to be optimal in the sense that it can be achieved for some explicit functions $F$ for any $\gamma <1$. As a consequence one cannot expect a $o(t^{-\frac{2\gamma\alpha}{\gamma+2}})$ decay when $\alpha <1+\frac{2}{\gamma}$.

Let us now consider the case when $\alpha > 1+\frac{2}{\gamma}$. The second result in this paper provides optimal convergence rates for functions whose geometry is sharp, with a large friction coefficient:
\begin{theorem}\label{Theo1b}
Let $\gamma\geqslant  1$ and
$\alpha >0$. If $F$ satisfies $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(2)$ for some $\gamma\leqslant 2$, if $F$ has a unique minimizer and if $\alpha>1+\frac{2}{\gamma}$ then
\begin{equation*}\label{eqTheo1}
F(x(t))-F^*=O\left(\frac{1}{t^{\frac{2\gamma\alpha}{\gamma+2}}}\right).
\end{equation*}
Moreover this decay is optimal in the sense that for any $\gamma\in(1,2]$ this rate is achieved for the function $F(x)=\vert x\vert^\gamma$.
\end{theorem}
\begin{figure}[h]
\includegraphics[width=\textwidth]{RateSharp.png}
\caption{Decay rate $r(\alpha,\gamma)=\frac{2\alpha\gamma}{\gamma+2}$ depending on the value of $\alpha$ when $F$ satisfies $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(2)$ (as in Theorem \ref{Theo1b}) with
$\gamma\leqslant 2$ for two values $\gamma$ :
$\gamma_1=1.5$ dashed line, $\gamma_2=2$,  solid line.}
\end{figure}
Note that Theorem~\ref{Theo1b} only applies for $\gamma\leqslant 2$, since there is no function that satisfies both conditions $\textbf{H}_1(\gamma)$ with $\gamma>2$ and $\textbf{H}_2(2)$ (see Lemma \ref{lem:geometry2}). The optimality of the convergence rate result is precisely stated in the next Proposition:
\begin{proposition} \label{prop_optimal}
Let $\gamma\in (1,2]$.
Let us assume that $\alpha>0$. Let $x$ be a solution of \eqref{ODE} with $F(x)=\vert x\vert^{\gamma}$, $|x(t_0)|<1$ and $\dot{x}(t_0)=0$ where $t_0>\sqrt{\max(0,\frac{\alpha \gamma(\alpha -1-2/\gamma)}{(\gamma +2)^2})}$. There exists $K >0$ such that for any $T>0$, there exists $t \geqslant T$ such that
\begin{equation}
F(x(t))-F^* \geqslant \frac{K}{t^{\frac{2 \gamma \alpha}{\gamma +2}}}.
\end{equation}
\end{proposition}

Let us make several observations: first, to apply Theorem~\ref{Theo1b}, more conditions are needed than for Theorem~\ref{Theo1}: the hypothesis $\textbf{H}_2(2)$ and the uniqueness of the minimizer are needed to prove a decay faster than $O(\frac{1}{t^2})$, which is the uniform rate than can be achieved with $\alpha\geqslant  3$ for convex functions \cite{su2016differential}. The uniqueness of the minimizer is crucial in the proof of Theorem~\ref{Theo1b}, but it is still an open problem to know if this uniqueness is a necessary condition. In particular, observe that if $\dot x(t_0)=0$, then for all $t \geqslant t_0$, $x(t)$ belongs to $x_0+ {\rm Im} (\nabla F) $
where ${\rm Im} (\nabla F) $ stands for the vector space generated by $\nabla F (x)$ for all $x$ in $\mathbb{R}^n$.
As a consequence, Theorem~\ref{Theo1b} still holds true as long as the assumptions are valid in  $x_0+ {\rm Im} (\nabla F) $.
\begin{remark}[The Least-Square problem]
Let us consider the
classical Least-Square problem defined by:
$$\displaystyle\min_{x\in \mathbb{R}^n}F(x):=\frac{1}{2}\|Ax-b\|^2,$$
where $A$ is a linear operator and $b\in \mathbb{R}^n$.
If $\dot x(t_0)=0$, then for all $t \geqslant t_0$, we have thus that
$x(t)$ belongs to the affine subspace $x_0+{\rm Im}(A^*)$. Since we have uniqueness of the solution on $x_0+{\rm Im}(A^*)$, Theorem~\ref{Theo1b} can be applied.
\end{remark}

We can also remark that if $F$ is a quadratic function in the neighborhood of $x^*$,  then $F$ satisfies $\textbf{H}_1(\gamma)$ for any $\gamma \in [1,2]$. Consequently,
Theorem~\ref{Theo1b}  applies with $\gamma=2$ and thus:
\begin{equation*}
F(x(t))-F^*=O\left(\frac{1}{t^{\alpha}}\right).
\end{equation*}
Observe that the optimality result provided by the Proposition~\ref{prop_optimal} ensures that we cannot expect an exponential decay of $F(x(t))-F^*$ for quadratic functions whereas this exponential decay can be achieved for the ODE associated to Gradient descent or Heavy ball method \cite{polyak2017lyapunov}.

Likewise, if $F$ is a convex differentiable function with a Lipschitz continuous gradient, and if $F$ satisfies the growth condition $\textbf{H}_2(2)$, then $F$ automatically satisfies the assumption $\textbf{H}_1(\gamma)$ with some $1<\gamma\leqslant 2$ as shown by Lemma~\ref{lem:Lipschitz}, and Theorem~\ref{Theo1b} applies with $\gamma>1$.

Finally if $F$ is strongly convex or has a strong minimizer, then $F$ naturally satisfies $\textbf{H}_1(1)$ and a global version of $\textbf{H}_2(2)$. Since we prove the optimality  of the decay rates given by Theorem~\ref{Theo1b}, a consequence of this work is also the optimality of the power $\frac{2\alpha}{3}$ in \cite{attouch2016fast} for strongly convex functions and functions having a strong minimizer.

In both cases, we thus obtain convergence rates which are strictly better than $O(t^{-\frac{2\alpha}{3}})$ that is proposed for strongly convex functions by Su et al. \cite{su2016differential} and Attouch et al. \cite{attouch2018fast}. Finally it is worth noticing that the decay for strongly convex functions is not exponential while it is the case for the classical gradient descent scheme (see e.g. \cite{garrigos2017convergence}). This shows that applying the classical Nesterov acceleration on convex functions without looking more at the geometrical properties of the objective functions may lead to sub-optimal algorithms.

Let us now focus on flat geometries i.e. geometries associated to $\gamma>2$. Note that the uniqueness of the minimizer is not need anymore:
\begin{theorem}\label{Theo2}
Let $\gamma_1>2$ and $\gamma_2 >2$. Assume that $F$ is coercive and satisfies $\textbf{H}_1(\gamma_1)$ and $\textbf{H}_2(\gamma_2)$ with $\gamma_1\leqslant \gamma_2$. If $\alpha\geqslant  \frac{\gamma_1+2}{\gamma_1-2}$  then we have:
\begin{equation*}\label{eqTheo2}
F(x(t))-F^*=O\left(\frac{1}{t^{\frac{2\gamma_2}{\gamma_2-2}}}\right).
\end{equation*}
\end{theorem}
In the case when $\gamma_1= \gamma_2$, we have furthermore the convergence of the trajectory:
\begin{corollary}\label{Corol2}
Let $\gamma>2$. If $F$ is coercive and satisfies $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(\gamma)$, and if
$\alpha\geqslant  \frac{\gamma+2}{\gamma-2}$ then we have:
\begin{equation*}\label{eqCorol2}
F(x(t))-F^*=O\left(\frac{1}{t^{\frac{2\gamma}{\gamma-2}}}\right),
\end{equation*}
and
\begin{equation}
\norm{\dot x(t)}=O\left(\frac{1}{t^{\frac{\gamma}{\gamma-2}}}\right).
\end{equation}
Moreover the trajectory $x(t)$ has a finite length and it converges to a minimizer $x^*$ of $F$.
\end{corollary}
\begin{figure}[h]
\includegraphics[width=\textwidth]{RateFlat.png}
\caption{Decay rate $r(\alpha,\gamma)=\frac{2\gamma}{\gamma-2}$ depending on the value of $\alpha$ when $\alpha\geqslant \frac{\gamma+2}{\gamma-2}$
when $F$ satisfies $\textbf{H}_1(\gamma)$ (as in Theorem \ref{Theo2}) for two values $\gamma$: $\gamma_3=3$ dotted line and $\gamma_4=5$ dashed-dotted line.}\label{fig:flat1}
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{RateAll.png}
\caption{Decay rate $r(\alpha,\gamma)$ depending on the value of $\alpha$ if $F$  satisfies $\textbf{H}_1(\gamma)$ and $\textbf{H}_2(r)$ with $r=\max(2,\gamma)$ for four values $\gamma$ :
$\gamma_1=1.5$ dashed line, $\gamma_2=2$,  solid line, $\gamma_3=3$ dotted line and $\gamma_4=5$ dashed-dotted line.}\label{fig:flat2}
\end{figure}
Observe that the decay obtained in Corollary \ref{Corol2} is optimal since Attouch et al. proved that it is achieved for the function $F(x)=\vert x\vert^\gamma$ in \cite{attouch2018fast}.\\

From Theorems \ref{Theo1}, \ref{Theo1b} and \ref{Theo2}, we can make the following comments: first in Theorems~\ref{Theo1b} and \ref{Theo2}, both conditions $\textbf{H}_1$ and $\textbf{H}_2$ are used to get a decay rate and it turns out that these two conditions are important.

With the sole hypothesis $\textbf{H}_2(\gamma)$ it seems difficult to establish optimal rate. Consider for instance the function $F(x)=|x|^3$ which satisfies $\textbf{H}_1(3)$ and $\textbf{H}_2(3)$. Applying Theorem \ref{Theo2} with $\gamma_1=\gamma_2=3$, we know that for this function with $\alpha=\frac{\gamma_1+2}{\gamma_1-2}=5$, we have $F(x(t))-F^*=O\left(\frac{1}{t^6}\right)$. But, with the sole hypothesis $\textbf{H}_2(3)$, such a decay cannot be achieved. Indeed,
the function $F(x)=|x|^{2}$ satisfies $\textbf{H}_2(3)$, but from the optimality part of Theorem \ref{Theo1b} we know that we cannot achieve a decay better than $\frac{1}{t^{\frac{2\alpha \gamma}{\gamma+2}}}=\frac{1}{t^5}$ for $\alpha=5$.

Consider now a convex function $F$ behaving like $\norm{x-x^*}^{\gamma}$ in the neighborhood of its unique minimizer $x^*$. The decay of $F(x(t))-F^*$ then depends directly on $\alpha$ if $\gamma\leqslant 2$, but it does not depend on $\alpha$ for large $\alpha$ if $\gamma>2$. Moreover for such functions the best decay rate of $F(x(t))-F^*$ is $O\left(\frac{1}{t^{\alpha}}\right)$ and is achieved for $\gamma=2$ i.e. for quadratic like functions around the minimizer. If $\gamma<2$, it seems that the oscillations of the solution $x(t)$ prevent us from getting an optimal decay rate.
The inertia seems to be too large for such functions. If $\gamma>2$, for large $\alpha$, the decay is not as fast because the gradient of the functions decays too fast in the neighborhood of the minimizer. For these functions a larger inertia could be more efficient.

Finally, observe that as shown in Figures \ref{fig:flat1} and \ref{fig:flat2}, the case when $1+\frac{2}{\gamma}<\alpha<\frac{\gamma+2}{\gamma-2}$ is not covered by our results. Although we did not get a better convergence rate than $\frac{1}{t^2}$ in that case, we can prove that there exist some initial conditions for which the convergence rate can not be better than $t^{-\frac{2\gamma\alpha}{\gamma+2}}$:
\begin{proposition}\label{PropOpt2}
Let $\gamma >2$ and $1+\frac{2}{\gamma}<\alpha<\frac{\gamma+2}{\gamma-2}$. Let $x$ be a solution of \eqref{EDO} with $F(x)=|x|^\gamma$, $|x(t_0)|<1$ and $\dot x(t_0)=0$ for any given $t_0>0$. Then there exists $K>0$ such that for any $T>0$, there exists $t\geqslant T$ such that:
$$F(x(t))-F^* \geqslant \dfrac{K}{t^{\frac{2\gamma\alpha}{\gamma+2}}}.$$\label{prop:gap}
\end{proposition}

\paragraph{Numerical Experiments}
In the following numerical experiments, the optimality of the decays given in all previous theorems, are tested for various choices of $\alpha$ and $\gamma$.

More precisely we use a discrete Nesterov scheme to approximate the solution of \eqref{ODE} for $F(x)=|x|^{\gamma}$ on the interval $[t_0,T]$ with $t_0=0$ and $\dot{x}(t_0)=0$, see \cite{su2016differential}.

If $\gamma\geqslant 2$, $\nabla F$ is a Lipschitz function and we define the sequence $(x_n)_{n\in\mathbb{N}}$ as follows:
\begin{equation*}
x_n=y_n-h\nabla F(y_n)\text{ with }y_n=x_n+\frac{n}{n+\alpha}(x_n-x_{n-1}),
\end{equation*}
where $h\in(0,1)$ is a time step.

If $\gamma<2$, we use a proximal step :
\begin{equation*}
x_n=prox_{h F}(y_n)\text{ with }y_n=x_n+\frac{n}{n+\alpha}(x_n-x_{n-1}).
\end{equation*}
It has been shown that $x_n\approx x(n\sqrt{h})$ where the function $x$ is a solution of the ODE \eqref{ODE}.

In the following numerical experiments the sequence $(x_n)_{n\in\mathbb{N}}$ is computed for various pairs $(\gamma,\alpha)$. The step size is always set to $h=10^{-7}$.

We define the function $rate(\alpha,\gamma)$ as the expected rate given in all the previous theorems and Proposition \ref{PropOpt2}, that is:
\begin{eqnarray*}
rate(\alpha,\gamma)&:=&\left\{\begin{array}{ll}
\dfrac{2\alpha \gamma}{\gamma+2} &\text{ if } \gamma\leqslant 2 \text{ or if }\gamma>2\text{ and }\alpha\leqslant 1+\frac{2}{\gamma}, \\
\dfrac{2\gamma}{\gamma-2}& \text{ if } \gamma>2\text{ and }\alpha\geqslant \frac{\gamma+2}{\gamma-2}, \\
\dfrac{2\alpha \gamma}{\gamma+2} &\text{ if } \gamma>2 \text{ and } \alpha\in(1+\frac{2}{\gamma}, \frac{\gamma+2}{\gamma-2}).
\end{array}\right.
\end{eqnarray*}

If the function $z(t):=\left(F(x(t))-F(x^*)\right)t^{\delta}$ is bounded but does not tend to 0, we can deduce that
$\delta$ is the largest value such that
$F(x(t))-F(x^*)=O\left(t^{-\delta}\right)$.
We define
\begin{equation*}
z_n:=(F(x_n)-F(x^*))\times (n\sqrt{h})^{rate(\alpha,\gamma)}\approx (F(x(t))-F(x^*))t^{rate(\alpha,\gamma)},
\end{equation*}
and if the function $rate(\alpha,\gamma)$ is optimal we expect that the sequence $(z_n)_{n\in\mathbb{N}}$ is bounded but do not decay to 0.
The following figures give for various choices of $(\alpha,\gamma)$ the trajectory of the sequence $(z_n)_{n\in\mathbb{N}}$. The values are re-scaled such that the maximum is always $1$. In all these numerical examples, we will observe that the sequence $(z_n)_{n\in\mathbb{N}}$ is bounded and does not tend to $0$.

\begin{figure}[h]
\includegraphics[width=0.495\textwidth]{Trajgam1dot5alpha1r0dot86.png}
\includegraphics[width=0.495\textwidth]{Trajgam1dot5alpha6r5dot14.png}
\caption{Case when $\gamma=1.5$. On the left $\alpha=1$ and $rate(\alpha,\gamma)=\frac{2\alpha\gamma}{\gamma+2}=\frac{6}{7}$.
On the right $\alpha=6$ and $rate(\alpha,\gamma)=\frac{2\alpha\gamma}{\gamma+2}=\frac{36}{7}$}\label{fig:gamma1.5}
\end{figure}
\begin{figure}[h]
\includegraphics[width=0.495\textwidth]{Trajgam2alpha1r1.png}
\includegraphics[width=0.495\textwidth]{Trajgam2alpha6r6.png}
\caption{Case when $\gamma=2$. On the left $\alpha=1$ and $rate(\alpha,\gamma)=\frac{2\alpha\gamma}{\gamma+2}=1$.
On the right $\alpha=6$ and $rate(\alpha,\gamma)=\frac{2\alpha\gamma}{\gamma+2}=6$}\label{fig:gamma2}
\end{figure}
\begin{figure}[h]
\includegraphics[width=0.495\textwidth]{Trajgam3alpha1r1dot2.png}
\includegraphics[width=0.495\textwidth]{Trajgam3alpha4r4dot8.png}
\includegraphics[width=0.495\textwidth]{Trajgam3alpha6r6.png}
\includegraphics[width=0.495\textwidth]{Trajgam3alpha8r6.png}
\caption{Case when $\gamma=3$. On the top left $\alpha=1$ and $rate(\alpha,\gamma)=\frac{2\alpha\gamma}{\gamma+2}=1.2$,
on the top right $\alpha=4$ and $rate(\alpha,\gamma)=\frac{2\alpha\gamma}{\gamma+2}=4.8$,
on bottom left $\alpha=6$ and $rate(\alpha,\gamma)=\frac{2\gamma}{\gamma-2}=6$,
on bottom right $\alpha=8$ and $rate(\alpha,\gamma)=\frac{2\gamma}{\gamma-2}=6$}\label{fig:gamma3}
\end{figure}

\begin{itemize}
\item The Figures \ref{fig:gamma1.5} and \ref{fig:gamma2} with $\gamma=1.5$ and $\gamma=2$ illustrate Theorem \ref{Theo1}, Theorem \ref{Theo1b} and Proposition \ref{prop_optimal}. Indeed for sharp functions (i.e for $\gamma\leqslant 2$) the rate is proved to be optimal.
\item In the case $\gamma=3$ and $\alpha=1$, the fact that $(F(x(t))-F(x^*))t^{rate(\alpha,\gamma)}$ is bounded is also a consequence of Theorem \ref{Theo1}.  The optimality of this rate is not proven but the experiments show that it numerically is.
\item In the case $\gamma=3$ and $\alpha=4$,  $\alpha\in(\frac{\gamma+2}{\gamma},\frac{\gamma+2}{\gamma-2})$ then
the fact that $(F(x(t))-F(x^*))t^{rate(\alpha,\gamma)}$ is bounded is not proved but the experiments from Figure \ref{fig:gamma3} show that it numerically is. However Proposition \ref{PropOpt2}
 proves that the sequence $(z_n)_{n\in\mathbb{N}}$ does not tend to 0, which is illustrated by the experiments.
\item When $\gamma=3$ and $\alpha=6$ or $\alpha=8$, Theorem \ref{Theo2} ensures that the sequence $(z_n)_{n\in\mathbb{N}}$ is bounded. This rate is proved to be optimal and the numerical experiments from Figure \ref{fig:gamma3} show that this rate is actually achieved for this specific choice of parameters.
\end{itemize}

\section{Proofs}\label{sec_proofs}

In this section, we detail the proofs of the results presented in Section~\ref{sec_contrib}, namely Theorems \ref{Theo1}, \ref{Theo1b} and \ref{Theo2}, Propositions~\ref{prop_optimal} and \ref{prop:gap},  Corollary~\ref{Corol2}.

The proofs of the theorems rely on Lyapunov functions $\mathcal{E}$ and $\mathcal{H}$ introduced by Su, Boyd and Candes \cite{su2016differential}, Attouch, Chbani, Peypouquet and Redont \cite{attouch2018fast} and Aujol-Dossal \cite{AujolDossal} :
\begin{equation*}
\mathcal{E}(t)=t^2(F(x(t))-F^*)+\frac{1}{2}
\norm{\lambda(x(t)-x^*)+t\dot{x}(t)}^2+\frac{\xi}{2}\norm{x(t)-x^*}^2,
\end{equation*}
where $x^*$ is a minimizer of $F$ and $\lambda$ and $\xi$ are two real numbers.
The function $\mathcal{H}$ is defined from $\mathcal{E}$ and it depends on another real parameter $p$ :
\begin{equation*}
\mathcal{H}(t)=t^p\mathcal{E}(t).
\end{equation*}
Using the following notations:
\begin{align*}
a(t)&=t(F(x(t))-F^*),\\
b(t)&=\frac{1}{2t}\norm{\lambda(x(t)-x^*)+t\dot{x}(t)}^2,\\
c(t)&=\frac{1}{2t}\norm{x(t)-x^*}^2,
\end{align*}
we have:
\begin{equation*}
\mathcal{E}(t)=t(a(t)+b(t)+\xi c(t)).
\end{equation*}
From now on we will choose
\begin{equation*}
\xi=\lambda(\lambda+1-\alpha),
\end{equation*}
and we will use the following Lemma whose proof is postponed to Appendix~\ref{appendix}:
\begin{lemma}\label{LemmeFonda}
If $F$ satisfies $\textbf{H}_1(\gamma)$ for any $\gamma\geq 1$, and if $\xi=\lambda(\lambda-\alpha+1)$ then
 \begin{equation*}
 \mathcal{H}'(t)\leqslant t^{p}\left((2-\gamma\lambda+p)a(t)+(2\lambda+2-2\alpha+p)b(t)+\lambda(\lambda+1-\alpha)(-2\lambda+p)c(t)\right).
 \end{equation*}
\end{lemma}
Note that this inequality is actually an equality for the specific choice $F(x)=\vert x\vert^\gamma$, $\gamma>1$.
\subsection{Proof of Theorems \ref{Theo1} and \ref{Theo1b}}
In this section we prove Theorem \ref{Theo1} and Theorem \ref{Theo1b}. Note that a complete proof of Theorem~\ref{Theo1}, including the optimality of the rate, can be found in the unpublished report \cite{AujolDossal} under the hypothesis that $(F-F^*)^\frac{1}{\gamma}$ is convex. The proof of both Theorems are actually similar. The choice of $p$ and $\lambda$ are the same but, to prove the first point, due to the value of $\alpha$, the function $\mathcal{H}$ is non-increasing and sum of non-negative terms, which simplifies the analysis and necessitates less hypotheses to conclude.

We choose here $p=\frac{2\gamma \alpha}{\gamma+2}-2$ and $\lambda=\frac{2\alpha}{\gamma+2}$
and thus
\begin{equation*}
\xi=\frac{2\alpha\gamma}{(\gamma+2)^2}(1+\frac{2}{\gamma}-\alpha).
\end{equation*}
From Lemma \ref{LemmeFonda}, it appears that:
\begin{equation}\label{ineqH1}
\mathcal{H}'(t)\leqslant K_1t^{p}c(t
\end{equation}
where the real constant $K_1$ is given by:
\begin{eqnarray*}
K_1 & = & \lambda(\lambda+1-\alpha)(-2\lambda+p)\\
&=& \frac{2\alpha}{\gamma+2} \left(\frac{2\alpha}{\gamma+2}+1-\alpha\right)
\left(-2 \frac{2\alpha}{\gamma+2}+\frac{2\gamma \alpha}{\gamma+2}-2
\right)
\\ & = &
\frac{4\alpha}{(\gamma+2)^3}
\left(2\alpha+\gamma+2-\alpha \gamma -2 \alpha \right)
\left(-2\alpha+\gamma \alpha - \gamma -2
\right)
\\ & = &
\frac{4\alpha}{(\gamma+2)^3}
\left(\gamma+2-\alpha \gamma\right)
\left(\alpha(-2+\gamma) - \gamma -2
\right).
\end{eqnarray*}
Hence:
\begin{equation}\label{eqdefK1}
K_1=\frac{4\alpha\gamma}{(\gamma+2)^3}
\left(1+\frac{2}{\gamma}-\alpha\right)
\left(\alpha(-2+\gamma) - \gamma -2
\right).
\end{equation}

Consider first the case when: $\alpha\leqslant 1+\frac{2}{\gamma}$. In that case, we observe that: $\xi\geq 0$, so that the energy $\mathcal{H}$ is actually a sum of non-negative terms. Coming back to \eqref{ineqH1}, we have:
\begin{equation}
    \mathcal{H}'(t)\leqslant K_1t^{p}c(t).\label{ineqH1b1}
\end{equation}
Since $\alpha\leqslant 1+\frac{2}{\gamma}$, the sign of the constant $K_1$ is the same as that of $\alpha(-2+\gamma) - \gamma -2$, and thus $K_1\leqslant 0$ for any $\gamma\geqslant 1$. According to \eqref{ineqH1b1}, the energy $\mathcal{H}$ is thus non-increasing and bounded i.e.:
$$\forall t\geqslant t_0,~\mathcal{H}(t)\leqslant \mathcal{H}(t_0).$$
Since $\mathcal{H}$ is a sum of non-negative terms, it follows directly that:
$$\forall t\geqslant t_0,~t^{p+2}(F(x(t))-F^*)\leqslant \mathcal{H}(t_0),$$
which concludes the proof of Theorem~\ref{Theo1}.

Consider now the case when: $\alpha > 1+\frac{2}{\gamma}$. In that case, we first observe that: $\xi<0$, so that $\mathcal{H}$ is not a sum of non-negative functions anymore, and an additional growth condition $\textbf{H}_2(2)$ will be needed to bound the term in $\norm{x(t)-x^*}^2$. Coming back to \eqref{ineqH1}, we have:
\begin{equation}
    \mathcal{H}'(t)\leqslant K_1t^{p}c(t).\label{ineqH1b2}
\end{equation}
Since $\alpha > 1+\frac{2}{\gamma}$, the sign of the constant $K_1$ is the opposite of the sign of $\alpha(\gamma -2)-(\gamma+2)$. Moreover, since $\gamma\leqslant 2$, then $\alpha(\gamma -2)-(\gamma+2)<0$ and thus $K_1 >0$.

Using Hypothesis $\textbf{H}_2(2)$ and the uniqueness of the minimizer, there exists $K>0$ such that:
\begin{equation*}
Kt\norm{x(t)-x^*}^2\leqslant t(F(x(t))-F^*)=a(t),
\end{equation*}
and thus
\begin{equation}
c(t)\leqslant \frac{1}{2Kt^2}a(t).\label{eqct}
\end{equation}
Since $\xi<0$ with our choice of parameters, we get:
\begin{eqnarray}
\mathcal{H}(t) &\geqslant & t^{p+1}(a(t) + \xi c(t))
\geqslant  t^{p+1}(1+\frac{\xi}{2Kt^2})a(t).\label{H:bound}
\end{eqnarray}
It follows that there exists $t_1$ such that for all $t\geqslant  t_1$, $\mathcal{H}(t)\geqslant  0$ and:
\begin{equation}
\mathcal{H}(t) \geqslant \frac{1}{2}t^{p+1}a(t).\label{eqat}
\end{equation}
From \eqref{ineqH1b2}, \eqref{eqct} and \eqref{eqat}, we get:
\begin{equation*}
\mathcal{H}'(t)\leqslant \frac{K_1}{K}\frac{\mathcal{H}(t)}{t^3}.
\end{equation*}
From the Gr\"onwall Lemma in its differential form, there exists $A>0$ such that for all $t\geqslant  t_1$, we have: $\mathcal{H}(t)\leqslant A$. According to \eqref{eqat},
we then conclude that $t^{p+2}(F(x(t))-F^*)=t^{p+1}a(t)$ is bounded which concludes the proof of Theorem \ref{Theo1b}.

\subsection{Proof of Proposition~\ref{prop_optimal} (Optimality of the convergence rates)}
Before proving the optimality of the convergence rate stated in Proposition~\ref{prop_optimal}, we need the following technical lemma:
\begin{lemma} \label{lemmatech}
Let $y$ a continuously differentiable function with values in $\mathbb R$.
Let $T>0$ and $\epsilon > 0$. If $y$ is bounded, then there exists $t_1 >T$ such that:
\begin{equation*}
|\dot{y}(t_1) | \leqslant \frac{\epsilon}{t_1}.
\end{equation*}
\end{lemma}

\begin{proof}

We split the proof into two cases.
\begin{enumerate}
\item
There exists $t_1 >T$ such that $\dot{y}(t_1)=0$.
\item
$\dot{y}(t)$ is of constant sign for $t> T$. For instance we assume $\dot{y}(t)>0$.
By contradiction, let us assume that $\dot{y}(t)>\frac{\epsilon}{t}$ $\forall t > T$.
Then $y(t)$ cannot be a bounded function as assumed.
\end{enumerate}
\end{proof}

Let us now prove the Proposition~\ref{prop_optimal}: the idea of the proof is the following: we first show that $\mathcal{H}$ is bounded from below. Since $\mathcal{H}$ is a sum of 3 terms including the term $F-F^*$, we then show that given $t_1 \geq t_0$, there always exists a time $t \geq t_1$ such that the value of $\mathcal{H}$ is concentrated on the term $F-F^*$.

We start the proof by using the fact that, for the function $F(x)=\vert x\vert^{\gamma}$, $\gamma>1$, the inequality of Lemma~\ref{LemmeFonda} is actually an equality. Using the values $p=\frac{2\gamma \alpha}{\gamma+2}-2$ and $\lambda=\frac{2\alpha}{\gamma+2}$ of Theorems~\ref{Theo1} and \ref{Theo1b}, we have a closed form for the derivative of function $\mathcal{H}$:
\begin{equation}\label{eqH1}
\mathcal{H}'(t) = K_1 t^{p}c(t) =\frac{K_1}{2} t^{p-1}\vert x(t)\vert^{2},
\end{equation}
where $K_1$ is the constant given in \eqref{eqdefK1}.
We will now prove that it exists $\ell>0$ such that for $t$ large enough:
\begin{equation*}
\mathcal{H}(t) \geqslant \ell.
\end{equation*}
To prove that point we consider two cases depending on the sign of
$\alpha-(1+\frac{2}{\gamma})$.
\begin{enumerate}
\item Case when $\alpha\leqslant 1+\frac{2}{\gamma}$, $\xi\geqslant 0$ and $K_1\leqslant 0$. We can first observe that $\mathcal{H}$ is a non negative and non increasing function.
Moreover it exists $\tilde t\geqslant t_0$ such that for $t\geqslant \tilde t$, $|x(t)|\leqslant 1$ and:
\begin{equation*}
t^pc(t)\leqslant \frac{t^pa(t)}{2t^2}\leqslant \frac{\mathcal{H}(t)}{t^3},
\end{equation*}
which implies using \eqref{eqH1} that:
\begin{equation*}
|\mathcal{H}'(t)|\leqslant |K_1|\frac{\mathcal{H}(t)}{t^3}.
\end{equation*}
If we denote $G(t)=\ln (\mathcal{H}(t))$ we get for all $t\geqslant \tilde t$,
\begin{equation*}
|G(t)-G(\tilde t)|\leqslant \int_{\tilde t}^t\frac{|K_1|}{s^3}ds.
\end{equation*}
We deduce that $|G(t)|$ is bounded below and then that it exists $\ell>0$ such that for $t$ large enough:
\begin{equation*}
\mathcal{H}(t) \geqslant \ell,
\end{equation*}
\item Case when $\alpha> 1+\frac{2}{\gamma}$, $\xi< 0$ and $K_1> 0$. This implies in particular that $\mathcal{H}$ is non-decreasing. Moreover, from Theorem~\ref{Theo1b}, $\mathcal{H}$ is bounded above. Coming back to the inequality \eqref{H:bound}, we observe that $\mathcal{H}(t_0)>0$ provided that $1+\frac{\xi}{2t_0^2}>0$, with $K=1$ and $\xi = \lambda(\lambda-\alpha+1)$, i.e.:
$$t_0>\sqrt{\frac{\alpha\gamma}{(\gamma +2)^2}(\alpha-(1+\frac{2}{\gamma}))}.$$

In particular, we have that for any $t\geqslant t_0$
\begin{equation*}
\mathcal{H}(t) \geqslant \ell,
\end{equation*}
with $\ell=\mathcal{H}(t_0)$
\end{enumerate}
Hence for any $\alpha>0$ and for $t$ large enough
\begin{equation*}
a(t)+b(t)+ \xi c(t) \geqslant \frac{\ell}{t^{p+1}}.
\end{equation*}
Moreover, since $c(t)=o(a(t))$ when $t \to + \infty$, we have that for $t$ large enough,
\begin{equation*}
a(t)+b(t) \geqslant \frac{\ell}{2 t^{p+1}}.
\end{equation*}

Let $T>0$ and $\epsilon >0$. We set:
\begin{equation*}
y(t):=t^{\lambda} x(t),
\end{equation*}
where: $\lambda =\frac{2 \alpha}{\gamma+2}$. From the Theorem~\ref{Theo1} and Theorem \ref{Theo1b}, we know that $y(t)$ is bounded.
Hence, from Lemma~\ref{lemmatech}, there exists $t_1 > T$ such that
\begin{equation} \label{eqt1}
|\dot{y}(t_1) |\leqslant \frac{\epsilon}{t_1}.
\end{equation}
But:
\begin{equation*}
\dot{y}(t) = t^{\lambda-1} \left(\lambda x(t)
+t \dot{x}(t)
\right).
\end{equation*}
Hence using \eqref{eqt1}:
\begin{equation*}
t_1^{\lambda} \left|\lambda x(t_1)
+t_1 \dot{x}(t_1)
\right|
\leqslant \epsilon.
\end{equation*}
We recall that: $b(t)=\frac{1}{2t}\norm{\lambda(x(t)-x^*)+t\dot{x}(t)}^2$.
We thus have:
\begin{equation*}
b(t_1) \leqslant \frac{\epsilon^2}{2 t_1^{2\lambda+1}}.
\end{equation*}
Since $\gamma \leqslant 2$, $\lambda =\frac{2 \alpha}{\gamma+2}$ and $p=\frac{2\gamma \alpha}{\gamma+2}-2$, we have
$
2\lambda+1
\geq
p+1
$, and thus
\begin{equation*}
b(t_1) \leqslant \frac{\epsilon^2}{2 t_1^{p+1}}.
\end{equation*}

For $\epsilon =\sqrt{\frac{
\ell}{2}}$ for example, there exists thus some $t_1 >T$ such that $b(t_1) \leqslant \frac{\ell}{4 t_1^{p+1}}$.
 Then $a(t_1) \geqslant  \frac{\ell}{4 t_1^{p+1}}$, i.e.
 $F(x(t_1))-F^* \geqslant  \frac{\ell}{4 t_1^{p+2}}$.
 Since $p+2= \frac{2 \gamma \alpha}{\gamma +2}$, this concludes the proof.

\subsection{Proof of Theorem \ref{Theo2}}
We detail here the proof of Theorem \ref{Theo2}.

Let us consider $\gamma_1>2$, $\gamma_2 >2$, and $\alpha\geqslant  \frac{\gamma_1+2}{\gamma_1-2}$. We consider here functions $\mathcal{H}$ for all $x^*$ in the set $X^*$ of minimizers of $F$ and prove that these functions are uniformely bounded. More precisely for any $x^*\in X^*$ we define $\mathcal{H}(t)$ with $p=\frac{4}{\gamma_1-2}$ and $\lambda=\frac{2}{\gamma_1-2}$.
With this choice of $\lambda$ and $p$, using Hypothesis $\textbf{H}_1(\gamma_1)$ we have from Lemma~\ref{LemmeFonda}:
\begin{equation*}
\mathcal{H}'(t)\leqslant 2t^{\frac{4}{\gamma_1-2}}\left(\frac{\gamma_1+2}{\gamma_1-2}-\alpha\right)b(t).
\end{equation*}
which is non-positive when $\alpha\geqslant\frac{\gamma_1+2}{\gamma_1-2}$, which implies that the function $\mathcal{H}$ is bounded above. Hence for any choice of $x^*$ in the set of minimizers $X^*$, the function $\mathcal{H}$ is bounded above and since the set of minimizers is bounded (F is coercive), there exists $A>0$ and $t_0$ such that for all choices of $x^*$ in $X^*$,
\begin{equation*} \label{inegHHAtz}
\mathcal{H}(t_0)\leqslant A,
\end{equation*}
which implies that for all $x^* \in X^{*}$ and
for all $t\geqslant  t_0$
\begin{equation*} \label{inegHHA}
\mathcal{H}(t)\leqslant A.
\end{equation*}
Hence for all $t\geqslant  t_0$ and for all $x^*\in X^*$
\begin{equation*}\label{BoundWold}
t^{\frac{4}{\gamma_1-2}}t^2(F(x(t))-F^*)\leqslant \frac{\vert \xi\vert}{2}
t^{\frac{4}{\gamma_1-2}}\norm{x(t)-x^*}^2
+A,
\end{equation*}
which implies that
\begin{equation}\label{BoundW}
t^{\frac{4}{\gamma_1-2}}t^2(F(x(t))-F^*)\leqslant \frac{\vert \xi\vert}{2}
t^{\frac{4}{\gamma_1-2}}d(x(t),X^{*})^2
+A.
\end{equation}

We now set:
\begin{equation}\label{defv}
v(t):=t^{\frac{4}{\gamma_2-2}}d(x(t),X^*)^2.
\end{equation}
Using  \eqref{BoundW} we have:
\begin{equation}\label{Boundu1bos}
t^{\frac{2 \gamma_1}{\gamma_1-2}}(F(x(t))-F^*)\leqslant \frac{\vert \xi\vert}{2}
t^{\frac{4}{\gamma_1-2}-\frac{4}{\gamma_2-2}}
v(t)+A.
\end{equation}
Using the hypothesis $\textbf{H}_2(\gamma_2)$ applied under the form given by Lemma \ref{lem:H2} (since $X^*$ is compact), there exists $K>0$ such that
\begin{equation*}
K\left(t^{-\frac{4}{\gamma_2-2}}v(t)\right)^{\frac{\gamma_2}{2}}\leqslant F(x(t))-F^*,
\end{equation*}
which is equivalent to
\begin{equation*}
Kv(t)^{\frac{\gamma_2}{2}}
t^{\frac{-2\gamma_2}{\gamma_2-2}}
\leqslant F(x(t))-F^*.
\end{equation*}
Hence:
\begin{equation*}
K
t^{\frac{2\gamma_1}{\gamma_1-2}} t^{\frac{-2\gamma_2}{\gamma_2-2}}
v(t)^{\frac{\gamma_2}{2}}\leqslant
t^{\frac{2\gamma_1}{\gamma_1-2}} (F(x(t))-F^*).
\end{equation*}
Using \eqref{Boundu1bos}, we obtain:
\begin{equation*}
K
t^{\frac{2\gamma_1}{\gamma_1-2}-\frac{2\gamma_2}{\gamma_2-2}}
v(t)^{\frac{\gamma_2}{2}}\leqslant \frac{|\xi|}{2}
t^{\frac{4}{\gamma_1-2}-\frac{4}{\gamma_2-2}}
v(t)+A,
\end{equation*}
i.e.:
\begin{equation}\label{vbounded}
K
v(t)^{\frac{\gamma_2}{2}}\leqslant \frac{|\xi|}{2}
v(t)+A
t^{\frac{4}{\gamma_2-2}-\frac{4}{\gamma_1-2}}.
\end{equation}
Since $2<\gamma_1 \leqslant \gamma_2$, we deduce that $v$ is bounded.
Hence, using  \eqref{Boundu1bos} there exists some positive constant $B$ such that:
\begin{equation*}
F(x(t))-F^*\leqslant
B t^{\frac{-2 \gamma_2}{\gamma_2-2}}
+A t^{\frac{-2 \gamma_1}{\gamma_1-2}}.
\end{equation*}
Since $2<\gamma_1 \leqslant  \gamma_2$, we have $\frac{-2 \gamma_2}{\gamma_2-2} \geqslant  \frac{-2 \gamma_1}{\gamma_1-2}$.
Hence we deduce that  $F(x(t))-F^* = O \left( t^{\frac{-2 \gamma_2}{\gamma_2-2}}\right)$.\\

\subsection{Proof of Corollary~\ref{Corol2}}
We are now in position to prove Corollary~\ref{Corol2}. The first point of Corollary~\ref{Corol2} is just a particular instance of Theorem~\ref{Theo2}.
In the sequel, we prove the second point of Corollary~\ref{Corol2}.

Let $t\geqslant  t_0$ and $\tilde x\in X^*$ such that
\begin{equation*}
\|x(t)-\tilde x\|=d(x(t),X^*).
\end{equation*}
We previously proved that there exists $A>0$ such that for any $t\geqslant  t_0$ and any $x^*\in X^*$,
\begin{equation*}
\mathcal{H}(t)\leqslant A.
\end{equation*}
For the choice $x^*=\tilde x$ this inequality ensures that
\begin{equation*}
\frac{t^{\frac{4}{\gamma-2}}}{2}\norm{\lambda (x(t)-\tilde x)+t\dot x(t)}^2+t^{\frac{4}{\gamma-2}}\frac{\xi}{2}d(x(t),\tilde x)^2\leqslant A,
\end{equation*}
which is equivalent to
\begin{equation*}
\frac{t^{\frac{4}{\gamma-2}}}{2}\norm{\lambda (x(t)-\tilde x)+t\dot x(t)}^2\leqslant \frac{|\xi|}{2}v(t)+A,
\end{equation*}
where $v(t)$ is defined in \eqref{defv} with $\gamma=\gamma_2$.
Using the fact that the function $v$ is bounded (a consequence of \eqref{vbounded}) we deduce that there exists a positive constant $A_1>0$ such that:
\begin{equation*}
\norm{\lambda (x(t)-\tilde x)+t\dot x(t)}\leqslant \frac{A_1}{t^{\frac{2}{\gamma-2}}}.
\end{equation*}
Thus:
\begin{equation*}
t\norm{\dot x(t)}\leqslant \frac{A_1}{t^{\frac{2}{\gamma-2}}}+|\lambda|d(x(t),\tilde x)=\frac{A_1+|\lambda|\sqrt{v(t)}}{t^{\frac{2}{\gamma-2}}}.
\end{equation*}
Using once again the fact that the function $v$ is bounded we deduce that there exists a real number $A_2$ such that
\begin{equation*}
\norm{\dot x(t)}\leqslant \frac{A_2}{t^{\frac{\gamma}{\gamma-2}}},
\end{equation*}
which implies that $\norm{\dot x(t)}$ is an integrable function. As a consequence, we deduce that the trajectory $x(t)$ has a finite length.

\subsection{Proof of Proposition \ref{prop:gap}}
The idea of the proof is very similar to that of Proposition~\ref{prop_optimal} (optimality of the convergence rate in the sharp case i.e. when $\gamma \in (1,2]$).

For the exact same choice of parameters $p=\frac{2\gamma\alpha}{\gamma+2}-2$ and $\lambda=\frac{2\alpha}{\gamma+2}$ and assuming that $1+\frac{2}{\gamma}<\alpha < \frac{\gamma+2}{\gamma-2}$, we first show that the energy $\mathcal H$ is non-decreasing and then:
\begin{equation}
\forall t\geqslant t_0,~\mathcal H(t) \geqslant \ell,\label{infH}
\end{equation}
where: $\ell=\mathcal H(t_0)>0$. Indeed, since $\gamma>2$ and $\alpha <\frac{\gamma+2}{\gamma-2}$, a straightforward computation shows that: $\lambda^2 -|\xi|>0$, so that:
\begin{eqnarray*}
\mathcal H(t_0)
&=& t_0^{p+2}|x(t_0)|^\gamma + \frac{t_0^p}{2}\left(|\lambda x(t_0)+t_0\dot x(t_0)|^2 -|\xi||x(t_0)|^2\right)\\
&=& t_0^{p+2}|x(t_0)|^\gamma + \frac{t_0^p}{2}\left( \lambda^2 -|\xi|\right)|x(t_0)|^2 >0,
\end{eqnarray*}
without any additional assumption on the initial time $t_0>0$.

Let $T>t_0$. We set: $y(t)=t^\lambda x(t)$. If $y(t)$ is bounded as it is in Proposition~\ref{prop_optimal}, by the exact same arguments, we prove that there exists $t_1>T$ such that: $b(t_1) \leq \frac{\ell}{4t_1^{p+1}}$. Moreover since $\xi<0$ we deduce from \eqref{infH} that:
$$t_1^{p+1}(a(t_1)+b(t_1)) \geqslant \ell.$$Hence:
$$a(t_1)=t_1(F(x(t_1)-F^*) \geqslant \frac{\ell}{4t_1^{p+1}},$$
i.e.: $F(x(t_1))-F^* \geqslant \frac{\ell}{4t_1^{p+2}} = \frac{\ell}{4t_1^\frac{2\alpha\gamma}{\gamma+2}}$.

If $y(t)$ is not bounded, then the proof is even simpler: indeed, in that case, for any $K>0$, there exists $t_1\geqslant T$ such that: $y(t_1)\geq K$, hence:
$$F(x(t_1))-F^*=|x(t_1)|^\gamma \geqslant \frac{K}{t_1^{\lambda\gamma}}=\frac{K}{t_1^{\frac{2\alpha\gamma}{\gamma+2}}},$$
which concludes the proof.

\section*{Acknowledgement}
This study has been carried out with financial support from the French state, managed by the French National Research Agency (ANR GOTMI)  (ANR-16-VCE33-0010-01) and partially supported by ANR-11-LABX-0040-CIMI within the program
ANR-11-IDEX-0002-02. J.-F. Aujol is a member of Institut Universitaire de France.


\section*{TOC Graphic}
\begin{figure}%
\centering
\includegraphics[width=5.1cm]{graph_abs.pdf}
\end{figure}

\textbf{Keywords}: Thiophene, P3HT, fullerene, blend, C$_{60}$, Classical Molecular Dynamics, UFF
\newpage
Organic photovoltaics is a rich and growing area of research, and among the most promising device architectures we find those in which the active layer contains thiophene polymers and fullerenes, more specifically poly(3-hexyl-thiophene) P3HT and [6,6]-Phenyl-C61-butyric acid methyl ester PCBM.\cite{brab+10am,verp+10afm,dang+11am,chen+11nl,holm+13semsc,chin+16nsc,busb+17acsami,rich+17cr}  This active layer is known as a bulk heterojunction (BHJ), in the sense that we will find regions (domains) where there is prevalence of P3HT mass proportion, others where the prevalence is of PCBM. The nature of the frontier between domains can be well-defined (bicontinuous heterojunction) or not, as well as the prevalence inside a domain, that can be absolute (bicontinuous) or not, in the sense that the material, in domains or in the frontier, is formed by a blend of the two components. There is still ongoing discussion on these topics, however it is agreed that the photo-conversion efficiency is related to charge transfer, exciton migration and recombination, internally in the domains and in the frontier between domains; a plethora of experimental and theoretical works concerning these issues is available, and it emerges that one of the most important features regards the blend morphology.\cite{schi+05cm,guo+10jacs} While PCBM is a reasonably simple molecule, consisting of a fullerene C$_{60}$ molecule with a methyl-ethyl ester side chain, due to the lower cost of C$_{60}$ production, blends consisting of P3HT and C$_{60}$ (or a mixture of the two molecules) are also interesting\cite{rait+07semsc,li+09oe,tada-onod12semsc,comi+15mcp,mori+17jpst}. Since the alkyl side chains can strongly interact, the morphology of the final complete blend, polymer and molecule, can be more or less complex, more homogeneous or phase-separated, depending on the average length of the P3HT main chain. The molecules can segregate and form molecular almost crystalline domains, or migrate into the polymer region, depending wether the vicinal polymer region is more or less crystalline.

P3HT belongs to the second generation of electronic polymers\cite{brab+10am}, and is one of the most studied to this day\cite{busb+17acsami}. The hexyl functionalization enhances solubility and, when regioregular, may contribute\cite{nort07prb,guo+10jacs} to the formation of stacked crystalline domains depending on the growth method; it is however usually found that in devices the polymer is present in complex morphology, with important proportion of amorphous domains. As a semiconducting polymer, the optical and transport properties are defined by the character and distribution of electron and hole states in the bulk, which in turn are defined by the \textit{local} morphology. The exciton splitting and charge transfer properties when considering the blend are also defined by local morphology, and the investigation of such transitions, theory or experiment, must take that into account. In order to study theoretically such properties we must have access to a reliable description of the structural phases, which is only accessible through Classical Molecular Dynamics (CMD) -- since now we are not describing periodic crystals, with a number of atoms-per-cell that could be studied through atomistic first-principles calculations \cite{buss+02apl,ruin+03sm,ferr+04prb,humm+05pssb}.

Concerning CMD calculations, it can be done through simplified formulations, usually referred to as coarse-grained molecular dynamics, which allows for treatment of systems with a very large number of atoms; this procedure has been recently applied to P3HT and blends\cite{negi+16mts,root+16mm,ales+17jacs,yoo+17cms}, and contributes to the understanding of morphology in these complex organic compounds. On the other hand, when we need to look more deeply into the local structural characteristics of these compounds, we must go to atomistic simulations which, even in the classical formulation\cite{luko+17ms}, restricts the number of included atoms to some tens of thousands.

We present here an atomistic CMD simulation of condensed P3HT, and P3HT blends not with PCBM but with simple fullerene C$_{60}$; the models are built with not-so-short oligomers for the thiophene unit (30 3HT mers) and with P3HT:C$_{60}$ blend mass proportion very close to 1:1, as often found\cite{dang+11am,holm+13semsc,busb+17acsami} in the experimental literature for P3HT:PCBM blends. We apply for this a well-tuned force-field, which we describe explicitly, based on the Universal Force Field of Rapp\'{e} and coll.\cite{rapp+92jacs} We start from two different initial models, random or phase separated P3HT/C$_{60}$ spatial distribution. We find that after reaching equilibrium conditions, the P3HT chains present mostly non-linear structure, and concomitantly the polymer domains present disordered amorphous character. We see weak intermixing of C$_{60}$ and P3HT for the phase-separated, and strong clustering of C$_{60}$ molecules for the random starting point.  Our main conclusions are that for blends at room temperature, even for these non-crystalline polymer domains, phase separation is dominating.

\section{Methodology}

One of the most used and reliable force-fields for CMD of polymers is the Universal Force Field UFF of Rapp\'{e} and collaborators\cite{rapp+92jacs}, which however presents unusual problems  in the case of thiophenes. Specifically, it was found\cite{alve-cald09sm} that the standard charge assignment scheme of UFF (charge equilibration) for sulfur S atoms results in negative partial charge $Q_S<0$, as frequently happening in other molecules, which is not correct for thiophene (T) since the special [$=HC-S-CH=$] sequence in the T rings results in a positive $Q_S>0$ effective charge. Furthermore, the dihedral ring-to-ring torsion angle in oligothiophenes (OTs) was also found not to be properly described, which lead us to adapt the related parameters.

Here we describe only briefly the methodology for reparametrization of the force field. In short, the atom-atom interaction in the UFF can be broadly separated in bonded and non-bonded functions. For bonded-functions we have 2, 3 and 4-body potentials: bond-length (2), direct angle (3), dihedral and inversion angles (4). For non-bonded functions the FFs normally adopt pair-potentials including electrostatic Coulomb long-range interactions, van der Waals attraction and Pauli repulsion; these last two are grouped in the Lennard-Jones (LJ) format in the UFF. Adequate LJ parameters are  crucial for the reliable description of the condensate morphology.

Starting with the non-bonded potentials, as mentioned above we need reliable values for atomic charges. We will adopt fixed atomic charges and define specific atomic types also for the C- and H-atoms in the T-ring and side-chains.  To arrive at the atomic type charges ATC (more than one ATC for the same atom type of the UFF) we perform calculations with Density Functional Theory (DFT), using the FHI-aims code\cite{blum+09cpc} with the PBE exchange-correlation functional\cite{perd+96prl}, known to provide reliable structural properties for organic compounds. The partial charges for all atoms in our chosen prototype-molecules ensemble were calculated through the Hirschfeld method\cite{hirs77tca}  and averaged in order to achieve the series of ATCs. For the Lennard-Jones potentials we calculate the dispersion coefficients $D_{IJ}$ and equilibrium distance $R_{IJ}$ for atom pairs of the ensemble using the Tkatchenko-Scheffler formalism\cite{tkat-sche09prl}. Our final values for ATCs, Ds and Rs are tabulated in the Supplementary Information.

We check our ATC results with experimental data\cite{bak+61jms,harr+53jcs,frin+77ahc,spoe-cols98cp} for the electric dipole of the single thiophene molecule T1, that range from 0.46 to 0.60D; our PBE result adopting the TS approach is 0.50D, in excellent agreement with the experimental data, and from our Nanomol-FF ATCs we arrive at 0.52D. With respect to the complete reparametrization of the dihedral T-T angle and all non-bonded parameters for the HT molecules and polymers, we selected 13 different\cite{nist} thiophene-based molecular crystals,  from clean T2 to longer oligomer chains, including also linked oligomers, alkyl-terminated oligomers and polymers (details in the Supplementary Information); for this ensemble we realized geometry optimization through Molecular Mechanics within the conjugate gradient procedure and with rigid convergence criteria. The maximal deviations in lattice constants and angles with respect to experimental data are below 10$\%$, again indicating that our parameters allow for reliable description of condensed systems.

Moving to the bonded, dihedral inter-ring torsion angle parametrization, we base our procedure on experimental data\cite{bucc+jacs74} for bi-thiophene T2, which state that at room-temperature in liquid crystalline solvent, 70$\%$ of the molecules present quasi-antiparallel arrangement with dihedral angle of 140$^{\circ}$, while the remaining 30$\%$  equilibrate at 40$^{\circ}$, quasi-parallel. This is in very good agreement with many-body theoretical calculations which allows us to adopt the full theoretical potential curve\cite{alve06phd} for the parametrization. To simulate longer polymers we perform calculations for the T4 oligomer, and adopt the displaced-dihedral form for the 4-body potential. The equation and related parameters are shown in the Supplementary Information. The other bonding 2-body parameters from the original UFF are kept.

All CMD and Molecular Mechanics (MM, temperature 0K) calculations were performed with the Cerius$^2$ package, Accelrys Inc.\cite{cerius2}. We adopt periodic boundary conditions in order to simulate condensates, with fixed number of particles N. For CMD we use fixed or variable volume V and pressure P, and fixed temperature T=300K. The time step for the sequential resolution of the MD equations is 10$^{-3}$ps, using the Verlet integrator\cite{verl67pr}. Depending on the sequence of steps, we adopt microcanonical NVE, canonical NVT  or isothermal-isobaric NPT ensembles\cite{alle-tild05book}. For NVT we use the Berendsen scheme\cite{bere+84jpc}, and for NPT the Parrinello-Rahman\cite{parr-rahm81jap}. The relaxation time is 0.1ps.

For MM we adopt different energy minimization protocols, conjugate-gradient for the crystalline structures used for definition of the Nanomol FF parameters, and the Smart Minimizer present in the Cerius$^2$ package for the condensates. This last protocol incorporates sequential steps of steepest descent, quasi-Newton and truncated-Newton minimization\cite{rapp-case97book}.

\begin{figure}[!htb]
\begin{center}
\begin{tabular}{cc}
\hline\\
\vspace{0.0cm}\\
\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{laminar.pdf}}\\
\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{mod1A.pdf}}\\
\hline\\
\vspace{0.0cm}\\
\includegraphics[width=0.35\textwidth]{mod2.pdf} & \includegraphics[width=0.4\textwidth]{mod3.pdf}\\
\hline
\end{tabular}
\caption{Condensed amorphous models obtained in this work; we show the initial random distribution, and the final equilibrated distribution in unit-by-unit and full cell view. Top, P3HT laminar deposition, side and top view; Center, P3HT isotropic distribution;  Bottom left, segregated P3HT:C$_{60}$ blend (different viewpoints for the unit-by-unit and full cell); Bottom right, isotropic P3HT:C$_{60}$ blend. }
\label{condensation}
\end{center}
\end{figure}

We simulate condensates of regioregular P3HT, pure and in blends with fullerene C$_{60}$. We use not-so-short oligomers with 30 3HT-units O3HT$_{30}$, molecular weight $\sim$5kg$\cdot$mol$^{-1}$, feasible for our atomistic molecular simulation and small but comparable to regular experimentally-grown samples\cite{dang+11am,holm+13semsc,spol+15oe}. The simulation is performed for four different initial configurations, namely pure P3HT in laminar and isotropic distributions and P3HT:C$_{60}$ in segregated and isotropic blends. In the case of pure P3HT these two distributions simulate the first a patterned deposition, and the second a spin-cast deposition. For pure P3HT we include 40 chains of O3HT$_{30}$ for the laminar distribution (30080 atoms), and 50 chains for the isotropic (37600 atoms); for the blends we include 40 chains of O3HT$_{30}$ and 250 C$_{60}$ molecules (45080 atoms). The initial and final atomic distributions are illustrated in \ref{condensation}. The initial spatial distribution of units in done in a very disperse molecular packing, with large inter-molecular distances, via the Packmol package.\cite{packmol} A smooth, cautious procedure is then adopted in order to allow for realistic arrangement of the molecules and oligomers, without unnatural high temperature or pressure effects. We first reduce slowly, by hand, the cell parameters, and for each fixed-cell parameters perform CMD at 300K; this slow cell-reduction scheme continues until reaching the minimum of the van der Waals energy for the cell. With these optimal cell parameters we start the free NPT CMD minimization until convergence is attained.

\section{Results and Discussion

The formation of heterojunctions in P3HT:PCBM films has been extensively investigated from the experimental side, and more than one detail of the growth method has been highlighted as a factor to be considered\cite{schi+05cm,agos+11afm,dang+11am,chen+11nl,holm+13semsc,spol+15oe,chin+16nsc,rich+17cr}. When the complete film is grown by deposition and then annealing of a in-solution mixture, as normally done, the factors can be, at least: for P3HT, molecular weight or average molecular mass (it is expected that crystalline regions will form\cite{schi+05cm} if it arrives at $\sim$ 10 kg$\cdot$mol$^{-1}$), and degree of regioregularity; for the blend, mass proportion, solvation of each compound in the deposition solution (choice of solvent), annealing time and temperature, and finally even the architecture of the monitored sample\cite{busb+17acsami}.  A general understanding is that PCBM will diffuse (with thermal annealing) only to disordered regions of P3HT, will not migrate into already crystallized regions of the polymer, and on the other hand more probably will segregate forming the junction structure. When the junction is built from bilayer deposition of the two compounds the process is different,\cite{trea+11aem} since the P3HT original layer already has a good fraction of crystalline regions; it is found that with temperature there is fast diffusion of PCBM, however only into the disordered regions of P3HT.

The adoption of C$_{60}$ for the blend is highly desirable, and experimental investigations of the P3HT:C$_{60}$ blend have been carried out. The blend properties have also been shown to depend on the growth methodology\cite{rait+07semsc,li+09oe,tada-onod12semsc,comi+15mcp,mori+17jpst} and, as expected, the behavior of C$_{60}$ is slightly different from that of PCBM, however the main trends are retrieved: dependence on the solvent, best efficiency for 1:1 mass proportion, and so forth. P3HT, C$_{60}$ or PCBM are soluble in polar solvents and usually, for the blend, polymer and molecule weights are mixed to arrive at a desired mass proportion, that can vary\cite{maye+09afm,dang+11am,geli+14sci} from 1:4 to 3:2 (P3HT:molecule), being this last possibly the maximum proportion allowing for efficient exciton charge-transfer splitting.

In this work, we use the NanomolFF to investigate phase segregation in a P3HT/C$_{60}$ blend. We first simulate pure systems, and in sequence the blends. We implement here for the blend simulations a mass proportion 11:10, very close to the suggested\cite{rait+07semsc,dang+11am,holm+13semsc} proportion 1:1.

\subsection{P3HT properties in films and blends}

We first analyse the P3HT single-chain morphological distribution for the different condensates, using for that the ring-to-ring angular distribution. For each T-ring characteristic vector axes can be defined: $\vec{c}$, linking the two C-atoms bonded to the S-atom, which defines the bonding or chain direction; a normal vector $\vec{n}$ perpendicular to the T-plane; and a basal `` dipole'' $\vec{d}$-vector, $\vec{n}=\vec{c}\times\vec{d}$ in-plane and pointing from the 2 other C-atoms to the S-atom. In this way, for each single chain we can follow the orientation of one ring to the next T$_{i+1}$-T$_{i}$, through the angles
$\varphi_{i+1,i}=\arccos({\vec{n}_{i+1}}\cdot{\vec{n}_i})$; $\theta_{i+1,i}=\arccos[(\vec{n}_{i+1}\cdot\vec{c}_i) / (\sqrt{(\vec{n}_{i+1}\cdot\vec{d}_i)^2+(\vec{n}_{i+1}\cdot\vec{n}_i)^2})]$. In this, $\varphi$ describes mostly the dihedral torsion angle between neighbor units while $\theta$ describes roughly the concavity at that segment.

We show in  \ref{map} the results for the four condensates. As a first output, we see that for the laminar-deposition condensate we mostly find, as the final geometry for the individual chains, the alternate $\varphi > 90^{\circ}$ ring-to-ring pattern, even so with a non-zero signal for $\varphi<90^\circ$, and still some remaining traces of linear (straight ideal  $\theta=90^{\circ}$) ring-to-ring orientation. For the pure P3HT isotropic condensate we see a ``butterfly'' pattern characteristic of torsion angles maxima around 130$^{\circ}$ (quasi-antiparallel), and with a smaller but definite proportion around 50$^{\circ}$ (quasi-parallel); the proportion of linear ring-to-ring orientation is almost null. For the blend simulations, for which the polymer region is isotropically initialized, we also find the butterfly pattern now with mainly the same percentage of parallel and anti-parallel ring-to-ring pattern. At the same time, we see that the full majority of chains in these three phases show concavity, that is, linearity is not a relevant characteristic, which is a clear indication of the disordered, amorphous character of the domains. We should point out here that this is expected for the molecular weight we simulate.\cite{schi+05cm}

\begin{figure}[!htb]
\begin{center}

\includegraphics[width=\textwidth]{graphic.pdf}

\caption{(\textit{color online}) Map of torsion and concavity angles $(\varphi, \theta)$ for the condensates: Laminar P3HT (left); Isotropic P3HT (middle left); Segregated P3HT:C$_{60}$ (middle right); Isotropic P3HT:C$_{60}$ (right). Extreme values $\theta=0^{\circ},\ 180^{\circ}$ would correspond to extreme kinks, not realistic, moderate angles correspond to chain concavity; $\varphi<90^{\circ}$ non-alternate, $\varphi>90^{\circ}$ alternate ring-to-ring torsion angle. }
\label{map}
\end{center}
\end{figure}

Summarizing our results for the polymer configurations in the different simulation schemes, we see that in the laminar structure the individual chains mostly maintain linearity and ring-to-ring alternation. In the next three cases, for pure or blended P3HT, linearity is not a signature anymore, and we find a strong presence of chiral mer-to-mer angles (butterfly pattern). More significant is that for the two blend simulations, initially segregated or isotropic, the final individual polymer structures show strong similarity.

\subsection{C$60$ and P3HT-C$60$ properties in the blends}

We now analyse the difference between the two blend models focusing on the C$_{60}$ distribution, and to do that we use the radial distribution function RDF of fullerene molecules, shown in \ref{RDF-C-C} for the two blend simulations. In the figure we show also, as (blue) vertical lines, the results obtained through Nanomol-FF at 0K (geometry minimization) for the ideal crystal, starting with the \textit{fcc} ideal lattice; we find the structure to be very close to the \textit{fcc}, with one lattice constant slightly larger than the other two, and thus the second-neighbor distances result very close but not equal, so the lines are not summed with the resolution used in the figure. Focusing on the blend results, the RDF's are  built from 100 snapshots of each CMD simulation, and we see that the first-neighbor distance is a little larger than in the 0K structure, as expected, due to temperature effects; the value is in very good accord with available experimental data\cite{heba93arm} for the pure fullerene compound at normal temperatures. For the segregated model we find that the RDF peaks are clearly defined up to the fourth neighbor distances, and specially up to the second neighbor (with a spread), which indicates the prevalence of C$_{60}$ local ordered domains. It should be noticed that also for the initially isotropic blend we see (less defined) RDF peaks to the third neighbor, which points to  a natural segregation of the C$_{60}$ molecules. The main conclusions here are that we see a tendency to formation of separated polymer and fullerene domains in both simulations, that is, even with the relatively low polymer molecular weight we adopt, that as we find does not favor formation of crystalline P3HT domains, our results indicate segregation of C$_{60}$ molecules.

\begin{figure}[!htb]
\begin{center}

\includegraphics[width=\textwidth]{rdf-CC-fig.pdf}

\caption{(\textit{color online}) Radial distribution function RDF for  fullerene-fullerene molecules, 100 snapshots, in the segregated (left) and isotropic (right) blends P3HT:C$_{60}$. Arbitrary units, same normalization ratio for the two systems. The vertical blue lines correspond to first- to third-neighbor RDF signals for ideal \textit{fcc} crystalline C$_{60}$, see text, obtained with the same Nanomol-FF at 0K. }
\label{RDF-C-C}
\end{center}
\end{figure}

We finally analyse the RDF, shown in \ref{RDF-T-C}, for the distance from a fullerene molecule surface to the central point of a thiophene ring. For both types of blend, we see in the figure a broad peak at $\sim$4{\AA} measuring the distance of the closest neighbor T-rings from a fullerene surface; the decrease at $\sim$5{\AA} indicates the curvature of the polymer chain when adjacent to the molecule, that is, a T-chain will not be linear when adjacent to a $C_{60}$ molecule.

\begin{figure}[!htb]
\begin{center}

\includegraphics[width=\textwidth]{rdf-CT-fig.pdf}

\caption{Radial distribution function RDF for fullerene molecule-thiophene unit, 100 snapshots, in the segregated (left) and isotropic (right) blends P3HT:C$_{60}$. Arbitrary units, same normalization ratio for the two systems. The distance is measured from the center of a C$_{60}$ molecule to the center of the thiophene rings, subtracted the normal fullerene molecular radius.}
\label{RDF-T-C}
\end{center}
\end{figure}

We now focus on the presence of thiophene-fullerene T-C$_{60}$ compared to C$_{60}$-C$_{60}$ first-neighbor pairs from \ref{RDF-C-C} and \ref{RDF-T-C}. We note that we find a $\sim$60$\%$ higher occurrence of T-C$_{60}$ pairs in the initially isotropic compared to the initially segregated blend, and the opposite proportion for C$_{60}$-C$_{60}$ pairs: these proportions are not so high considering the difference is the initial structures, and again point to the natural segregation of $C_{60}$ molecules.

Summarizing our results, overall we see that for the molecular weight we adopt and in the spin-cast type of deposition, there is no formation of crystalline P3HT domains. On the other hand, even so we see a tendency to segregation of fullerene molecules, indicating formation of nano-domains typical of bulk heterojunctions.

\section{Conclusions}

As said in the Introduction, one of the more important factors affecting efficiency in bulk heterojunction polymer/molecule photovoltaic cells is the probability of exciton migration against recombination, and the ease of charge transfer. This will be governed by the morphology of the junctions, coming from different factors including blend mass proportion, and average molecular weight of the polymer. We simulated, through our finely tuned force-field based on the UFF\cite{rapp+92jacs} and cautious CMD procedure, different blend mixing for P3HT/C$_{60}$. We find that, even with the relatively low molecular weight of $\sim$5kg$\cdot$mol$^{-1}$ where there will be no polymer crystallization, fullerene segregation is dominant. This indicates that the use of simple fullerene for the blend should bring high performance to photovoltaic cells.

\begin{acknowledgement}
The authors are grateful to M. Alves-Santos for useful discussions.
We acknowledge support from INEO, FAPESP and CNPq (Brazil).
\end{acknowledgement}