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| -rw-r--r-- | 2-point.bib | 30 | ||||
| -rw-r--r-- | 2-point.tex | 58 |
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diff --git a/2-point.bib b/2-point.bib index 06da15a..63ec0db 100644 --- a/2-point.bib +++ b/2-point.bib @@ -28,7 +28,7 @@ @article{Biroli_1999_Dynamical, author = {Biroli, Giulio}, - title = {Dynamical TAP approach to mean field glassy systems}, + title = {Dynamical {TAP} approach to mean field glassy systems}, journal = {Journal of Physics A: Mathematical and General}, publisher = {IOP Publishing}, year = {1999}, @@ -290,6 +290,20 @@ doi = {10.1103/physrevb.36.5388} } +@article{Kirkpatrick_2015_Colloquium, + author = {Kirkpatrick, T. R. and Thirumalai, D.}, + title = {\textit{Colloquium}: Random first order transition theory concepts in biology and physics}, + journal = {Reviews of Modern Physics}, + publisher = {American Physical Society (APS)}, + year = {2015}, + month = {3}, + number = {1}, + volume = {87}, + pages = {183--209}, + url = {https://doi.org/10.1103%2Frevmodphys.87.183}, + doi = {10.1103/revmodphys.87.183} +} + @article{Kurchan_1996_Phase, author = {Kurchan, Jorge and Laloux, Laurent}, title = {Phase space geometry and slow dynamics}, @@ -360,3 +374,17 @@ doi = {10.1088/1751-8121/ab73ac} } +@article{Stillinger_2013_Glass, + author = {Stillinger, Frank H. and Debenedetti, Pablo G.}, + title = {Glass Transition Thermodynamics and Kinetics}, + journal = {Annual Review of Condensed Matter Physics}, + publisher = {Annual Reviews}, + year = {2013}, + month = {4}, + number = {1}, + volume = {4}, + pages = {263--285}, + url = {https://doi.org/10.1146%2Fannurev-conmatphys-030212-184329}, + doi = {10.1146/annurev-conmatphys-030212-184329} +} + diff --git a/2-point.tex b/2-point.tex index 22cce5b..21d7dbf 100644 --- a/2-point.tex +++ b/2-point.tex @@ -110,11 +110,10 @@ Many systems exhibit ``glassiness,'' characterized by rapid slowing of dynamics over a short parameter interval. These include actual (structural) glasses, -spin glasses, certain inference and optimization problems, and more -\cite{lots}. Glassiness is qualitatively understood to arise from structure of +spin glasses, certain inference and optimization problems, and more. Glassiness is qualitatively understood to arise from structure of an energy or cost landscape, whether due to the proliferation of metastable -states, or to the raising of barriers which cause effective dynamic constraints \cite{Cavagna_2001_Fragile}. -However, in most models there is no quantitative correspondence between these +states, or to the raising of barriers which cause effective dynamic constraints \cite{Cavagna_2001_Fragile, Stillinger_2013_Glass, Kirkpatrick_2015_Colloquium}. +However, in most models there is no known quantitative correspondence between these landscape properties and the dynamic behavior they are purported to describe. There is such a correspondence in one of the simplest mean-field model of @@ -124,7 +123,7 @@ the energy level at which thermodynamic states attached to marginal inherent sta \emph{minimum} or equivalently \emph{inherent state} and not a thermodynamic state. Any discussion of thermodynamic or equilibrium states will explicitly specify this. -}dominate the free energy +} dominate the free energy \cite{Castellani_2005_Spin-glass}. At that level, called the \emph{threshold energy} $E_\mathrm{th}$, slices of the landscape at fixed energy undergo a percolation transition. In fact, this threshold energy is significant in other @@ -172,7 +171,7 @@ are not distinguished by this measure. Moreover, we show that the usual picture marginal `manifold' of inherent states separated by subextensive barriers is only true at the threshold energy, while at other energies marginal minima are far apart and separated by extensive barriers \cite{Kurchan_1996_Phase}. Therefore, with respect to the problem of -dynamics this paper merely deepens the outstanding problems. +dynamics this paper merely deepens the outstanding issues. In \S\ref{sec:model}, we introduce the mixed spherical models and discuss their properties. In \S\ref{sec:results}, we share the main results of the paper. In @@ -316,11 +315,11 @@ In this study, we focus exclusively on the model studied in f_{3+4}(q)=\frac12\big(q^3+q^4\big) \end{equation} First, it has convex $f''(q)^{-1/2}$, so at least the ground state complexity -must be replica symmetric, and second, properties of its long-time dynamics -have been extensively studied. The annealed one-point complexity of these -models was calculated in \cite{BenArous_2019_Geometry}, and for this model the -annealed is expected to be correct. The complexity of this model is plotted in -Fig.~\ref{fig:complexities}. +must be replica symmetric, as in the pure spherical models. Second, properties +of its long-time dynamics have been extensively studied. The annealed one-point +complexity of these models was calculated in \cite{BenArous_2019_Geometry}, and +for this model the annealed is expected to be correct. The complexity of this +model is plotted in Fig.~\ref{fig:complexities}. \section{Results} \label{sec:results} @@ -378,7 +377,7 @@ the pure $p$-spin model continue to hold, with some small modifications \cite{Ros_2019_Complexity}. First, the nearest neighbor points are always oriented saddles, sometimes -extensive saddles and sometimes index-one saddles (Fig.~\ref{fig:spectra}(d, +saddles with an extensive index and sometimes index-one saddles (Fig.~\ref{fig:spectra}(d, f)). Like in the pure models, the emergence of oriented index-one saddles along the line of lowest-energy states at a given overlap occurs at the local minimum of this line. Unlike the pure models, neighbors exist for independent $\mu_1$ @@ -407,8 +406,8 @@ lowest-energy states. This is seen in Fig.~\ref{fig:franz-parisi}. The set of marginal states is of special interest. First, it has more structure than in the pure models, with different types of marginal states being found at -different energies. Second, these states attract the dynamics, and so are the -inevitable end-point of equilibrium and algorithmic processes. We find, +different energies. Second, these states attract the dynamics (as evidenced by power-law relaxations), and so are the +inevitable end-point of equilibrium and algorithmic processes \cite{Folena_2023_On}. We find, surprisingly, that the properties of marginal states pivot around the threshold energy. @@ -1376,7 +1375,7 @@ block constant matrices, things can be worked out from here. For instance, the second term in $M_{11}$ contributes nothing once the appropriate limits are taken, because each contribution is proportional to $n$. -The contribution con be written as +The contribution can be written as \begin{equation} \label{eq:inverse.quadratic.form} \begin{bmatrix} X_0\\i\hat X_0 @@ -1662,13 +1661,36 @@ found something striking: only those at the threshold energy have other marginal minima nearby. For the many marginal minima away from the threshold (including the exponential majority), there is a gap in overlap between them. +This has implications for pictures of dynamical relaxation. In most $p+s$ +models studied, quenches from infinite to zero temperature (gradient descent +starting from a random point) relax towards marginal states with energies above +the threshold energy \cite{Folena_2023_On}, while at least in some models a +quench to zero temperature from a temperature around the dynamic transition +relaxes towards marginal states with energies below the threshold energy +\cite{Folena_2020_Rethinking, Folena_2021_Gradient}. We found (see especially +Figs.~\ref{fig:marginal.prop.below} and \ref{fig:marginal.prop.above}) that the +neighborhoods of marginal states above and below the threshold are quite +different, and yet the emergent aging behaviors relaxing to states above and +below the threshold seem to be the same. Therefore, this kind of dynamics +appears to be insensitive to the neighborhood of the marginal state being +approached. To understand something better about why certain states attract the +dynamics in certain situations, nonlocal information, like the +structure of their entire basin of attraction, seems vital. + The methods developed in this paper are straightforwardly (if not easily) -generalized to landscapes with replica symmetry broken complexities \cite{Kent-Dobias_2023_How}. +generalized to landscapes with replica symmetry broken complexities +\cite{Kent-Dobias_2023_How}. We suspect that many of the qualitative features +of this study would persist, with neighboring states being divided into +different clusters based on the \textsc{rsb} order but with the basic presence +or absence of overlap gaps and the nature of the stability of near-neighbors +remaining unchanged. Interesting structure might emerge in the arrangement of +marginal states in \textsc{frsb} systems, where the ground state itself is +marginal and coincides with the threshold. \paragraph{Acknowledgements} -The author would like to thank Valentina Ros, Giampaolo Folena, and Chiara -Cammarota for useful discussions related to this work. +The author would like to thank Valentina Ros, Giampaolo Folena, Chiara +Cammarota, and Jorge Kurchan for useful discussions related to this work. \paragraph{Funding information} |