Nobel Lecture: Multiple equilibria^*

This is an extended version of my Nobel Lecture, delivered on December 8, 2021. I will recall the genesis of the concept of multiple equilibria in natural sciences. I will then describe my contribution to the development of this concept in the framework of statistical mechanics. Finally, I will briefly mention the cornucopia of applications of these ideas both in physics and in other disciplines.

Figure 1

The magnetic susceptibility of spin glasses: the results for Au-Fe alloys in the region of low Fe concentration. From [39].

Figure 2

An example of the tree of states in the case where the overlap takes only three possible values. The reader should notice that the tree is upside down: the root is at the top and the leaves are at the bottom. From [166].

Figure 3

The analytic results of the mean-field approximation for the linear response susceptibility (${\chi }_{\mathrm{LR}}$, lower curve) and the field-cooled susceptibility (${\chi }_{\mathrm{eq}}$, upper curve). They coincide in the high-temperature region.

Figure 4

The linear response susceptibility (${\chi }_{\mathrm{LR}}$, lower curve) and the field-cooled susceptibility (${\chi }_{\mathrm{eq}}$, upper curve). The experimental results are taken from [67].

Figure 5

The results from the Janus Collaboration ([18], [19]) for the average function $P(Q)$ in three dimensions for $L=8$, 12, 16, 24, 32 in the low-temperature regions.

Figure 6

Four different examples of the $P_J(q)$ function in three dimensions for $L=32$ corresponding to different samples in the low-temperature region ([18], [19]). The average over 1000 different samples gives the smooth function $P(q)$ of Fig. 5.

Figure 7

Experimental raw results (full symbols) and extrapolations to the infinite time limit (open symbols) for the response function [defined in Eq. (30)] vs the correlation [defined in Eq. (29)]. The different curves span the waiting times studied: $t_w=100\mathrm{s}$, $t_w=200\mathrm{s}$, $t_w=500\mathrm{s}$, $t_w=1000\mathrm{s}$, $t_w=2000\mathrm{s}$. From [98].