# The relation between Parisi scheme and multi-thermalized dynamics in finite
dimensions

Silvio Franz  LPTMS, UMR 8626, CNRS, Univ. Paris-Sud, Université Paris-Saclay,
91405 Orsay, France  J Kurchan  Laboratoire de Physique de l’Ecole normale
supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de
Paris, F-75005 Paris, France

###### Abstract

In this note we summarize the connections between equilibrium and slow out of
equilibrium dynamics in finite dimensional glasses, such as we understand them
today. If we assume that a finite-dimensional system is stable with respect to
a family of weak random perturbations (stochastic stability), then its
dynamics have a ‘Multithermalization’ structure if and only if the Boltzmann-
Gibbs distribution obeys an Ultrametric Parisi distribution.

###### Contents

  1. I  Introduction
  2. II Equilibrium: Ergodicity Breaking and the Parisi scheme
    1. II.1 Overlap Statistics
    2. II.2 The method of random perturbations
    3. II.3 Self-averageness and higher order overlap statistics
    4. II.4 Triangles
    5. II.5 Overlap locking
    6. II.6 Marginal stability and Stochastic Stability in Mean-Field
  3. III Dynamics and Multithermalization
    1. III.1 Slow dynamics
    2. III.2 Reparametrization softness
    3. III.3 Reparametrization-invariant quantities
    4. III.4 Time scales
    5. III.5 Bringing a system to a stationary form
    6. III.6 The Multithermalization scheme
  4. IV Susceptibility Lego
    1. IV.1 Triangles
  5. V The nucleation argument
    1. V.1 Overlap equivalence if and only if timescale-separation
  6. VI Commuting limits
  7. VII The whole Parisi $\leftrightarrow$ multithermalization scheme needs time reparametrization invariance
  8. VIII Reparametrization invariance and replica space
    1. VIII.1 A consequence of ultrametricity
  9. IX Two, perhaps three finite-dimensional puzzles
    1. IX.1 Is one-step RSB unstable in finite dimensions?
    2. IX.2 Dynamic timescales in three and four dimensions
    3. IX.3 Aging strongly far from equilibrium
  10. X Conclusions

##  I  Introduction

A system whose equilibrium measure follows the Parisi scheme Mézard  _et al._
(1987), commonly known as Replica Symmetry Breaking (RSB), will take an
infinite time to reach its equilibrium starting form random configurations, a
process called ‘aging’. It may also be driven into an out of equilibrium
steady-state by an infinitesimal drive, such as shear Thalmann (2001);
Berthier  _et al._ (2000a), or time-dependence of disorder Horner (1992). If
the relaxation times are long, or, in a steady-state if the drive is weak, the
dynamics are slow: this is the regime we are interested in.

The out of equilibrium dynamics under these circumstances is a very specific
one Sompolinsky and Zippelius (1982, 1981); Horner (1992); Cugliandolo and
Kurchan (1993, 1994); Franz and Mézard (1994). At given times one may define
an effective temperature with a thermodynamic meaning Cugliandolo  _et al._
(1997), it is the same for all observables. Different temperatures are
possible, but in different ‘time scales’, a notion one has to define. We refer
to this situation as ‘multithermalization’ Contucci  _et al._ (2019, 2020).

In finite dimensional systems, neither the Parisi scheme nor the multi-
thermalization scenario are proven to hold. However, the Parisi kind of
ordering appears as the only possible non ergodic state which is stable
against small random perturbations in the Hamiltonian.

Moreover, it turns out that Parisi RSB and multi-thermalization are deeply
related: one of them holds if and only if the other also does. The
temperatures involved in the slow dynamics then relate to the average overlap
distribution computed for equilibrium in the Parisi scheme. This double
implication may be argued on the basis of a strategy devised by Franz, Mézard,
Parisi and Peliti (FMPP) Franz  _et al._ (1998, 1999) and later generalized in
Kurchan (2021) that uses linear response theory and equilibration of bulk
quantities to compare dynamical quantities to equilibrium one. An extra
element that is of central importance for the validity of both schemes, as we
now realize, is the fact that the slow glassy dynamics is time-
reparametrization soft: small perturbations dramatically stretch the
timescales. Given the relation described above between static replica theory
and slow dynamics, it is natural to search for reparametrization invariances
in the Parisi replica context, and indeed questions like these have appeared
in the literature. The whole matter deserves further clarification.

##  II Equilibrium: Ergodicity Breaking and the Parisi scheme

Replica Symmetry Breaking is ubiquitous in disordered systems with long range
interactions Mézard  _et al._ (1987). Its successful implementation by Giorgio
Parisi dates back to the study of the Sherrington-Kirkpatrick model of a Spin
Glass Sherrington and Kirkpatrick (1975), in the effort of parametrizing a
$n\times n$ matrix for $n\to 0$ Parisi (1979). It describes the low
temperature disordered frozen state with broken ergodicity in systems where
ordinary ordering is not possible. For large samples, the Gibbs measure is
dominated by a few ’valleys’ unrelated by symmetry and in random positions in
configuration space. The distances between these valleys have broad
distributions, but they are strongly constrained, in fact only isosceles
triangles, with the different side the smallest are possible in the
thermodynamic limit, the property of ‘ultrametricity’. The mathematical
analysis of long range systems Talagrand  _et al._ (2003); Panchenko (2013a)
has fully confirmed and considerably enriched the picture coming from the
physicist’s analysis based on the iconoclastic replica method and the more
conventional, but still non rigorous Cavity Method. The Parisi picture is also
a natural candidate to describe ideal thermodynamic glassy phases of systems
with short range interactions. However, its validity in three and more in
general in any finite dimensional systems has been seriously questioned Fisher
and Huse (1986); Bray and Moore (1987). The best available numerical
simulations give strong indications in favor of RSB above a critical dimension
that is somewhere above two and below four.
The least one can say is that RSB provides a very good statistical description
of samples of finite sizes, or, equivalently, of local properties of infinite
samples. Local RSB has indeed been proven to hold in models with large but
finite interaction range Franz and Toninelli (2004). But we do not know, with
a convincing level of theoretical confidence, and even less of mathematical
rigour, if RSB really survives as long range order in the thermodynamic limit.
What indeed we know, is that under some rather mild assumption (see later), if
random broken ergodicity holds, the structure of the probability measure on
random Gibbs measures, also called metastate, should be qualitatively
described by the RSB scenario. In fact, the mathematicians were able to prove
that under the hypothesis that the overlap distributions verify the so called
Ghirlanda-Guerra identities (GGI) Ghirlanda and Guerra (1998) then RSB
ultrametric statistics follows Panchenko (2013b). In turn, the GGE follow from
the stability of Gibbs measure against small perturbation Ghirlanda and Guerra
(1998); Aizenman and Contucci (1998), an hypothesis that holds true in mean-
field and, we will argue, can be generically accepted in finite dimension.
Within Mean-Field, it soon appeared that the dynamics of RSB systems presents
anomalies with respect to the one in ordinary ergodic systems Sompolinsky and
Zippelius (1981, 1982). Recognizing the full fledged off-equilibrium character
of these anomalies led to the surprise that aging phenomena, which are
commonly observed in laboratory glasses, also appeared in mean-field models,
giving rise to a coherent and detailed theory of dynamic ergodicity breaking
Cugliandolo and Kurchan (1993, 1994). This is obtained by the asymptotic
analysis of exact Dynamical Mean Field Theory (DMFT) equations, which display
a low temperature solution violating basic equilibrium properties as time
translation invariance and the relations between responses and correlation
implied by the fluctuation dissipation theorem. It soon appeared that RSB and
aging are intertwined phenomena in mean-field. Two classes of models were
identified: p-spin like models and SK like models Cugliandolo and Kurchan
(1993, 1994). In the former, one observes a decoupling between equilibrium and
dynamics. The low temperature equilibrium measure is typically dominated by
metastable states (ergodic components) with high stability, while the dynamics
starting from a random configuration, tends to a marginal manifold which has
higher energy density respect to the equilibrium states. In the latter,
instead, the states dominating equilibrium are marginally stable, and the
asymptotic dynamical solution predicts asymptotic values of extensive
observables coinciding with the equilibrium ones. Unfortunately, the results
of asymptotic dynamics have been rather elusive to rigorous mathematical
analysis. While Parisi picture for equilibrium has been made fully rigurous by
mathematicians, not much has been done in dynamics, where the only progress
has been the proof of the validity of the physicist’s effective one body
dynamical equations and their self-consistent Dynamical Mean Field Theory
closure Arous and Guionnet (1995); Ben Arous  _et al._ (2006). As far as
finite dimension is concerned the situation is similar to the one of
equilibrium: the Cugliandolo-Kurchan dynamical scenario provides an excellent
description of finite time spin-glass dynamics Barrat and Kob (1999); Berthier
_et al._ (1999), but we cannot exclude that the aging responses eventually
become trivial and the system crosses over to complete thermalization.

###  II.1 Overlap Statistics

In the physics of the disordered systems in equilibrium a crucial role is
played by the statistics of the overlap between replicas induced by the
thermal and disorder fluctuations Mézard  _et al._ (1987). What we call
replicas here are configurations extracted independently from the Boltzmann
measure for the same disorder. The overlap $Q[S,S^{\prime}]$ between two
configurations is usually defined as
$Q[S,S^{\prime}]=\frac{1}{N}\sum_{i=1}^{N}S_{i}S_{i}^{\prime}$.

A first characterization of the statistics of the overlap is given by its
’average histogram’

$\displaystyle
P(q)=\mathbb{E}\left(\langle\delta(Q[S_{1},S_{2}]-q))\rangle_{12}\right)$ (1)

where we denoted as $\langle\cdot\rangle_{12}$ the average over the (product)
Gibbs measure of the two replicas, and with $\mathbb{E}$ the average over
their common disorder. It was shown by Parisi a long time ago, that a non
trivial $P(q)$ (i.e. different from a single $\delta$ function) is associated
to breakdown of ergodicity Parisi (1983). This is trivially true for example
in systems with symmetry breaking in absence of a source, where it just
reflects that the measure is uniform over the different possible phases of the
system related to one another by symmetry. In that case, the function can be
simply collapsed to a $\delta$ by the introduction of an infinitesimal
symmetry-breaking source that selects one among the possible phases, and the
function $P(q)$ does not play any physically important role in this case. In
disordered systems $P(q)$ is not related to a any symmetry, and the situation
can be much more interesting: a small perturbation unrelated to the original
Hamiltonian cannot be expected to project on a single state, and modulo
possible removal of accidental degeneracies of the original Hamiltonian, the
function $P(q)$ of a perturbed system should stay close to the same function
in the unperturbed one. This has deep consequences, since the multiplicity of
states will affect the response properties of the system and $P(q)$ can be
interpreted as the generating function of the responses to a family of random
perturbations.

###  II.2 The method of random perturbations

The study of the overlap statistics requires access to microscopic
configurations. This is of course easy in computer studies, but is more
problematic in experiments. Fortunately one can use linear response theory
(LRT) to give a different characterization of $P(q)$, as the generating
function of a families of susceptibilities. Starting from a model with
Hamiltonian $H$, one considers a perturbed system where random $p-spin$ like
Hamiltonian are added to the energy of the system

$\displaystyle H_{pert}[S]=H[S]+\sum_{p=1}^{\infty}\epsilon_{p}H_{p}[S]$ (2)
111The $\epsilon_{p}$ have to decrease fast enough with $p$ in order for the
system to be well defined.

where the perturbing terms are defined as

$\displaystyle
H_{p}[S]=-\sum_{i_{1},...,i_{p}}J_{i_{1},...,i_{p}}S_{i_{1}}...S_{i_{p}}$ (3)

The couplings $J_{1_{i},...,i_{p}}$ are usually taken as centered i.i.d.
Gaussian variables scaled as
$\mathbb{E}\left(J_{1_{i},...,i_{p}}^{2}\right)=\frac{1}{N^{p-1}}$. If the
parameters $\epsilon_{p}$ are chosen to be independent of $N$, the perturbing
Hamiltonian involves extensive energy in equilibrium; it is straightforward to
see that

$\displaystyle I^{(p)}\equiv\frac{1}{N}\mathbb{E}[\langle H_{p}\rangle]$
$\displaystyle=$ $\displaystyle T\frac{\partial}{\partial\epsilon_{p}}\log
Z=N\beta\epsilon_{p}(1-\langle q^{p}\rangle)$ (4) $\displaystyle\langle
q^{p}\rangle$ $\displaystyle=$ $\displaystyle\int dq\;P_{\epsilon}(q)q^{p}$
(5)

that can be readily obtained by integration by parts. Values of $\epsilon_{p}$
vanishing with $N$, namely $\epsilon_{p}\sim N^{-a}$ with $0<a<1$, have also
been considered, where (4) continues to hold, but the energy absorbed is sub-
extensive. This suggests we define Stochastic Stability as the property that
$P_{\epsilon}(q)\to P_{0}(q)$ (modulo symmetry or accidental degeneracy of the
unperturbed Hamiltonian).

The reader may object that in an experimental system the susceptibilities
defined by (4) are not more accessible than the microscopic configurations.
The crucial point that we discuss later in the paper, however, is that, in
turn, these quantities can be related to dynamical susceptibilities whose
generator function can be in principle measured through response experiments
in ageing dynamics.

###  II.3 Self-averageness and higher order overlap statistics

Higher order statistics of the overlaps, where one considers those between
more than two replicas, are also important. Their distributions appears when
formally considering the quadratic fluctuations of the perturbing terms
$H_{p}$. For example, it is easy to show that the average square Hamiltonian
$\mathbb{E}(\langle H_{p}[S]^{2}\rangle)$ is related to the statistics of two
overlaps $P(q_{12},q_{13})$

$\displaystyle\mathbb{E}(\langle
H_{p}[S]^{2}\rangle)=\epsilon_{p}^{2}\beta^{2}N^{2}\left(1-2\langle
q^{p}\rangle-\langle q^{2p}\rangle+\langle
q_{12}^{p}q_{13}^{p}\rangle\right).$ (6)

But $H$ is a self-averaging quantity, and at order $N^{2}$ $\mathbb{E}(\langle
H_{p}[S]^{2}\rangle)=(\mathbb{E}(\langle H_{p}[S]\rangle))^{2}$. This implies
a strong constraint on $P(q_{12},q_{13})$ which needs to verify

$\displaystyle
P(q_{13},q_{23})=\frac{1}{2}\delta(q_{13}-q_{23})P(q_{13})+\frac{1}{2}P(q_{12})P(q_{13}),$
(7)

A similar analysis can be applied to $\mathbb{E}(\langle
H_{p}[S]\rangle^{2})$, resulting in

$\displaystyle
P(q_{12},q_{34})=\frac{1}{3}\delta(q_{12}-q_{34})P(q_{12})+\frac{2}{3}P(q_{12})P(q_{34}).$
(8)

Notice that $P(q_{12},q_{34})$ represents the average over disorder of the
product of the disorder dependent overlap distribution of four independent
replicas. Equation (8) tells us that this is different from
$P(q_{12})P(q_{34})$: self-averageness of the energy, from which (8) is
derived implies that the overlap probability (unless trivial) is instead non-
self averaging. Also notice that once $P(q)$ is know, the higher order
statistics of two overlaps are fixed.

###  II.4 Triangles

Equations (7,8) are examples of the celebrated Ghirlanda-Guerra identities
Ghirlanda and Guerra (1998) that have played a crucial role in the
mathematical analysis of Spin Glasses. It has been shown by Panchenko
Panchenko (2013b) that in their most general form they imply ultrametricity
(see Figs 1 and2) for any three states at mutual overlaps
$q_{12},q_{23},q_{31}$: with probability one in the thermodynamic limit the
two smallest overlaps are equal (all triangles are isosceles)
$q_{13}=\min(q_{12};q_{23})$. Moreover, if we consider the conditional
probability Mézard  _et al._ (1987) we find:

$\displaystyle P(q_{13}|q_{12}=q_{23}=q)$ $\displaystyle=$
$\displaystyle\frac{P(q_{13},q_{12}=q,q_{23}=q)}{P(q_{12}=q,q_{23}=q)}$ (9)
$\displaystyle=$
$\displaystyle\theta(q_{13}-q)P(q_{13})+\delta(q_{13}-q)x(q).$ (10)

where $x(q)=\int_{0}^{q}dq^{\prime}\;P(q^{\prime})$ is the cumulative
probability. Notice that $q_{13}$ is independent of $q$ when it is larger than
$q$, and that, as in the case of the overlap pairs, the triangle probabilities
are fully specified by the function $P(q)$, or equivalently by its primitive,
$x(q)$. Indeed, this generalizes to arbitrary number of replicas: thanks to
self-averageness, $x(q)$ provides a sufficient statistics for the overlaps.

The picture is rather remarkable: in any system where extensive quantities are
self-averaging and the system is stable with respect to small long-range
random interactions, the only possibility besides triviality is ergodicity
breaking according to the Parisi scheme. Originated in an unconventional
mathematically analysis of the famous $0\times 0$ replica matrix, the Parisi
RSB scheme has revealed a deep, and inescapable symmetry of nature,
transcending the replica trick.

###  II.5 Overlap locking

It can be easily realized that if RSB is present in a system, then the overlap
between any two random configurations should be homogeneous in space. This is
true in long range systems, where by space we mean the complete graph (SK
model) or the Bethe lattice (Viana-Bray model), but also, and especially, in
any finite dimensional system with short range interaction that could possibly
display RSB. Let us specifically refer to this case. Overlap homogeneity means
that given two random equilibrium configurations, if one measures the value of
the overlap in a given large enough region of space, the same value would be
found by an analogous measure in any other region, no matter how far away from
the original one. This is a very strong correlation property of equilibrium
states.

Suppose you divide the system in two adjacent parts. The surface interaction
term could be disconnected, without any appreciable change in the expectation
values of bulk quantities. On the other hand, this term is crucial for the
overlap statistics: in absence of the surface interaction the overlaps in the
two halves are independent variables; in presence of it, the overlaps are
locked to take the same values. We can generalize this consideration to more
general weakly coupled disordered systems. The locking of the overlaps to
identical values in the parts of the same system occurs because of statistical
homogeneity of a given sample. An analogous locking however also occurs in
absence of homogeneity, whenever we weakly couple systems with RSB. Suppose we
have two systems ${\cal S}_{1}$ ${\cal S}_{2}$, whose overlap statistics is
specified by $x_{1}(q)$ and $x_{2}(q)$ respectively, not necessarily
identical. As before, as long as the systems are not interacting the overlaps
are independent. Stability of the RSB state on the other hand requires that
the as soon as a weak interaction is switched on, the overlaps do lock, and
one can define a function $Q_{2}(q_{1})$, which allows to predict the overlap
between two replicas in the system $S_{2}$, when the overlap is $q_{1}$ in the
system $S_{1}$. It is clear that this relation should be specified by the
condition $P_{1}(q_{1})dq_{1}=P_{2}(q_{2})dq_{2}$, or equivalently
$x_{1}(q_{1})=x_{2}(Q_{2}(q_{1}))$ Franz  _et al._ (1992). The actual
functions $P_{1}(q)$ and $P_{2}(q)$ on the other hand will be only linearly
affected by the coupling, and coincide in a fist approximation with the
overlap probability distribution in the unperturbed systems. Here we encounter
a first instance of ‘softness’ of the solution: the presence or absence of
small, thermodynamically irrelevant terms, leads to a complete rearrangement
of the Gibbs weights, without affecting the values of self-averaging
quantities. Below we shall find the same property in the context of dynamics.

###  II.6 Marginal stability and Stochastic Stability in Mean-Field

Within mean-field, two classes of models are known: Models with ’full-RSB’, as
the SK model, where the support $Supp(P)$ is a continuous interval, say
$[q_{0},q_{1}]$ and models where the support has ’holes’. The most common case
of this scenario is the case of ’one-step RSB’ (1RSB) where only two values of
the overlap support the whole probability, this is the case of Potts-glasses
and $p$-spin models and for simplicity we will refer to it here. We shall
address these support values as ‘skeleton’ values of the correlations, and
discuss how they play an important role in the correspondence between
equilibrium and dynamics.

The physical picture between continuous and 1RSB cases is very different: in
the first case the glassy phase is only marginally stable. This can be easily
realized: the supports of the overlaps distributions are self-averaging Parisi
and Talagrand (2004), implying that it is possible to go from a given state to
any other crossing only sub-extensive barriers Franz and Parisi (1995, 1998).
Within the replica method one finds families of Hessian zero modes associated
to small variations $P(q)\to P(q)+\delta P(q)$ Temesvári  _et al._ (2000); De
Dominicis and Giardina (2006), a reparametrization transformation that has a
deep dynamical analogue as we will see in the following sections. In the
second case, conversely, equilibrium states are absolutely stable and are
separated by extensive free-energy barriers. The effect of a small but
extensive random perturbation is very different in the two cases. In the 1RSB
cases the states are reshuffled: their weight changes but they keep their
identity and continue to exist. In the full-RSB case the effect is more
dramatic, any $O(N)$ perturbation destabilizes the unperturbed equilibrium
states and produce new ones in random positions. A crucial property is that
the probability distribution of the disorder-dependent Gibbs state is only
affected in a regular way by the perturbation (i.e. linearly for small
perturbations). This regularity property, called Stochastic Stability is very
natural and at the same time important, it tells us that the original model is
generic, sharing the same physics with a broad class of similar systems.

Figure 1: The famous Parisi hierarchical overlap matrix, encoding the full
overlaps’ statistics of disordered systems with broken ergodicity. Figure 2:
The possible overlaps and the triangles they determine, according to the
ultrametric ansatz.

##  III Dynamics and Multithermalization

###  III.1 Slow dynamics

In the dynamic approach we consider a system evolving according to local
reversible dynamics. To fix the ideas we can consider the Langevin equation

$-m_{i}{\ddot{s}}_{i}-\frac{\partial H({\bf s})}{\partial
s_{i}}=\underbrace{\Gamma_{0}{\dot{s}}_{i}-\eta_{i}}_{bath}$ (11)

where $\eta_{i}$ are uncorrelated Gaussian white noises with variance
$2\Gamma_{0}T$ and $\Gamma_{0}$ is the strength of the coupling to the ‘white’
bath. This is guaranteed to reach eventually equilibrium, although in the
systems that concern us, in times that may diverge with the system’s size $N$.
For the considerations that follow, any stochastic reversible dynamics whose
update rule just depends on the force $f_{i}=-\frac{\partial H({\bf
s})}{\partial s_{i}}$ as e.g. Metropolis or Glauber dynamics would give rise
to the same results. We shall be concerned with the limit of slow dynamics,
which appears, in glassy systems, in different ways:

  * •

Aging Castellani and Cavagna (2005): Following a quench of the system from a
high to a low temperature, at which the equilibration time is infinite, the
system ‘ages’: it evolves slower and slower as the time since the quench
elapses. The two-time correlations never become an exclusive function of time-
differences. The large parameter is the smallest ‘waiting’ time since the
quench $t^{\prime}=t_{w}$, that modulates the decay at further time
$C(t,t^{\prime})=C\left(\frac{\tau}{t_{w}}\right)$: the older (large $t_{w}$)
the system, the slower the decay. In such conditions of long time scales local
observables depending on configurations at a single time, such as the energy
or specific heat, magnetization etc., only undergo small variations, and in
what follows we will consider the ideal situation where they can be taken to
be essentially equal to their asymptotic values.

  * •

Driven system Thalmann (2001); Berthier  _et al._ (2000a) When the system is
subjected to forces not deriving from a potential – shear, for example – it is
an experimental fact that aging is interrupted, in the sense that all
functions become time-translational invariant, but slow. Their timescale of
decay of correlation then is controlled by the driving rate $\sigma$, the
slower the weaker the drive: $C(t-t^{\prime})=C\left({\tau}{\sigma}\right)$

  * •

Time-dependent disorder Horner (1992).  Another way to make a system with
disorder time-translational invariant is to change the disorder slowly : the
small parameter is the timescale $\tau_{0}$ of change of disorder:
$C(t,t^{\prime})=C\left(\frac{\tau}{\tau_{0}}\right)$.

In what follows, we refer as slow dynamics to the limit of either long waiting
times, small shear strains or slow variation of parameters. The latter two
cases are stationary, all dependence on history is lost, but quite
surprisingly they are not equilibrium systems, as we shall see.

In all these systems we consider correlation $C_{AB}(t,t^{\prime})$ and linear
response function $R_{AB}(t,t^{\prime})$ (here $t\geq t^{\prime}$), the
average response of $A$ at time $t$ to a kick of $B$ at time $t^{\prime}$. In
the case of spin variables above, correlations and response functions are
given by:

$C_{ij}(t,t^{\prime})=\langle s_{i}(t)s_{j}(t^{\prime})\rangle\qquad;\qquad
R_{ij}(t,t^{\prime})=\frac{\delta}{\delta h_{j}(t^{\prime})}\langle
s_{i}(t)\rangle|_{h=0}$ (12)

where $h_{i}(t)$ is the value at time $t$ of a field conjugated to $S_{i}$ in
the Hamiltonian. One in general is interested to the global correlations and
response, defined as
$C(t,t^{\prime})=\frac{1}{N}\sum_{i}C_{ii}(t,t^{\prime})\qquad;\qquad
R(t,t^{\prime})=\frac{1}{N}\sum_{i}R_{ii}(t,t^{\prime})$. It is useful to
consider the response function in its integral form:

$\chi(t,t^{\prime})=\int_{t^{\prime}}^{t}dt^{\prime\prime}\;R(t,t^{\prime\prime})$
(13)

representing the response at time $t$ to a field acting on the system from
$t^{\prime}$ to time $t$

In an equilibrium situation, we have for all $A,B$ time translation invariance
$C_{AB}(t,t^{\prime})=C_{AB}(t-t^{\prime})$ and the Fluctuation-Dissipation
relation:

$TR_{AB}(t,t^{\prime})=\frac{\partial C_{AB}(t,t^{\prime})}{\partial
t^{\prime}}$ (14)

Glassy dynamics, even when it is rendered stationary as above, violates these
relations. In the spirit of the above, it makes sense to define effective
temperatures Cugliandolo  _et al._ (1997) as:

$T_{AB}(t,t^{\prime})R_{AB}(t,t^{\prime})=\frac{\partial
C_{AB}(t,t^{\prime})}{\partial t^{\prime}}\qquad\qquad;\qquad\qquad
T_{AB}(t,t^{\prime})=\frac{T}{X_{AB}(t,t^{\prime})}$ (15)

In equilibrium $X=1$ and where $T_{AB}(t,t^{\prime})=T$, the bath’s
temperature. It turns out that these effective temperatures are exactly what a
thermometer would measure when tuned to respond to the corresponding
observables at the corresponding temperatures Cugliandolo  _et al._ (1997).

###  III.2 Reparametrization softness

Glassy dynamics have a remarkable emergent property that has long been
recognized: their evolutions may be drastically modified by small
perturbations. For example, a static perturbation may lead the system to
explore regions of the configuration space very different from the one reached
in absence of the perturbations; the introduction of a perturbation in a well-
aged system leads to rejuvenation, modifying the rhythm of relaxation. Even
more dramatically, as we have mentioned above, the presence of a small shear
or evolution of disorder leads to interruption of aging and the system is
rendered stationary.

All these are highly nonlinear effects on the system’s dynamics.
Paradoxically, this is does not mean that linear response breaks down: in
spite a very modified time-dependence, there are families of relations where
time is gauged out that are only linearly affected by the perturbations. When
one looks more closely at how this happens, it turns out that this ‘softness’
concerns the pace of the dynamics, where two point correlations become
modified by a time reparametrization $t\to h(t)$ as follows:

$\displaystyle C(t,t^{\prime})\rightarrow C(h(t),h(t^{\prime}))$
$\displaystyle
R(t,t^{\prime})\rightarrow\dot{h}(t^{\prime})R(h(t),h(t^{\prime}))$

Or, in general, for a multi-point average $O(t_{1},...,)=\langle
s_{i}(t_{1})\eta_{j}(t_{2})s_{k}(t_{3})...\rangle$ a reparametrization acts on
each factor as $s_{i}(t)\rightarrow s_{i}(h(t))$ and
$\eta_{i}(t)\rightarrow\dot{h}(t)\eta_{i}(h(t))$

###  III.3 Reparametrization-invariant quantities

Reparametrization softness means that small perturbations affect, as it were,
the speed of the film, but leave the mutual relations ‘within a scene’
invariant. It thus follows that there is a particularly significant sub-
ensemble of dynamic quantities: those where time is factored out Cugliandolo
and Kurchan (1994), and thus in the limit of large times they are invariant
under reparametrizations $t\rightarrow h(t)$ ($t\to\infty$), being $h$ any
monotonously increasing function of time. The first example is rather trivial,
but important for the following: for single time quantities $O(t)$,
reparametrization invariance implies that they must become independent of
time, i.e. to be asymptotically close to their infinite time value. More
significant are the following examples:

  * •

Given three long, successive times $t_{1}<t_{2}<t_{3}$, and the corresponding
correlations $C_{21},C_{32},C_{31}$, define for large times (see Fig. 3).
$C_{31}=f(C_{21};C_{32})=\lim_{t_{1}\rightarrow\infty}f(C_{21};C_{32},t_{1})$,
a ‘triangle relation’.222More precisely the relation should be thought to be
valid in the large time limit according to
$f(C_{21};C_{32})=\lim_{t_{1}<t_{2}<t_{3}\to\infty\atop
C(t_{2},t_{1})=C_{21};C(t_{3},t_{2})=C_{32}}C(t_{3},t_{1})$ and similarly for
the quantities that we discuss in what follows. Similarly for the integrated
responses: $\chi(C_{31})=\tilde{f}[\chi(C_{21});\chi(C_{32})]$, i.e. the
triangle relations $\tilde{f}$ and $f$ are isomorphic.

  * •

Given any dynamic parameter $X_{AB}(t,t^{\prime})$ define for large times
$X_{AB}(t,t^{\prime})\rightarrow X_{AB}\left[C_{AB}(t,t^{\prime})\right]$. We
shall focus on cases in which this limit is non-trivial $X\neq 1$. This also
implies that the integrated response becomes a function of the correlation:
$\chi(t,t^{\prime})\rightarrow\chi(C(t,t^{\prime}))$.

  * •

Given any two correlations of the system $\bar{C}(t,t^{\prime})$ and
$C(t,t^{\prime})$ we write, again in the large-time limit, $\bar{C}\rightarrow
g(C)$ for some $g$.

A way of putting this is as follows: a reparametrization-invariant quantity is
a function in which times have been eliminated in favor of a correlation at
those times: the particular one used acts as a ‘clock’ with respect to which
all other quantities can be referred.

It is very natural to separate those quantities that are reparametrization-
invariant from the reparametrizations themselves. A procedure for this may be
implemented numerically: see Castillo  _et al._ (2002, 2003); Chamon  _et al._
(2004, 2002); Chamon and Cugliandolo (2007); Chamon  _et al._ (2011).

Figure 3: The triangle relation $f(a,b)$ is associative:
$C_{41}=f[C_{43},f(C_{32},C_{21})]=f[f(C_{43},C_{32}),C_{21}]$: the function
$f$ is associative.

###  III.4 Time scales

The reparametrizations we consider are such that they leave the triangle
function $C_{13}=f(C_{12},C_{23})$ invariant. The function $f$ allows us to
define, in a natural way, time scales in a reparametrization-invariant way.

The function $f(a,b)$ may be easily seen to be associative (see Fig 3). It is
possible to classify all associative functions, it turns out that the
correlation has special ‘skeleton’ values, that for simplicity we write here
as a discrete and growing series, $q_{1},...,q_{k}$ that divide the possible
correlation in intervals $(q_{1},q_{2}),(q_{2},q_{3})...(q_{k-1},q_{k})$ with
the property that if $C_{12}$ and $C_{23}$ belong to two different intervals
then $f(C_{12},C_{23})=\min(C_{12},C_{23})$. If, on the contrary, $a$ and $b$
are in the same interval, another law applies. Each interval corresponds to a
time scale, defined in a reparametrization-invariant way. The correspondence
between Parisi and a dynamic, ‘multi-thermalization’ scheme (to be defined
below) holds at the level of these ‘skeleton’ values.  Let us see some
examples of two, three and infinitely many scales where the set of skeleton
values corresopndo to a continuous interval.

#### Two scales

This is the most usual case. An example is when the correlation $0<C\leq 1$
and there is a value $q$ such that for the interval $q\leq C\leq 1$ the
correlation is much faster than for the interval $0\leq C<q$. We have, for
example:

  * •

For a stationary case
$C(t-t^{\prime})=(1-q)\;\bar{A}(t-t^{\prime})+q\;\bar{B}\left(\frac{t-t^{\prime}}{H(\tau_{0})}\right)$,
where $H$ is a growing function of $\tau_{0}$.

  * •

For an aging case $t>t^{\prime}$:
$C(t,t^{\prime})=(1-q)\;A(t-t^{\prime})+q\;B\left(\frac{L(t^{\prime})}{L(t)}\right)=(1-q)\;A(t-t^{\prime})+q\;B\left(e^{h(t)-h(t^{\prime})}\right)$

where $(A,B,\bar{A},\bar{B})$ are functions decreasing from one to zero as
their argument goes from zero to infinity. The time reparametrization
$h(t)=\ln L(t)$ brings the aging form into a time-translational invariant
form.

#### Three scales

Again, the correlation $0<C\leq 1$ and there two values $q_{0}$ and $q_{1}$
such that for the interval $q_{1}\leq C\leq 1$ the correlation is much faster
than for the interval $q_{0}\leq C<q_{1}$, itself much faster than $0\leq
C<q_{0}$ We have, for example, for the stationary state:

  * •

$C(t-t^{\prime})=(1-q_{1})\;\bar{A}(t-t^{\prime})+(q_{1}-q_{0})\;\bar{B}\left(\frac{t-t^{\prime}}{H(\tau_{0})}\right)+q_{0}\;\bar{\bar{B}}\left(\frac{t-t^{\prime}}{\bar{H}({\tau_{0}})}\right)$

where $\bar{\bar{B}}$ is also decreasing from one to zero as their argument
goes from zero to infinity. The timescales are nested as
$\tau_{0}\rightarrow\infty$: $\bar{H}(\tau_{0})\gg H(\tau_{0})\gg 1$.

The function $f$ is the function $\min$ for correlations in any two different
scales.

#### A continuum of scales

An important case is when there is a dense set of values of correlation in
which for all values of correlation

$f(C_{21},C_{32})=\min[g(C_{21}),g(C_{32})]$ (17)

holds. An example is:

$C(t,t^{\prime})={\cal{C}}\left(\frac{\ln(t-t^{\prime}+t_{0})}{\ln\tau_{0}}\right)$
(18)

where $t_{0}$ is a constant. This form satisfies (17) when
$\tau_{0}\rightarrow\infty$ 333Note however that, confusingly,
$C(t,t^{\prime})={\cal{C}}\left(\frac{\ln t^{\prime}}{\ln t}\right)$ is only
one scale!. Note how simple ultrametricity appears in dynamics. The
Sherrington-Kirkpatrick model (stationarized by shear or by parameter
evolution, see next subsection) follows this law with astonishing precision.
Contucci  _et al._ (2020).

###  III.5 Bringing a system to a stationary form

Shear and evolving disorder render a system stationary. It came as a surprise
that for small drive the resulting stationary system is by no means close to
an equilibrium one (as in usual systems with small currents) but rather a
time-reparametrized version of the aging system.

For many technical purposes it is convenient then, at least in numerical
computations, to stationarize’ the system. The shear strength or the speed of
evolution of the disorder becomes then the control parameter, substituting the
self-generated waiting time of the aging situation.

###  III.6 The Multithermalization scheme

Having defined timescales and effective temperatures, we are ready to state
the multithermalization scheme. For all observables $A,B$ the long-time
response and correlation functions define an effective temperature $A,B$ as
$T_{AB}(t,t^{\prime})$ which we may write using a correlation as a ‘clock’
$T_{AB}(t,t^{\prime})\rightarrow T_{AB}(C_{AB})$.

$\bullet$ Timescales defined on the basis of the correlation of any pair of
observables are the same.

$\bullet$ The effective temperatures are, within a timescale, the same for all
pairs of observables.

##  IV Susceptibility Lego

In this section we discuss naive linear response theory (LRT), and show that
both in statics and in dynamics, the non-ergodic order parameter, $x(q)$ and
$X(q)$ can be viewed as the generating functions, respectively in equilibrium
and in dynamics of a set of responses to random perturbation to the
Hamiltonian. Let us add a to the Hamiltonian a perturbation of the form of a
$p$-spin long range Hamiltonian

$\displaystyle H\rightarrow H+\epsilon_{p}H_{p}$ (19) $\displaystyle
H_{p}=-\frac{1}{N^{(p-1)/2}}\sum_{i_{1},...,i_{p}}h_{i_{1},...,i_{p}}S_{i_{1}}...S_{i_{p}}$
(20)

The perturbations $H_{p}$ are bulk self-averaging quantities, and we have seen
that in equilibrium take values
$I^{(p)}_{eq}=\mathbb{E}(\omega(H_{p})/N=\beta\epsilon_{p}(1-\int
dq\;P_{\epsilon}(q)q^{p})$.

As usual in linear response theory we can consider the limit of small
$\epsilon$, $\chi^{(p)}_{eq}=\lim_{\epsilon_{p}\to
0}I^{(p)}/\epsilon_{p}=T\lim_{\epsilon_{p}\to
0}\lim_{N\to\infty}\frac{1}{N}\frac{\partial^{2}}{\partial\epsilon_{p}^{2}}\log
Z$. If this limit is regular, simple measures of the the energy absorbed by
the perturbation inform us on the correlations in the unperturbed system.

Analogously, in dynamics, we can write for the finite time expected value of
$H_{p}$

$\displaystyle I^{(p)}(t)$ $\displaystyle=$ $\displaystyle\mathbb{E}\langle
H_{p}(t)\rangle_{dyn}$ (21) $\displaystyle=$
$\displaystyle\epsilon_{p}\int_{0}^{t}ds\;R(t,s)pC(t,s)^{p-1}$ (22)

and introducing $\beta X(q)=\lim_{t,s\rightarrow\infty\atop
C(t,s)=q}\frac{R(t,s)}{\partial C(t,s)/\partial s}$ we have

$\displaystyle\lim_{t\to\infty}I^{(p)}(t)$ $\displaystyle=$
$\displaystyle\beta\epsilon_{p}\int dq\;X(q)\;pq^{p-1}$ (23)

As in the case of equilibrium $X$ is the generator function of response in
presence of the perturbation and if its $\epsilon\to 0$ limit is regular, we
have information about the response in the unperturbed system.

###  IV.1 Triangles

One can define higher order susceptibilities that in equilibrium allow one to
reconstruct the probability distribution of several replicas, and in dynamics
reflect the relations of correlations at different times. Particularly
interesting are the ones associated to the probability of triangles
$P(q_{12},q_{13},q_{23})$, which within Parisi RSB is concentrated on
ultrametric triangles. In FMPP the triangle probabilities were reconstructed
by considering average values of mixed combinations of the two different
$H_{p}$, where the tensors defining the interactions are contracted over a
subset of indexes, namely, in presence of the perturbations $H_{n+m}$ and
$H_{n+l}$ in the Hamiltonian one considers the average value of the observable

$\displaystyle
O_{n;m,l}=\sum_{i_{1},...,i_{n+m+l}}J^{(n+m)}_{i_{1},...i_{n},j_{n+1},...,j_{n+m}}J^{(n+l)}_{i_{1},...i_{n},k_{n+1},...,k_{n+l}}S_{j_{n+1}}...S_{j_{n+m}}S_{k_{n+1}}...S_{k_{n+l}}.$
(24)

where $n$ among the indexes of the two tensors, of rank $n+m$ and $n+l$
respectively, are contracted between them, and the others with the spins of
the system.

Straightforward integration by part allows to see that in equilibrium

$\displaystyle\frac{1}{N}\mathbb{E}\langle
O_{n;m,l}\rangle=\beta^{2}\epsilon_{n+m}\epsilon_{n+l}\left(1-\langle
q^{n+m}+q^{m+l}+q^{n+l}\rangle+2\langle
q_{12}^{n}q_{13}^{m}q_{23}^{l}\rangle\right)$ (25)

while in dynamics

$\displaystyle\frac{1}{N}\mathbb{E}\langle\delta
O_{n,m,l}(t)\rangle=\epsilon_{n+m}\epsilon_{n+l}\int_{0}^{t}dv\int_{0}^{t}du$
$\displaystyle([lC(t,v)^{l-1}R(t,v)][C(v,u)^{m}][nC(t,u)^{n-1}R(t,u)]+$ (26)
$\displaystyle[lC(t,v)^{l-1}R(t,v)][mC(v,u)^{m-1}R(v,u)][C(t,u)n]+$
$\displaystyle[C(t,v)l][mC(u,v)^{m-1}R(u,v)][nC(t,u)^{n-1}R(t,u)])$

Since $\langle O_{n;m,l}\rangle$ is not directly a response function (i.e. a
derivative of the free-energy) its equilibration was considered plausible, but
the relation between triangles was not-completely proven within LRT. To remedy
to this weakness, in Kurchan (2021) it was shown that in fact the triangular
correlations could also be generated as responses to perturbations: it
suffices to perturb the Hamiltonian with the following term

$\displaystyle\delta
H_{n,m,l}=(\gamma_{ml}O_{n;m,l}+\gamma_{nl}O_{m;n,l}+\gamma_{nm}O_{l;n,m})$
(27)

that generates the triangles at the cubic order in the $\gamma$’s.

#### 2\. Overlap equivalence between sublattices

Another generating function that has proven useful is the following Let us
first consider a lattice system, which we divide in four sublattices. whose
components we shall call $s^{(1)}_{i}$, $s^{(2)}_{i}$ and $s^{(3)}_{i}$ and
$s^{(4)}_{i}$. Adding to the energy a term

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\gamma}{N}\sum_{ij}\left(h^{(1)}_{ij}h^{(2)}_{jk}s^{(1)}_{k}s^{(2)}_{i}+h^{(2)}_{ij}h^{(3)}_{jk}s^{(2)}_{i}s^{(3)}_{k}+h^{(3)}_{ij}{h}^{(4)}_{jk}s^{(3)}_{i}s^{(4)}_{k}\right)$

with the $h^{(\ell)}_{ij}$ random Gaussian variables and computing the
following correlation and its associated response:

$C^{(4)}(t,t^{\prime})=\frac{1}{N^{2}}\sum
h^{(1)}_{ij}h^{(4)}_{jk}\left\langle
s^{(1)}_{i}(t)s^{(3)}_{k}(t^{\prime})\right\rangle_{\gamma}\qquad$ (29)

This generating function may be generalized to any number of sublattices, and
it is useful to show that there is overlap equivalence if and only if the
timescales at different effective temperatures are widely separated.

##  V The nucleation argument

As we have stressed, in the large limit, even in presence of ageing, the
values of extensive and local single time observables should tend to
asymptotic values. In mean-field systems with long range interactions these
may or may not coincide with the corresponding values in equilibrium. In
systems with finite range interactions in finite dimensions, on the other
hand, these values should coincide with their Boltzmann-Gibbs equilibrium
values. The usual argument which is advanced is that in finite dimensions
there are always local relaxation mechanisms allowing for equilibration. As a
consequence, for the bulk free-energy, the infinite-time limit and the
thermodynamic limits commute. The same should be generically true for its
derivatives, such as internal energy and static responses. Notice that this
argument tells that any possible broken ergodicity state should be only
marginally stable, since the introduction of a small extensive random
perturbation destabilizes the old equilibrium states to the advantage of new
ones with a more favorable energetic balance between old and new energy.

This does not of course prevent the practical existence of systems such as
structural glasses where the relaxation times at low temperature are so high
that the energy density remains off-equilibrium on geological scales.
Notwithstanding this, the limiting situation of aging close to equilibrium is
a useful conceptualization to understand how glasses explore the configuration
space. Whether in actual experimental systems on lab time scale one is close
or not to this situation has to be decided case by case, depending on the
system the preparation and the external parameters. We believe that this
should be the case for example of spin glasses below, but close enough to the
critical temperature. In these systems local equilibrium develops on growing
scales $\xi(t)$ and the asymptotic situation may be observed if the growth of
this scale is not too slow. We have seen that a possible way to measure $P(q)$
function consists in introducing long range random perturbations. If one
admits that these perturbations indeed equilibrate (at the level $\epsilon$)
in off equilibrium dynamics, then one could reconstruct the Parisi function
from simple off-equilibrium measures of the response. Unfortunately, the
nucleation argument does not hold in presence of long range interaction, and
one could cast doubts that small long range perturbations would induce
metastable states with free-energy density $O(\epsilon)$ higher than the
equilibrium one. These could trap the off-equilibrium dynamics, thus spoiling
the static dynamic equivalence. Thinking about what it could go wrong, this
appears as a very remote possibility for small $\epsilon$. Within mean-field,
with a long range unperturbed Hamiltonian, where inducing metastability is
certainly easier that in finite D, we do not know any case where the
introduction of a perturbation induces a difference of order $\epsilon$
between the value of the asymptotic dynamical energy and its equilibrium
counterpart.

To obviate to this problem however, one can look for local perturbations
giving rise to the same correlation functions within Linear Response. In FMPP
it was suggested that this could be achieved diluting the $p$-spin
perturbations so that they reduce to finite range interactions.

Let see how this works for $p=2$, the generalization to arbitrary $p$ being
straightforward (and discussed in detail in FMPP). To fix the idea consider a
spin system on a $D$ dimensional square lattice of size $L$,
$\Lambda=\\{1,...,L\\}^{D}$. One can divide the lattice in two halves,
$\Lambda_{l}$ and $\Lambda_{r}$, where the one of the coordinates, say in
direction of the x-axes $e$, take the values $\\{1,...,L/2\\}$ and
${L+1,...,L}$ respectively. Let us add to the original Hamiltonian a
perturbation of the form $\epsilon_{2}H_{2}$, with with

$\displaystyle H_{2}=\sum_{x\in\Lambda_{-}}h_{x}S_{x}S_{T(x)}$ (30)

where the $h_{x}$ are independent Gaussian random variables with unit
variance, and $T(x)$ is a translation of $x$ of length $L/2$ in direction $e$.
The thermal expectation value of the perturbation $H_{2}$ gives a contribution
to the internal energy of the system which is extensive and self-averaging,
i.e., independent (in the thermodynamical limit) of the particular realization
of the disorder contained in either $H$ or $H_{2}$. The interaction $H_{2}$,
which looks long range, is, in fact, a local perturbation in a different
space. Let us rename the spins in the right-hand part so that if
$x\in\Lambda_{l}$ then $T(x)\in\Lambda_{r}$ and $S_{T(x)}=S^{\prime}_{x}$. The
total Hamiltonian can now be written as

$\displaystyle
H[S,S^{\prime}]=H_{l}[S]+H_{r}[S^{\prime}]+B[S,S^{\prime}]+\epsilon_{2}\sum_{x\in\Lambda_{-}}h_{x}S_{x}S^{\prime}_{x}$
(31)

The Hamiltonian $H_{l}$ and $H_{r}$ refer, respectively, to the spins in
$\Lambda_{l}$ and $\Lambda_{r}$. The term $B[S,S^{\prime}]$ is a surface term
whose presence does not affect the average of $H_{2}$. Dropping it, the
Hamiltonian (31) characterizes a spin system of size $L/2$ with two spins
$S_{x}$ and $S_{x}^{\prime}$ on each site, and a purely local interaction. We
have traded the long range interactions with a local coupling between among
non-interacting or weakly coupled systems with different disorder and equal
number of spins. We would like now to show that within LRT $H_{2}$ continues
to be associated to the second moment of the overlap PDF. Simple integration
by part in equilibrium reveals that

$\displaystyle\langle
H_{2}/N\rangle=\beta\epsilon_{2}\frac{1}{N}\sum_{x\in\Lambda_{l}}\left[1-\mathbb{E}\left(\langle
S_{x}S_{T(x)}\rangle^{2}_{\epsilon}\right)\right].$ (32)

In FMPP it was argued that the limit for $\epsilon\to 0$ of
$\mathbb{E}\left(\langle S_{x}S_{T(x)}\rangle^{2}_{\epsilon}\right)$ both in
presence or in absence of the boundary term $B[S,S^{\prime}]$ should be equal
to the value that this correlation function takes for strictly zero coupling
$\epsilon=0$ in presence of $B[S,S^{\prime}]$. The small coupling between the
systems must have the same locking effect on the overlap as the surface
interaction that we have discussed in the introduction.

###  V.1 Overlap equivalence if and only if timescale-separation

The property of overlap equivalence is very natural in the Parisi scheme. One
has a distribution of states with their Gibbs weights. One now assumes that
two pairs of states $(a,b),(c,d)$ having the same mutual overlap measured with
one observable $q_{AA}(a,b)=q_{AA}(c,d)=q$, will have also the same mutual
overlap measured with any other observable $q_{BB}(a,b)=q_{BB}(c,d)=\bar{q}$.
This means that there is a universal function that gives, for every state, the
second overlap if one knows the first:

$\bar{q}=g(q)$ (33)

Clearly, we have:

$\bar{P}(\bar{q})d\bar{q}=P(g(q))dq\qquad\rightarrow\qquad\bar{P}(\bar{q})=P(g(q))\frac{1}{g^{\prime}(q)}$
(34)

We may rewrite this putting $\frac{dx}{dq}=P(q)$ and
$\frac{d\bar{x}}{d\bar{q}}=\bar{P}(\bar{q})$ as:

$x(q)=\bar{x}(\bar{q})$ (35)

The property (35) has the dynamic counterpart:

$X(C)=\bar{X}(\bar{C})\qquad\rightarrow\qquad\bar{T}(\bar{C}(t,t^{\prime}))=T(C(t,t^{\prime}))$
(36)

i.e. the effective temperature obtained from the fluctuation-dissipation
relation of two observables at given times coincide.

This is very striking: a very similar relation arises from two apparently
different arguments. And yet, using the second of the generating functions
above, we find that there is a complete correspondence between $g,X,\bar{X}$
in the dynamic and in the equilibrium context.

In the dynamic case, we may also prove a strong result Kurchan (2021): overlap
equivalence holds if and only if there is a wide separation of timescales
associated to two different temperatures for each observable. The proof is not
complicated: suppose we have two observables that have the same
$X(C)=\bar{X}(\bar{C})$ at all times, but the times for different $X$’s are
not really separate. Then, one can show that by coupling the observables one
may construct a new correlation $\bar{\bar{C}}$ that does not have
$\bar{\bar{X}}$ at the same times. In other words, Multithermalization is only
consistent if the timescales of different temperatures are widely separated.

##  VI Commuting limits

We have seen that within naive perturbation theory, under the hypothesis of
equilibration of bulk quantities, dynamics and equilibrium are in strong
correspondence and the study of the dynamical response can be used to
reconstruct the equilibrium Parisi function, which is an average over disorder
realizations. This is at first sight strange, one has a relation between
conceptually different kinds of objects that are relevant in different time
regimes (in and out of equilibrium, respectively), when the system explores
very different regions of phase-space.

In usual applications of LRT thermodynamic states are supposed to be stable
against the introduction of a perturbation. For example if we introduce a
small positive magnetic field $h$ in the low temperature Ising model we have

$\displaystyle\langle S_{i}\rangle_{h}=\langle
S_{i}\rangle_{+}+h\beta\sum_{j}(\langle S_{i}S_{j}\rangle_{+}-\langle
S_{i}\rangle_{+}\langle S_{j}\rangle_{+})$ (37)

Typical configurations of the perturbed systems are close to the unperturbed
ones in the phase chosen by the field.

In disordered system by contrast, when a random perturbation is introduced
this does not hold, the Gibbs state is marginally stable and ’chaotic’ Fisher
and Huse (1986); Kondor (1989): the perturbation implies a complete
reorganization of the Gibbs measure, which concentrates on regions with zero
overlap with the unperturbed states

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}(\langle S_{i}\rangle\langle
S_{i}\rangle_{0})=0.$ (38)

Yet these regions should correspond to close values of the bulk quantities and
similar statistical properties. The marginal stability of the Gibbs state
should then correspond to the stochastic stability of the metastate and
$P_{\epsilon}(q)\to P(q)$. In dynamics the situation is similar: the chaotic
property imply that no matter how small a perturbation, a perturbed trajectory
evolving with the same thermal noise as an unperturbed one, will stay close
for a certain time, but will eventually diverge when time in large enough.
Moreover, any random perturbation introduced after evolution up to a waiting
time $t_{w}$ leads to a free-energy increase and hence to a rejuvenation of
the system. This means that if we consider a field acting from $t_{w}$ to $t$,
the limit for $h\to 0$ of the magnetization at time $t$,
$\chi(t,t_{w})=\lim_{h\to 0}m(t,t_{w},h)/h$ is not uniform in time. As in the
case of statics this does not prevent to use LRT. We just have to consider
reparameterization invariant quantities (e.g.
$\chi_{dyn}(q)=\int_{q}^{1}dq\;X_{\epsilon}(q)$), whose limit $\epsilon\to 0$
should be regular as the perturbation tends to zero.

##  VII The whole Parisi $\leftrightarrow$ multithermalization scheme needs
time reparametrization invariance

All reparametrization-invariant relations between macroscopic quantities
should coincide in dynamics and statics, where in statics only the ‘skeleton’
values are present. Ultrametricity, overlap equivalence and $x(q)$ are three
examples. As we have mentioned above, the reciprocal is also true: if there is
no timescale separation between different temperatures , there is no
multithermalization, and no equilibrium overlap equivalence. There is however
a problem. Consider two systems brought into weak contact from the beginning,
for example two different lattice models, coupled locally so as to obtain a
single model with two sublattices. Assume that at the same times the separate
systems have non-coincident temperatures. This is perfectly possible, since
the systems are independent. Now, couple weakly the two systems: should we
conclude that the coupled system, for which there is no distinction between
observables of one or the other system, violates the multithermalization
scenario - and, a fortiori, the Parisi scheme? If this were so, both would be
fragile to the point of irrelevance. The answer is surprising: the timescales
of the systems rearrange so that different temperatures happen at different
scales: thus, the combined system conforms to the scenario. But this needs to
happen even for infinitesimal coupling, and it can only be possible if the
system is ‘soft’ with respect to time-rearrangements of each temperature
separately: in other words, it has to have independent time-reparametrization
invariances in the slow dynamics limit. Such invariances, which we have
already described above, where first reported by Sompolinsky and Zippelius
Sompolinsky and Zippelius (1982, 1981) some forty years ago, and have recently
had a crucial role in the interpretation of the SYK model Sachdev and Ye
(1993) as a toy model of holography Kitaev (2015); Maldacena and Stanford
(2016).

Figure 4: Caption

In conclusion, we realize that reparametrization softness is crucial for the
consistency of the multi-thermalization scenario. But we have seen here that
this scenario is in a one-to-one correspondence with the Parisi construction
with replicas, at least in finite dimensions. We are thus led to ask
ourselves, how do these reparametrizations appear in a replica treatment,
where there is no time? To this we turn in the next section.

##  VIII Reparametrization invariance and replica space

###  VIII.1 A consequence of ultrametricity

In aging experiment the actual perturbation allowing to measure the function
$X(q)$ is just an applied magnetic field. A small magnetic field $h$ acting
from time $t_{w}$ to time $t$ induces a magnetization at time $t$ which is

$\displaystyle
m(t,t_{w})/h=\chi^{+}(t,t_{w})=\int_{t_{w}}^{t}ds\;R(t,s){\rightarrow\atop
t,t_{w}\to\infty;C(t,t_{w})=q}\int_{q}^{1}dq^{\prime}X(q^{\prime}).$ (39)

This suggests in equilibrium to consider the response to a magnetic field of a
system $S_{1}$ which is kept to a fixed overlap $q\in{\mbox{Supp}}(P)$ with a
configuration $S_{0}$ which is well equilibrated in absence of the
perturbation.

Namely, considering an external random field term $\delta
H=-\epsilon_{1}\sum_{i}h_{i}S_{i}$ with $\mathbb{E}h^{2}=1$ we would like to
compute

$\displaystyle\chi^{+}(q)=\frac{1}{\epsilon_{1}}\mathbb{E}\sum_{S_{0}}\frac{e^{-\beta
H{[S_{0}]}}}{Z}\sum_{S}\frac{e^{-\beta(H{[S]}+\delta
H[S])}\delta(Q(S,S_{0})-q)\sum_{i}h_{i}S_{i}}{Z_{1}[S_{0}]}$ (40)

It is simple to see that within Linear Response Theory the response this is
given by

$\displaystyle\chi^{+}(q)=\beta\left(1-\mathbb{E}\;\left\langle\frac{\langle
q_{12}\mathbbm{1}_{q_{10}=q_{20}=q}\rangle_{12}}{[\langle\mathbbm{1}_{q_{10}=q}\rangle]^{2}}\right\rangle_{0}\right)$
(41)

where we have denoted as $\langle\cdot\rangle_{a,...}\cdot)$ the Boltzmann
average with respect to the replicas $a,...$ Remarkably, using the Ghirlanda-
Guerra identities, it is possible to show that this last quantity is equal to

$\displaystyle\mathbb{E}\;\left\langle\frac{\langle
q_{12}\mathbbm{1}_{q_{10}=q_{20}=q}\rangle_{12}}{[\langle\mathbbm{1}_{q_{10}=q}\rangle]^{2}}\right\rangle_{0}=\frac{\mathbb{E}\left({\langle
q_{12}\mathbbm{1}_{q_{10}=q_{20}=q}\rangle_{012}}\right)}{\mathbb{E}\;\langle{[\langle\mathbbm{1}_{q_{10}=q}\rangle_{1}]^{2}}\rangle_{0}}=\int
dq_{12}\;P(q_{12}|q_{10}=q_{20}=q)q_{12}$ (42)

The conditional probability $P(q_{12}|q_{10}=q_{20}=q)$ is fixed by
ultrametricity
$P(q_{12}|q_{10}=q_{20}=q)=x(q)\delta(q_{12}-q)+\theta(q_{12}-q)P(q_{12})$.
Which in turn implies that

$\displaystyle\chi^{+}(q)=\beta\int_{q}^{1}dq\;x(q).$ (43)

This has to be compared with the dynamical susceptibility at time $t$ to a
field acting from time $t_{w}$ to $t$, in the long time limit having fixed
$C(t,t_{w})=q$, namely

$\displaystyle\chi_{dyn}^{+}(q)=\beta\int_{q}^{1}dq\;X(q).$ (44)

The identity $X(q)=x(q)$ shows that the dynamical susceptibility tends to the
constrained equilibrium susceptibility at large times. The result is simple
and it has a clear interpretation. It means that asymptotically, during aging,
given the configuration reached at a time $t_{w}$ partial equilibrium within
the subspace with $C(t,t_{w})=q$ is essentially achieved before the system can
relax to $q-\delta q$.

This deep property suggests that the reparametrization invariant part of the
dynamics can be described through a fictive quasi-equilibrium process where
the physical time is completely elimitated. The system is supposed to be in
equilbrium on a correlation scale $q$ and to evolve to a scale $q^{-}$
choosing configurations according to the Gibbs distribution. Formally this
correspond to the Markov chain defined by the following transition matrix

$\displaystyle M_{q}({\bf S^{\prime}}|{\bf S})=\frac{1}{Z_{q^{-}}[{\bf
S}]}\exp(-\beta H[{\bf S^{\prime}}])\delta(Q({\bf S^{\prime}},{\bf S})-q^{-})$
(45)

where $Z_{q^{-}}[{\bf S}]$ is a restricted partition function normalizing the
probability. The chain, that was called Boltzmann pseudo-dynamics in Franz and
Parisi (2013); Franz  _et al._ (2015) does not correspond to any physical
dynamics, it implies macroscopic jumps from one configuration to the next one
without solution of continuity. In Franz and Parisi (2013); Franz  _et al._
(2015) however it was suggested that in a system with aging if one chooses
$q=q_{EA}$ this fictitious dynamical process provides a coarse-grained
reparamtrization invariant effective representation of the slow part of the
relaxation below $q_{EA}$ of true dynamics: the system fully equilibrates in
’dynamical quasi-states’ Franz and Virasoro (2000) of amplitude $q_{EA}$
before relaxing below this value; in this relaxation the system explores the
configuration space choosing new quasi-states at random with Boltzmann
probability among the available ones, i.e. those lying at overlap $q_{EA}$. It
is clear from the considerations of the previous section that thanks to
Ultrametricity we can go further with the effective description and instead of
coarse graining only the short times corresponding to $q_{EA}$, we could
choose any arbitrary skeleton value of $q\in Supp(P)$, and the corresponding
pseudo-dynamics (45) would just provide a reparametrization invariant
representation of the relaxation below $q$ with the same physical content as
the true dynamics.

##  IX Two, perhaps three finite-dimensional puzzles

###  IX.1 Is one-step RSB unstable in finite dimensions?

As we argued above, a system in finite dimensions must be reparametrization-
soft and marginally stable, otherwise two systems with similar timescales and
different effective temperatures would be unable to become a two-step RSB when
weakly coupled. Analogously, a 1RSB system coupled with a fullRSB one must
develop a continuum of effective temperatures and hierarchical time scales to
become fullRSB itself.

In the mean-field case, however, the equilibrium of a one-step systems is not
marginal. The situation is analogous to the ferromagnetic case, which is not
marginal within mean-field (there is an extensive barrier between minima),
while it becomes so in finite dimensions, being unstable with respect to a
small uniform field 444This however does not give rise to divergent
susceptibility, as the free-energy has only an essential singularity in $h=0$,
and $m(h)=\operatorname{sgn}(h)m^{+}+h\chi$. A hypothetical dynamical 1RSB
state in finite dimension should be at least marginally stable against
continuous RSB, a situation that could be called weak fullRSB. Such a
situation would seem rather unnatural in a disordered system, by its nature
heterogeneous, where the effective temperature would have spatial
fluctuations. It is thus natural to conjecture that close to equilibrium, one-
step, or more generally discontinuous solutions are unstable against higher
RSB, thus leading to a situation where all the values of the correlations
except the ones corresponding to the trivial equilibrium regime are ’skeleton’
values and full ultrametricity holds.

###  IX.2 Dynamic timescales in three and four dimensions

The experimental evidence of spin-glasses seems to point to only two
timescales Berthier  _et al._ (2000b); Vincent  _et al._ (1997), and a non-
trivial $P(q)$ Hérisson and Ocio (2002) with what appears to be many
temperatures – i.e. the $\chi$ vs $C$ plot appears to have a curved section,
apparently violating the Multithermalization. This however should be taken
with a grain of salt as it could be preasymptotic effect. Perhaps a clarifying
step would be to see whether the correlation decay in times of hours and days
is fitted with the same time-dependence of the one in microseconds, a
comparison that has not, to our knowledge, been done. The numerical evidence
in three dimensions seems to point to overlap equivalence between site and
link overlap Contucci  _et al._ (2006), although we know that this is not
possible if there is only one slow timescale and many temperatures. The
probable explanation is that the violation is, however, too small to be
observed numerically.

In four dimensions the situation is much more clear: the little evidence we
have Stariolo (2001) seems to point resolutely to dynamic ultrametricity with
many scales, just as in the Sherrington-Kirkpatrick model.

###  IX.3 Aging strongly far from equilibrium

Structural glasses as we know them in the laboratory find themselves in
extremely long lived metastable states. The internal energy density, specific
volume etc. persist to off-equilibrium values high above the equilibrium ones
for at least geological times. Yet these systems slowly age. The response
during physical aging in such conditions verifies to an excellent
approximation scaling laws with a single effective time scale Struik (1977).
Numerical simulations of model glasses indicate that a description in terms of
a single effective temperature, reminiscent of Mean-field 1RSB systems is
appropriate Parisi (1997); Barrat and Kob (1999). One can ask

One can conjecture that in such conditions, which are formally very far from
the asymptotic situation needed to have quasi-equilibrium sampling and
reparametrization invariance, the dynamics in the aging regime can still be
approximately described in term of these concepts, as a quasi-equilibrium
process where degrees of freedom that evolve on the same scales are in mutual
‘multithermalized’ equilibrium, and that a quasi-equilibrium process as in
Franz and Parisi (2013); Franz  _et al._ (2015) with an properly chosen $q$
would still be an appropriate quasi-reparameterization invariant description
of the slow dynamics.

##  X Conclusions

We have discussed the deep connection between dynamic and equilibrium
theoretical constructions as it is realized whenever during slow dynamics bulk
expectation values are close to equilibrium, as it should asymptotically be
the case in finite dimensional systems. In that case Linear Response Theory
allows one to relate dynamical response functions to equilibrium correlations.
Linear Response in disordered systems is, however, subtle.

The main elements at play are the softness displayed by these systems both in
equilibrium and off-equilibrium with respect to random perturbations, and
somewhat paradoxically, the stability of appropriately defined responses and
correlations with respect to the same perturbations.

It seems fair to say that the evolution of the subject has been from an almost
miraculous ansatz to the gradual understanding of the questions it raises on
more robust and method-independent properties.

Acknowledgments We acknowledge the support of the Simons Foundation, Grants No
454941 S. Franz No 454943 J. Kurchan.

## References

  * Mézard  _et al._ (1987) M. Mézard, G. Parisi, and M. A. Virasoro,  _Spin Glass Theory and Beyond_ (World Scientific, Singapore, 1987).
  * Thalmann (2001) F. Thalmann, The European Physical Journal B-Condensed Matter and Complex Systems 19, 65 (2001).
  * Berthier  _et al._ (2000a) L. Berthier, J.-L. Barrat, and J. Kurchan, Physical Review E 61, 5464 (2000a).
  * Horner (1992) H. Horner, Zeitschrift für Physik B Condensed Matter 86, 291 (1992).
  * Sompolinsky and Zippelius (1982) H. Sompolinsky and A. Zippelius, Physical Review B 25, 6860 (1982).
  * Sompolinsky and Zippelius (1981) H. Sompolinsky and A. Zippelius, Physical Review Letters 47, 359 (1981).
  * Cugliandolo and Kurchan (1993) L. F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. 71, 173 (1993).
  * Cugliandolo and Kurchan (1994) L. F. Cugliandolo and J. Kurchan, Journal of Physics A: Mathematical and General 27, 5749 (1994).
  * Franz and Mézard (1994) S. Franz and M. Mézard, EPL (Europhysics Letters) 26, 209 (1994).
  * Cugliandolo  _et al._ (1997) L. F. Cugliandolo, J. Kurchan, and L. Peliti, Phys. Rev. E 55, 3898 (1997).
  * Contucci  _et al._ (2019) P. Contucci, J. Kurchan, and E. Mingione, Journal of Physics A: Mathematical and Theoretical 52, 324001 (2019).
  * Contucci  _et al._ (2020) P. Contucci, F. Corberi, J. Kurchan, and E. Mingione, arXiv preprint arXiv:2012.03922 (2020).
  * Franz  _et al._ (1998) S. Franz, M. Mézard, G. Parisi, and L. Peliti, Physical Review Letters 81, 1758 (1998).
  * Franz  _et al._ (1999) S. Franz, M. Mezard, G. Parisi, and L. Peliti, Journal of statistical physics 97, 459 (1999).
  * Kurchan (2021) J. Kurchan, arXiv preprint arXiv:2101.12702 (2021).
  * Sherrington and Kirkpatrick (1975) D. Sherrington and S. Kirkpatrick, Physical review letters 35, 1792 (1975).
  * Parisi (1979) G. Parisi, Physical Review Letters 43, 1754 (1979).
  * Talagrand  _et al._ (2003) M. Talagrand _et al._ ,  _Spin glasses: a challenge for mathematicians: cavity and mean field models_ , Vol. 46 (Springer Science & Business Media, 2003).
  * Panchenko (2013a) D. Panchenko,  _The sherrington-kirkpatrick model_ (Springer Science & Business Media, 2013).
  * Fisher and Huse (1986) D. S. Fisher and D. A. Huse, Physical review letters 56, 1601 (1986).
  * Bray and Moore (1987) A. J. Bray and M. A. Moore, Physical review letters 58, 57 (1987).
  * Franz and Toninelli (2004) S. Franz and F. L. Toninelli, Journal of Physics A: Mathematical and General 37, 7433 (2004).
  * Ghirlanda and Guerra (1998) S. Ghirlanda and F. Guerra, Journal of Physics A: Mathematical and General 31, 9149 (1998).
  * Panchenko (2013b) D. Panchenko, Annals of Mathematics , 383 (2013b).
  * Aizenman and Contucci (1998) M. Aizenman and P. Contucci, Journal of statistical physics 92, 765 (1998).
  * Arous and Guionnet (1995) G. B. Arous and A. Guionnet, Probability Theory and Related Fields 102, 455 (1995).
  * Ben Arous  _et al._ (2006) G. Ben Arous, A. Dembo, and A. Guionnet, Probability theory and related fields 136, 619 (2006).
  * Barrat and Kob (1999) J.-L. Barrat and W. Kob, EPL (Europhysics Letters) 46, 637 (1999).
  * Berthier  _et al._ (1999) L. Berthier, J.-L. Barrat, and J. Kurchan, The European Physical Journal B-Condensed Matter and Complex Systems 11, 635 (1999).
  * Parisi (1983) G. Parisi, Physical Review Letters 50, 1946 (1983).
  * Franz  _et al._ (1992) S. Franz, G. Parisi, and M. Virasoro, EPL (Europhysics Letters) 17, 5 (1992).
  * Parisi and Talagrand (2004) G. Parisi and M. Talagrand, Comptes Rendus Mathematique 339, 303 (2004).
  * Franz and Parisi (1995) S. Franz and G. Parisi, Journal de Physique I 5, 1401 (1995).
  * Franz and Parisi (1998) S. Franz and G. Parisi, Physica A: Statistical Mechanics and its Applications 261, 317 (1998).
  * Temesvári  _et al._ (2000) T. Temesvári, I. Kondor, and C. De Dominicis, The European Physical Journal B-Condensed Matter and Complex Systems 18, 493 (2000).
  * De Dominicis and Giardina (2006) C. De Dominicis and I. Giardina,  _Random fields and spin glasses: a field theory approach_ (Cambridge University Press, 2006).
  * Castellani and Cavagna (2005) T. Castellani and A. Cavagna, J. Stat. Mech. 2005, P05012 (2005).
  * Castillo  _et al._ (2002) H. E. Castillo, C. Chamon, L. F. Cugliandolo, and M. P. Kennett, Phys. Rev. Lett. 88, 237201 (2002).
  * Castillo  _et al._ (2003) H. E. Castillo, C. Chamon, L. F. Cugliandolo, J. L. Iguain, and M. P. Kennett, Phys. Rev. B 68, 134442 (2003).
  * Chamon  _et al._ (2004) C. Chamon, P. Charbonneau, L. F. Cugliandolo, D. R. Reichman, and M. Sellitto, J. Chem. Phys. 121, 10120 (2004).
  * Chamon  _et al._ (2002) C. Chamon, M. P. Kennett, H. E. Castillo, and L. F. Cugliandolo, Phys. Rev. Lett. 89, 217201 (2002).
  * Chamon and Cugliandolo (2007) C. Chamon and L. F. Cugliandolo, J. Stat. Mech. 2007, P07022 (2007).
  * Chamon  _et al._ (2011) C. Chamon, F. Corberi, and L. F. Cugliandolo, J. Stat. Mech. 2011, P08015 (2011).
  * Kondor (1989) I. Kondor, Journal of Physics A: Mathematical and General 22, L163 (1989).
  * Sachdev and Ye (1993) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
  * Kitaev (2015) A. Kitaev, “A simple model of quantum holography,” (2015),  _A simple model of quantum holography_ , http://online.kitp.ucsb.edu/online/entangled15/kitaev/, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
  * Maldacena and Stanford (2016) J. Maldacena and D. Stanford, Phys. Rev. D 94, 106002 (2016).
  * Franz and Parisi (2013) S. Franz and G. Parisi, Journal of Statistical Mechanics: Theory and Experiment 2013, P02003 (2013).
  * Franz  _et al._ (2015) S. Franz, G. Parisi, F. Ricci-Tersenghi, and P. Urbani, Journal of Statistical Mechanics: Theory and Experiment 2015, P10010 (2015).
  * Franz and Virasoro (2000) S. Franz and M. A. Virasoro, Journal of Physics A: Mathematical and General 33, 891 (2000).
  * Berthier  _et al._ (2000b) L. Berthier, J.-L. Barrat, and J. Kurchan, Physical Review E 63, 016105 (2000b).
  * Vincent  _et al._ (1997) E. Vincent, J. Hammann, M. Ocio, J.-P. Bouchaud, and L. F. Cugliandolo, in  _Complex Behaviour of Glassy Systems_ (Springer, 1997) pp. 184–219.
  * Hérisson and Ocio (2002) D. Hérisson and M. Ocio, Physical Review Letters 88, 257202 (2002).
  * Contucci  _et al._ (2006) P. Contucci, C. Giardina, C. Giberti, and C. Vernia, Physical review letters 96, 217204 (2006).
  * Stariolo (2001) D. A. Stariolo, EPL (Europhysics Letters) 55, 726 (2001).
  * Struik (1977) L. C. E. Struik, (1977).
  * Parisi (1997) G. Parisi, Physical review letters 79, 3660 (1997).