Bachelor Individual Project
Imperial College London
Department of Computing

Maximum Return on Investment for a Domestic Photovoltaic Installation

_Author:_

Tom Nonnenmacher

_Supervisor:_
Prof. Jenny Nelson
_Co-supervisor:_
Dr. Benedict Winchester

June 9, 2023

###### Abstract

The rising energy prices in Europe and the urgent need to address global
warming have sparked a significant increase in the installation of domestic
photovoltaic systems to harness solar energy. However, since solar energy is
available only during daytime hours and its availability varies daily,
effectively shifting energy use becomes crucial. Whilst batteries can assist
in storing excess energy, their high prices hinder their widespread adoption.
In this study, we explore the importance of load to maximise return on
investment.

We propose an incremental approach to fitting load profiles into the
production envelope, allowing for practical implementation. We compare
different meter resolutions: 1 second, 5 minutes, 15 minutes, and 1 hour. Our
analysis reveals that making real-time decisions (per second) leads to
significant energy savings of 16% compared to hourly decisions.

Furthermore, we explore three types of device management strategies:

  1. 1.

ON/OFF management independent of PV production,

  2. 2.

ON/OFF management based on the current PV production,

  3. 3.

ON/OFF management based on both current and forecasted PV production,
utilising an optimal fit algorithm.

Through our study, we demonstrate that our implementation of the third
approach outperforms a standard management approach, resulting in more than
17% cost savings. This study provides insights into the optimisation of load-
shifting strategies in domestic photovoltaic installations, highlighting the
importance of load control and the potential benefits in maximising the
utilisation of solar energy while minimising energy costs and environmental
impact.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Practical considerations and context
    3. 1.3 Literature overview
  2. 2 Methodology
    1. 2.1 The estimation of future solar production
    2. 2.2 The estimation of each load profile
      1. 2.2.1 The different controllable devices
      2. 2.2.2 Estimation of each device
      3. 2.2.3 Turbulent periods and base load
    3. 2.3 Optimal fitting of load profiles into the production envelope
    4. 2.4 Simulations
      1. 2.4.1 Time decision
      2. 2.4.2 Comparison of device management algorithms
  3. 3 Results
    1. 3.1 Time decision
    2. 3.2 Comparison of device management algorithms
      1. 3.2.1 Comparison of smart and bruteforce management simulation
      2. 3.2.2 Comparison of smart and Adaptative management simulation
  4. 4 Conclusion
    1. 4.1 Summarisation
    2. 4.2 Limitations
    3. 4.3 Extensions

##  Chapter 1 Introduction

###  1.1 Motivation

Rising energy prices in Europe in 2022 and global warming causes the
installation of domestic photovoltaic (PV) installations to harvest lost solar
energy [1, 2, 3]. Solar energy is available only during daytime hours and
varies day by day. It is therefore key to shift energy use (load) to the time
when solar power is available. Batteries can help to move energy availability
but have a prohibitive high price at the time of the writing [4]. We take the
perspective of domestic use, where the production and consumption of energy
happen at home, thereby avoiding the cost and loss of transporting energy
across distance.

Allocation of load of solar energy is paramount to optimise the use of the
“free” solar energy. We define typical loads of the home as:

  * •

Electric car

  * •

Domestic hot water

  * •

Pool pump

  * •

Feed energy back into the grid (sell energy)

  * •

other (lighting, washing machines, cooking, PCs, etc.)

The problem becomes an optimisation problem that includes the learning of the
parameters. We are trying to fit the various loads into the solar production
curve of the day. At the time at which we make a decision for each and all
loads, we do not have certainty about the coming solar energy available, and
we do not know the energy use of each load. By using these assumptions we
ensure that the practical implementation of our work fits universally and can
be implemented without prior knowledge, or manual re-adjustment which itself
is prone to error.

###  1.2 Practical considerations and context

Our data from the 1st of January 2023 to the 10th of April 2023 is from a 14.7
kWp PV installation in the South of France. The time is in UTC. In our study,
we will concentrate on developing algorithms tailored to this particular
household, utilising its specific data and focusing on the control of three
devices: the electric car, the pool pump, and the domestic hot water system.

France, as a highly industrialised nation, relies heavily on its energy
infrastructure to meet the growing demands of its population and economy. The
energy system in France predominantly comprises a mix of nuclear, renewable,
and fossil fuel-based power generation sources [5]. While nuclear energy has
historically played a significant role in the country’s energy production,
there has been an increasing focus on incorporating renewable energy sources,
such as solar, to diversify the energy mix and reduce carbon emissions [6].

The varying availability of solar energy, coupled with fluctuating electricity
demand patterns, necessitates the development of innovative strategies to
optimise energy consumption and enhance the flexibility of the grid. Load-
shifting, as a demand-side management technique, offers a promising solution
to effectively balance electricity supply and demand while considering the
constraints of the energy system.

###  1.3 Literature overview

Load shifting, a key aspect of demand-side management, has gained significant
attention in recent years due to its potential to optimise energy consumption,
reduce peak loads, and enhance overall grid efficiency. Load shifting involves
adjusting the timing of energy usage, particularly in residential households,
to take advantage of variable electricity prices, renewable energy
availability, and grid demand patterns. Several studies have explored the
application of load-shifting techniques in households equipped with energy
storage units and electric mobility options, and have investigated the impact
of demand-side management strategies on reducing household electricity
consumption.

One notable research paper from Di Giorgio et al. presents a model predictive
control framework for optimising load shifting in residential settings with
energy storage systems [7]. The study focuses on developing strategies to
effectively utilise the energy storage capacity to shift loads and minimise
electricity costs.

Another research paper from Paetz et al. examines the load-shifting potentials
in households that incorporate electric vehicles [8]. The study compares the
behavior of users with the modeling results to evaluate the feasibility and
effectiveness of load-shifting strategies in the context of electric mobility.

In the context of demand-side management, home appliance scheduling and peak
load reduction play crucial roles in reducing household electricity
consumption. Laicane et al. explore the impact of effective scheduling
techniques on optimising energy usage and reducing overall electricity
consumption in households [9].

Furthermore, Fischer et al. focus on smart household usage profiling to
recommend suitable energy tariffs and load-shifting strategies. The study
utilises smart meter data and advanced profiling techniques to tailor load-
shifting recommendations based on individual household energy consumption
patterns [10].

Lastly, Farzambehboudi et al. investigate the economic implications of load-
shifting strategies in smart households [11]. The study assesses the potential
cost savings and benefits achieved through load-shifting techniques, shedding
light on the economic feasibility and advantages of implementing such
strategies.

In this paper, we aim to build upon the insights provided by these studies and
contribute to the understanding of load-shifting methodologies in residential
households. Our approach involves estimating future solar production,
estimating each load profile, and optimally fitting load profiles within the
production envelope.

##  Chapter 2 Methodology

In order to optimise the return on investment for our photovoltaic
installation, we show our approach structured into three key sections:
forecasting future solar production, estimating load profiles, and determining
the optimal fitting of load profiles within the production envelope.

###  2.1 The estimation of future solar production

The PV output of each installation is different. Orientation, location, and
obstacles vary the production profile. We want our results to be universally
applicable anywhere in the world (southern or northern hemisphere, any
location, any installation). Therefore, we proceed by measuring the solar
production of the last 7 days and estimate future production. Each day,
network connectivity issues and outages caused a few instances of missing
data. To ensure data completeness, we utilised linear interpolation to fill in
the gaps and obtained a total of 86400 data items per day, representing every
second of the day. To estimate solar production, the starting point is the
maximum possible production available for a given installation - the clear-sky
model [12].

First, the real production data from every day of January 2023 is being
visualised. At first glance, the production curves resemble a normal
distribution. We validate via the means squared error (MSE) how well the
normal distribution approximates the maximum possible production (the clear
sky model). If the normal distribution approximates well the clear sky model
it will approximate also well the real production. We, therefore, start by
approximating the real production data with a normal distribution to either
validate or abandon the normal distribution as an approximation of the clear
sky model.

We tried first to find the closest normal distribution for PV production of
each given day. The normal distribution is defined by three parameters: the
maximum, the mean, and the standard deviation. We selected the closest normal
distribution for every production day in January 2023.

To increase the speed of finding the closest distribution we optimised the
search in the following way: first, we selected the maximum value of the
normal distribution in this range (in Watt):

$[max(PV)-2000,max(PV)+2000],$ (2.1)

second, we selected the mean of the normal distribution to be in the range
from 8:20 am to 4:40 pm, and third the standard deviation in the range of from
4000 seconds to 20000 seconds. Within this search space, we chose the normal
distribution with the smallest Mean Squared Error (MSE). Even the best normal
distribution didn’t well approximate the production data, as we see on the
following graph. The Figure 2.1 shows the day with the smallest MSE between a
normal distribution and production data (31st January 2023). The reader sees
immediately that the start and the end of the PV production are far
overestimated. The MSE, computed using the W unit, is 362893.

We conclude that the normal distribution is not a good approximation of the
production data, and therefore is also not a good approximation for the clear
sky model.

In the paper [13] a method of estimating clear-sky PV electricity production
without exogenous data is given. This article adopts a direct forecasting
approach and thus, the proposed data-driven approach is dependent solely on
historical time-series power output data from grid-connected without having to
rely on the technical characteristics of the PV or on other energy or
meteorological-related data, equipment, models, sky/satellite images, etc.

Their method to estimate the clear sky is following:

  1. 1.

The quantities that characterise the shape and the distribution of PV power
output time series at a given time are calculated first. The three criteria
used are the mean value, the maximum value, and the standard deviation of the
PV power output time series.

  2. 2.

The calculated quantities were used as limits to the PV power output at each
second. All the PV power output values outside the limits were excluded.
Different limits were studied and tested in their study.

  3. 3.

The predicted clear-sky signal is calculated by the resulting average of the
PV power output time series within the limits for the corresponding month.

The best method (with the smallest MSE) describes:

$\begin{split}upperlimit&=A+1.5SD\\\ lowerlimit&=B.\end{split}$ (2.2)

A is the mean value of the examined PV power output time series at a specific
time.
SD is the Standard Deviation of the examined PV power output time series at a
specific time.
B is the average value of PV power output, excluding all points outside the
upper limit and the lower limit band.
If we use the previous example of the 31st of January, with the use of the PV
production of the last seven days (from the 24th of January 2023 to the 30th
of January 2023), the algorithm outputs a predicted production for the 31st of
January 2023 with a sufficient MSE of 68340, which is an 81% decrease compared
to the MSE of the normal distribution method (that knows the current PV
production). The result is shown in Figure 2.2.

To measure the performance of the PV forecasting, we measured all the clear
sky data we had: from the 8th of January (because we need 7 past days) to the
10th of April 2023. We computed the mean per second of the difference between
the real PV production and the PV production forecasted for every day. The
result is: the forecast of the PV production overestimates the real PV
production per second with a mean of 2.12%. Please find the following graphs
for the 10 best PV forecasts with 2.5.

Figure 2.1: Normal distribution estimation for the PV production on the 31st
of January 2023 Figure 2.2: PV production forecasted and PV production on the
31st of January 2023

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

Figure 2.5: The 10 best PV production forecasts from the 8th January to the
10th April 2023

###  2.2 The estimation of each load profile

We assume that we have control over the power we give to each device and we
can observe only the total energy load of all consumer loads in the home. We
filter and estimate the load of a device by observing the total consumption of
energy and the impact when we switch the device ON and OFF. By doing this
multiple times, during multiple days we are able to distill a load profile
within reasonable confidence intervals.

####  2.2.1 The different controllable devices

##### Electric car

The majority of the electric-car charging is at constant power, e.g. cars come
with a charger that allows for household plug charging (2.3 kW - 10 A at 230
V), or with a dedicated Wallbox at a given power, oftentimes this is 7.36 kW
or 11 kW (corresponding to 32A, 48 A, respectively with 230 V) [14]. However,
charging at very high power (fast charging) reduces the battery longevity
[15].

In France, the average consumption is 50 minutes per person and per day, for
an average distance of 36km [16]. An electric car uses in average 0.161 kWh
per km [17]. Therefore in this study, we assume the daily power requirement
for the car is: 5.796 kWh.

Hence, given this information, we fix the charging time for the car at 2 hours
30 minutes, because it is the charging time needed to satisfy the average
consumption in France if we use a household plug charging of 2.3 kW.
Practically, a longer trip with the car can be foreseen and may require manual
intervention to boost the electric-car battery to maximum capacity.

##### Domestic hot water

There are a vast amount of water boilers deployed in homes today, and many use
electricity as the primary source to heat hot water until the target
temperature is reached (typically 60 °C). When the target temperature is
reached the boiler’s security system shuts the heating OFF, and no electricity
is consumed anymore beyond that point.

Per person and year, 800 kWh is the required energy for hot water in a
household [18]. In France, there are an average of 2.19 people per household
[19]. So the estimated daily energy used in the household for hot water is 4.8
kWh.

Therefore, an average 2 kW water boiler requires 2.4 h of heating daily with 2
kWh of electricity. During the day, this form of water heating allows for load
shifting as showers and hot water consumption is concentrated at the beginning
and the end of the day.

Usually, the typical pattern is full charge for 2 and half hours followed by
brief intervals of full charging for a few minutes to maintain the water
temperature at the desired level. In this study, we only include the main big
initial heating period of 2 hours 30 minutes and neglect the small periods.

##### Pool pump

A pool pump is a device used to circulate and filter water in a swimming pool.
It is responsible for pulling water from the pool, passing it through a
filtration system to remove debris and contaminants, and then returning the
clean water back to the pool. The rule of thumb used for the running time of a
pool pump that runs water through the filtration system is water temperature
divided by 2, the result is converted in hours.

So at 20 degrees water temperature, the pool pump is supposed to run for 10
hours every day. Pool filtration and solar power go nicely hand-in-hand.

####  2.2.2 Estimation of each device

After having computed the PV forecasting, we need to estimate the consumption
of a device. For perfect device management, the highest level of automation,
and the avoidance of human errors, we use the automatically estimated device
profiles to do the device and load management in the next chapter.

We want to estimate the consumption of a device, we have as input the time
when the device is switched ON and the time when the device is switched OFF.
Additionally, we have access to the global load of all devices during these
periods. We assume the most common case, which is the presence of only one
energy meter for the household. The problem can therefore be stated as:

Separating the energy consumed by one device from the sum of energy consumed
by all devices.

With the available ON/OFF timing information and the meter reading at all
times, we search for the best estimate of the device consumption. We separate
2 cases:

  * •

when the device switches ON, and

  * •

when the device switches OFF.

When the device switches ON, the load increases with a magnitude corresponding
approximately to the consumption of the device. However, sometimes, the change
is not instant, for example with the car (2.6). It often takes up to one
minute after switching ON until the load becomes active and apparent on the
meter reading.

(a)

(b)

Figure 2.6: The first ON and OFF of the electric CAR on the 1st April of 2023

When the device switches OFF, the load decreases with a magnitude
corresponding approximately to the consumption of the device. The response is
quicker after an OFF event than an ON event (see Figure 2.6).

To find the consumption of the device we do edge detection on the global load
with a window of 10 seconds: we shift a window of 10 seconds over the load,
and we save the absolute value of the difference between the maximum and
minimum inside the window as the device loads. To determine if it is an ON or
an OFF event we verify the order, i.e. the minimum needs to be after the
maximum for the ON event, while for the OFF event, it is the inverse.

For an ON event, we search for the maximum load difference within a two-minute
interval starting from the time when the device is switched ON (t to t + two
minutes). Similarly, for an OFF event, we search for the maximum load
difference within a one-minute interval starting from the time when the device
is switched OFF (t to t + one minute). Furthermore, we apply exponential
weighting to the window, meaning that an absolute difference measured between
t and t + size of the window carries twice the weight of the absolute
difference measured between t + size of the window and t + two times the size
of the window. In summary, the consumption of a device is determined by the
maximum weighted absolute difference, following the aforementioned conditions.

####  2.2.3 Turbulent periods and base load

A "calm" period is a 10 minutes period of load that has a standard deviation
corresponding to the 10% of the lowest standard deviations of all the 10
minutes periods of load of any given day. A turbulent period is defined as a
period that is not a "calm" period. As we know, devices are sometimes switched
ON/OFF during turbulent periods and thus can lead to wrong results. In fact,
two devices can be launched at the same time, and it will be very difficult to
find the consumption of each device. To enhance the accuracy of our device
estimation, we considered the possibility of measuring the load change when
activating a device during regular "calm" periods, aiming for the highest
level of precision possible. For $n$ devices, we choose $n\times(2+1)+1$
minutes to define a calm period. Three minutes per device (composed of two
minutes for the ON period, and one minute for the OFF period) and one minute
as a safety buffer for all devices.

To forecast a calm period in a day, we compute all of the calm periods during
the past 7 days and require that at least 5 of them overlap. All these points
forecasted the start of the calm periods for our current day. The graphs in
Figure 2.9 show that the two calm periods forecasted for the 26th of March
2023 start at 01:00 am and 10:22 pm, based on the data of the last 7 days
(from the 19th of March 2023 to the 25th of March 2023). Therefore, on the
26th of March 2023, the measurement and device load estimation happens from
01:00 am to 01:10 am and from 10:22 pm to 10:32 pm.

(a)

(b)

(c)

(a)

(b)

(c)

(a) Figure 2.9: The calm periods found from the 19th March 2023 to the 25th
March 2023

Calm periods are often during times outside of the PV production window (most
of the calm periods found are during the night), which means that we will need
to pay electricity from the grid to test and to estimate the device loads.
Devices might change, therefore device estimation is done on a continuous
basis, always using the information from the past seven days. Once seven days
of data have been, we have enough data to even avoid testing outside the PV
production window. By taking the median on the data measured, we avoid the
outliers caused during turbulent periods and maintain good results. The
algorithm with the seven past days found the correct device consumption of
each device for every day by taking the median of all the consumption measured
for ON/OFF events without using the "calm" periods.

Then we needed to find the “base load” to simulate the load of the devices
that we do not have control over, like the washing machine, the fridge, the
lights, etc. The “base load” is the load when all the ON/OFF controllable
devices are OFF. To compute it, we just computed the median of all the loads
measured when all the controllable devices (the car, the pool pump, and the
domestic hot water) were OFF.

###  2.3 Optimal fitting of load profiles into the production envelope

We use an incremental approach and fit one additional load into an existing
load of various other profiles into the production envelope. As the
incremental approach can be repeated several times we have a practical method
for all loads. The incremental deployment also reflects the reality of
implementation.

The problem now is to fit the load profile into the production envelope to
maximise return on investment. First, money is saved by the consumption of
cheap energy. Second, money is gained by selling excess capacity to the
electricity provider.

Currently, the price to buy 1 kWh in France is 0.2€, and the price to sell 1
kWh to the electricity provider is 0.13€ [20]. Furthermore, we can not sell
more than 6 kW per second.

With this information, we deduct the following constants for a day $j$:

$\displaystyle E_{max,sell}$ $\displaystyle=6kWs$ $\displaystyle C_{e,sell}$
$\displaystyle=0.13EUR/kWh$ $\displaystyle C_{e,buy}$
$\displaystyle=0.2EUR/kWh$

We name the 2 following functions for the load and the PV production:

  * •

P(j, t): "The PV production in kW for day $j$ at time $t$ seconds"

  * •

L(j, t): "The load in kW for day $j$ at time $t$ seconds"

We, therefore, have one combined optimisation function of money that comes
from saving and selling:

$M(j)=S(j)-B(j)$

which is the money in € for day $j$.

$S(j)=\sum_{t=0}^{86399}min(E_{max,sell},max(P(j,t)+L(j,t),0))\times\frac{C_{e,sell}}{3600s}$

which is the money in € we gain by selling energy on day $j$.

$B(j)=\sum_{t=0}^{86399}max(0,-P(j,t)-L(j,t))\times\frac{C_{e,buy}}{3600s}$

which is the money in € spent by buying energy on day $j$.

Our first idea was to work with basic conditions on thresholds (adaptative
mode described in section 2.4.1), for example, if the PV production deducted
by the existing loads is greater than the consumption device $i$ at time $t$,
then we can switch device $i$ ON. If the PV production deducted by the
existing loads is less than the consumption device $i$ at time $t$, then we
switch device $i$ OFF.

The problem with this algorithm is that it does not maximise electricity sales
on sunny days, as sales have an upper bound. In fact, because it will turn on
all the devices linearly with the beginning of the PV production, and then at
the peak of the PV production, we can only sell 6 kW per second, so we will
lose some PV production.

Then, we tried a different approach given the PV forecasting by maximising the
function $M$. The idea is to define the loads of each device as a rectangle.
In fact, the height of the rectangle is the consumption of the device, and its
length is the time the device needs to be on. We then place the rectangle of
each device by the reverse order of length (rectangle with maximum length is
placed first) to maximise the function $M$, shown in Figure 2.10.

Then, when we have our forecasting of when our device $i$ should start:
($\mathbf{T}_{device,ON_{i}}$), and when our device $i$ should end:
($\mathbf{T}_{device,OFF_{i}}$). We define an algorithm to be in concordance
with the current production. This algorithm will thus adapt our original
rectangles by creating different rectangles of different lengths (but with the
same height) to adapt to the current production if the forecasted production
is different (see Figure 2.11).

1

Input: $\mathbf{S}_{\text{device}}$: Device state (ON or OFF)

$\mathbf{T}_{\text{device, ON}}$: Forecasted device ON time

$\mathbf{T}_{\text{device, OFF}}$: Forecasted device OFF time

$\mathbf{t}$: Current second of the day

$\mathbf{N}$: Number of devices

$\mathbf{T}_{\text{device, remaining}}$: Remaining ON time for each device

2

3for  _$i\leftarrow 1$ to $N$_ do

4 if  _$S_{\text{device}_{i}}$ is $OFF$_ then

5 if  _$t\geq T_{\text{device, ON}_{i}}$ and PV production is sufficient_ then

6 $S_{\text{device}_{i}}\leftarrow ON$

7 $T_{\text{device, OFF}_{i}}\leftarrow t+T_{\text{device, remaining}_{i}}$

8

9 else

10 if  _$t\geq T_{\text{device, OFF}_{i}}$_ then

11 $S_{\text{device}_{i}}\leftarrow OFF$

12 $T_{\text{device, remaining}_{i}}\leftarrow 0$

13 if  _$t <T_{\text{device, OFF}_{i}}$ and PV production is insufficient_
then

14 $S_{\text{device}_{i}}\leftarrow OFF$

15 $T_{\text{device, remaining}_{i}}\leftarrow T_{\text{device, OFF}_{i}}-t$

16

17

Algorithm 1 Adaptive Device Algorithm

Figure 2.10: Forecasted load and PV production for the 1st April of 2023
Figure 2.11: The load and PV production for the 1st April of 2023

###  2.4 Simulations

####  2.4.1 Time decision

We wanted to see the influence of the time frame for making decisions. For
that, we implemented a device management algorithm that works with decisions
based on seconds, 5 minutes, 15 minutes, and 1 hour.

The algorithm works with the mode ‘adaptative’. In the adaptive mode, we
implement a dynamic threshold approach to optimise energy usage based on the
PV production. A predefined threshold value is set, such as 4 kWs, which
represents the maximum power consumption allowed from the PV production. To
illustrate the adaptive mode, let’s consider a scenario where the PV
production is 3.9 kWs at 1 pm, 4.1 kWs at 2 pm, 4.5 kWs at 3 pm, and 3.8 kWs
at 4 pm. At each time interval, we compare the PV production with the
threshold value. For instance, at 1 pm, the PV production of 3.9 kWs is below
the threshold of 4 kWs, so no device will be activated during this period.
However, at 2 pm, when the PV production reaches 4.1 kWs, which exceeds the
threshold, we initiate the activation of a controllable device. Once the
device is turned ON at 2 pm, it will consume the power from the PV production,
in this case, 4 kWs per second, for the duration of the activation period. In
our example, the device will remain ON until 4 pm, utilising the available 4
kWs per second during this time frame. Finally, at 4 pm, when the PV
production drops below the threshold to 3.8 kWs, we deactivate the device,
turning it OFF as it can no longer operate at the desired power consumption
level.

This adaptive mode allows us to dynamically adjust device activation based on
real-time PV production, ensuring optimal energy utilisation within the
constraints of the available power. By aligning device activation with higher
PV production periods, we maximise the utilisation of clean energy and
minimise reliance on the grid or other energy sources. The adaptative mode can
make devices work in parallel and can combine multiple devices. For example,
for every second, we can say that the car takes 3 kWs, and the pool pump
consumes 2 kWs.

To measure the difference between 15-minute, 15-minute, and hour-based
decisions, we decided to launch an adaptive mode with a fixed threshold of 4
kWs.

####  2.4.2 Comparison of device management algorithms

To assess the efficacy of our device management algorithm, we conducted three
simulations spanning from January 8th (considering the need for seven previous
days) to April 10th, 2023, using 9 kWp solar panels.

The first simulation employed the smart device management algorithm described
in this report, based on both current and forecasted PV production. Our
automatic system prioritise the three controllable devices in the following
order: the pool pump, the domestic hot water, and the car. Device A is
considered to have higher priority than device B if, under the same conditions
for activation, device A is switched ON first. Similarly, if given the same
conditions for deactivation, device A is switched OFF last.

In the second simulation, we scheduled the activation of the pool pump at 12
pm, domestic hot water at 7 pm, and the electric car at 6 pm each day. Each
device was switched off after running for the same duration as in the first
simulation. We chose these specific starting times based on our analysis of
load data before the installation of the solar panels. This device management
strategy is referred to as "bruteforce". This automatic system is independent
of PV production.

The third simulation utilised the adaptive mode outlined in section 2.4.1. In
this mode, the threshold for each device’s activation is based on its
consumption level. This automatic system is based on the current PV
production.

It should be noted that the car was installed on March 25th, and, therefore,
its consumption did not need to be managed before that date.

##  Chapter 3 Results

###  3.1 Time decision

In order to evaluate the disparity in energy utilisation between decisions
made at different time intervals (5 minutes, 15 minutes, and 1 hour) using the
adaptive algorithm, we present the percentage of energy consumption relative
to real-time decisions (measured in seconds) for each day throughout January
2023. Due to network connectivity issues and outages, we didn’t have enough
data to make the measurements desired with a high level of confidence on the
1st, 6th, 8th, and 23rd of January. The results are shown in Figure 3.1.

Figure 3.1: Proportion of energy used from PV production relative to the real-time decision using adaptive mode at a given 4 kW threshold for each day of January 2023 | 5min | 15min | 1hour
---|---|---|---
mean | 98% | 95% | 84%
median | 99% | 97% | 88%
min | 91% | 71% | 28%
Table 3.1: Statistics on the proportion of energy used from PV production
relative to the real-time decision using adaptive mode at a given 4 kW
threshold for each day of January 2023

We notice that on the worst day of January 2023, an hour-based decision is
using 28% of the available production from PV (3.1). Then, we wanted to see
how much we will pay for energy from the grid to satisfy the threshold of 4
kW, at different time intervals (5 minutes, 15 minutes, and 1 hour), relative
to real-time decisions (measured in seconds) for each day throughout January
2023 (3.2).

Figure 3.2: Energy paid from grid relative to the real-time decision using adaptive mode at a given 4 kW threshold for each day of January 2023 | 5min | 15min | 1hour
---|---|---|---
mean | 714.237kW | 1495.218kW | 7286.616kW
median | 581.408kW | 966.566kW | 6518.619kW
max | 3905.480kW | 4708.951kW | 17035.819kW
Table 3.2: Statistics on the energy paid from grid relative to the real-time
decision using adaptive mode at a given 4 kW threshold for each day of January
2023

We notice that on the worst day of January 2023, hour-based decisions will
cause us to pay 17035.819 kWs from the grid (3.2). 1 kWh costs 0.2€ in France
[20]. 17035.819 kWs equals approximately 4.73 kWh which makes approximately
0.95€ for the worst day of January 2023.

Over the first month period of real data, taking second-level decisions
outperformed hourly decisions during 29 days. Over the whole period, taking
second-based decisions saved 16% of energy compared to hour-based decisions.
In the following simulations of the report, we base our study purely on
decisions by second.

###  3.2 Comparison of device management algorithms

####  3.2.1 Comparison of smart and bruteforce management simulation

The simulation results presented in Figure 3.3 depict that the bruteforce
device management algorithm consistently achieves higher daily sales compared
to the smart management algorithm. This disparity can be attributed to the
fact that the smart management algorithm prioritises maximising the
consumption of solar production overselling it, considering that the cost of
purchasing energy exceeds the price of selling energy.

The concept of maximising the utilisation of PV production is further
substantiated by Figure 3.4, which demonstrates that the bruteforce device
management algorithm incurs greater daily purchases compared to the smart
management algorithm.

Figure 3.3: Sales of smart management and bruteforce management algorithms
Figure 3.4: Purchases of smart management and bruteforce management algorithms

To assess the overall performance of each device management approach, we
compute the daily balance by summing the sales and purchases. Figure 3.5
displays the cumulative balance for each management algorithm throughout the
given period. Notably, the cumulative balance for the smart management
algorithm consistently surpasses that of the bruteforce management algorithm.

Furthermore, by analysing the statistics for the daily balance of each device
management approach, we find that, on average, our algorithm achieves savings
of approximately 0.43€ per day, which translates to approximately 157€ per
year. Consequently, our algorithm yields 17.3% higher cost savings compared to
the bruteforce approach. However, it is important to note that the difference
between the cumulative balance of the bruteforce management and the smart
management varies over time, as depicted in Figure 3.6, indicating a non-
linear relationship.

In summary, our simulation-based comparison demonstrates the superior
performance of the smart management algorithm in optimising energy consumption
and financial savings. By maximising the utilisation of PV production, our
algorithm consistently outperforms the bruteforce approach, resulting in
significant cost reductions.

Figure 3.5: Cumulative balance of smart management and brute-force management
algorithms Figure 3.6: Difference of cumulative balance between smart
management and bruteforce management algorithms

####  3.2.2 Comparison of smart and Adaptative management simulation

Next, we proceed to compare the results of the first simulation (Smart
management) with the third simulation (Adaptive management). Similar to the
previous comparison, the outcomes presented in Figure 3.7 demonstrate that the
adaptive device management algorithm consistently achieves higher daily sales
compared to the smart management algorithm. This can be attributed to the fact
that the smart management algorithm possesses an optimal solution for
maximising PV consumption compared to the adaptive management algorithm. The
notion of maximising the utilisation of PV production is further validated by
Figure 3.8, which illustrates that the daily purchases made by the adaptive
device management algorithm surpass those of the smart management algorithm.

Figure 3.7: Sales of smart management and adaptive management algorithms
Figure 3.8: Purchases of smart management and adaptive management algorithms

As in the previous comparison, we calculate the daily balance by summing the
sales and purchases, and the results are shown in Figure 3.9. Notably, the
cumulative balance for the smart management algorithm consistently surpasses
that of the adaptive management algorithm.

Moreover, by analysing the statistics for the daily balance of each device
management approach, we observe an average savings of approximately 0.07€ per
day, equivalent to approximately 26€ per year. Consequently, our algorithm
achieves cost savings that are 2.8% higher than those of the adaptive
management algorithm. The difference in cumulative balances between the two
device management algorithms is illustrated in Figure 3.10.

In summary, our comparative simulation of smart and adaptive device management
algorithms demonstrates that the smart management algorithm outperforms the
adaptive management algorithm in maximising energy consumption and achieving
financial savings. By optimising PV utilisation, our algorithm consistently
yields higher sales and cumulative balance, resulting in significant cost
reductions compared to the adaptive management approach.

Figure 3.9: Cumulative balance of smart management and adaptive management
algorithms Figure 3.10: Difference of cumulative balance between smart
management and adaptive management algorithms

##  Chapter 4 Conclusion

###  4.1 Summarisation

In conclusion, this project has made significant advancements in energy
forecasting and management. Firstly, in Chapter 2, we established the
consumption time of each device, laying the foundation for further analysis.

We then succeed to forecast the solar production without any exogenous data by
using a statistical method on the data of the seven past days. Our forecast of
the PV production overestimates the real PV production per second with a mean
of approximately 2.12% from the 8th of January to the 10th of April 2023.

In Chapter 4, we succeeded to find the consumption of each controllable device
(the electric car, the pool pump, and the domestic hot water) for every day
from the 8th of January to the 10th of April 2023 by using the median of the
results of the seven past days for each day of our algorithm defined in
section 2.2.2 without using the "calm periods" defined in section 2.2.3. We
found a way to define the consumption of the other devices by defining the
"base load" in section 2.2.3.

In Chapter 5, we proved that real-time energy measures results in 2% higher
daily charge (in kWh) of the car compared to 5 minutes resolution, 16%
improvement for hourly measurement and reaction for the month of January 2023.
We furthermore defined optimal fit algorithms to shift the loads into the
solar production envelope we forecasted in Chapter 3. We didn’t forget to
adapt our forecasting loads with the current real data per second.

Comparing the brute force management (section 2.4.2) with the smart management
algorithm developed in this project, we found that the smart management
approach resulted in 17.3% more cost savings from the 8th of January to the
10th of April 2023. This result can be compared to the study conducted by
Paetz et al., where they observed a cost saving of 17% during their experiment
involving a semi-automatic shift-loading system in a household located in
Germany [8]. Furthermore, when comparing the adaptive management (section
2.4.2) with the smart management algorithm, the latter achieved an 2.8% in
cost savings over the same period.

These results highlight the effectiveness of our forecasting and management
approaches in optimising energy consumption and cost savings. The findings
serve as a valuable contribution to the field of energy management and lay the
groundwork for future research and implementation in real-world scenarios.

###  4.2 Limitations

This report is subject to certain limitations that should be acknowledged.
Firstly, the data used for analysis and evaluation were limited to the period
from the 1st of January to the 10th of April 2023. As a result, the findings
and conclusions may not fully capture the variations and patterns that could
emerge during the summer and autumn seasons. Future studies should consider
expanding the data collection period to include a more comprehensive
representation of seasonal changes.

Secondly, the analysis conducted in this report was based solely on the data
collected from a single household located in the South of France. Although
this dataset provided valuable insights into the energy consumption patterns
and solar power generation of this specific household with a 14.7 kWc PV
installation, it may not be fully representative of households in other
regions or with different PV system configurations. Therefore, caution should
be exercised when generalising the findings to broader contexts.

To address these limitations and ensure the robustness and applicability of
future research, it is recommended to gather data from a more diverse range of
households in different geographical locations, with varying PV installation
capacities. By incorporating a broader dataset, researchers can gain a more
comprehensive understanding of the factors influencing energy consumption and
solar power generation across different contexts.

Moreover, extending the data collection period to include all four seasons
would provide a more comprehensive analysis of how energy consumption and
solar power generation patterns evolve throughout the year. This expanded
timeframe would allow for a better understanding of the seasonal variations
and their impact on the proposed forecasting algorithms and energy management
strategies.

###  4.3 Extensions

To extend the work accomplished in this project, our objective is to develop
an algorithm capable of accurately forecasting the energy consumption patterns
of controllable devices, without relying on user input. This holds significant
importance as it can lead to optimised energy usage, cost reduction, and
improved energy efficiency in households worldwide.

One of the key challenges in achieving this goal is to create an algorithm
that can adapt to any household, irrespective of the specific devices used. By
eliminating the need for defining the consumption time of each controllable
device, we can ensure the algorithm’s applicability in diverse settings.

Furthermore, it is essential to enhance the accuracy of the energy production
forecast by considering external factors such as weather conditions. While our
study utilised data from a household in the South of France, where many days
were sunny and closely aligned with the clear sky model, incorporating
numerical weather predictions (NWPs) can significantly improve the forecast
accuracy.

NWPs, including factors such as Direct Normal Irradiance (DNI), Diffuse
Horizontal Irradiance (DHI), and Global Horizontal Irradiance (GHI), play a
vital role in solar power forecasting. By integrating NWPs as inputs into the
forecasting algorithm, we can account for variations in solar radiation and
adjust the forecasted production accordingly. The paper [21] used these NWPs
as input and achieved an improvement of around 35% with his ARX model. DNI,
DHI and GHI can be estimated using the REST2 model described in this paper
[22].

## References

  * [1] Giovanni Sgaravatti, Simone Tagliapietra and Georg Zachmann  “National policies to shield consumers from rising energy prices”  In _Bruegel Datasets_ , 2022
  * [2] John Houghton  “Global warming”  In _Reports on progress in physics_ 68.6 IOP Publishing, 2005, pp. 1343
  * [3] Prashant V Kamat  “Meeting the clean energy demand: nanostructure architectures for solar energy conversion”  In _The Journal of Physical Chemistry C_ 111.7 ACS Publications, 2007, pp. 2834–2860
  * [4] Catherine Lane  “Solar batteries: Are they worth the cost?”  In _SolarReviews_ URL: https://www.solarreviews.com/blog/is-solar-battery-storage-worth-it-given-current-solar-battery-cost
  * [5] MD Mathew  “Nuclear energy: A pathway towards mitigation of global warming”  In _Progress in Nuclear Energy_ 143 Elsevier, 2022, pp. 104080
  * [6] Badr Eddine Lebrouhi et al.  “Energy Transition in France”  In _Sustainability_ 14.10 MDPI, 2022, pp. 5818
  * [7] Alessandro Di Giorgio, Laura Pimpinella and Francesco Liberati  “A model predictive control approach to the load shifting problem in a household equipped with an energy storage unit”  In _2012 20th Mediterranean Conference on Control & Automation (MED)_, 2012, pp. 1491–1498  IEEE
  * [8] Alexandra-Gwyn Paetz, Thomas Kaschub, Patrick Jochem and Wolf Fichtner  “Load-shifting potentials in households including electric mobility-A comparison of user behaviour with modelling results”  In _2013 10th International Conference on the European Energy Market (EEM)_ , 2013, pp. 1–7  IEEE
  * [9] Ilze Laicane, Dagnija Blumberga, Andra Blumberga and Marika Rosa  “Reducing household electricity consumption through demand side management: the role of home appliance scheduling and peak load reduction”  In _Energy procedia_ 72 Elsevier, 2015, pp. 222–229
  * [10] Joel E Fischer et al.  “Recommending energy tariffs and load shifting based on smart household usage profiling”  In _Proceedings of the 2013 international conference on Intelligent user interfaces_ , 2013, pp. 383–394
  * [11] Yasaman Farzambehboudi et al.  “Economic impact analysis of load shifting in a smart household”  In _2018 International conference on smart energy systems and technologies (SEST)_ , 2018, pp. 1–5  IEEE
  * [12] F Antonanzas-Torres et al.  “Clear sky solar irradiance models: A review of seventy models”  In _Renewable and Sustainable Energy Reviews_ 107 Elsevier, 2019, pp. 374–387
  * [13] Stefani Peratikou and Alexandros G Charalambides  “Estimating clear-sky PV electricity production without exogenous data”  In _Solar Energy Advances_ 2 Elsevier, 2022, pp. 100015
  * [14] EVBOX  “MG ZS charging specifications”  URL: https://evbox.com/en/electric-cars/mg/mg-zs
  * [15] Mohammed Al-Saadi et al.  “Fast charging impact on the lithium-ion batteries’ lifetime and cost-effective battery sizing in heavy-duty electric vehicles applications”  In _Energies_ 15.4 MDPI, 2022, pp. 1278
  * [16] Sandra Zilloniz  “Les temps de déplacement entre domicile et travail”  In _Dares_ URL: https://dares.travail-emploi.gouv.fr/publications/les-temps-de-deplacement-entre-domicile-et-travail-des-disparites-selon-l
  * [17] Electric Vehicle Database  “MG ZS EV Standard Range”  URL: https://ev-database.org/car/1540/MG-ZS-EV-Standard-Range
  * [18] “Quelle est la consommation d’un chauffe-eau électrique?”  In _Butagaz_ URL: https://www.butagaz.fr/espace-energies/energie/eau-chaude/quelle-est-la-consommation-dun-chauffe-eau-electrique
  * [19] “Average Size of Households in France (2010 - 2021)”  In _GlobalData_ URL: https://www.globaldata.com/data-insights/macroeconomic/average-household-size-in-france-2096123/
  * [20] Eurostat  “Electricity price statistics”  URL: https://ec.europa.eu/eurostat/statistics-explained/index.php?title=Electricity_price_statistics
  * [21] Peder Bacher, Henrik Madsen and Henrik Aalborg Nielsen  “Online short-term solar power forecasting”  In _Solar energy_ 83.10 Elsevier, 2009, pp. 1772–1783
  * [22] Christian A Gueymard  “REST2: High-performance solar radiation model for cloudless-sky irradiance, illuminance, and photosynthetically active radiation–Validation with a benchmark dataset”  In _Solar Energy_ 82.3 Elsevier, 2008, pp. 272–285


# Is glass a state of matter?

Benjamin Guiselin  ENSL, CNRS, Laboratoire de physique, Université de Lyon,
F-69342 Lyon, France  Gilles Tarjus  LPTMC, CNRS-UMR 7600, Sorbonne
Université, 4 Pl. Jussieu, F-75005 Paris, France  Ludovic Berthier
Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS, 34095
Montpellier, France  Yusuf Hamied Department of Chemistry, University of
Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

###### Abstract

Glass is everywhere. We use and are surrounded by glass objects which make
tangible the reality of glass as a distinct state of matter. Yet, glass as we
know it is usually obtained by cooling a liquid sufficiently rapidly below its
melting point to avoid crystallisation. The viscosity of this supercooled
liquid increases by many orders of magnitude upon cooling, until the liquid
becomes essentially arrested on experimental timescales below the “glass
transition” temperature. From a structural viewpoint, the obtained glass still
very much resembles the disordered liquid, but from a mechanical viewpoint, it
is as rigid as an ordered crystal. Does glass qualify as a separate state of
matter? We provide a pedagogical perspective on this question using basic
statistical mechanical concepts. We recall the definitions of states of matter
and of phase transitions between them. We review recent theoretical results
suggesting why and how an “ideal glass” can indeed be defined as a separate
equilibrium state of matter. We discuss recent success of computer simulations
trying to analyse this glass state. We close with some experimental
perspectives.

##  I What is a glass?

We possess an intuitive notion of what a state of matter is. For a pure
substance, there are usually three such states: gas, liquid, and crystal Tabor
(1991). The gas is found at high temperatures and transforms into a liquid
upon cooling. For an infinitely slow cooling process, where thermal
equilibrium is maintained all along, condensation at the boiling point is a
first-order equilibrium phase transition associated with a discontinuous jump
in various thermodynamic quantities, such as the specific volume shown in Fig.
1 (blue curve). The liquid itself transforms abruptly into a crystalline solid
when cooled below the melting temperature $T_{\mathrm{m}}$ through another
first-order equilibrium phase transition. These states of matter are
characterised and distinguished by their macroscopic properties. For example,
the gas occupies the entire available volume, unlike the crystalline solid
which retains its shape.

Glass is absent from this description because we have so far focused on
equilibrium properties. If a liquid is cooled quickly enough below
$T_{\mathrm{m}}$ (red curve in Fig. 1), crystallisation can be avoided and the
system becomes a supercooled liquid Cavagna (2009). This supercooled state can
exist over very long times because the formation of a crystal is very slow for
a large number of substances. In this supercooled regime, the system appears
for all practical purposes at thermal equilibrium although it is metastable
with respect to the crystal. For instance, its physical properties do not
depend on time and can easily be reproduced in independent experiments.

Figure 1: Phases of matter. A gas transforms into a liquid below the boiling
temperature $T_{\mathrm{b}}$ through an equilibrium first-order phase
transition. A liquid crystallises below the melting temperature
$T_{\mathrm{m}}$ through another first-order phase transition. A metastable
supercooled liquid can persist below $T_{\mathrm{m}}$ and gradually transforms
into a solid at the glass transition temperature $T_{\mathrm{g}}$. Below
$T_{\mathrm{g}}$, the glass is frozen in a rigid disordered structure which
depends on the cooling rate. This is the common glass, which slowly ages
towards equilibrium. An equilibrium “ideal glass” state would be obtained
across a first-order phase transition at $T_{\mathrm{K}}$ provided thermal
equilibrium could be maintained down to $T_{\mathrm{K}}$.

While the structure of the supercooled liquid changes very little with
decreasing the temperature, its transport properties, such as viscosity or
diffusivity, change by many orders of magnitude over a small temperature
range. At the microscopic scale, molecular motion slows down dramatically and
becomes so slow that very little motion can be detected over timescales of
several hundreds of seconds when the temperature falls below the glass
transition temperature $T_{\mathrm{g}}$. Below $T_{\mathrm{g}}$, the
supercooled liquid is out of equilibrium, and molecules stop diffusing over
experimentally accessible timescales Berthier and Biroli (2011). This nearly-
arrested liquid behaves macroscopically like a solid even though its
microscopic structure still resembles the one of a completely disordered
liquid. The “glass transition” temperature at $T_{\mathrm{g}}$ is not a well-
defined temperature and depends on the protocol used to cool the liquid. As a
consequence, the glass that we use in our everyday life is a non-equilibrium
material with physical properties that depend on its preparation history, see
Fig. 1, and may vary with time as the system slowly ages towards equilibrium.
Therefore, a conventional glass is not a state of matter, as it is not an
equilibrium material with crisply defined and fully reproducible thermodynamic
properties.

This is of course not the end of our story. The idea that a glass state of
matter can exist in equilibrium conditions has a long history, dating back at
least to the work of Kauzmann in 1948 Kauzmann (1948). Kauzmann collected
calorimetric data measured at equilibrium above $T_{\mathrm{g}}$ for a number
of molecular liquids, and by extrapolation below $T_{\mathrm{g}}$ noted the
possibility that a thermodynamic phase transition could take place at a
temperature $T_{\mathrm{K}}<T_{\mathrm{g}}$, now called the Kauzmann
temperature. Although Kauzmann himself favoured an interpretation in terms of
a “pseudocritical point” marking the limit of stability of the liquid with
respect to crystallization, his observation raised the prospect that the
system below $T_{\mathrm{K}}$ could be in an equilibrium glass state. We shall
call this state an “ideal glass”, to make the distinction with conventional
non-equilibrium glasses very clear. The Kauzmann temperature and the
equilibrium ideal glass phase are shown in Fig. 1.

The hypothesis that an equilibrium phase transition at $T_{\mathrm{K}}$
underlies the formation of conventional glasses has given rise to a large
litterature Berthier and Biroli (2011). Our understanding of its nature has
significantly evolved over the years since the early work of Gibbs and
diMarzio who first described a simple lattice model with a thermodynamic
liquid-glass transition Gibbs and DiMarzio (1958). Here, we offer a
pedagogical discussion of the ideal glass and the equilibrium glass transition
relying on recent theoretical and numerical developments. Our aim is to make
this discussion accessible to anyone with an undergraduate background in
statistical mechanics at the level of well-known textbooks Landau and Lifshitz
(2013); Callen (1998); Sethna (2021).

In Sec. II, we review the concepts of states of matter and phase transitions
to precisely define the question addressed in our title. In Sec. III, we
summarise the mean-field theory of the glass transition and explain why, in
this limit, glass does correspond to a state of matter. In Sec. IV, we present
numerical results of glass-forming liquids which are consistent with the
existence of an underlying liquid-to-glass equilibrium phase transition. We
conclude in Sec. V.

##  II What is a state of matter?

Following the conventional Landau approach to phase transitions Landau and
Lifshitz (2013) and without delving into all the subtleties, a state or a
phase of matter is commonly defined as a homogeneous region of space in which
all intensive thermodynamic variables are well-defined and uniform. We
distinguish among these variables the control parameters (temperature,
pressure, magnetic field, etc.) which can be imposed by an external operator,
and others, which are densities associated with extensive quantities (specific
mass, magnetisation, massic energy, etc.). The order parameter is one of these
variables, but it plays a special role since it takes different values in the
different phases, and its behaviour makes it possible to distinguish between
them.

The different phases are usually represented by domains in a phase diagram,
which is the space of control parameters. These domains are limited by phase
transition lines through which the order parameter can vary continuously
(second-order phase transitions) or discontinuously (first-order phase
transitions). To determine the domain of stability of the different phases,
Landau proposed a systematic method based on statistical mechanics Landau and
Lifshitz (2013), which now corresponds to the starting point of most
introductory courses on phase transitions Callen (1998). Having identified a
good order parameter, one has to express the equilibrium free energy of the
system as a function of this order parameter for fixed values of the control
parameters (for example the temperature $T$), and then determine its absolute
minimum for each value of $T$. The computed values of the order parameter and
of the free energy at this minimum represent the value that would be measured
in a macroscopic system at thermal equilibrium.

Some phase transitions are associated with a spontaneous symmetry breaking,
corresponding to a change in the degree of symmetry of the system Goldenfeld
(2018). The variation of the order parameter from one phase to another
reflects this loss of symmetry. For example, in the case of the transition
between paramagnetic and ferromagnetic phases, the order parameter is the
magnetisation vector which has a non-zero value in the ferromagnetic phase.
The magnetisation then favours a single direction of space and thus breaks
rotational invariance. Similarly, the crystallisation of a liquid into an
ordered periodic lattice decreases the degree of symmetry of the system. Some
phase transitions are not associated with a symmetry breaking, such as the
liquid-gas transition.

To describe glass as an equilibrium phase along similar lines, a legitimate
objection emerges immediately. Glass as we know it refers to a liquid that has
left thermal equilibrium at $T_{\mathrm{g}}$ and can no longer explore its
configuration space ergodically. This means that an equilibrium free energy
cannot even be defined for a conventional glass and the above discussion
appears irrelevant. Therefore, the questions we wish to address are: assuming
thermal equilibrium can be maintained at arbitrary low temperatures and
configuration space can be explored in equilibrium conditions, does the free
energy show a transition from a liquid to a glass state, what is the order
parameter that would then reveal the corresponding phase transition, and what
is the corresponding symmetry breaking?

##  III Landau theory of the glass transition in the mean-field limit

A series of works initiated in 2012 has led to the complete derivation of an
analytic solution of the glass transition for simple models of glass-forming
liquids which becomes mathematically exact in the limit of a large number of
space dimensions, $d\to\infty$ Parisi  _et al._ (2020). This non-physical
mathematical limit makes the statistical mechanics of supercooled liquids
exactly solvable. Such a situation is very common in statistical mechanics
where it is often found that mean-field approximate solutions in finite
dimensions become exact when $d$ becomes large Callen (1998). In large
dimensions, the interactions between a given particle and its neighbours can
be replaced by an average force which can be self-consistently determined.
Despite this immense simplification, the calculation remains mathematically
involved because it requires the explicit description of the aperiodic
structure characterising the glass phase. The results found in large
dimensions confirm a long series of earlier results which hinted at the nature
of the glass transition when using mean-field approximations Singh  _et al._
(1985); Kirkpatrick and Thirumalai (1987); Biroli and Mézard (2001).

It is possible to outline the main results obtained in this framework
following the Landau approach described in Sec. II. To this end, the first
step is to identify the good order parameter, $Q$, which can distinguish
between the equilibrium liquid and glass phases, and then to express the free
energy of the system as a function of $Q$, $V(Q)$. In this context, a suitable
choice for $Q$ is what is called the overlap and for the associated $V(Q)$ the
Franz-Parisi potential Franz and Parisi (1997).

What indeed is a good order parameter $Q$? Clearly, neither the average
density nor its spatial fluctuations can be used because if the glass looks
like an arrested liquid then its density, $\rho$, its pair correlation
function $g(r)$ and its static structure factor $S(k)$ also look similar.
Therefore, contrary to the description of the liquid state where they play a
key role Hansen and McDonald (2013), density fluctuations are uninteresting
spectators of the glass transition.

The correct answer originates from another physics problem where glassy
behaviour is also found, namely, spin glasses Mézard  _et al._ (1987);
Castellani and Cavagna (2005). Although spin glass physics has a much shorter
history than structural glass physics, the formulation of statistical
mechanics models for spin glasses and a deep analytic understanding of their
properties went comparatively much faster. Spin glass models describe spins
with interactions mediated by random coupling constants, which makes it
possible to describe the properties of dilute magnetic alloys Edwards and
Anderson (1975). The approriate order parameter to distinguish the spin glass
and paramagnetic phases is the overlap $Q$ between magnetisation profiles of
two independent copies (usually called “replicas” in the field of disordered
systems) of the same system Edwards and Anderson (1975). For Ising spins, this
is simply expressed as $Q=(1/N)\sum_{i=1}^{N}S_{i}^{(1)}S_{i}^{(2)}$, where
$S_{i}^{(1)}=\pm 1$ is the value of spin $i=1,\dots,N$ in one copy and
$S_{i}^{(2)}$ the value of the same spin in the second copy. Clearly, the
overlap $Q\simeq 1$ if copies 1 and 2 are highly correlated (spin glass),
whereas for uncorrelated configurations $Q\simeq 0$ (paramagnet). The overlap
$Q$ then quantifies the degree of similarity of the local degrees of freedom
(here, the spins) in two configurations of the system drawn from the
equilibrium Boltzmann distribution.

Figure 2: Overlap order parameter $Q$ of the liquid-glass transition. It
quantifies the degree of similarity between the density profiles of two
independent equilibrium configurations (here obtained from numerical
simulations of a two-dimensional glass-forming liquid Guiselin  _et al._
(2022a)). For $T>T_{\mathrm{K}}$, the liquid explores a large number of
equilibrium structures and $Q\simeq 0$. Instead, in the equilibrium glass for
$T<T_{\mathrm{K}}$, all equilibrium configurations are very similar and $Q>0$.

For supercooled liquids, the overlap quantifies the degree of similarity
between the density profiles measured in two independent configurations. As
sketched in Fig. 2, for two copies in which the density profiles are
uncorrelated, $Q\simeq 0$ but the overlap becomes close to 1 for correlated
configurations. The order parameter $Q$ can also be interpreted as a distance
between the two copies in configuration space Parisi (1983).

Figure 3: Mean-field theory of glass formation. (a) The Landau free energy
$V(Q)$ develops a secondary minimum corresponding to the metastable glass
phase approaching $T_{\mathrm{K}}$ from above. (b) The free energy difference
between the stable liquid and the metastable glass defines the configurational
entropy $S_{\mathrm{conf}}(T)$, which vanishes continuously at
$T_{\mathrm{K}}$, (c) where the overlap order parameter $Q$ jumps
discontinuously to a finite value. (d) Increasing the thermodynamic field
$\epsilon$ conjugate to $Q$ induces a first-order transition line for
$T>T_{\mathrm{K}}$ ending at a critical point in the universality class of the
random-field Ising model. Above the critical point, the line of maximum
overlap fluctuations is marked as dash-dotted line.

In practice the calculation of $Q$ requires the introduction of a microscopic
length scale $a$ corresponding to the amplitude of the thermal vibrations
around the average positions of the particles, roughly corresponding to the
cage size Guiselin  _et al._ (2020a). If one considers two copies of the
liquid $\mathcal{C}_{\alpha}=\\{\bm{r}_{i}^{(\alpha)}\\}_{i=1\dots N}$ defined
from the particle positions $\bm{r}_{i}^{(\alpha)}$ for $\alpha=1$, 2, the
microscopic density profiles are defined as
$\rho^{(\alpha)}(\bm{r})=\sum_{i=1}^{N}\delta(\bm{r}-\bm{r}_{i}^{(\alpha)})$
with $\delta(x)$ the Dirac distribution. The overlap then reads

$\displaystyle Q[\mathcal{C}_{1},\mathcal{C}_{2}]$
$\displaystyle=\frac{1}{N}\iint\mathrm{d}\bm{r}\mathrm{d}\bm{r}^{\prime}\rho^{(1)}(\bm{r})\rho^{(2)}(\bm{r}^{\prime})w(|\bm{r}-\bm{r}^{\prime}|)$
(1)
$\displaystyle=\frac{1}{N}\sum_{i,j=1}^{N}w(|\bm{r}_{i}^{(1)}-\bm{r}_{j}^{(2)}|),$

with $w(x)=1$ if $x\leq a$ and 0 otherwise. From Eq. (1) it is clear that
$Q[\mathcal{C}_{1},\mathcal{C}_{2}]\simeq 0$ for uncorrelated configurations,
while $Q[\mathcal{C}_{1},\mathcal{C}_{2}]\simeq 1$ for strongly correlated
ones. Just as for spin glasses, it is an intensive, normalised function which
quantifies the degree of similarity of the main microscopic degrees of freedom
in two independent configurations.

The calculation of the Landau free energy $V(Q)$ proceeds in two separate
steps. First, one chooses and fixes a reference configuration
$\mathcal{C}_{0}$ for the system at thermal equilibrium. The overlap between
$\mathcal{C}_{0}$ and any other configuration $\mathcal{C}$ is
$Q[\mathcal{C},\mathcal{C}_{0}]$, defined by Eq. (1). The equilibrium
probability distribution function $P(Q,\mathcal{C}_{0})$ of this overlap is
obtained by performing a canonical average

$P(Q,\mathcal{C}_{0})=\sum_{\mathcal{C}}p_{\mathrm{eq}}(\mathcal{C})\delta\left(Q-Q[\mathcal{C},\mathcal{C}_{0}]\right),$
(2)

where the statistical weight of any configuration $\mathcal{C}$ is controlled
by the Boltzmann factor at temperature $T$ ($k_{\mathrm{B}}T=1/\beta$),

$p_{\rm
eq}(\mathcal{C})=\frac{e^{-\beta\mathcal{H}[\mathcal{C}]}}{\sum_{\mathcal{C}^{\prime}}e^{-\beta\mathcal{H}[\mathcal{C}^{\prime}]}}.$
(3)

By definition, the free energy $V(Q,\mathcal{C}_{0})$ is the thermodynamic
potential controlling the fluctuations of the overlap $Q$ for a fixed
reference configuration Landau and Lifshitz (2013),

$P(Q,\mathcal{C}_{0})\sim\exp(-\beta NV(Q,\mathcal{C}_{0})),$ (4)

so that the most probable value of the overlap indeed corresponds to the
minimum of the free energy in the thermodynamic limit, $N\to\infty$. In a
second step one needs to perform a final average over all possible choices for
the reference configuration to get the desired averaged free energy

$V(Q)=\sum_{\mathcal{C}_{0}}p_{\rm eq}(\mathcal{C}_{0})V(Q,\mathcal{C}_{0}),$
(5)

where the statistical weight of a given reference configuration is again
controlled by the Boltzmann factor. The amorphous structure of the glass
directly enters the canonical average in Eq. (2) which contains quenched
disorder (the reference configuration $\mathcal{C}_{0}$) requiring a separate
average in Eq. (5). It is this double averaging procedure which makes the
analytic calculation of $V(Q)$ a difficult mathematical problem, even in the
limit of infinite dimensions Parisi  _et al._ (2020).

The temperature evolution of $V(Q)$ in the mean-field limit is shown in Fig.
3(a). At sufficiently high temperatures, the free energy has a unique minimum
for $Q=0$. This equilibrium state corresponds to the liquid phase in which
there is a large number of distinct accessible configurations, so that two
randomly chosen configurations are very different. At lower temperature, the
free energy develops a secondary minimum for $Q>0$, which therefore represents
a metastable phase. This corresponds to a phase with strong correlations
between equilibrium configurations which are all amorphous. When the
temperature decreases further, the secondary minimum becomes more and more
stable. When the Kauzmann temperature $T_{\mathrm{K}}$ is reached, the two
minima have the same free energy and an equilibrium phase transition takes
place between the liquid with $Q=0$ and the ideal glass at $Q>0$. This is a
first-order phase transition because the overlap $Q$ jumps discontinuously at
$T_{\mathrm{K}}$, as shown in Fig. 3(c). The central conclusion for our
discussion is that, in the mean-field limit, an equilibrium glass exists as a
genuine state of matter for $T<T_{\mathrm{K}}$.

The discontinuity of the order parameter $Q$ at the transition indicates that
equilibrium configurations in the ideal glass phase are strongly correlated.
More fundamentally, the transition reflects a spontaneous symmetry breaking,
which is that of the underlying replica symmetry, and the discontinuity is
associated with a specific breaking scheme of this symmetry Mézard  _et al._
(1987). The liquid is symmetric with respect to the replica symmetry, because
if several replicas (copies) are sampled in the liquid, no density field is
privileged. Conversely, the ideal glass breaks this symmetry, as now all
replicas possess a similar amorphous density profile. Physically, the ideal
glass is a disordered state of matter in real space (the distribution of the
average positions of the particles is amorphous), but ordered in configuration
space (all equilibrium configurations look similar).

For $T>T_{\mathrm{K}}$, the free energy difference $\Delta V(T)$ between the
two minima of $V(Q)$ represents the thermodynamic cost to force the system to
remain localised near a given configuration in order to maintain a high value
of $Q$, preventing the system to explore the much larger configuration space
available in the liquid. The free energy cost is therefore entropic in nature.
This free energy difference $\Delta V(T)$ in fact represents a natural
definition of the configurational entropy $S_{\mathrm{conf}}(T)\equiv\Delta
V(T)/T$ with the consequence that it vanishes continuously when $T\to
T_{\mathrm{K}}$ as shown in Fig. 3(b). This last result implies that the
liquid-glass transition has no latent heat as the latent heat is normally
proportional to the entropy jump at a first-order phase transition. The ideal
glass transition is thus an unconventional example of a discontinuous
transition Kirkpatrick  _et al._ (1989). More strikingly perhaps, the
continuous vanishing of the configurational entropy in Fig. 3(b) exactly
realises the scenario in which the ideal glass transition corresponds to the
entropy crisis first put forward by Kauzmann Kauzmann (1948). With hindsight,
Kauzmann’s work appears truly visionary.

This exposition of the mean-field theory of the glass transition closely
follows traditional descriptions of the van der Waals model for the liquid-gas
transition or the Curie-Weiss model to describe ferromagnetism. Pushing these
analogies further, one can introduce the field $\epsilon$ that is
thermodynamically coupled to the order parameter $Q$ Franz and Parisi (1997).
This field plays a role similar to the magnetic field $H$ conjugate to the
magnetisation $M$ in the Curie-Weiss model. It introduces a new contribution
$-\epsilon Q$ in the Hamiltonian which for $\epsilon>0$ makes large $Q$ values
energetically more favourable. In the presence of a field, equilibrium is
found by looking for the minimum of the tilted free energy $V(Q)-\epsilon Q$
for fixed ($T$, $\epsilon$). When the glass is metastable for
$T>T_{\mathrm{K}}$, increasing $\epsilon$ triggers a first-order phase
transition towards a high-overlap state, see Fig. 3(d). Physically, this can
be understood from an energy-entropy argument, as for many equilibrium phase
transitions Goldenfeld (2018). When $\epsilon$ is increased, the system gains
energy by adopting a large value of $Q$ but loses configurational entropy
because it only visits a set of highly-correlated configurations. Beyond a
threshold value $\epsilon^{*}(T)\simeq TS_{\mathrm{conf}}(T)=\Delta V(T)$, the
system ends up in a high-overlap phase. As a result, the existence of a
discontinuous phase transition at the Kauzmann temperature $T_{\mathrm{K}}$
implies the existence of a first-order transition line in the ($T$,
$\epsilon$) plane. This line ends at a second-order critical point, once again
in full analogy with the liquid-gas transition in the van der Waals
description.

The behaviour of the order parameter in the vicinity of a critical point
defines universality classes. Each universality class is characterised by a
different set of critical exponents. For glasses, the presence of disorder in
the various canonical averages in Eqs. (2, 5) implies that the critical end-
point at a finite applied field $\epsilon$ belongs to the universality class
of the random-field Ising model Biroli  _et al._ (2014); Franz and Parisi
(2013). (See Ref. [Nattermann, 1998] for an introduction to the random-field
Ising model.) By contrast, the critical points of the liquid-gas and
paramagnetic-ferromagnetic phase transitions both belong to the universality
class of the Ising model with no disorder. The transition between a paramagnet
and a spin glass is also controlled by the presence of quenched disorder but
it forms a universality class on its own Mézard  _et al._ (1987). Despite the
mathematical similarities between molecular glasses and conventional spin
glasses, in particular the definition of the order parameter, the emergence of
a glassy phase is qualitatively distinct in these two systems.

Before closing this section, it is instructive to pause and ask: should one be
satisfied that after decades of theoretical work on the problem of the glass
transition, the only solid set of results is about the mean-field theory?
There are several answers. First, the study of phase transitions and critical
phenomena in the presence of quenched disorder is notoriously more complicated
than for pure, conventional systems De Dominicis and Giardina (2006). It
should thus come as no surprise that the statistical mechanics of glasses is a
very difficult analytic problem, and the existence of a solid starting point
is an important step. However, past experience with disordered systems also
suggests that including corrections to mean-field is going to be an even
harder problem. At present it is not known whether and how the mean-field
construction can survive the inclusion of fluctuations that are neglected in
the mean-field theory. It is conceivable that the very existence of an ideal
glass as a phase of matter, established within the mean-field approximation,
becomes impossible when the relevant fluctuations are understood and included
in the calculations.

##  IV Does the ideal glass exist beyond mean-field?

The ideal glass, which is defined in full equilibrium conditions, differs from
the conventional material prepared in the laboratory that has fallen out of
equilibrium near $T_{\mathrm{g}}$ while cooling. How can one observe an ideal
glass? The most direct way to detect the ideal glass consists in the quasi-
static cooling of two copies of the supercooled liquid to maintain thermal
equilibrium. A discontinuous jump of their relative overlap $Q$ would be
observed at the Kauzmann temperature $T_{\mathrm{K}}$. This is conceptually
very simple! This seems however impossible due to the dramatic slowing down of
the dynamics of the supercooled liquid at low temperature which prevents an
ergodic exploration of the configuration space below $T_{\mathrm{g}}$.

The difference between ideal and conventional glasses appears subtle: the
conventional glass is confined in configuration space because it moves too
slowly to explore the many different available states, while the ideal glass
is confined in configuration space because there is nowhere to go even given
an infinite amount of time. Establishing this difference for temperatures
where equilibration is extremely long is then obviously a challenge. In
addition, the first-order nature of the equilibrium liquid-glass transition
implies that it cannot be simply revealed by the growth of some critical
fluctuations, as would be the case when approaching a second-order phase
transition. More subtle signs of an incipient phase transition need to be
detected.

Major progress has recently been made to tackle these two distinct problems in
computer simulation studies. To solve the first problem, efficient Monte Carlo
algorithms have been developed which are able to easily sample the
configuration space in equilibrium conditions employing non-physical particle
motions Grigera and Parisi (2001); Berthier  _et al._ (2016a); Ninarello  _et
al._ (2017); Berthier  _et al._ (2019a); Parmar  _et al._ (2020). As a result,
equilibrium thermodynamic measurements can be performed at low temperatures
where this task would be impossible by simply simulating the physical
dynamics. In essence, these algorithms solve the first problem and maintain
equilibrium at temperatures much lower than in conventional simulations. For
certain numerical models, it is estimated that temperatures smaller than the
experimental glass temperature $T_{\mathrm{g}}$ can be equilibrated, thus
provinding access to the experimental temperature gap between $T_{\mathrm{g}}$
and $T_{\mathrm{K}}$.

Figure 4: Numerical results of three-dimensional systems reveal features of
the mean-field theory. (a) Temperature evolution of Landau free energy $V(Q)$
which develops a singularity at large $Q$ consistent with the existence of a
metastable glass phase. The dashed line highlights the non-convexity of the
free energy at the lowest temperature. (b) Configurational entropy measured in
two glass-forming models. The free energy difference between the equilibrium
liquid and metastable glass decreases rapidly with decreasing $T$ and seems to
extrapolate to a finite Kauzmann temperature $T_{\mathrm{K}}$. (c) The growth
of the overlap upon application of a field $\epsilon$ is smooth at large $T$
but becomes discontinuous at low $T$, revealing a first-order equilibrium
transition line $\epsilon^{*}(T)$. (d) Equilibrium phase diagram ($T$,
$\epsilon$). Dashed lines are temperatures shown in (a, c). The dash-dotted
line represents the location of the inflection point of $Q(\epsilon)$ which
transforms in a discontinuity (full line) below a second-order critical point
(dot).

The second problem concerns numerical signatures of the approach to a first-
order phase transition. The most direct measurement amounts to directly
estimating the Landau-Franz-Parisi thermodynamic potential $V(Q)$ across a
range of temperatures. This program was started long ago Franz and Parisi
(1998); Cardenas  _et al._ (1999) but was plagued by both equilibration
problems and sampling issues. More recently, improved methods to assess $V(Q)$
from Eq. (2) based on large deviations techniques have been proposed Berthier
(2013) and combined with enhanced sampling techniques Berthier and Coslovich
(2014). As a result, the Landau free energy $V(Q)$ has been measured in
several numerical models of glass-forming liquids in dimension $d=2$ and $d=3$
down to very low temperatures Berthier (2013); Berthier and Jack (2015);
Berthier  _et al._ (2017); Guiselin  _et al._ (2022a). The expected behaviour
for $V(Q)$ has been confirmed directly in these computer simulations, as
illustrated in Fig. 4(a), with the emergence of singularities at large $Q$
when $T$ is low enough. Since these measurements are performed in equilibrium
conditions and do not require any extrapolation, they are direct proof that
the system is on its way to an equilibrium first-order phase transition with
an emerging metastable glass phase at large $Q$.

Glass metastability can be established even more clearly by coupling the
overlap $Q$ to its conjugate field $\epsilon$ Berthier (2013); Berthier and
Jack (2015); Guiselin  _et al._ (2020b, 2022a, 2022b). In three dimensions,
one finds a discontinuous jump of the overlap as $\epsilon$ is increased at
constant $T$ beyond a critical value $\epsilon^{*}(T)$, see Fig. 4(c). This
discontinuous jump disappears when the temperature is too large, as expected
from mean-field theory.

As a result, a line of first-order transition $\epsilon^{*}(T)$ can be
determined in the phase diagram ($T$, $\epsilon$), see Fig. 4(d), which ends
at a critical point whose universality class was recently analysed by detailed
finite-size scaling analysis Guiselin  _et al._ (2020b); Jack and Garrahan
(2016). (At the critical point, the amplitude of the order parameter
fluctuations only diverges in the thermodynamic limit, so that analysis of
this divergence with system size yields information on the critical exponents
Newman and Barkema (1999).) Results in three-dimensional systems appear
compatible with the predicted random-field Ising model universality class.

Interestingly, the connection with the random-field Ising model implies that
neither the critical point nor the first-order transition line can exist in
two dimensions at any finite temperature. This is because the lower critical
dimension Goldenfeld (2018) of the random-field Ising model is $d=2$, so that
for $d\leq 2$ there is no longer a finite transition temperature in this model
Imry and Ma (1975). This non-trivial prediction was recently confirmed in
computer simulations of a two-dimensional glass-forming model Guiselin  _et
al._ (2022a).

Using similar techniques Berthier  _et al._ (2019b), the configurational
entropy $S_{\mathrm{conf}}(T)$ was recently estimated in computer models down
to extremely low temperatures, including below the experimental
$T_{\mathrm{g}}$, see Fig. 4(b). These measurements on the one hand confirm
and amplify the results obtained from calorimetric experiments, such as the
results discussed in Kauzmann’s original work Kauzmann (1948). Whereas the
nature of the configurational entropy obtained experimentally is subject to
discussions Tatsumi  _et al._ (2012), in the simulations $S_{\rm conf}(T)$ can
be interpreted as the free energy difference between the equilibrium liquid
and metastable glass. The steep decrease observed in Fig. 4(b) at low
temperatures shows that the glass could become the equilibrium phase below a
finite temperature $T_{\mathrm{K}}$. Two-dimensional simulations Berthier  _et
al._ (2019c) are compatible with a value $T_{\mathrm{K}}=0$, again suggesting
that $d=2$ plays a special role with a zero-temperature critical point and no
equilibrium glass phase at any $T>0$.

Finally, metastability near a first-order phase transition is generically
associated with an important length scale related to the physics of
nucleation. If a system is prepared in a metastable phase, the emergence of
the stable phase proceeds by first nucleating a critical droplet of the stable
phase which can then invade the entire system. The minimal size of the
nucleating droplet is inversely proportional to the free energy difference
between the two phases and diverges at the transition. This behaviour is
generic to first-order phase transitions and describes for instance how
crystals form below $T_{\mathrm{m}}$. In the present case, one would
essentially like to estimate the size of the critical nucleus of the stable
liquid from the metastable glass above $T_{\mathrm{K}}$. However, considering
the unconventional first-order nature of the transition at $T_{\mathrm{K}}$,
this is far from trivial and special computational tools are needed. Such
tools have been proposed Bouchaud and Biroli (2004) and implemented Biroli
_et al._ (2008); Berthier  _et al._ (2016b); Yaida  _et al._ (2016); Berthier
_et al._ (2017), allowing one to estimate this important length scale,
$\xi(T)$, which is also known as the point-to-set length scale Montanari and
Semerjian (2006).

Numerical measurements show that $\xi(T)$ increases when the temperature $T$
decreases, as expected when approaching a first-order phase transition. The
temperature dependence of $\xi(T)$ is compatible with the above nucleation
argument, which suggests that $\xi(T)\sim 1/S_{\rm conf}(T)$. The growth of
the point-to-set length scale $\xi(T)$ therefore accompanies the steep
decrease in the configurational entropy which signals that glass and liquid
free energies are getting closer as $T$ decreases towards $T_{\mathrm{K}}$.

##  V Conclusion

State-of-the-art computer studies provide solid confirmations that supercooled
liquids carefully maintained in equilibrium conditions at low temperatures
(even below $T_{\mathrm{g}}$) display thermodynamic fluctuations that are
fully compatible with the mean-field description of an incipient discontinuous
liquid-glass phase transition. These observations invalidate the well-known
motto that that glass formation is essentially a dynamic process accompanied
by little structural change. Of course, the existence of the transition at a
finite Kauzmann temperature $T_{\mathrm{K}}>0$ remains to be demonstrated by
direct measurements. This would represent a major objective for future work.
From a computational viewpoint, this requires the development of numerical
algorithms more sophisticated than current ones in order to achieve
equilibration even closer to the putative $T_{\mathrm{K}}$.

What about experiments? Seventy years after Kauzmann Kauzmann (1948),
calorimetric measurements provide estimates of a configurational entropy which
robustly extrapolate to a finite Kauzmann temperature Tatsumi  _et al._
(2012). In recent years, indirect experimental estimates of the point-to-set
correlation length scale $\xi(T)$ were also obtained via non-linear dielectric
susceptibilities Albert  _et al._ (2016); Biroli  _et al._ (2021). (If the
frequency of the imposed oscillatory electric field is larger than the typical
relaxation frequency of the supercooled liquid, then the liquid approximately
behaves like a mosaic of independent glasses of typical size $\xi(T)$
Kirkpatrick  _et al._ (1989), each one of them responding with an induced
dipolar moment that depends on their typical size $\xi(T)$, which can then be
inferred.)

The seemingly impossible task to achieve equilibration below $T_{\mathrm{g}}$
was overcome after the discovery that amorphous glassy films prepared by
physical vapour deposition are much closer to equilibrium than their liquid-
cooled counterparts Swallen  _et al._ (2007). When the deposition rate is
sufficiently low (of the order of 1 nm/s), and the temperature judiciously
chosen (about 15 % lower than the experimental glass transition temperature
$T_{\mathrm{g}}$), it has become possible to create glassy films that seem to
remain in equilibrium much closer to $T_{\mathrm{K}}$ than in any previous
work Beasley  _et al._ (2019), thus providing novel support for previous
extrapolations of the configurational entropy from above $T_{\mathrm{g}}$.
These new glassy materials are “ultrastable”, because they correspond to
thermodynamic conditions impossible to achieve using bulk cooling Ediger
(2017).

In addition to providing equilibrium materials much closer to
$T_{\mathrm{K}}$, the transformation kinetics of an ultrastable glass into a
liquid upon heating closely resembles the nucleation and growth process
commonly observed near a first-order phase transition Kearns  _et al._ (2010);
Jack and Berthier (2016). For thin ultrastable films, the transformation is
initiated at the free surface and propagates as a front. This is reminiscent
of heterogeneous nucleation. For thicker samples, liquid droplets first
nucleate in the bulk and their size then increases until complete
transformation, reminiscent of homogeneous nucleation Vila-Costa  _et al._
(2020). These analogies reinforce the idea that ultrastable glasses behave
very much like a distinct phase of matter accessed via a discontinuous phase
transition. Computer simulations of this transformation process Fullerton and
Berthier (2017); Flenner  _et al._ (2019) could help unravel the main
mechanisms at play, and could allow direct quantitive comparisons with
predictions from classical nucleation theory Kalikmanov (2013).

The construction of a complete theoretical description of the nature of the
glass remains an active research area. The progress summarised here results
from a synergy between mathematical work, experimental discoveries and the
development of novel computational tools. This quest already gave rise to many
fruitful developments, as recognized by the 2021 Nobel Prize in Physics
awarded to Giorgio Parisi who has been a driving force in this field. In
addition to understanding glass more deeply, tools and concepts developed for
glass physics are at the core of deep analogies between a multitude of complex
systems, such as neural networks in biology Mézard  _et al._ (1987); Amit  _et
al._ (1985), algorithms for constrained optimisation Antenucci  _et al._
(2019), or, more recently, machine learning Zdeborová and Krzakala (2016);
Bahri  _et al._ (2020).

###### Acknowledgements.

This work was supported by a grant from the Simons Foundation (Grant No.
454933, L.B.).

## References

  * Tabor (1991) D. Tabor,  _Gases, liquids and solids: and other states of matter_ (Cambridge University Press, 1991).
  * Cavagna (2009) A. Cavagna, Physics Reports 476, 51 (2009).
  * Berthier and Biroli (2011) L. Berthier and G. Biroli, Reviews of Modern Physics 83, 587 (2011).
  * Kauzmann (1948) W. Kauzmann, Chemical Reviews 43, 219 (1948).
  * Gibbs and DiMarzio (1958) J. H. Gibbs and E. A. DiMarzio, The Journal of Chemical Physics 28, 373 (1958).
  * Landau and Lifshitz (2013) L. D. Landau and E. M. Lifshitz,  _Statistical Physics_ , Vol. 5 (Elsevier, 2013).
  * Callen (1998) H. B. Callen,  _Thermodynamics and an Introduction to Thermostatistics_ (American Association of Physics Teachers, 1998).
  * Sethna (2021) J. Sethna,  _Statistical mechanics: entropy, order parameters, and complexity_ (Oxford University Press, 2021).
  * Goldenfeld (2018) N. Goldenfeld,  _Lectures on phase transitions and the renormalization group_ (CRC Press, 2018).
  * Parisi  _et al._ (2020) G. Parisi, P. Urbani, and F. Zamponi,  _Theory of simple glasses: exact solutions in infinite dimensions_ (Cambridge University Press, 2020).
  * Singh  _et al._ (1985) Y. Singh, J. Stoessel, and P. Wolynes, Physical Review Letters 54, 1059 (1985).
  * Kirkpatrick and Thirumalai (1987) T. R. Kirkpatrick and D. Thirumalai, Physical Review B 36, 5388 (1987).
  * Biroli and Mézard (2001) G. Biroli and M. Mézard, Physical Review Letters 88, 025501 (2001).
  * Franz and Parisi (1997) S. Franz and G. Parisi, Physical Review Letters 79, 2486 (1997).
  * Hansen and McDonald (2013) J.-P. Hansen and I. R. McDonald,  _Theory of simple liquids: with Applications to Soft Matter_ (Academic Press, 2013).
  * Mézard  _et al._ (1987) M. Mézard, G. Parisi, and M. Virasoro,  _Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications_ (World Scientific, 1987).
  * Castellani and Cavagna (2005) T. Castellani and A. Cavagna, Journal of Statistical Mechanics: Theory and Experiment 2005, P05012 (2005).
  * Edwards and Anderson (1975) S. F. Edwards and P. W. Anderson, Journal of Physics F: Metal Physics 5, 965 (1975).
  * Guiselin  _et al._ (2022a) B. Guiselin, L. Berthier, and G. Tarjus, SciPost Physics 12, 091 (2022a).
  * Parisi (1983) G. Parisi, Physical Review Letters 50, 1946 (1983).
  * Guiselin  _et al._ (2020a) B. Guiselin, G. Tarjus, and L. Berthier, The Journal of Chemical Physics 153, 224502 (2020a).
  * Kirkpatrick  _et al._ (1989) T. R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes, Physical Review A 40, 1045 (1989).
  * Biroli  _et al._ (2014) G. Biroli, C. Cammarota, G. Tarjus, and M. Tarzia, Physical Review Letters 112, 175701 (2014).
  * Franz and Parisi (2013) S. Franz and G. Parisi, Journal of Statistical Mechanics: Theory and Experiment 2013, P11012 (2013).
  * Nattermann (1998) T. Nattermann, “Theory of the random field ising model,” in  _Spin glasses and random fields_ (World Scientific, 1998) pp. 277–298.
  * De Dominicis and Giardina (2006) C. De Dominicis and I. Giardina,  _Random fields and spin glasses: a field theory approach_ (Cambridge University Press, 2006).
  * Grigera and Parisi (2001) T. S. Grigera and G. Parisi, Physical Review E 63, 045102 (2001).
  * Berthier  _et al._ (2016a) L. Berthier, D. Coslovich, A. Ninarello, and M. Ozawa, Physical Review Letters 116, 238002 (2016a).
  * Ninarello  _et al._ (2017) A. Ninarello, L. Berthier, and D. Coslovich, Physical Review X 7, 021039 (2017).
  * Berthier  _et al._ (2019a) L. Berthier, E. Flenner, C. J. Fullerton, C. Scalliet, and M. Singh, Journal of Statistical Mechanics: Theory and Experiment 2019, 064004 (2019a).
  * Parmar  _et al._ (2020) A. D. S. Parmar, M. Ozawa, and L. Berthier, Physical Review Letters 125, 085505 (2020).
  * Franz and Parisi (1998) S. Franz and G. Parisi, Physica A: Statistical Mechanics and its Applications 261, 317 (1998).
  * Cardenas  _et al._ (1999) M. Cardenas, S. Franz, and G. Parisi, The Journal of Chemical Physics 110, 1726 (1999).
  * Berthier (2013) L. Berthier, Physical Review E 88, 022313 (2013).
  * Berthier and Coslovich (2014) L. Berthier and D. Coslovich, Proceedings of the National Academy of Sciences 111, 11668 (2014).
  * Berthier and Jack (2015) L. Berthier and R. L. Jack, Physical Review Letters 114, 205701 (2015).
  * Berthier  _et al._ (2017) L. Berthier, P. Charbonneau, D. Coslovich, A. Ninarello, M. Ozawa, and S. Yaida, Proceedings of the National Academy of Sciences 114, 11356 (2017).
  * Guiselin  _et al._ (2020b) B. Guiselin, L. Berthier, and G. Tarjus, Physical Review E 102, 042129 (2020b).
  * Guiselin  _et al._ (2022b) B. Guiselin, G. Tarjus, and L. Berthier, The Journal of Chemical Physics 156, 194503 (2022b).
  * Jack and Garrahan (2016) R. L. Jack and J. P. Garrahan, Physical Review Letters 116, 055702 (2016).
  * Newman and Barkema (1999) M. E. Newman and G. T. Barkema,  _Monte Carlo methods in statistical physics_ (Clarendon Press, 1999).
  * Imry and Ma (1975) Y. Imry and S.-K. Ma, Physical Review Letters 35, 1399 (1975).
  * Berthier  _et al._ (2019b) L. Berthier, M. Ozawa, and C. Scalliet, The Journal of Chemical Physics 150, 160902 (2019b).
  * Tatsumi  _et al._ (2012) S. Tatsumi, S. Aso, and O. Yamamuro, Physical Review Letters 109, 045701 (2012).
  * Berthier  _et al._ (2019c) L. Berthier, P. Charbonneau, A. Ninarello, M. Ozawa, and S. Yaida, Nature Communications 10, 1 (2019c).
  * Bouchaud and Biroli (2004) J.-P. Bouchaud and G. Biroli, The Journal of Chemical Physics 121, 7347 (2004).
  * Biroli  _et al._ (2008) G. Biroli, J.-P. Bouchaud, A. Cavagna, T. S. Grigera, and P. Verrocchio, Nature Physics 4, 771 (2008).
  * Berthier  _et al._ (2016b) L. Berthier, P. Charbonneau, and S. Yaida, The Journal of Chemical Physics 144, 024501 (2016b).
  * Yaida  _et al._ (2016) S. Yaida, L. Berthier, P. Charbonneau, and G. Tarjus, Physical Review E 94, 032605 (2016).
  * Montanari and Semerjian (2006) A. Montanari and G. Semerjian, Journal of Statistical Physics 125, 23 (2006).
  * Albert  _et al._ (2016) S. Albert, T. Bauer, M. Michl, G. Biroli, J.-P. Bouchaud, A. Loidl, P. Lunkenheimer, R. Tourbot, C. Wiertel-Gasquet, and F. Ladieu, Science 352, 1308 (2016).
  * Biroli  _et al._ (2021) G. Biroli, J.-P. Bouchaud, and F. Ladieu, The Journal of Physical Chemistry B 125, 7578 (2021).
  * Swallen  _et al._ (2007) S. F. Swallen, K. L. Kearns, M. K. Mapes, Y. S. Kim, R. J. McMahon, M. D. Ediger, T. Wu, L. Yu, and S. Satija, Science 315, 353 (2007).
  * Beasley  _et al._ (2019) M. S. Beasley, C. Bishop, B. J. Kasting, and M. D. Ediger, The Journal of Physical Chemistry Letters 10, 4069 (2019).
  * Ediger (2017) M. D. Ediger, The Journal of Chemical Physics 147, 210901 (2017).
  * Kearns  _et al._ (2010) K. L. Kearns, M. Ediger, H. Huth, and C. Schick, The Journal of Physical Chemistry Letters 1, 388 (2010).
  * Jack and Berthier (2016) R. L. Jack and L. Berthier, The Journal of Chemical Physics 144, 244506 (2016).
  * Vila-Costa  _et al._ (2020) A. Vila-Costa, J. Ràfols-Ribé, M. González-Silveira, A. Lopeandia, L. Abad-Muñoz, and J. Rodríguez-Viejo, Physical Review Letters 124, 076002 (2020).
  * Fullerton and Berthier (2017) C. J. Fullerton and L. Berthier, EPL (Europhysics Letters) 119, 36003 (2017).
  * Flenner  _et al._ (2019) E. Flenner, L. Berthier, P. Charbonneau, and C. J. Fullerton, Physical Review Letters 123, 175501 (2019).
  * Kalikmanov (2013) V. I. Kalikmanov, “Classical nucleation theory,” in  _Nucleation Theory_ (Springer Netherlands, Dordrecht, 2013) pp. 17–41.
  * Amit  _et al._ (1985) D. J. Amit, H. Gutfreund, and H. Sompolinsky, Physical Review A 32, 1007 (1985).
  * Antenucci  _et al._ (2019) F. Antenucci, S. Franz, P. Urbani, and L. Zdeborová, Physical Review X 9, 011020 (2019).
  * Zdeborová and Krzakala (2016) L. Zdeborová and F. Krzakala, Advances in Physics 65, 453 (2016).
  * Bahri  _et al._ (2020) Y. Bahri, J. Kadmon, J. Pennington, S. S. Schoenholz, J. Sohl-Dickstein, and S. Ganguli, Annual Review of Condensed Matter Physics 11 (2020).