# Multi-Channel MOSRA: Mean Opinion Score and Room Acoustics Estimation Using
Simulated Data and a Teacher Model

###### Abstract

Previous methods for predicting room acoustic parameters and speech quality
metrics have focused on the single-channel case, where room acoustics and Mean
Opinion Score (MOS) are predicted for a single recording device. However,
quality-based device selection for rooms with multiple recording devices may
benefit from a multi-channel approach where the descriptive metrics are
predicted for multiple devices in parallel. Following our hypothesis that a
model may benefit from multi-channel training, we develop a multi-channel
model for joint MOS and room acoustics prediction (MOSRA) for five channels in
parallel. The lack of multi-channel audio data with ground truth labels
necessitated the creation of simulated data using an acoustic simulator with
room acoustic labels extracted from the generated impulse responses and labels
for MOS generated in a student-teacher setup using a wav2vec2-based MOS
prediction model. Our experiments show that the multi-channel model improves
the prediction of the direct-to-reverberation ratio, clarity, and speech
transmission index over the single-channel model with roughly 5$\times$ less
computation while suffering minimal losses in the performance of the other
metrics.

Index Terms—  Speech quality assessment, joint learning, room acoustics,
neural networks

##  1 Introduction

The Mean Opinion Score (MOS) is one of the simplest yet effective metrics of
subjective audio quality and can be evaluated using a subjective listening
test based on the ITU-T P.800 recommendation [1], or its crowdsourcing
approach described in the ITU-T P.808 [2].

However, given the costly nature of carrying out subjective listening tests,
many methods have been developed to estimate this subjective metric for
unevaluated audio recordings blindly. Most methods are based on neural
networks trained to map audio to MOS scores. Nevertheless, many of these
systems lack robustness due to the lack of large human-labeled datasets.
Therefore, the ConferencingSpeech 2022 challenge [3] recently released a
larger dataset consisting of 86k audio files with crowdsourced MOS labels,
with top performers in the challenge achieving a Pearson correlation of
roughly 0.8 with human labels.

An overall speech quality metric is very useful but does not provide insights
into the causes of degraded speech quality. To counteract this, multi-valued
non-intrusive speech quality assessment (NISQA) methods have been proposed [4,
5]. Characterization of the listening environment also provides such insights.
Therefore, efforts have been made in the blind estimation of room acoustic
descriptors, where deep neural networks (DNNs) estimate a single acoustic
descriptor, such as reverberation time measured in seconds, T60 [6], or the
speech transmission index [7]. Besides, joint prediction of more acoustic
parameters using recurrent neural networks can estimate reverberation time
(T60), clarity metrics in decibels (C50 and C80), direct to reverberant ratio
(DRR) also in decibels [8]. Finally, Lopez et al. [9] further expanded the
joint prediction by adding the signal-to-noise ratio (SNR, in decibels) and
the speech transmission index (STI, ranging from 0 to 1). A lightweight model
that jointly predicted MOS and room acoustic parameters was presented in [10].
Later, Hajal et al. investigated self-supervised learning methods for the same
joint prediction and achieved state-of-the-art results [11]. More recently,
Sarabia et al. [12] released a corpus of simulated audio data with
corresponding acoustic parameters where the authors showed that models trained
on simulated data were able to generalize to real world conditions.

Fig. 1: Overview of the proposed framework. On the left, a high-level overview
of the data generation process is given. On the right, the details of the
model architecture are shown, with the number of parameters shown in the red
boxes.

The recent work on speech quality assessment and room acoustics estimation has
focused on the single-device case where descriptive metrics are predicted for
a single recording device in a one-to-one fashion. However, there has recently
been interest in a many-to-one setup where multiple input channels map to a
single metric. One such setting is the clarity prediction challenge [13],
where the challenge entrants are tasked with predicting the intelligibility of
binaural hearing aid recordings achieved by human listeners.

Another application of a many-to-one setup is the task of device selection.
The rise of smart home devices and personal digital devices led to the
situation where multiple spatially disjoint microphones can simultaneously
record a speaker. This naturally leads to the challenge of selecting which
audio stream to transmit. Thus, in previous work on device selection in the
context of smart home devices, Barber et al. [14] designed a multi-channel
system to predict the closest device to the current speaker, where an
improvement was found over a signal processing baseline. Similarly, Yoshioka
et al. [15] presented PickNet, a system to predict the closest device, where
they also note an improvement in word error rate (WER) when using a multi-
channel model over their single-channel baseline.

As noted, these previous works use the heuristic that the best device is the
closest to the current speaker. However, choosing the closest device may not
be optimal regarding audio quality, as many factors besides distance can
influence the quality of an audio stream. For example, the closest device may
be placed in a region with many reflections, decreasing DRR and clarity. It
may also be placed next to a noise source or run a less powerful denoising
model than other devices in the room.

Inspired by these previous works, we aim to further extend the task setup to a
many-to-many case where the model is tasked to predict multiple sets of
metrics given multiple input channels. Specifically, we aim to train a model
that predicts speech quality MOS and a set of room acoustic parameters (STI,
T60, DRR, and C50) for five channels in parallel. Our motivation for this is
two-fold. Firstly, having access to MOS and room acoustics will improve the
interpretability of the device selection, thus aiding in making informed
decisions. Secondly, we conjecture that a model trained to predict the metrics
of interest may benefit from multi-channel training, as information contained
in the different audio streams may give the model global information about the
overall acoustic environment.

Thus, in this paper, we present multi-channel MOSRA, a model that predicts the
MOS and room acoustics for five channels in parallel. Given the lack of multi-
device data with ground truth labels, we simulate it using an acoustic
simulator where the ground truth room acoustic labels are extracted from the
room impulse responses, while the labels for MOS are generated by student-
teacher setup using a wav2vec2 (XLSR)-based model trained to predict MOS
scores [10]. Our experiments show that the multi-channel model improves the
prediction of DRR, C50, and STI over the single-channel model with roughly
5$\times$ less computation while suffering minimal losses in the performance
of the other metrics.

##  2 Methods

This section proposes an extension to the MOSRA model that allows it to
generate predictions for five channels in parallel. The overall model
architecture is shown on the right in Fig. 1, and has a total number of 411k
trainable parameters. Moreover, the used data simulation pipeline is described
in detail.

###  2.1 Multi-channel feature extractor

The multi-channel feature extractor is an adapted version of the one used in
[11], where the first convolutional neural network (CNN) layer is modified to
take as input five mel-spectrogram segments instead of a single one, with the
rest of the network left unchanged.

These multi-channel Mel-spectrogram segments are obtained by Mel transforming
the STFT representations of the five audio files. We use 48 Mel bands with an
FFT window size of 20ms and a hop size of 10ms. Super wideband audio is
supported as the maximum Mel band frequency is 16kHz. The resulting Mel-
spectrogram is further divided into overlapping frames, each spanning 150ms
with a hop size of 40ms.

Consequently, the Mel-spectrogram segments are passed to the CNN, which
generates an embedding per segment. These multi-channel embeddings are then
further processed by a transformer encoder [11], which accounts for temporal
dependencies.

###  2.2 Multi-channel prediction heads

Similarly to the feature extractor, we adapt the original prediction heads to
output a metric estimate per channel. This is done by changing the final fully
connected layer to have an output size of 5 instead of 1.

###  2.3 Multi-channel meeting data generation

Given the lack of suitable training data, we leverage acoustic simulation to
generate it. An overview of the proposed simulation pipeline is shown on the
left in Fig. 1. PyRoomAcoustics [16] is used to generate the room impulse
responses (RIRs) which are convolved with the speech and noise sources. We use
a combination of the image-source method (ISM), and ray tracing to generate
the RIRs, where the maximum order for the ISM is set to 3 and the number of
rays is computed automatically following Vorländer 2008, Eq. (11.12) [17].

Then, for every unique scenario, we randomly generate a shoebox-shaped room
between 2.1-10m in width/length and 2-4m in height. The absorption
coefficients of the room materials are randomized to reflect a realistic
acoustic environment. This results in a mean T60 of 0.41s with a standard
deviation of 0.18. We then place a speech source at a height between 1.3-2m at
least 10cm from a wall. Then, one or two noises are added to the room in a
random location with the constraint that they are at least 10cm from any wall.
Finally, each of the five microphones is placed in one of two configurations,
either being mounted on a wall or in approximately the center of the room,
reflecting a device placed on a table.

We use clean speech from the LibriSpeech train-clean-100, dev-clean, test-
clean sets [18] as source data. The noise data is taken from the DNS challenge
[19], which we randomly partitioned in train, validation, and test splits at
fractions of 0.8, 0.1, and 0.1, leading to 43902, 5487, 5489 noise files per
subset respectively.

Thus, after generating the room impulse responses, we randomly choose a clean
speech file and up to two noise files and resample them from 16kHz to 32kHz.
The noise and speech samples are then repeated to have a length of 10 seconds.
Then, the dB full scale (dBFS) levels of the speech and noise are sampled as
defined in Eq. 1, which is done to obtain a desirable distribution of SNR at
the outputs.

$\begin{split}\text{dBFS(Noise)}\sim(Beta(1.5,1.5)\cdot-40)-20\\\
\text{dBFS(Speech)}\sim(Beta(1.5,1.5)\cdot-30)-10\end{split}$ (1)

The scaled audio segments are then convolved with the RIRs to obtain the five
output mixtures. These are then uniformly scaled such that the loudest signal
in the mixtures has a dBFS between -20 and 0. To obtain the final audio, the
wet mixtures are passed through three different internal Logitech denoising
models to simulate the case where the recording devices apply denoising before
transmitting the signal to a central hub.

To obtain the labels, we compute the room acoustic parameters from the RIRs:
STI [20], DRR [21], T60 and C50 [22], . Since we do not have access to MOS
labels for our generated data, we leverage a student-teacher setup.
Consequently, the audio outputs are passed to a larger single-channel network
trained to estimate MOS. We used an XLSR-based model trained to predict MOS
scores on single-channel recordings as our teacher model [10].

In the end we obtain three multi-channel datasets with a size in hours of
79.3, 7.5, and 6.3 for training, validation, and testing, respectively. The
length of every individual audio clip is slightly longer than 10 seconds due
to the convolution with the RIRs. Finally, the simulation resulted in the
distributions of T60 and DRR versus distance to the source, shown in Fig. 2.
The figure highlights that distance is only loosely correlated with acoustic
parameters.

###  2.4 Loss function

To train the multi-channel MOSRA network, we used mean squared error (MSE)
with the same loss weights as the original MOSRA paper [11]. The loss used for
training is shown in Eq. 2.

$\begin{split}\text{MSE}&=\frac{1}{N}\Sigma_{1}^{N}(y_{pred_{n}}-y_{true_{n}})^{2}\\\
\mathcal{L}_{\text{Room
acoustics}}&=\text{MSE}_{T60}+\text{MSE}_{C50}+\text{MSE}_{STI}+\text{MSE}_{DRR}\\\
\mathcal{L}_{overall}&=2\cdot\text{MSE}_{\text{MOS}}+0.2\cdot\mathcal{L}_{\text{Room
acoustics}}\end{split}$ (2)

Similarly to the original training scheme, we choose to normalize our labels
with the mean and standard deviations to ensure that all the model outputs
have the same range.

##  3 Experimental setup

Fig. 2: T60 and DRR versus distance to the active speaker.

###  3.1 Baseline system

To show the benefit of multi-channel training and inference, we compare the
proposed model against a baseline. The chosen baseline model is the original
single-channel MOSRA model [11] trained on the individual channels of our
simulated data. Both the single and multi-channel models have a similar number
of parameters with 411k and 413K parameters, respectively.

###  3.2 Training and evaluation

Both models were trained using the ADAM [23] optimizer, a batch size of 32, a
learning rate of 5e-4, and early stopping with patience of 30 on the Pearson
correlation of the MOS predictions. Furthermore, the learning rate was
decreased by a factor of 10 if the validation loss did not decrease for 15
epochs. Note that the single-channel baseline model had 5$\times$ more
training steps per epoch, one for each channel per unique simulated room.

During the multi-channel model training, we sample five devices from a room
with replacement to simulate the case that less than five microphones are
present in a given space i.e. some channels are repeated in the input to the
model. This also ensures that the model does not learn any patterns from the
ordering of the channels. Note that during testing, all channels in the room
are passed to the model.

For evaluation purposes, RMSE will be used for the acoustic parameters (STI,
T60, DRR, and C50). For MOS, the Pearson correlation metric is used.

##  4 Results

###  4.1 Acoustic parameters

The results for the prediction of acoustic parameters are shown in Table 1. We
can observe that the multi-channel system outperforms the baseline single-
channel system in STI, DRR, and C50. The confidence intervals also show that
the improvement in the three acoustic parameters is significant.

Model | Multi-channel | Single-channel
---|---|---
| RMSE | (baseline) RMSE
STI |  0.017 [0.0170, 0.0176] | 0.018 [0.0178, 0.0184]
DRR [dB] |  2.78 [2.736, 2.839] | 3.23 [3.190, 3.267]
T60 [s] | 0.11 [0.105, 0.112] |  0.09 [0.090, 0.097]
C50 [dB] |  1.80 [1.757, 1.834] | 1.87 [1.835, 1.912]
Table 1: RMSE ($\downarrow$) of the baseline and multi-channel models on
acoustic parameters using the simulated test set. The confidence intervals are
obtained by bootstrapping with 1000 repetitions. Fig. 3: Multi-channel MOSRA
predictions using a circular buffer of roughly 4 seconds of audio. The audio
is recorded in a real room where the speaker is crossfaded between recording
devices placed in three spatially disjoint locations. The model makes
predictions on three channels, where the first two are repeated, e.g., the
input to the model is channel [1,2,3,1,2]. Note that the scores are
standardized across channels for each time step to aid interpretability.

###  4.2 Speech quality

Table 2 shows the performance of the baseline single-channel and multi-channel
systems when trained on the simulated training data. The results show that in
terms of MOS, the baseline single-channel system slightly but significantly
outperforms the multi-channel system in Pearson correlation coefficient and
RMSE metrics.

###  4.3 Simulated dataset generalization

To show the generalization capability of models trained on the simulated
training data, we compare the original single-channel MOSRA model trained on
human-labeled data [11] with the multi-channel model trained on our simulated
data. To generate results with the multi-channel model on the single-channel
data, we repeat the single-channel audio five times and take the average
prediction at the output. We show results for in-distribution data, i.e., data
generated in the same fashion where speech is convolved with an RIR and noise
is added. We consider the NISQA TEST LIVETALK dataset as also in-distribution,
as it contains only near and far field loud and quiet speech spoken in various
noisy environments. The results in Table 3 show that for the in-distribution
data, the model performs at least equally well as the original model. Out-of-
distribution data is also shown in the form of the other NISQA TEST datasets,
where the performance of the model trained on synthetic data is lower than the
model trained on human-labeled data. These test data include distortions not
seen during simulation (such as clipping, packet loss, and various codec
distortions).

MOS on the test set |  | 95 % CI
---|---|---
PCC | Single-channel (baseline) | 0.97 | [0.971, 0.973]
Multi-channel | 0.96 | [0.957, 0.960]
RMSE | Single-channel (baseline) | 0.21 | [0.207, 0.213]
Multi-channel | 0.28 | [0.277, 0.285]
Table 2: PCC ($\uparrow$) and RMSE ($\downarrow$) of the single-channel baseline and multi-channel models on MOS labels using the simulated test set. The confidence intervals are obtained by bootstrapping with 1000 repetitions. Pearson correlation |  Original [11] | Simulated
---|---|---
after third-order mapping | MOSRA | data
In-distribution data
ReverbSpeechQuality[24] | 0.85 | 0.87
NISQA TEST LIVETALK[4] | 0.68 | 0.71
Out-of-distribution data
NISQA TEST FOR[4] | 0.87 | 0.53
NISQA TEST P501[4] | 0.89 | 0.60
Table 3: PCC ($\uparrow$) after third order mapping on NISQA and internal test
sets for the original MOSRA model trained on real human-labeled data vs. the
multi-channel model trained on our simulated data.

##  5 Discussion and Conclusion

Given that the architecture of our model allows for any length of audio input,
the model could be used to give near real-time predictions on multiple
channels using a circular buffer. Fig. 3 shows the model predictions for MOS
and DRR on three-channel audio where the speech is crossfaded between the
different input channels. The figure shows that the model can detect the
transitions of the speaker between channels and thus provides a proof of
concept for the use of quality-based audio stream selection. Besides, Fig. 2
underscores the limited correlation between DRR and the distance to speakers.
This observation emphasizes the drawback of distance as a criterion for
selecting the optimal device regarding audio quality, as it may overlook
crucial factors such as DRR and T60.

Our experimental findings confirm the advantages of a multi-channel model in
predicting acoustic parameters, surpassing the single-channel counterpart.
Notably, the multi-channel model improves prediction of STI, DRR, and C50,
demonstrating a clear advantage over the single-channel model while utilizing
5$\times$ less computational resources per channel. Lower T60 prediction
compared to the single-channel baseline might be caused by the model lacking
capacity, as it is predicting five channels in parallel with roughly the same
number of parameters and computational cost. However, we have not seen
improvement in multi-channel quality (MOS) prediction, speculating the
slightly worse performance against the single-channel version caused by
missing multi-channel human speech quality labels.

On the other hand, the experiment on generalization shows that simulated data
can allow for generalization to real data. Specifically, we observe that the
model performs well on the NISQA TEST LIVETALK and ReverbSpeechQuality
datasets but lacks performance on the two other NISQA datasets. Our conjecture
is that the observed performance degradation is due to a mismatch in data
distribution, as the high performance on the ReverbSpeechQuality could be
attributed to the fact that data is generated in the same fashion. In line
with previous findings on denoising [25], we emphasize that future work should
extend the types of degradations applied to speech to improve the
generalization capability of MOS estimation models.

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# Kibble-Zurek mechanism with time-averaged work

Pierre Nazé<EMAIL_ADDRESS>Departamento de Física, Instituto de
Geociências e Ciências Exatas, Universidade Estadual Paulista “Júlio de
Mesquita Filho”, 13506-900, Rio Claro, SP, Brazil

###### Abstract

Like many other thermally isolated systems performing an adiabatic driving
process, the quantum Ising chain presents an ill-defined decorrelation time.
On the other hand, Kibble-Zurek mechanism uses a heuristic relaxation time to
achieve its famous scaling. In previous work, a successful connection between
this relaxation time and the upper bound of the oscillatory decorrelation time
was made in the context of weak drivings. The problem however of having a
well-defined decorrelation time, derived from first principles, that describes
Kibble-Zurek mechanism is still open. A solution to this conundrum is proposed
here by measuring the work using the time-averaged relaxation function of the
system. This quantity offers a new and well-defined decorrelation time for
thermally isolated systems by turning them similarly into systems performing
an isothermal process. I recover with this the Kibble-Zurek mechanism in the
context of the finite-time and weak drivings, and new features in the context
of slowly-varying processes.

##  I Introduction

Out-of-equilibrium Thermodynamics naturally extended its equilibrium
counterpart to processes occurring at a finite time. Notions of how fast or
how slow is a process are now pertinent in the thermodynamic analysis, and
simple parameters that express such ideas become fundamental. For instance, in
the context of isothermal processes, such “velocity” is represented using the
ratio between two characteristic times: the natural decorrelation timescale of
the system and the inverse of the rate of the process [1]. In this manner,
fast processes occur faster than the relaxation of the process into
equilibrium, while the slower ones the opposite happens.

However, such a decorrelation timescale is not always well-defined in some
systems, which turns the thermodynamic analysis very difficult. This is what
happens for example to thermally isolated systems performing adiabatic driven
processes. For a variety of systems, such as the quantum harmonic oscillator
or Landau-Zener model, one can interpret such timescale as a random one [2].

The paradigmatic example of the quantum Ising model, very studied today by its
applicability in adiabatic quantum computing or quantum annealing [3], has
such characteristics. Even though, its phenomenology has been largely
elucidated over the years with the formulation of the Kibble-Zurek mechanism
[4, 5, 3]. In this description, it is used a heuristic relaxation time that
will dictate its non-equilibrium effects due to the quantum phase transition
that the system passes through when it crosses the critical point. The
following question is then established: how can one bring into line such
different aspects?

In a recent work [6], my co-workers and I have shown that, in the context of
finite-time and weak driving processes, the upper bound of the oscillatory
decorrelation timescale has the same diverging behavior at the critical point
assumed by the Kibble-Zurek mechanism. In another work [2], again in the same
context, I proposed a solution to capture a decorrelation timescale of
thermally isolated systems, where such quantity naturally appears if the time
average of the relaxation function is taken. In this work, I combine both
ideas of the two mentioned papers, taking the time average of the relaxation
function of the quantum Ising chain and expecting the same diverging behavior
of the correlation time close to the critical point.

The main results are the following: the time-averaging proceeding delivers
what was expected and, with it, I established clear regimes where the process
is fast and slow, called respectively finite-time and weak processes and
slowly-varying ones. In this manner, I verify that, in the regime where the
process is fast, the system behaves as predicted by Kibble-Zurek mechanism,
having the same impulse window and same scaling with the rate of the process
for the excess work calculated in the impulse part. Also, for the regime where
the process is slow, the system presents new features, with a fixed impulse
window and a new scaling, now calculated at the excess work in the adiabatic
part.

##  II Excess work in linear response theory

Consider a quantum system with a Hamiltonian $\mathcal{H}(\lambda(t))$, where
$\lambda(t)$ is a time-dependent external parameter. Initially, this system is
in contact with a heat bath of temperature $\beta\equiv{(k_{B}T)}^{-1}$, where
$k_{B}$ is Boltzmann’s constant. The system is then decoupled from the heat
bath and, during a switching time $\tau$, the external parameter is changed
from $\lambda_{0}$ to $\lambda_{0}+\delta\lambda$. The average work performed
on the system during this process is

$W\equiv\int_{0}^{\tau}\left\langle\partial_{\lambda}\mathcal{H}(t)\right\rangle\dot{\lambda}(t)dt,$
(1)

where $\partial_{\lambda}$ is the partial derivative for $\lambda$ and the
superscripted dot is the total time derivative. The generalized force
$\left\langle\partial_{\lambda}\mathcal{H}(t)\right\rangle$ is calculated
using the trace over the density matrix $\rho(t)$

$\left\langle A(t)\right\rangle=\mathrm{tr}\left\\{A\rho(t)\right\\}$ (2)

where $A$ is some observable. The density matrix $\rho(t)$ evolves according
to Liouville equation

$\dot{\rho}=\mathcal{L}\rho:=-\frac{1}{i\hbar}[\rho,\mathcal{H}],$ (3)

where $\mathcal{L}$ is the Liouville operator, $[\cdot,\cdot]$ is the
commutator and $\rho(0)=\rho_{c}$ is the initial canonical density matrix.
Consider also that the external parameter can be expressed as

$\lambda(t)=\lambda_{0}+g(t)\delta\lambda,$ (4)

where to satisfy the initial conditions of the external parameter, the
protocol $g(t)$ must satisfy the following boundary conditions

$g(0)=0,\quad g(\tau)=1.$ (5)

Linear response theory aims to express the average of some observable until
the first order of some perturbation considering how this perturbation affects
the observable and the non-equilibrium density matrix [7]. In our case, we
consider that the parameter considerably does not change during the process,
$|g(t)\delta\lambda/\lambda_{0}|\ll 1$, for all $t\in[0,\tau]$. Using the
framework of linear-response theory, the generalized force can be approximated
until the first-order as

$\begin{split}\left\langle\partial_{\lambda}\mathcal{H}(t)\right\rangle=&\,\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}+\delta\lambda\left\langle\partial_{\lambda\lambda}^{2}\mathcal{H}\right\rangle_{0}g(t)\\\
&-\delta\lambda\int_{0}^{t}\phi_{0}(t-t^{\prime})g(t^{\prime})dt^{\prime},\end{split}$
(6)

where the $\left\langle\cdot\right\rangle_{0}$ is the average over the initial
canonical density matrix. The quantity $\phi_{0}(t)$ is the so-called response
function, which can be conveniently expressed as the derivative of the
relaxation function $\Psi_{0}(t)$

$\phi_{0}(t)=-\frac{d\Psi_{0}}{dt},$ (7)

where

$\Psi_{0}(t)=\beta\langle\partial_{\lambda}\mathcal{H}(t)\partial_{\lambda}\mathcal{H}(0)\rangle_{0}+\mathcal{C}$
(8)

being the constant $\mathcal{C}$ calculated via the final value theorem [7].
In this manner, the generalized force, written in terms of the relaxation
function, is

$\begin{split}\left\langle\partial_{\lambda}\mathcal{H}(t)\right\rangle=&\,\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}-\delta\lambda\widetilde{\Psi}_{0}g(t)\\\
&+\delta\lambda\int_{0}^{t}\Psi_{0}(t-t^{\prime})\dot{g}(t^{\prime})dt^{\prime},\end{split}$
(9)

where
$\widetilde{\Psi}_{0}(t)\equiv\Psi_{0}(0)-\left\langle\partial_{\lambda\lambda}^{2}\mathcal{H}\right\rangle_{0}$.
Combining Eqs. (1) and (9), the average work performed at the linear response
of the generalized force is

$\begin{split}W=&\,\delta\lambda\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}-\frac{\delta\lambda^{2}}{2}\widetilde{\Psi}_{0}\\\
&+\delta\lambda^{2}\int_{0}^{\tau}\int_{0}^{t}\Psi_{0}(t-t^{\prime})\dot{g}(t^{\prime})\dot{g}(t)dt^{\prime}dt.\end{split}$
(10)

We remark that in thermally isolated systems, the work is separated into two
contributions: the quasistatic work $W_{\rm qs}$ and the excess work $W_{\rm
ex}$. We observe that only the double integral on Eq. (10) has “memory” of the
trajectory of $\lambda(t)$. Therefore the other terms are part of the
contribution of the quasistatic work. Thus, we can split them as

$W_{\rm
qs}=\delta\lambda\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}-\frac{\delta\lambda^{2}}{2}\widetilde{\Psi}_{0},$
(11)
$\begin{split}W_{\text{ex}}=\delta\lambda^{2}\int_{0}^{\tau}\int_{0}^{t}\Psi_{0}(t-t^{\prime})\dot{g}(t^{\prime})\dot{g}(t)dt^{\prime}dt.\end{split}$
(12)

In particular, the excess work can be rewritten using the symmetry property of
the relaxation function, $\Psi(t)=\Psi(-t)$ (see Ref. [7]),

$\begin{split}W_{\text{ex}}=\frac{\delta\lambda^{2}}{2}\int_{0}^{\tau}\int_{0}^{\tau}\Psi_{0}(t-t^{\prime})\dot{g}(t^{\prime})\dot{g}(t)dt^{\prime}dt.\end{split}$
(13)

We remark that such treatment can be applied to classic systems, by changing
the operators to functions, and the commutator by the Poisson bracket [7].

##  III Time-averaged excess work

Thermally isolated systems performing an adiabatic driven process can be
interpreted as having a random decorrelation time [2]. Therefore, at each
instant of time that the process is performed, the relaxation function changes
with it. This is very similar to what happens with systems performing an
isothermal process, where the stochastic aspect of the dynamics changes the
relaxation function. In this case, we take a stochastic average on the work to
correct such an effect. In the case of thermally isolated systems, I propose
as a solution the following time-averaging

$\overline{W}(\tau)=\frac{1}{\tau}\int_{0}^{\tau}W(t)dt.$ (14)

Such quantity can be measured in the laboratory considering an average in the
data set of processes executed in the following way: first, we choose a
switching time $\tau$. After, we randomly choose an initial condition from the
canonical ensemble and a time $t$ from a uniform distribution, where
$0<t<\tau$. Removing the heat bath, we perform the work by changing the
external parameter and collecting then its value at the end. The data set
produced will furnish, on average, the time-averaged work.

In the following, I present how time-averaged work can be calculated using
linear-response theory and how one can calculate the decorrelation time of the
system. To do so, we define the idea of time-averaged excess work

$\overline{W}_{\rm ex}=\frac{1}{\tau}\int_{0}^{\tau}W_{\rm ex}(t)dt,$ (15)

where $W=W_{\rm ex}+W_{\rm qs}$.

Now we observe how the time-averaged excess work can be calculated using
linear-response theory. In Ref. [6], I have shown that

$\overline{W}_{\text{ex}}(\tau)=\delta\lambda^{2}\int_{0}^{\tau}\int_{0}^{t}\overline{\Psi}_{0}(t-t^{\prime})\dot{g}(t)\dot{g}(t^{\prime})dtdt^{\prime},$
(16)

where

$\overline{\Psi}_{0}(t)=\frac{1}{t}\int_{0}^{t}\Psi_{0}(u)du,$ (17)

is the time-averaged relaxation function. This means that calculating the
time-averaged excess work is the same as calculating the averaged excess work,
but with a time-averaged relaxation function. Again, this is quite similar to
what happens to systems performing isothermal processes, where a stochastic
average is taken on the relaxation function.

Now, when measured with time-averaged work, the thermally isolated system
presents a decorrelation time. Indeed, the conditions such that linear-
response theory is compatible with the Second Law of Thermodynamics are [1]

$\widetilde{\overline{\Psi}}_{0}(0)<\infty,\quad\hat{\overline{\Psi}}_{0}(\omega)\geq
0.$ (18)

Therefore, analogously to what happens in an isothermal process, we define a
new decorrelation time

$\overline{\tau}_{c}:=\int_{0}^{\infty}\frac{\overline{\Psi}_{0}(t)}{\overline{\Psi}_{0}(0)}dt=\frac{\widetilde{\overline{\Psi}}_{0}(0)}{\overline{\Psi}_{0}(0)}<\infty.$
(19)

##  IV Kibble-Zurek mechanism

###  IV.1 Phenomenology

Consider the transverse-field quantum Ising chain, whose Hamiltonian is

$\mathcal{H}=-J\sum_{i=1}^{N}\sigma_{i}^{x}\sigma_{i+1}^{x}-\Gamma\sum_{i=1}^{N}\sigma_{i}^{z}.$
(20)

where each one of the $N$ spins has a vector
$\vec{\sigma}_{i}:=\sigma_{i}^{x}{\bf x}+\sigma_{i}^{y}{\bf
y}+\sigma_{i}^{z}{\bf z}$ composed by the Pauli matrices. The parameter $J$ is
the coupling energy and $\Gamma$ is the transverse magnetic field. We assume
for simplicity that $N$ is an even number, and the spins obey a periodic
boundary condition, $\hat{\vec{\sigma}}_{N+1}=\hat{\vec{\sigma}}_{1}$.

The Kibble-Zurek mechanism is a phenomenological theory that describes the
non-equilibrium dynamics of the transverse-field quantum Ising chain around
the critical point $\Gamma=J$. It predicts as well the scaling behavior of
observables in the driving rate $\tau$ when the system crosses the critical
point [4, 5, 3].

To have a better understanding of this theory, suppose that the magnetic field
$\Gamma$ is driven by a linear protocol

$\Gamma(t)=J\left|1-r(t)\right|,\quad r(t)=\frac{t}{\tau}.$ (21)

Figure 1 illustrates the Kibble-Zurek mechanism. When the system is far enough
from the critical point, the dynamics are adiabatic, and the excitations and
topological defects heal faster than when they are created. That particular
region is called adiabatic. However, when the system approaches the critical
point, the relaxation time of the system increases dramatically, and the
capacity for healing is lost. This phenomenon is manifested by the appearance
of finite-sized magnetic domains on the system. That particular region around
the critical point is called the impulse one. After the system crosses the
impulse region, it enters again into a new adiabatic region.

Figure 1: Illustration of the Kibble-Zurek mechanism. Far from the critical
point, the dynamics of the system are essentially adiabatic, meaning that the
system recovers from the defects of the driving faster than the inverse of the
driving rate. Close to the critical point, the situation changes dramatically.
The healing capacity is lost and there is a creation of finite-size magnetic
domains.

The instants when the system passes from adiabatic to impulse regime, denoted
by $\pm\hat{t}$, are defined as the times when the rate time $r(t)/\dot{r}(t)$
is equal to the rate of the relaxation time $\tau_{R}(t)$. The latter quantity
is defined as the time interval for a quantum system to decrease its energy by
one energy gap $\Delta$ when it is driven by some protocol at some particular
instant $t$

$\tau_{R}(t):=\frac{\hbar}{\Delta(t)},$ (22)

where $\hbar$ is Planck constant. In the particular case of the Hamiltonian
(20), in the thermodynamic limit, $N\rightarrow\infty$, the gap is

$\Delta(t):=2|J-\Gamma(t)|.$ (23)

Therefore, solving the equation

$r(\hat{t})/\dot{r}(\hat{t})=\tau_{R}(\hat{t}),$ (24)

the instants $\pm\hat{t}$ are

$\hat{t}=\pm\sqrt{\frac{\hbar\tau}{2J}},$ (25)

which depends on the driving rate by which the system crosses the critical
point.

###  IV.2 Kibble-Zurek scaling

After describing the phenomenology of the system when it crosses the critical
point, the Kibble-Zurek mechanism predicts how observables scale concerning
the driving rate when it crosses the critical point. It says that there is a
universal exponent $\gamma_{\rm KZ}$ that rules that phenomenon, which depends
on the equilibrium critical exponents of the system. In particular, for
systems of infinite size, the work $W_{\rm im}$ calculated in the impulse part
scales as

$W_{\rm im}\propto\tau^{-\gamma_{\rm KZ}},\quad\gamma_{\rm
KZ}=\frac{z\nu}{z\nu+1},$ (26)

where $z$ the dynamical critical exponent and $\nu$ the spatial critical
exponent. It was assumed that the contributions of excess work due to the
adiabatic regions are negligible. In particular, for the transverse-field
quantum Ising chain driven in the magnetic field, $z=1$ and $\nu=1$. Therefore

$\gamma_{\rm KZ}=1/2.$ (27)

###  IV.3 Weak drivings

In Ref. [6], my co-workers and I have shown that the relaxation function per
number of spins for the transverse-field quantum Ising chain is

$\Psi_{N}(t)=\frac{16}{N}\sum_{n=1}^{N/2}\frac{J^{2}}{\epsilon^{3}(n)}\sin^{2}{\left(\left(\frac{2n-1}{N}\right)\pi\right)}\cos{\left(\frac{2\epsilon(n)}{\hbar}t\right)},$
(28)

where

$\epsilon(n)=2\sqrt{J^{2}+\Gamma_{0}^{2}-2J\Gamma_{0}\cos{\left(\left(\frac{2n-1}{N}\right)\pi\right)}},$
(29)

being $\Gamma_{0}$ the initial value of the magnetic field. The time-averaged
relaxation function per number of spins will be

$\overline{\Psi}_{N}(t)=\frac{16}{N}\sum_{n=1}^{N/2}\frac{J^{2}}{\epsilon^{3}(n)}\sin^{2}{\left(\left(\frac{2n-1}{N}\right)\pi\right)}{\rm
sinc}{\left(\frac{2\epsilon(n)}{\hbar}t\right)},$ (30)

where

${\rm sinc}(x)=\frac{\sin{(x)}}{x}.$ (31)

##  V Time-averaged decorrelation time

Given the time-averaged relaxation function per number of spins (30), and
using Eq. (19), the time-averaged decorrelation time will be

$\overline{\tau}_{c}(\Gamma_{0})=\frac{\sum_{i=1}^{N/2}\frac{\pi\hbar}{\epsilon^{4}(n)}\sin^{2}{\left(\left(\frac{2n-1}{N}\right)\pi\right)}}{\sum_{i=1}^{N/2}\frac{4}{\epsilon^{3}(n)}\sin^{2}{\left(\left(\frac{2n-1}{N}\right)\pi\right)}},$
(32)

which is naturally measured in units of $\hbar/J$. For a large number of
spins, Fig. 2 shows that

$\overline{\tau}_{c}(\Gamma_{0})\propto\frac{\hbar}{|J-\Gamma_{0}|},$ (33)

which agrees with the heuristic of the Kibble-Zurek mechanism in respect of
its relaxation time. Therefore, we have now a decorrelation time derived from
first principles and agreeing with Kibble-Zurek mechanism.

Now one can identify the non-equilibrium regimes of the process. This is done
using the ratio between the decorrelation time and switching time, which
informs how fast the driving is performed, and the ratio
$\delta\Gamma/\Gamma_{0}$, which informs how strong the process is. One can
create a diagram of non-equilibrium regions illustrating that. See Fig. 3. In
region 1, the so-called finite-time and weak processes, the ratio
$\delta\Gamma/\Gamma_{0}\ll 1$, while $\overline{\tau}_{c}/\tau$ is arbitrary.
By contrast, in region 2, the so-called slowly-varying processes, the ratio
$\delta\Gamma/\Gamma_{0}$ is arbitrary, while $\overline{\tau}_{c}/\tau\ll 1$.
In region 3, the so-called arbitrarily far-from-equilibrium processes, both
ratios are arbitrary. Linear-response theory can calculate the time-averaged
excess works of regions 1 and 2 [8].

Indeed, for region 1, the time-averaged excess work per number of spins is
given by

$\overline{W}_{\text{ex}}^{1}(\tau)=\int_{0}^{\tau}\int_{0}^{t}\overline{\Psi}_{N}(t-t^{\prime})\dot{\Gamma}(t)\dot{\Gamma}(t^{\prime})dtdt^{\prime},$
(34)

For region 2, using the asymptotic approximation of decorrelation of the time-
averaged relaxation function per number of spins [8]

$\lim_{\overline{\tau}_{c}/\tau\ll
1}\overline{\Psi}_{N}(t)=2\overline{\tau_{c}}\overline{\Psi}_{N}(0)\delta(t),$
(35)

the time-averaged excess work per number of spins will be

$\overline{W}_{\rm
ex}^{2}(\tau)=\int_{0}^{\tau}\overline{\tau}_{c}[\Gamma(t)]\overline{\chi}_{N}[\Gamma(t)]\Gamma(t)^{2}dt,$
(36)

where

$\overline{\chi}_{N}[\Gamma_{0}]=\overline{\Psi}_{N}(0).$ (37)

An important observation is necessary to be made: the distinction between
systems of finite size and infinite size. In the first case, the regime of
region 2 is well-defined, while in the second one, only the one of region 1
exists. Indeed, Eq. (36) is only valid for $\tau\gg\tau_{c}(\Gamma(t))$, whose
highest value occurs at $\Gamma=J$. For systems of finite size, such value is
finite, while for infinite size, is not. Therefore, we can only find suitable
switching times for the regime of region 2 in the first case. Also, since
Kibble-Zurek mechanism describes systems of infinite size, every process is
been performed outside the slowly-varying processes regime, that is, is a fast
process. Also, as Ref. [6] has presented, the range of validity of numerical
precision of linear-response theory of the regime of region 1 is only valid
for systems of finite size and very small perturbations, although it predicts
the scaling behaviors. We assume that such behavior will happen here in the
time-averaged context. Finally, in our simulations, we are going to work only
with systems of finite size, which will allow us to detect the effects of
Kibble-Zurek mechanism in finite-time and weak processes and observe the new
features in the regime of slowly-varying processes.

Figure 2: Time-averaged decorrelation time according to Eq. (19). It presents
a good agreement with Kibble-Zurek mechanism prediction. It was used
$N=10^{5}$. Figure 3: Diagram of non-equilibrium regions. Region 1 corresponds
to finite-time and weak processes, region 2 to slowly-varying processes, and
region 3 to far-from-equilibrium processes. Linear-response theory can
describe regions 1 and 2.

##  VI Adiabatic and impulse regions

To evaluate if the time-averaged excess work per number of spins calculated in
the impulse region is much bigger than its adiabatic counterparts, we have to
calculate first the interval of time where this impulsive part occurs. To do
so, we evaluate Eq. (24) with our analogous quantities. In this case,
considering a linear driving, $\Gamma(t)=J-\Gamma_{0}+2\Gamma_{0}t/\tau$, we
need to solve

$\frac{\Gamma(t)-J}{\dot{\Gamma}(t)}=\left|\frac{\tau}{2}-t\right|=\overline{\tau}_{c}(\Gamma(t)).$
(38)

The graphic of $\hat{t}/\overline{\tau}_{c}$, plotted against
$\tau/\overline{\tau}_{c}$, is depicted in Fig. 4. It was used $N=10^{5}$ and
$\Gamma_{0}=0.5J$. As predicted by Kibble-Zurek mechanism, for
$\tau\leq\tau_{c}(\Gamma_{0})$, $\hat{t}=\sqrt{\hbar\tau/2J}$. Also, for
$\tau\gg\tau_{c}$, $\hat{t}$ achieves a plateau. This means that the window
for the impulse part remains with the same size, even though the duration of
the process becomes larger for slower rates.

Figure 4: Illustration of $\hat{t}/\overline{\tau}_{c}$ for different
$\tau/\overline{\tau}_{c}$. For $\tau\ll\overline{\tau}_{c}$,
$\hat{t}\propto\tau^{1/2}$, as predicted by Kibble-Zurek mechanism. For
$\tau\gg\overline{\tau}_{c}$, $\hat{t}$ achieves a plateau. It was used
$N=10^{5},\Gamma_{0}=0.5J$.

The next step is to calculate the impulse and adiabatic time-averaged excess
work per number of spins in both regimes. For finite-time and weak processes,
and $\tau\ll\overline{\tau}_{c}(\Gamma_{0})$, we have in the impulse part

$\overline{W}_{\rm
im}^{1}(\tau)=\int_{\tau/2-\sqrt{\hbar\tau/2J}}^{\tau/2+\sqrt{\hbar\tau/2J}}\int_{0}^{t}\overline{\Psi}_{N}(t-t^{\prime})\dot{\Gamma}(t)\dot{\Gamma}(t^{\prime})dtdt^{\prime},$
(39)

while its adiabatic counterpart is

$\overline{W}_{\rm ad}^{1}(\tau)=\overline{W}^{1}(\tau)-\overline{W}_{\rm
im}^{1}(\tau).$ (40)

For slowly-varying processes, and $\tau\gg\overline{\tau}_{c}(J)$, we have in
the impulse part

$\overline{W}_{\rm
im}^{2}(\tau)=\int_{\tau/2-c}^{\tau/2+c}\overline{\tau}_{c}(\Gamma(t))\overline{\chi}(\Gamma(t))\dot{\Gamma}(t)^{2}dt,$
(41)

while its adiabatic counterpart is

$\overline{W}_{\rm ad}^{2}(\tau)=\overline{W}^{2}(\tau)-\overline{W}_{\rm
im}^{2}(\tau).$ (42)

Here, the constant $c$ will be evaluated from the solution of Eq. (38).

From Fig. 4 it is possible to evaluate the proportion between the adiabatic
and impulse parts in regime 1, for $\tau\ll\overline{\tau}_{c}(\Gamma_{0})$,
$N=10^{5}$ and $\Gamma_{0}=0.5J$. For instance, for
$\tau=0.01\overline{\tau}_{c}$, $\hat{t}=0.1\overline{\tau}_{c}$, which
indicates that the whole driving occurs at the impulse region. Therefore, the
mentioned proportion is null. Indeed, as predicted by Kibble-Zurek mechanism,
the adiabatic part can be neglected without so much loss in the final result
of the time-averaged excess work. On the other hand, for
$\tau\gg\overline{\tau}_{c}$, the proportion diverges since the adiabatic part
goes to the quasistatic work and the impulse part to zero. In this situation,
is not more useful to calculate the impulse part, which is null, but the
adiabatic part only.

Figure 5: Scale $\tau^{-1/2}$ for time-averaged excess work per number of
spins in the impulse part for the regime of region 1. It agrees with the
prediction of Kibble-Zurek mechanism. It was used $N=10^{5}$ and
$\Gamma_{0}=0.5J$. Figure 6: Scale $\tau^{-1}$ for time-averaged excess work
per number of spins in the adiabatic part for the regime of region 2. It was
used $N=10^{5}$ and $\Gamma_{0}=0.5J$.

##  VII Kibble-Zurek scalings

Using a linear protocol, $\Gamma(t)=\Gamma_{0}+\delta\Gamma t/\tau$, we
explore the rate scalings of $\overline{W}_{\rm im}^{1}(\tau)$ and
$\overline{W}_{\rm ad}^{2}(\tau)$ respectively in the conditions of
$\tau\ll\overline{\tau}_{c}(\Gamma_{0})$ and $\tau\gg\overline{\tau}_{c}(J)$.
In this manner, for the first case, Fig. 5 depicts the scaling $\tau^{-1/2}$.
Again, such an effect is predicted by Kibble-Zurek mechanism. In the second
case, Fig. 6 depicts the scaling $\tau^{-1}$.

It is interesting to remark that, with our new framework about decorrelation
time and out-of-equilibrium regimes, the scale $\tau^{-1}$ measured in Ref.
[6] was made considering the impulse excess work measured in
$\tau\gtrsim\tau_{c}(J)$. Evaluating the scale in the same range of switching
times with our new results of $\hat{t}$, the scale presents a deviation to
$\tau^{-1.1}$, which shows that the assumption $\hat{t}=\sqrt{\hbar\tau/2J}$
made in the previous work is not so good for this case (see Fig. 4). Here, I
assumed that the normal and time-averaged cases should have the same scaling.

##  VIII Conclusion

Using the time-averaged relaxation function, I calculated a well-defined
decorrelation time for the quantum Ising chain, such that two different
regimes, the finite-time and weak processes and slowly-varying ones, were
plenty established. In the first regime, I found the effects predicted by the
Kibble-Zurek mechanism, while, in the second one, two interesting behaviors: a
$\hat{t}$ achieving a plateau and a time-averaged excess work calculated in
the adiabatic part of the process scaling with $\tau^{-1}$. These results show
the force of the new concept of measuring the work by taking the time average
of its relaxation function, as it was a typical example of a system performing
an isothermal process.

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