# Activity-Induced Annealing leads to Ductile-to-Brittle Transition in
Amorphous Solids

Rishabh Sharma  Smarajit Karmakar<EMAIL_ADDRESS>Tata Institute of
Fundamental Research, 36/P, Gopanpally Village, Serilingampally Mandal,Ranga
Reddy District, Hyderabad, 500046, Telangana, India

###### Abstract

Investigating the behavior of amorphous solids under various external loading
conditions continues to be an intriguing area of research with significant
practical implications. In this study, we demonstrate the utilization of self-
motility as a means to anneal glasses and use that as a means to fine-tune the
failure mode of the system under uniaxial tensile deformation. We begin by
highlighting the annealing effects of activity and draw parallels with other
well-known mechanical annealing processes, such as oscillatory shearing (both
uni- and multi-directional). Furthermore, we explore the annealing effects in
the presence of open boundaries, observing enhanced surface relaxations due to
activity. By implementing various activity-induced annealing protocols, we
successfully induce a transition in the failure mode from ductile to brittle.
This is demonstrated via performing tensile tests on the glass samples
resulting from the active-annealing process. The intricate effects of geometry
on the formation of shear bands are also examined. We find that samples having
an aspect ratio greater than one fail via shear band formation, owing to their
formation angle of $45\degree$ from the strain axis. In conclusion, we
introduce a novel method for producing well-annealed glasses in silico and
establish a correspondence between sheared and actively driven glasses.

##  I Introduction

Glasses are out-of-equilibrium materials formed when a supercooled liquid is
cooled below its (empirically defined) glass transition temperature ($T_{G}$),
at which point the liquid viscosity exceeds $10^{12}$ Pa.s [1, 2, 3, 4, 5, 6].
This enormous change in viscosity occurs over a small temperature change,
usually just a few tens of kelvins. The central problem of glass transition is
understanding how such a large dynamical change can occur in a system without
any significant structural change. Additionally, when the temperature is kept
below $T_{G}$, the relaxation time becomes so large that, for all practical
purposes, one observes an amorphous solid. The mechanical properties of such
solids are a topic of significant research interest due to their importance in
various practical applications and in future material design. Because of the
inherent non-equilibrium nature of glass formation, the material properties
depend not only on their composition but also on the preparation protocols.
Owing to the lack of long-range order, glasses do not suffer from defects like
grain boundaries, which are present in their crystalline counterparts and can
lead to unwanted structural and optical features. Constant efforts to alter or
tune the material’s mechanical properties as desired have led to the
establishment of various annealing methods. Such methods aim at achieving
better material properties, with the recent discovery of ultra-stable glasses
being a notable development [7].

The mechanical failure mode of glasses can be tuned in various ways. For
instance, confining geometry can change how glasses break. In recent
experimental and simulation works [8, 9, 10], it has been found that amorphous
solids can show very different mechanical failure behavior under uniaxial
tensile deformation at bulk and nano-scale. For example, a material that can
withstand only 2% strain at bulk before complete brittle-like failure can be
deformed up to 200% if the dimension of the sample is at a few hundred nano-
meter scales [8], with neck-like failure. The thickness of the necking region
can become as small as a few atomic diameters just before the total failure.
This drastic change in failure mode from brittle to ductile with changing
sample size when going from bulk to nano-scale highlights the importance of
surface relaxation in the failure process. In [10], the existence of a
critical aspect ratio (ratio of height to width in a two-dimensional sample or
ratio of length to cross-sectional linear dimension for a three-dimensional
sample) is observed. Below this critical aspect ratio, the material shows
neck-like ductile failure, which then crosses over to cavity-dominated
brittle-like failure for aspect ratios larger than the critical value.

Similarly, one can modify the interaction range of the constituent particles
to transition from brittle (smaller range of inter-particle interaction) to
ductile (larger range of inter-particle interaction) failure [11]. Introducing
impurities or inclusions having different mechanical properties than the
embedding matrix can also lead to changes in failure mode[12]. For example,
one could cross over from a heterogeneous shear band-mediated yielding
behavior to homogeneous ductile yielding with increasing appropriate
inclusions, as is done in various micro-alloying methods. The degree of
annealing or changing the cooling rate during preparation has a huge impact on
the resulting glass’ mechanical properties. For example, a glass that is
prepared through a rather high cooling rate will be poorly annealed. If such a
glass is subjected to simple shear, it shows a more ductile yield. In
contrast, a glass that is prepared at a slow cooling rate or produced via
other methods that can anneal the solids much better will generally be more
well-packed and have much larger elastic moduli. Unfortunately, such a solid
will also show catastrophic brittle failure via shear band formation when
subjected to deformations [13]. In general, the better-annealed a glass is,
the more “brittle” it tends to be.

The pursuit of creating increasingly well-annealed glasses has implications
not only for exploring the mechanical properties of these systems but also for
getting closer to the proposed “ideal glass.” This ideal glass can be
considered the most well-packed yet disordered state of matter, whose
configurational entropy, much like that of a crystal, is zero. Grasping these
concepts is crucial for probing the nature of the glass transition at a deeper
level. Recently, significant breakthroughs have been made in developing
various experimental techniques, such as Physical Vapour Deposition (PVD)[7],
and computational techniques like swap Monte Carlo [14] and in silico vapour
deposition [15], to achieve ever-lower energy states in glasses. Each of these
methods comes with its advantages and limitations. Thus, an exploration of
other generic methods which can be employed for better annealing a wide
variety of disordered solids is certainly a compelling avenue of research. In
this regard, in computer simulations, people have investigated different forms
of mechanical deformations that can lead to the annealing of glasses.
Oscillatory shearing is a particularly well-researched method among these
techniques[16, 17, 18].

In this paper, we introduce a novel form of mechanical annealing: annealing
glasses through local internal perturbations facilitated by active particles.
Active particles have motility that derives from either their internal energy
reserves or from external sources [19, 20, 21, 22]. Such motility in the
context of glasses has been helpful in understanding many biological systems
where dense cell packing and ATP-driven transport is the norm [23, 24, 25, 26,
27, 28]. In our observations, we find that the annealing and yielding behavior
of glasses subjected to active driving closely resembles that seen in glasses
annealed using oscillatory shearing[16]. This helps to strengthen the
correspondence between actively driven and sheared glasses which has been
observed before [29]. We alluded to such similarities in an earlier work
dealing with the formation of cavities in glasses [30]. There, it was found
that oscillatory shear cycles can lead to cavitation at a density that is much
larger than the density at which one expects to see cavitation failure via
only uniform expansion protocols. It was also shown that qualitatively similar
results could be obtained if these samples were subjected to local deformation
via active particles instead of oscillatory shear. In this current work, we
take this correspondence further by focusing on the annealing effects of
active dynamics. After establishing the annealing effects, we further
demonstrate how this process can be leveraged to tune the failure mode from
ductile to brittle under tensile loading conditions. These results suggest a
deeper connection between annealing via oscillatory shear and active
annealing.

The organization of this paper is as follows: Section II provides details on
the simulation and model. Section III describes the protocols employed to
generate the initial states, as well as the methods used for activity-induced
annealing and tensile testing. In Section IV, we present our findings, and we
conclude the paper in Section V with the summary and future directions.

##  II Model and simulation details

For our MD simulations, we used a well-known model glass former, the binary
Kob-Anderson (KA) mixture [31]. It involves 2 species of particles interacting
via Leonard-Jones potential (eq.1), with the larger A-type and smaller B-tyle
present in the ratio of 80:20.

$V_{\alpha\beta}(r_{ij})=4\epsilon_{\alpha\beta}\left[\left(\frac{\sigma_{\alpha\beta}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{\alpha\beta}}{r_{ij}}\right)^{6}\right]+u(r_{ij})$
(1)

The energy and length scales are chosen such that $\epsilon_{AA}=1$ and
$\sigma_{AA}=1$ . The interactions between the other combinations of particles
are then given in terms of $\epsilon_{AA}$ as $\epsilon_{AB}=1.5\epsilon_{AA}$
and $\epsilon_{BB}=0.5\epsilon_{AA}$. Similarly for length scales, we have
$\sigma_{AB}=0.8\sigma_{AA}$ and $\sigma_{BB}=0.88\sigma_{AA}$. The cut-off
range is taken to be $2.5\sigma_{ij}$, and the potential is made to go to zero
smoothly at this point (by choosing $u(r_{ij})$ accordingly so as to make the
slope continuous at the potential cutoff).

For imparting activity, we add an additional force $\vec{f_{0}}$ (eq. 2) to
the smaller B-type particles in addition to the potential derived force.

$\vec{f_{0}}=f_{0}(k_{x}\hat{x}+k_{y}\hat{y}+k_{z}\hat{z}).$ (2)

The force is added along the eight diagonal directions, and the directions are
shuffled after a persistence time $\tau_{p}$ (here taken to be $\tau_{p}=4$).
This results in run and tumble dynamics, and in literature, it is referred to
as the 8-state clock model [32]. Momentum conservation is ensured by
maintaining the sum of $k$’s to be zero component-wise throughout the
simulations. Nosé-Hoover thermostat and barostat were used to maintain the
desired temperatures and pressures.

The typical system size considered is $N=10,000$, with additional simulations
performed for $N=32,000$, $N=64,000$, and $N=128,000$ particles to study
systematic finite size effects. A very low temperature of $T=0.01$ is
maintained throughout activity-driven annealing and during tensile testing.
Annealing was done under periodic boundary conditions, and open boundaries
were created for tensile testing and surface relaxation studies. All the
simulations were conducted using our custom parallel MPI C codes.

##  III Annealing and tensile Testing protocols

Initial states: The states for annealing were prepared by cooling a liquid
equilibrated at a high temperature of $T=1.0$ and high density $\rho=N/V=1.2$
to a temperature of $0.01$. The cooling rates were varied from
$\dot{T}=10^{-1}$ to $\dot{T}=10^{-6}$. These resulted in states with inherent
state per particle energies ranging from $-6.910$ (poorly annealed) to
$-7.035$ (well annealed). An ensemble average of $16$ independent samples are
taken for each case.

Annealing protocol: The obtained states, having a range of inherent state
energies, are then subjected to the local perturbations via active dynamics.
The protocol involved imparting activity to all the B-type particles in the
system and evolving such a system for $10^{5}$ time units (or $2\times 10^{7}$
MD steps) under isothermal conditions at a low temperature of $T=0.01$. Thus,
the system was left to age in the presence of various magnitudes of active
forcing ($f_{0}$), and the resulting states were then used to perform tensile
testing.

Energy minimization protocol: To understand where in the energy landscape the
system is during the annealing process, we sampled states at equal intervals
from the isothermal aging trajectory and performed energy minimization using
the well-known conjugate gradient (CG) method. The final energies plotted in
Fig. 2(a) are the average energies of the last 5 frames from the tail of the
energy-minimized trajectories (Fig. 2(b)).

Figure 1: Schematic of the tensile testing procedure carried out in the study.

Tensile testing protocol: To perform tensile testing, we utilized the various
annealed states and subjected them to $200,000$ Molecular Dynamics (MD) steps
under zero pressure conditions. This step was crucial to prevent any pressure
shock when transitioning from periodic to open boundary conditions.
Subsequently, we created two walls of width $2.5\sigma_{AA}$ by freezing the
particles’ degrees of freedom along the two faces of the containing box (along
the x - direction). Open boundary conditions were created along the other two
directions. This configuration was run for an additional $200,000$ MD steps to
allow the surface to “settle in” before applying a constant strain rate,
denoted by $\dot{\gamma}$, to the walls in opposing and outward directions.
This application induced tension in the system, and we studied the system’s
response to this tension. The system was pulled till complete failure. The
schematic for the process is shown in Fig. 1.

Figure 2: Annealing and yielding in glasses via local perturbations due to
activity. Fig. (a) shows the effects of active dynamics (carried out for a
time window of $t_{w}$) on glasses having various inherent state energies. Up
to a threshold of $f_{0}=1.9$ for this model, we see enhanced aging behavior,
as seen by the system’s progression towards progressively deeper minima.
Beyond this threshold, the various curves collapse to a single curve, denoting
a yielded state where the system no longer retains a memory of its initial
preparation. A very similar transition from absorbing to yielded state is seen
in oscillatory shearing, with increasing oscillation amplitude. (b) Annealing
of glass having $E_{IS}=-6.947$ (corresponding to the green curve in (a)) is
shown. Decaying behavior for states till $f_{0}=1.9$ can be seen, whereas, for
larger forcing amplitude, we observe a yielded steady state. System size of
$N=10,000$ averaged over $16$ ensembles. Figure 3: The states encircled on the
highlighted curve in (a) are sampled for tensile testing; the green circles
mark the aging (or annealing) states, and red is used for the yielded state.
The reason for choosing this curve is to span the largest range of inherent
energies reached through active annealing. In (b), we show the stress-strain
curves obtained when these states are pulled after turning off the activity.
This sequence somewhat resembles a Physical Vapour Deposition (PVD) process
where highly motile particles -responsible for annealing the surface- lose
their mobility after being trapped in additional layers of particles. From the
stress response of these states, we see better-annealed states showing a
greater stress drop, indicating an enhancement in the brittleness after an
active dynamics annealing treatment. In (c), we perform the same testing, but
with the annealing activity kept on. Here too, the larger stress drops are
preserved. Here we also note the yielded state is unable to withstand any
considerable amount of stress. These results are for a system size of
$N=10,000$ averaged over 16 ensembles and strain rate $\dot{\gamma}=5\times
10^{-5}$.

##  IV Results

Fig. 2 encapsulates the outcome of our annealing protocol. We report a trend
that is remarkably similar to the one observed in [33], where the annealing is
performed using oscillatory shearing and under athermal conditions. Active
glasses appear to age faster, enabling them to reach lower points in their
potential energy landscape than their inactive counterparts. Up to a certain
threshold $f_{0}$, we observe annealing beyond which the energies again begin
to rise and collapse to a single curve. Drawing an analogy with the yielding
amplitude of oscillation ($\gamma_{max}$) seen in oscillatory shear
simulations, we term this the active yielding amplitude ($f_{0}{{}^{Y}}$). In
our case, we find $f_{0}{{}^{Y}}\simeq 1.9$. The various curves collapsing to
a single curve beyond $f_{0}{{}^{Y}}$ signifies that the system no longer
remembers its original preparation history and thus can no longer be in the
aging regime. This rise in saturation energy beyond yielding amplitude is
illustrated in Fig. 2(a).

One also observes the degree of annealing via activity decreases as one goes
to lower and lower initial state energies. Thus, poorly annealed glass
responds much better to annealing via activity as compared to very well-
annealed glasses. This trend, too, resonates with that reported in oscillatory
shearing. One way to understand this is to realize that poorly annealed
glasses have more “soft spots” or Shear Transformation Zones (STZ’s) [34, 35,
36] that are available to be triggered by the microscopic perturbations caused
by active forcing. A dearth of such soft spots in well-annealed glasses also
explains their low sensitivity to both active and oscillatory annealing.

One crucial difference to note is that under athermal oscillatory annealing,
the system below the yielding amplitude always reaches a limit cycle in the
form of an absorbing state, and this limiting energy makes for a natural
“stopping point.” Such an absorbing state transition has also been observed in
aging studies of topologically constrained active matter [37] (again, only in
the absence of temperature). In active annealing, no such absorbing states are
present, partly due to the dynamics being run at a finite, albeit low,
temperature. The energy below the yield point decays logarithmically, and thus
we take a finite time window instead to get a sense of the limiting energies.
The exact values in Fig. 2 will depend on this waiting period ($t_{w}$). Such
waiting time dependence is a hallmark of aging systems, and active glasses
have been shown to display complex aging behaviors [38]. Still, the overall
figure’s character remains consistent, with poorly annealed glasses exhibiting
superior annealing within a given window of time compared to well-annealed
ones.

To understand these states’ mechanical responses, we selected the states along
the top-most (least annealed) curve in Fig. 2(a) and conducted the tensile
testing simulations under constant strain rates, the results of which are
summarised in Fig. 3. In Fig. 3(b), we see the stress response with the
activity turned off after annealing. As expected, we find the annealing
effects being reflected in the stress-strain curves as well, with the highest
stress peak observed for the $f_{0}=1.9$ case. The $f_{0}=0$ curve shows no
overshoot in stress, a signature of the ductile yielding process. These stress
overshoots are preserved even if the pulling simulations are performed in the
presence of the activity, as shown in Fig. 3(c). One crucial difference one
observes in the stress-strain curve with and without the presence of activity
during the shearing process is the nature of the steady state reached by the
system at large strain. Without any active force, the system reaches a unique
steady state for all the configurations irrespective of their annealing
history, a strong signature of attaining ergodicity after yielding, a well-
known feature of passive glasses. In contrast, the steady-state stress
($\sigma_{\infty}$) of the system with activity on is very different.
$\sigma_{\infty}$ decreases with increasing activity, although stress
overshoot increases with increasing activity. We discuss this in further
detail in subsequent paragraphs. A state much above the yielding amplitude
shows nearly liquid-like behavior indicating fluidization due to the active
forces. Thus, somewhat counterintuitively, we see that introducing self-
motility can lead to a system favoring a brittle response over a ductile one
below a critical strength of self-motility.

Figure 4: Interplay between intrinsic time scales and imposed time scale (in
the form of strain rate) in well-annealed active glasses. A saturation to
various pseudo steady states depending on activity showing Herschel-Bulkley-
like signature. In the inset, we show the variation of the steady-state stress
$\sigma_{\infty}$ with different magnitudes of active forcing, $f_{0}$. System
size of $N=10,000$ averaged over $16$ ensembles. Figure 5: Analyzing the
effect of Aspect Ratio (AR) and system size on shear band formation: The
formation of shear bands is favored by both a higher aspect ratio and a larger
system size. The color bar denotes the magnitude of squared displacements. For
an aspect ratio less than 1, a shear band at 45°is geometrically impossible to
accommodate. For aspect ratios $\leq 1$, hindrance from the walls is
significant, and it goes down with larger AR. Thus the same system size
considered in (a), but with a higher AR, the system now does show a sharp
stress drop; see fig 7. Similarly, the larger system size used in (b), now
with decreased AR, ceases to form shear bands. See Fig 6.

###  IV.1 Effect of strain rate - faster intrinsic time-scales in active
glasses

To study the nature of steady states reached during tensile extension in
glasses with active driving present, we looked at the system’s response under
varying strain rates. For this, we took the initial state for testing to be
the best-annealed state having per particle inherent structure energy,
$E_{IS}=-7.035$, that we achieved via the slowest cooling employed in our
simulations. These states were not further subjected to any additional active
aging process. This strategy was adopted to separate any potential effects of
annealing imposed on the steady states, as we saw in Fig. 3(c). Regardless, as
seen from the bottom-most curve in Fig 2(a), an additional annealing treatment
for such well-annealed glasses would only have had a minute effect. In Fig. 4,
we show the stress-strain curve of the system by changing the strength of the
active forces for two different strain rates, $\dot{\gamma}=5\times 10^{-5}$
and $5\times 10^{-4}$. The inset of the respective panels shows how the steady
state stress, $\sigma_{\infty}$, decreases with increasing active force,
$f_{0}$. The qualitative nature of the results can be understood if we
consider that with increasing activity, the relaxation time of the system
decreases. One can effectively map this behavior to an effective temperature
description of a passive system [39]. In a passive system, if one maintains
the strain rate but increases the system’s bath temperature, then one expects
that the steady-state flow stress decreases [40, 41] as $\sigma_{\infty}\sim
T^{-2/3}$ across a wide variety of amorphous solids including metallic
glasses. Thus it seems that these results can be qualitatively understood
using an effective temperature description. In contrast, the annealing effect
can not be understood using the same effective temperature description as with
increasing the system’s effective temperatures; one should observe a reduction
in stress overshoot rather than an increase. This suggests that the mechanical
response of an active amorphous solid is rather complex, and it is not readily
understandable using the simple concept of effective temperature. Thus,
further studies along this direction will be needed to explore the full
complex rheological behavior of active glasses under various external loading
conditions.

As a side note, we want to highlight the effect of activity on the yield
strain itself. We observe that introducing activity during tensile testing
shifts the yield point towards a lower value, although the magnitude of the
stress drop remains fairly unaffected. This is completely different compared
to the effect of random pinning on the yield strain as reported in [12]. Thus
active particles can be considered as fluidizing or anti-pinning centers that
make the materials less rigid mechanically during deformation. However, it
helps to achieve stable mechanical states via annealing when it is allowed to
relax.

###  IV.2 Effect of geometry and larger system size

To see the catastrophic characteristic of a brittle failure in the stress-
strain curve, larger system size and lower inherent state energies are
required [13]. For this, system sizes of $N=32,000$ , $64,000$, and $128,000$
are considered. To study the effect of geometry and enhanced annealing due to
open surfaces, we changed the geometry from a cubical box to a cuboidal rod
shape [42]. This is done by changing the aspect ratio (defined as
$AR=\frac{L_{x}}{L_{y}=L_{z}}$, with x always being the long axis along which
the tension is applied) while keeping surface to volume ratio constant. We
have considered AR of $1$, $2,$ and $4$ respectively for the three system
sizes taken. This gives us an idea of the importance of system size, AR, and
absolute surface area during the deformation process. Here, in the original
protocol, we introduce an additional time window of $5\times 10^{6}$ MD steps.
During this period, we anneal the system in an open geometry while it is still
active - that is, after the periodic boundary conditions (PBC) have been
removed but before the activity is turned off. The importance of understanding
the surface dynamics of glassy systems has far-reaching effects on processes
like Physical Vapour Deposition (PVD). PVD utilizes the disparity in the
mobility of surface particles with orders of magnitude higher mobility than
the bulk particle to create ultra-stable glasses [7]. Thus the effects of
giving additional mobility are an intriguing avenue to explore.

In Fig. 5, we show the appearance of a shear band with increasing aspect ratio
and system size. In panel (a), we show that for a system size of $N=32,000$
particles with $AR=1$, there is no clear shear band, although, from the
overshoot in the stress-strain curve in panel (d), one clearly sees that a
well-annealed glass is achieved with $f_{0}=1.9$. In panel (b), the system
size is increased by two folds by increasing $L_{x}$. Once again, one clearly
sees a much sharper stress overshoot in a well-annealed sample prepared with
active force $f_{0}=1.9$, as well as a sharper shear band in the middle of the
sample at an angle of $45\degree$. With a further increase in system size by
increasing $L_{x}$ by a factor of $2$ again, the stress overshoot becomes
nearly discontinuous, as shown in panel (f), with a clear shear band seen in
panel (c).

Figure 6: Larger system size but with smaller AR does not form clean shear
bands.  Figure 7: Smaller system size but with higher AR, enough to resolve
the shear band.

To address the question of whether increasing system size or the increasing
aspect ratio plays a dominant role in forming shear bands during failure in
well-annealed glasses, we considered the following two scenarios. In the first
situation, we take a system size where we clearly observed shear bands for an
aspect ratio of 2 (panels b and e in Fig. 5) and decreased the aspect ratio to
1 as shown in Fig. 6. For an aspect ratio of $1$, even for a large system size
of $N=64,000$, the sharpness of the stress drop is decreased significantly
compared to when AR was $2$. A lack of a clear shear band in the system
accompanies this. This happens because, at this AR, the walls interfere with
the formation of the shear band, which happens at a $45\degree$ angle. In the
second scenario, we took a system size of $N=32,000$, which did not show a
shear band for an AR of $1$ (panel a and d in Fig. 5), but now increased the
aspect ratio to $4$, as shown in Fig. 7. In this case, one can see that the
stress overshoot becomes nearly discontinuous across the yielding transition,
and a clear shear band appears in the sample. This suggests that the geometric
shape of the sample, apart from its annealing history, will play a crucial
role in determining the eventual failure mode in amorphous solids with open
boundaries.

###  IV.3 Morphology of fractured states

Figure 8: Maximum relative elongation before complete failure. The sample in
(a) was treated with activity $f_{0}=1.9$, while the one in (b) was aged
without any activity. The aspect ratio is taken to be $AR=4$ with system size
$N=128,000$. The strain rate is $\dot{\gamma}=5\times 10^{-5}$. $\Gamma_{max}$
the maximum strain the system can withstand before complete failure. The data
is averaged over $3$ ensembles each.

The percentage elongation of material before fracture, as well as the cross-
section of fracture, both provide valuable insights into the failure
mechanism. A longer elongation of necking and a thinner cross-section are
reminiscent of a ductile failure. Here we show the morphology of the fractured
states for two of the samples considered. Sample (a) was treated with
activity, whereas sample (b) was just aged passively for the same duration of
time. We compute the fractional relative elongation as

$\Gamma_{max}=\frac{L_{x}^{f}-L_{x}^{0}}{L_{x}^{0}},$ (3)

where $L_{x}^{0}$ is the starting length of the sample in the pulling
direction, and $L_{x}^{f}$ is the length after the sample detaches into two
pieces, as shown in Fig. 8. We see the average relative elongation in our
samples (denoted by $\langle\Gamma_{max}\rangle$ ) after being aged with
activity is significantly (nearly 22% ) smaller than the passively aged case.
Thinner necks for the passive case are also observed. This result suggests
that activity induces brittleness in the materials. Note that this behavior is
completely different than what one expects if the effect of activity can be
simply understood as effective temperature. In passive amorphous solids, an
increase in temperature leads to more ductile failure. Some of these results
are very counter-intuitive, but if one looks at the system in terms of the
degree of annealing, then the observation is in agreement with the fact that
better-annealed solids will show more brittle type failure behavior.

##  V Conclusion and Perspectives

In conclusion, we have shown that active particles can effectively anneal
glasses to lower energies, just like the mechanical annealing process under
oscillatory shear. The similarities observed with the oscillatory shearing
suggest that thinking of activity as a form of local shearing rather than just
as an effective temperature might provide fruitful insights. By performing
extensive tensile testing, we also showed that the annealing effects of
activity could be used to change the mode of failure from a ductile to a
brittle type. This finding may seem counterintuitive if the activity is simply
viewed as a temperature-like phenomenon, considering that higher temperatures
generally favor ductile failure. However, it is crucial to recognize the
enhanced aging effect in active glasses, which, over time, can transform a
ductile active glass into a brittle one simply due to better annealing. Apart
from creating better materials, this work can have consequences for developing
techniques that are much better at annealing glasses than the usual ones; for
instance, a combination of cyclic shear with internal activity might prove to
be even better at annealing than any one of those individually. Some future
research directions could be to see the effect of slightly higher temperatures
on the annealing process. Is there an optimal combination of temperature and
activity for which a system would anneal the fastest, akin to an optimal
combination of shear amplitude and temperature reported in [43]? We also plan
to explore, in particular, the effect of persistence time in the annealing
process. Given the local shear interpretation of active forcing, it seems that
higher values of $\tau_{p}$ would mimic enhanced local random shearing,
whereas lower values should have temperature-like effects. Our preliminary
findings suggest the same. It would be intriguing to figure out the exact
range of $\tau_{p}$ over which one interpretation of activity should be
preferred over the other. Could there also be absorbing states if such a study
is done in the absence of temperature? This would open the possibilities for
encoding memory in such systems, similar to how memory can be encoded in
cyclically sheared glasses [44]. In addition, from an algorithmic perspective,
one would like to understand what aspect of activity helps to optimize the
search for minima in the complex glassy landscape. Is it the additional search
directions available during the persistence time, or is it the stochastic
nature of the activity? Answering such questions might help in developing
better optimization algorithms in the near future.

##  VI Acknowledgment

SK acknowledges fruitful discussions with Juergen Horbach during his visit to
The Heinrich Heine University, Dusseldorf, Germany. We acknowledge funding by
intramural funds at TIFR Hyderabad from the Department of Atomic Energy (DAE)
under Project Identification No. RTI 4007. Core Research Grant CRG/2019/005373
from Science and Engineering Research Board (SERB) is acknowledged for
generous funding. Most of the computations are done using the HPC clusters
bought using CRG/2019/005373 grant and Swarna Jayanti Fellowship, grants
DST/SJF/PSA01/2018-19, and SB/SFJ/2019-20/05 of SK.

## References

  * Arceri  _et al._ [2020] F. Arceri, F. P. Landes, L. Berthier, and G. Biroli, Glasses and aging: A Statistical Mechanics Perspective, arXiv 10.48550/arxiv.2006.09725 (2020), 2006.09725 .
  * Berthier and Biroli [2011] L. Berthier and G. Biroli, Theoretical perspective on the glass transition and amorphous materials, Reviews of Modern Physics 83, 587 (2011).
  * Karmakar  _et al._ [2014] S. Karmakar, C. Dasgupta, and S. Sastry, Growing length scales and their relation to timescales in glass-forming liquids, Annual Review of Condensed Matter Physics 5, 255 (2014).
  * Karmakar  _et al._ [2015] S. Karmakar, C. Dasgupta, and S. Sastry, Length scales in glass-forming liquids and related systems: a review, Reports on Progress in Physics 79, 016601 (2015).
  * Berthier and Ediger [2015] L. Berthier and M. D. Ediger, Facets of glass physics, arXiv 10.48550/arxiv.1512.03540 (2015), 1512.03540 .
  * Cavagna [2009] A. Cavagna, Supercooled liquids for pedestrians, Physics Reports 476, 51 (2009), 0903.4264 .
  * 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, Organic Glasses with Exceptional Thermodynamic and Kinetic Stability, Science 315, 353 (2007).
  * Luo  _et al._ [2010] J. H. Luo, F. F. Wu, J. Y. Huang, J. Q. Wang, and S. X. Mao, Superelongation and Atomic Chain Formation in Nanosized Metallic Glass, Physical Review Letters 104, 215503 (2010).
  * Volkert  _et al._ [2008] C. A. Volkert, A. Donohue, and F. Spaepen, Effect of sample size on deformation in amorphous metals, Journal of Applied Physics 103, 083539 (2008).
  * Paul  _et al._ [2020] K. Paul, R. Dasgupta, J. Horbach, and S. Karmakar, Cavity formation in deformed amorphous solids on the nanoscale, Physical Review Research 2, 042012 (2020).
  * Dauchot  _et al._ [2011] O. Dauchot, S. Karmakar, I. Procaccia, and J. Zylberg, Athermal brittle-to-ductile transition in amorphous solids, Physical Review E 84, 046105 (2011), 1105.2937 .
  * Bhowmik  _et al._ [2019] B. P. Bhowmik, P. Chaudhuri, and S. Karmakar, Effect of Pinning on the Yielding Transition of Amorphous Solids, Physical Review Letters 123, 185501 (2019), 1808.09723 .
  * Ozawa  _et al._ [2018] M. Ozawa, L. Berthier, G. Biroli, A. Rosso, and G. Tarjus, Random critical point separates brittle and ductile yielding transitions in amorphous materials, Proceedings of the National Academy of Sciences 115, 6656 (2018), 1803.11502 .
  * Berthier  _et al._ [2016] L. Berthier, D. Coslovich, A. Ninarello, and M. Ozawa, Equilibrium Sampling of Hard Spheres up to the Jamming Density and Beyond, Physical Review Letters 116, 238002 (2016), 1511.06182 .
  * Singh  _et al._ [2013] S. Singh, M. D. Ediger, and J. J. d. Pablo, Ultrastable glasses from in silico vapour deposition, Nature Materials 12, 139 (2013).
  * Leishangthem  _et al._ [2017] P. Leishangthem, A. D. S. Parmar, and S. Sastry, The yielding transition in amorphous solids under oscillatory shear deformation, Nature Communications 8, 14653 (2017), 1612.02629 .
  * Das  _et al._ [2022] P. Das, A. D. S. Parmar, and S. Sastry, Annealing glasses by cyclic shear deformation, The Journal of Chemical Physics 157, 044501 (2022).
  * Krishnan  _et al._ [2023] V. V. Krishnan, K. Ramola, and S. Karmakar, Annealing Effects of Multidirectional Oscillatory Shear in Model Glass Formers, Physical Review Applied 19, 024004 (2023), 2112.07412 .
  * Ramaswamy [2010] S. Ramaswamy, The Mechanics and Statistics of Active Matter, Annual Review of Condensed Matter Physics 1, 323 (2010), 1004.1933 .
  * Narayan  _et al._ [2007] V. Narayan, S. Ramaswamy, and N. Menon, Long-Lived Giant Number Fluctuations in a Swarming Granular Nematic, Science 317, 105 (2007), cond-mat/0612020 .
  * Wilson  _et al._ [2012] D. A. Wilson, R. J. M. Nolte, and J. C. M. v. Hest, Autonomous movement of platinum-loaded stomatocytes, Nature Chemistry 4, 268 (2012).
  * Walther and Muller [2013] A. Walther and A. H. E. Muller, Janus Particles: Synthesis, Self-Assembly, Physical Properties, and Applications, Chemical Reviews 113, 5194 (2013).
  * Marchetti  _et al._ [2013] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Reviews of Modern Physics 85, 1143 (2013).
  * Janssen [2019] L. M. C. Janssen, Active glasses, Journal of Physics: Condensed Matter 31, 503002 (2019), 1906.03678 .
  * Henkes  _et al._ [2011] S. Henkes, Y. Fily, and M. C. Marchetti, Active jamming: Self-propelled soft particles at high density, Physical Review E 84, 040301 (2011), 1107.4072 .
  * Bi  _et al._ [2014] D. Bi, J. H. Lopez, J. M. Schwarz, and M. L. Manning, Energy barriers and cell migration in densely packed tissues, Soft Matter 10, 1885 (2014), 1308.3891 .
  * Bi  _et al._ [2015] D. Bi, J. H. Lopez, J. M. Schwarz, and M. L. Manning, A density-independent rigidity transition in biological tissues, Nature Physics 11, 1074 (2015), 1409.0593 .
  * Nandi  _et al._ [2018] S. K. Nandi, R. Mandal, P. J. Bhuyan, C. Dasgupta, M. Rao, and N. S. Gov, A random first-order transition theory for an active glass, Proceedings of the National Academy of Sciences 115, 7688 (2018), 1605.06073 .
  * Morse  _et al._ [2021] P. K. Morse, S. Roy, E. Agoritsas, E. Stanifer, E. I. Corwin, and M. L. Manning, A direct link between active matter and sheared granular systems, Proceedings of the National Academy of Sciences 118, e2019909118 (2021), 2009.07706 .
  * [30] U. A. Dattani, R. Sharma, S. Karmakar, and P. Chaudhuri, [2303.04529] Cavitation instabilities in amorphous solids via secondary mechanical perturbations.
  * Kob and Andersen [1995] W. Kob and H. C. Andersen, Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture I: The van Hove correlation function, Physical Review E 51, 4626 (1995), cond-mat/9501102 .
  * Mandal  _et al._ [2016] R. Mandal, P. J. Bhuyan, M. Rao, and C. Dasgupta, Active fluidization in dense glassy systems, Soft Matter 12, 6268 (2016), 1412.1631 .
  * Bhaumik  _et al._ [2021] H. Bhaumik, G. Foffi, and S. Sastry, The role of annealing in determining the yielding behavior of glasses under cyclic shear deformation, Proceedings of the National Academy of Sciences 118, e2100227118 (2021).
  * Pan  _et al._ [2008] D. Pan, A. Inoue, T. Sakurai, and M. W. Chen, Experimental characterization of shear transformation zones for plastic flow of bulk metallic glasses, Proceedings of the National Academy of Sciences 105, 14769 (2008).
  * Langer [2008] J. S. Langer, Shear-transformation-zone theory of plastic deformation near the glass transition, Physical Review E 77, 021502 (2008), 0712.0399 .
  * Ding  _et al._ [2014] J. Ding, S. Patinet, M. L. Falk, Y. Cheng, and E. Ma, Soft spots and their structural signature in a metallic glass, Proceedings of the National Academy of Sciences 111, 14052 (2014).
  * Janssen  _et al._ [2017] L. M. C. Janssen, A. Kaiser, and H. Löwen, Aging and rejuvenation of active matter under topological constraints, Scientific Reports 7, 5667 (2017), 1611.03528 .
  * Mandal and Sollich [2020] R. Mandal and P. Sollich, Multiple Types of Aging in Active Glasses, Physical Review Letters 125, 218001 (2020), 1911.04558 .
  * Nandi and Gov [2018] S. K. Nandi and N. S. Gov, Effective temperature of active fluids and sheared soft glassy materials, The European Physical Journal E 41, 117 (2018).
  * Johnson and Samwer [2005] W. L. Johnson and K. Samwer, A Universal Criterion for Plastic Yielding of Metallic Glasses with a (T/Tg)2/3 Temperature Dependence, Physical Review Letters 95, 195501 (2005).
  * Dasgupta  _et al._ [2013] R. Dasgupta, A. Joy, H. G. E. Hentschel, and I. Procaccia, Derivation of the Johnson-Samwer T2/3 temperature dependence of the yield strain in metallic glasses, Physical Review B 87, 020101 (2013), 1210.7982 .
  * Shrivastav  _et al._ [2016] G. P. Shrivastav, P. Chaudhuri, and J. Horbach, Yielding of glass under shear: A directed percolation transition precedes shear-band formation, Physical Review E 94, 042605 (2016), 1506.03049 .
  * Das  _et al._ [2018] P. Das, A. D. S. Parmar, and S. Sastry, Annealing glasses by cyclic shear deformation, arXiv 10.48550/arxiv.1805.12476 (2018), 1805.12476 .
  * Fiocco  _et al._ [2014] D. Fiocco, G. Foffi, and S. Sastry, Encoding of Memory in Sheared Amorphous Solids, Physical Review Letters 112, 025702 (2014), 1312.3099 .


# On the Thermodynamics of Particles
Obeying Monotone Statistics

Fabio Ciolli  Dipartimento di Matematica e Informatica, Università della
Calabria, Via Pietro Bucci, Cubo 30B, I-87036 Rende, Italy
<EMAIL_ADDRESS>,  Francesco Fidaleo  Dipartimento di Matematica,
Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, I-00133
Roma, Italy<EMAIL_ADDRESS>and  Chiara Marullo  Dipartimento di
Matematica “Guido Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo
Moro 5, I-00185 Roma, Italy<EMAIL_ADDRESS>
###### Abstract.

The aim of the present paper is to provide a preliminary investigation of the
thermodynamics of particles obeying monotone statistics. To render the
potential physical applications realistic, we propose a modified scheme called
block-monotone, based on a partial order arising from the natural one on the
spectrum of a positive Hamiltonian with compact resolvent. The block-monotone
scheme is never comparable with the weak monotone one and is reduced to the
usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian
are non-degenerate. Through a detailed analysis of a model based on the
quantum harmonic oscillator, we can see that: (a) the computation of the
grand-partition function does not require the Gibbs correction factor $n!$
(connected with the indistinguishability of particles) in the various terms of
its expansion with respect to the activity; and (b) the decimation of terms
contributing to the grand-partition function leads to a kind of “exclusion
principle” analogous to the Pauli exclusion principle enjoined by Fermi
particles, which is more relevant in the high-density regime and becomes
negligible in the low-density regime, as expected.

###### Key words and phrases:

Monotone grand-canonical ensemble; thermodynamics of grand-canonical ensemble;
exclusion principle; high- and low-density regimes

###### 2010 Mathematics Subject Classification:

82B30, 58B34, 58J37, 46F10.

##  1\. Introduction

In recent years, the investigation of exotic models has significantly
increased in the hope of making some progress in solving long-standing
unsolved problems involved in the physics of complex models. In this regard,
we certainly must mention the question of providing a satisfactory
mathematical description of the quantum electrodynamics, which obtains
predictions via the renormalisation technique that are in surprisingly perfect
accordance with the experiments.

Along the same line, we can identify the models which aim to study and unify
the strong interactions (i.e., quantum chromodynamics) with the electroweak
ones. All these models are called “standard” models and have the same
strengths and weaknesses. That is, they are in good accordance with the
experiments but not satisfactory from the mathematical point of view. The
long-standing problem of unifying these three fundamental forces, which are
present in nature, with the remaining one, that is, the gravitation, which was
recently addressed through the use of the so-called noncommutative geometry
(e.g., [1]), is very far from being solved, even in a partial form.

We should also mention the potential relevance of the investigation of models
enjoying exotic commutation relations with some other disciplines, such as
information theory and quantum computing, both of which are connected with,
and relevant for, concrete applications.

Among such models, we can certainly find those associated with the so called
$q$-particles, or quons, $q\in(-1,0)\bigcup(0,1)$, from the perspective of the
extension to the so-called anyons, corresponding to the case when the
parameter $q$ assumes values in some roots of the unity, and plektons.

Such exotic $q$-particles are naturally associated with the following
commutation relations:

(1) ${\bf a}_{q}(f){\bf a}_{q}^{\dagger}(g)-q{\bf a}_{q}^{\dagger}(g){\bf
a}(f)=\langle g,f\rangle I_{\mathscr{H}}\,,\quad f,g\in{\mathscr{H}}\,,$

${\mathscr{H}}$ being the one-particle space, where the creators and
annihilators act on the corresponding Fock spaces. The quons can certainly be
seen as an interpolation between the particles obeying the Fermi statistics
(i.e., $q=-1$) and those obeying the Bose statistics (i.e., $q=1$), passing
through the $q=0$ value describing the classical particles and, thus, obeying
the Boltzmann statistics. We can observe that for $q=\pm 1$, the commutation
rules (1) are reduced to the well-known ones associated with the Bose and
Fermi particles, respectively (see, e.g., [2]).

All such models are relevant to the so-called quantum probability. In fact,
the Boltzmann case $q=0$, describing the statistics of classical particles
concerning the physical meaning, is also known as free because it naturally
arises from a particular case of quantum probability called free probability
(see, e.g., [3]).

In the setting of quantum probability, the various generalisations of the
commutation rules (1) allow one to introduce and investigate exotic quantum
stochastic processes (see, e.g., [4]).

As is well-known in the Bose and Fermi cases, all the above-mentioned
commutation rules naturally arise from the so called second quantisation,
which is associated with the so-called grand-canonical ensemble. The various
functors of the second quantisation allow one to construct the corresponding
Fock spaces. The Fock space encodes the statistics that the involved particles
obey and allows the involved commutation relations to be faithfully
represented.

We can also remark that the grand-canonical ensemble allows one to investigate
one of the most fascinating phenomena occurring in the condensate state of the
matter, involving bosons, that is, the Bose–Einstein Condensation in the
fundamental state (see, e.g., [2, 5]). In [6], it is shown that such a
phenomenon of condensation also appears in the case of Bose-like quons, that
is, when $q\in(0,1]$.

Returning to exotic commutation relations and their applications in quantum
probability, we can cite the Boolean and Monotone ones. As in the case of all
the models mentioned above, they satisfy commutation relations falling into
the general form described in [7], (Corollary 3.2), and are therefore
associated with a suitable Fock space. The Boolean Fock space is the simplest
non-trivial example of second quantisation, because only one particle can be
created and/or annihilated. In fact, it describes the absorption of a single
photon, at most, from an apparatus (see [8]).

The monotone statistics of particles, independently introduced in [9, 10], do
not seem to have any evident physical application. The arising Fock space is
easily constructed, as described in Section 2, using the monotonic
prescription induced by a totally ordered orthonormal basis of the one-
particle Hilbert space.

Now, we can also point out the well-known deep connection between the second
quantisation scheme and the equilibrium statistical mechanics (see, e.g., [2,
5]). Indeed, starting from a Fock space constructed by taking into account the
statistics of the involved particles, we can compute, at least in principle,
the so-called grand-partition function. Such a crucial function is supposed to
encode all the thermodynamic properties enjoyed by a large number (of the
order of the Avogadro number $N_{\rm A}\sim 10^{23}$) of involved particles.
This is certainly true for the Bose and Fermi cases and also, due to its
simplicity, for the boolean one, in which the indistinguishability of the
particles plays no role.

Due to the Gibbs paradox (cf. [5]), the computation of the grand-partition
function in the Boltzmann case, in terms of the associated full Fock space,
deserves a suitable correction due to the supposed indistinguishability of the
involved particles (see below). The general case of the quons (i.e.,
$q\in(-1,0)\cup(0,1)$) is differently solved in [11], since the necessarily
“deformed” statistics that such exotic particles obey are completely unknown.

The case of non-interacting particles obeying monotone statistics, simply
called monotone particles in the following pages, is unclear for two reasons.
The first one is that the concrete physical applications of such a model are
completely unknown. The second one is that we do not know whether the monotone
scheme directly encodes the principle of the indistinguishability of
particles, the latter being a fundamental prescription for the development of
equilibrium statistical mechanics.

Taking into account all the previous considerations, it becomes natural to
address the investigation of the thermodynamic properties of particles obeying
the monotone prescription, which are encoded in the monotone Fock space.
Unfortunately, since there is no natural, total order of the one-particle
subspace on an orthonormal basis, this investigation deserves an appropriate
preliminary analysis.

A simple one-particle physical system confined in a finite volume is
essentially described by a Hamiltonian $H$, which is a self-adjoint positive
operator with compact resolvent acting on a separable Hilbert space
${\mathscr{H}}$. The statistics (i.e., Bose/Fermi or Boltzmann) of very large
systems formed of a number of the order of the Avogadro number of non-
interacting particles is encoded in the corresponding Fock space.

Since there is a natural order of the eigenvectors of $H$ induced by the
corresponding eigenvalues (i.e., the energy levels of the system under
consideration), one is tempted to use such an order to implement the monotone
scheme for models of statistical mechanics. This can be carried out only when
the eigenvalues of $H$ all have a multiplicity of 1 or, in simple terms, when
any energy level of the model is non-degenerate.

Unfortunately, this is not the case for all concrete models when the
degeneracy of all the energy levels increases to infinity in the thermodynamic
limit (i.e., when, in particular, the volume of the system tends to occupy the
whole environment), in which case the so-called “passage to the continuum” can
be performed (see, e.g., [5, 12]).

The passage to the continuum is the fundamental tool used to investigate the
thermodynamic properties of more realistic models in which the one-particle
Hamiltonian has a continuum spectrum, such as a free particle living in
${\mathbb{R}}^{3}$.

In the present paper, we propose a method that can be used to overcome this
basic difficulty and, thus, take into account the possible degeneracy of the
energy levels. Indeed, we simply generalise the monotone prescription to index
sets, which are merely partially ordered according to those arising from the
spectrum of a positive, compact resolvent Hamiltonian with possible degenerate
energy levels. This model, called block-monotone in the following pages, which
is expected to be more suitable for physical applications, is described in
Section 3.

Since the grand-partition function of a system associated with
(block-)monotone particles is not directly computable in most infinite
dimensional cases, the remaining part of the paper is devoted to a detailed
analysis of a simple model formed of infinite uncoupled quantum harmonic
oscillators. Since the corresponding Hamiltonian has non-degenerate
eigenvalues, such a computation falls into the monotone scheme. The grand-
partition function for such a model is computed in Section 4.

Section 5 is then devoted to the explicit computation of the statistical
weights appearing in the expansion of the monotone grand-canonical partition
function and to a refined study of the high- and low-density regimes. Such an
investigation leads to the following relevant facts.

First of all, the computation of such statistical weights suggests that the
Gibbs correction factor $n!$, connected with the indistinguishability of
particles in the various terms of its expansion with respect to the activity,
directly appears in the low-density regime, that is, when the temperature of
the system becomes increasingly higher. This suggests that the monotone scheme
directly encodes the indistinguishability of the involved particles.

Secondly, the decimation of the terms contributing to the grand-partition
function provides a kind of “exclusion principle” analogous to the Pauli
exclusion principle observed for Fermi particles. Such an exclusion principle
appears to be more relevant to the high-density regime and becomes negligible
in the low-density regime, as expected.

The last part of Section 5 is devoted to a refined comparison between the
grand-partition functions relative to the Boltzmann and monotone models,
allowing us to estimate the correction of the relevant thermodynamic
quantities, such as the average number, in the low-density regime.

To conclude the present introduction, we point out that this preliminary
investigation seems to provide a promising perspective concerning the
potential physical applications of the block-monotone scheme, which we plan to
return to in a future work.

##  2\. Preliminaries

One-particle Hamiltonian. We start with a system whose Hamiltonian $H$ is a
self-adjoint positive (i.e., $\sigma(H)\subset[0,+\infty)$) operator with
compact resolvent, acting on a separable Hilbert space ${\mathscr{H}}$, called
the one-particle space.

In such a situation, the spectrum $\sigma(H)$ is formed of isolated points,
accumulating at $+\infty$ if ${\mathscr{H}}$ is infinite dimensional. In
addition, the multiplicity $g(\varepsilon)$ of each eigenvalue
$\varepsilon\in\sigma(H)$ is finite. In summary, by considering the resolution
of the identity of $H$, we obtain $I\equiv
I_{\mathscr{H}}=\sum_{\varepsilon\in\sigma(H)}P_{\varepsilon}$,

$H=\sum_{\varepsilon\in\sigma(H)}\varepsilon
P_{\varepsilon},\,\,\text{and}\,\,g(\varepsilon)=\text{dim}\big{(}\text{Ran}(P_{\varepsilon})\big{)}<\infty\,.$

Let $k_{\rm B}\approx 1.3806488\times 10^{-23}\,JK^{-1}$ be the Boltzmann
constant, and $\beta:=\frac{1}{k_{B}T}$ the “inverse temperature”. Assuming
that $e^{-\beta H}$ is trace class for each $\beta>0$, we can define the
partition function $\zeta:=\mathop{\rm{Tr}}(e^{-\beta H})$.

The grand-partition function. Here, we define the the grand-partition function
in a relatively general framework relative to a gas comprising non-interacting
particles obeying rather general statistics and thus potentially suitable for
physical applications. The knowledge of such grand-partition functions plays a
crucial role in the so-called equilibrium statistical mechanics. The standard
method for such an analysis is the so-called second quantisation, (see, e.g.,
[2, 5]).

Indeed, for the one-particle Hilbert space ${\mathscr{H}}$, we define the so-
called full Fock space ${\mathscr{F}}_{0}({\mathscr{H}})\equiv{\mathscr{F}}$,
given by

${\mathscr{F}}:=\bigoplus_{n=0}^{+\infty}\underbrace{{\mathscr{H}}\otimes\cdots\otimes{\mathscr{H}}}_{\textrm{n-times}}\,,$

with the convention that
$\underbrace{{\mathscr{H}}\otimes\cdots\otimes{\mathscr{H}}}_{\textrm{0-times}}:={\mathbb{C}}\equiv{\mathbb{C}}\Omega$,
where $\Omega$ is the so-called vacuum vector. The number operator $N$ has a
clear meaning (see e.g., [2]).

For a linear operator $A$ with domain ${\mathscr{D}}\subset{\mathscr{H}}$, we
define

$\displaystyle\text{d}\Gamma_{o}(A)\lceil_{{\mathscr{D}}\otimes\cdots\otimes{\mathscr{D}}}:=$
$\displaystyle A\otimes I\otimes\cdots\otimes I+I\otimes A\otimes\cdots\otimes
I$ $\displaystyle+\cdots+$ $\displaystyle I\otimes\cdots\otimes A\otimes
I+I\otimes\cdots\otimes I\otimes A\,,$

and extend it to the whole Fock space by linearity. If $A$ is self-adjoint,
the closure $\text{d}\Gamma(A)$ of $\text{d}\Gamma_{o}(A)$ will still be self-
adjoint (see, e.g., [2]). Note that $\text{d}\Gamma(I_{\mathscr{H}})$ provides
the number operator.

Now, let $P$ be a self-adjoint projection acting on ${\mathscr{F}}$. For a
fixed positive operator $H$ (i.e., a Hamiltonian) and the parameters $\beta>0$
(the inverse temperature) and $\mu\in{\mathbb{R}}$ (the chemical potential),
such that $Pe^{-\beta\text{d}\Gamma(H-\mu I)}P\equiv
Pe^{-\beta(\text{d}\Gamma(H)-\mu N)}P$ is trace class, we define the grand-
partition function as

(2) $Z_{P}\equiv
Z_{H,P}(\beta,\mu):=\textrm{Tr}\big{(}Pe^{-\beta\text{d}\Gamma(H-\mu
I)}P\big{)}\,.$

The most important cases, describing the thermodynamics of Bose and Fermi
gases, are those when $P$ is the self-adjoint projections onto the completely
symmetric and antisymmetric subspaces (with respect to the natural action of
the permutations on ${\mathscr{F}}$), respectively.

When $P$ is the identity operator $I\equiv I_{\mathscr{F}}$, the corresponding
grand-partition function will describe the thermodynamics of classical
particles, that is, those obeying the Boltzmann statistics. Unfortunately,
this is not the case, as explained in [11, 13]. Below, we outline how it is
possible to recover such a grand-partition function in the Boltzmann case.

We can also remark that the $q$-deformed Fock space
${\mathscr{F}}_{q}({\mathscr{H}})$ (e.g., [4] and [14] for the arising ergodic
properties) could be used to compute the grand-partition function for the so-
called quons and, thus, their thermodynamics. Unfortunately, in this case,
too, the second quantisation method does not work. The grand-partition
function for the free gas of quons is entirely computed in [11] without using
the $q$-deformed Fock space.

The so-called Boolean (e.g., [15]) and monotone (see below) models might also
be described as outlined above. This is certainly true for the Boolean case,
in which ${\mathscr{F}}_{\rm
boole}({\mathscr{H}})={\mathbb{C}}\bigoplus{\mathscr{H}}$ and, thus, there is
no question about the indistinguishability of the particles. We will see, in
the forthcoming analysis, that this is also the case for the monotone model
and its generalisations addressed in the present paper.

Monotone Fock space. For the reader’s convenience, we outline some basic facts
regarding monotone Fock spaces and their fundamental operators (see [9, 10,
16] for more details). Here, we define a generalisation of the monotone
particles which is suitable for physical applications. However, as a
particular case arising directly from quantum physics, we study, in some
detail, the thermodynamics of a simple model satisfying the monotone
statistics/commutation relations. For the interested reader, we point out the
existence of new investigations concerning exotic commutation relations that
are related to those previously mentioned and might have potential physical
applications (see [17]).

For $k\geq 1$, denoted by
$I_{k}:=\\{(i_{1},i_{2},\ldots,i_{k})|i_{1}<i_{2}<\cdots<i_{k},\,i_{j}\in\mathbb{N},\,j=1,\dots,k\\}$,
the class of all the ordered sequences of natural numbers are of length $k$.
For $k=0$, we take $I_{0}:=\\{\emptyset\\}$. If $k\geq 0$, the Hilbert space
${\mathscr{H}}_{k}:=l^{2}(I_{k})$ is called the $k$-particles space. In
particular, the $0$-particle space ${\mathscr{H}}_{0}=l^{2}(\emptyset)$ is
identified with the complex scalar field $\mathbb{C}$. The monotone Fock space
is then defined as ${\mathscr{F}}_{\rm
m}:=\bigoplus_{k=0}^{\infty}{\mathscr{H}}_{k}$.

With any increasing sequence $\alpha=\\{i_{1},i_{2},\ldots,i_{k}\\}$ of
natural numbers, we canonically associate the vector $e_{\alpha}$, which, for
all such sequences $\alpha$, provides the canonical basis of
${\mathscr{F}}_{\rm m}$. For each pair of such sequences
$\alpha=\\{i_{1},i_{2},\ldots,i_{k}\\}$,
$\beta=\\{j_{1},j_{2},\ldots,j_{l}\\}$, we state that $\alpha<\beta$ if
$i_{k}<j_{1}$. By convention, $I_{0}<\alpha$ for each $\alpha\neq I_{0}$.

In other words, if $\\{e_{n}\mid n\in{\mathbb{N}}\\}$ is the canonically
ordered basis of $\ell^{2}({\mathbb{N}})$ (or any ordered basis of a separable
Hilbert space), the $n$-particle space is generated by the vectors
$e_{\alpha}\equiv e_{j_{1}}\otimes e_{j_{2}}\otimes\cdots\otimes e_{j_{n}}$
whenever $\alpha=\\{j_{1},j_{2},\dots,j_{n}\\}$ with
$j_{1}<j_{2}<\cdots<j_{n}$. If we relax the last condition by merely assuming
that $j_{1}\leq j_{2}\leq\cdots\leq j_{n}$, we will obtain the so-called
weakly monotone Fock space (see, e.g., [16]). Note that ${\mathscr{F}}_{\rm
m}$ and ${\mathscr{F}}_{\rm wm}$ are the range of the self-adjoint projections
$P_{\rm m}$ and $P_{\rm wm}$ acting on the full Fock space ${\mathscr{F}}$:

${\mathscr{F}}_{\rm m}=P_{\rm
m}{\mathscr{F}}\,\,\,\text{and}\,\,\,{\mathscr{F}}_{\rm wm}=P_{\rm
wm}{\mathscr{F}}\,.$

Even if this is not used in the forthcoming analysis, we can report the
structure of the monotone creation and annihilation operators and the
generating commutation relations, which are nevertheless useful for
application to quantum probability. Indeed, the monotone creation and
annihilation operators are, respectively, for any $i\in\mathbb{Z}$, given by

(3)
$a^{\dagger}_{i}(e_{(i_{1},i_{2},\ldots,i_{k})}):=\left\\{\begin{array}[]{ll}e_{(i,i_{1},i_{2},\ldots,i_{k})}&\text{if}\,\\{i\\}<\\{i_{1},i_{2},\ldots,i_{k}\\}\,,\\\
0&\text{}\\\ \end{array}\right.$

otherwise,

(4)
$a_{i}(e_{(i_{1},i_{2},\ldots,i_{k})}):=\left\\{\begin{array}[]{ll}e_{(i_{2},\ldots,i_{k})}&\text{if}\,k\geq
1\,\,\,\,\,\,\text{and}\,\,\,\,\,\,i=i_{1}\,.\\\ 0&\text{}\\\
\end{array}\right.$

One can verify that $||a^{\dagger}_{i}||=||a_{i}||=1$ (see e.g., [16],
Proposition 8). Moreover, $a^{\dagger}_{i}$ and $a_{i}$ are mutually adjoint
and satisfy the following relations:

$\begin{array}[]{ll}a^{\dagger}_{i}a^{\dagger}_{j}=a_{j}a_{i}=0&\text{if}\,\,i\geq
j\,,\\\ a_{i}a^{\dagger}_{j}=0&\text{if}\,\,i\neq j\,.\end{array}$

In addition, the following commutation relation, designated in the weak
operator topology of ${\mathscr{B}}({\mathscr{F}}_{\rm m})$ and falling into
the general class of commutation relations managed in Corollary 3.2 of [7] for
applications in quantum probability,

$a_{i}a^{\dagger}_{i}={\bf 1}{{\rm I}}-\sum_{k\leq i}a^{\dagger}_{k}a_{k}$

is also satisfied.

For some properties of monotone systems, including their ergodic properties,
see, e.g., [18] and the literature cited therein.

The grand-partition function for the Boltzmann case. The Boltzmann (or
classical) case is very particular because, in Boltzmann statistics, the Gibbs
paradox (e.g., [5]) takes place and, consequently, we should suitably correct
the statistical weights.

As for the computation of $\mathop{\rm{Z}}_{\pm 1}$ in the Bose and Fermi
cases (e.g., [2, 5]), it might be natural to use the full Fock space
${\mathscr{F}}({\mathscr{H}})$ and the _grand-canonical Hamiltonian_
$K:=\textrm{d}\Gamma(H)-\mu N$, as explained above, provided that $K$ is trace
class for a fixed $\beta$ and $\mu$. This corresponds to taking
$P=I_{\mathscr{F}}$ in (2).

This easy computation is reported in [13], obtaining

$\textrm{Tr}\left(e^{-\beta K}\right)=\frac{1}{1-\zeta e^{\beta\mu}}\,,$

which still holds for $\mu<\min\sigma(H)$.

For the reasons explained in [11, 13], such a formula is unrealistic. However,
the correct formula should be $Z_{0}=e^{\zeta e^{\beta\mu}}$.

After defining the _fugacity_ , also denoted as the _activity_ , through
$z:=e^{\beta\mu}$, we have:

(5) $\textrm{Tr}\left(e^{-\beta
K}\right)=\sum_{n=0}^{+\infty}\zeta^{n}z^{n}\,,\quad 0\leq z<1\,.$

It is interesting to see that, if one corrects (5) with the weight $n!$ in the
denominator of the series, thus taking into account the indistinguishability
of particles, as it is customary to avoid the Gibbs paradox, we obtain the
correct formula:

(6) $Z_{0}=\sum_{n=0}^{+\infty}\frac{\zeta^{n}}{n!}z^{n}=e^{\zeta z}\,.$

Harmonic oscillator. Since we provide a detailed study of the simple model
formed of non-interacting harmonic oscillators, for the reader’s convenience,
we report some basic facts that are used in the following analysis.
Consequently, relative to the Boltzmann statistics, we compute the relative
grand-partition function at the inverse temperature $\beta$ and activity
$z=e^{\beta\mu}$, $\mu$ being the chemical potential.

Indeed, given that $K$ acts as the Hook strength of the spring and $m$ as the
mass of the involved particle, it is well-known that the spectrum of the
Hamiltonian of this model consists of non-degenerate eigenvalues, given by:

(7) $\sigma(H)=\Big{\\{}\hbar\omega(n+1/2)\mid n\in{\mathbb{N}}\Big{\\}}\,,$

where $\omega:=\sqrt{K/m}$ is the given frequency.

In this way, the partition function is given by:

$\zeta\equiv\textrm{Tr}\big{(}e^{-\beta
H}\big{)}=e^{-\frac{\beta\hbar\omega}{2}}\sum_{n=0}^{+\infty}\big{(}e^{-\beta\hbar\omega}\big{)}^{n}=\frac{e^{-\frac{\beta\hbar\omega}{2}}}{1-e^{-\beta\hbar\omega}}=\frac{e^{\frac{\beta\hbar\omega}{2}}}{e^{\beta\hbar\omega}-1}=\frac{1}{2\sinh\big{(}\frac{\beta\hbar\omega}{2}\big{)}}\,.$

After taking into account the Gibbs correction (e.g., [11, 5, 13]), for the
grand-partition function, we obtain:

$Z_{0}=e^{z\zeta}=\sum_{n=0}^{+\infty}\frac{z^{n}e^{\frac{n\beta\hbar\omega}{2}}}{n!}\Big{(}\frac{1}{e^{\beta\hbar\omega}-1}\Big{)}^{n}\,.$

##  3\. Block-Monotone Particles

The present section is devoted to a generalisation of the monotone
prescription for the statistics of the particles which, on the one hand, is
more suitable for potential physical applications and, on the other hand, is
always different from the weak monotone scheme briefly outlined above.

For such a purpose, we consider an index set $I$, being necessarily finite or
countable, which is a finite or countable disjoint union of finite sets.
Indeed, $I:=\bigsqcup_{j=0}^{+\infty}I_{j}$, where $|I_{j}|<+\infty$,
$j=0,1,\dots\,\,$. The set $I$ is naturally partially ordered, because if
$k_{j},l_{j}$ are in the same subset $I_{j}$, there is no pre-assigned order
between them. Conversely, if $k_{1}\in I_{j_{1}}$ and $k_{2}\in I_{j_{2}}$,
then $k_{1}\prec k_{2}\iff j_{1}<j_{2}$.

Such a picture is suggested by the potential physical applications. In fact, a
positive Hamiltonian $H$ with compact resolvent acting on a separable Hilbert
space ${\mathscr{H}}$, as described in Section 2, induces a natural order, as
shown above, on the natural basis of ${\mathscr{H}}$, formed of the
eigenvectors associated with the eigenvalues $\\{\varepsilon_{j}\\}$ of $H$,
where the finite cardinality of the involved subsets are given by the
degeneracies $g_{j}$ of the eigenvalues $\varepsilon_{j}$. The picture arising
from this analysis is defined as block-monotone. The corresponding block-
monotone Fock space ${\mathscr{F}}_{\rm bm}$ and the relative creation and
annihilator operators are easily constructed as follows below.

Let $\\{e_{j}\mid j\in I\\}$ be an orthonormal basis of ${\mathscr{H}}$
equipped with the previously described partial order. Typically, such a
partial order is induced by a positive Hamiltonian with compact resolvent. As
noted above, ${\mathscr{F}}_{\rm bm}$ is a subspace of the full Fock one
${\mathscr{F}}\equiv{\mathscr{F}}_{0}$, and on the $n$-particle subspace, the
$n$-particle block monotone subspace is generated as follows below.

Such an $n$-particle subspace is generated by all the sequences of the
elementary (orthonormal) tensors $e_{k_{1}}\otimes\cdots e_{k_{n}}$ with the
condition $k_{1}<k_{2}<\dots<k_{n}$ relative to the partial order defined
above. The block monotone creator and annihilator operators assume the same
form as in (3) and (4), respectively, according to the above partial order.

We denote $P_{\rm bm}$ as the self-adjoint projection acting on the full Fock
space projected onto ${\mathscr{F}}_{\rm bm}$. This allows us to compute the
grand-partition $Z_{\rm bm}$ according to (2). We can now explicitly compute
such a grand-partition for the simplest non-trivial finite dimensional case,
where ${\mathscr{H}}$ is generated by the orthonormal basis
$\\{(e_{1},e_{2}),e\\}$, the eigenvalues of a Hamiltonian whose eigenvalues
are $h,k$, with a multiplicity of 2 and 1, respectively. We express such a
grand-partition function in terms of the activity $z=e^{\beta\mu}$.

In this simple situation, the block-monotone Fock space ${\mathscr{F}}_{\rm
bm}$ ends with the two-particle subspace and is given by:

${\mathscr{F}}_{\rm
bm}={\mathbb{C}}\bigoplus\Big{(}\big{(}{\mathbb{C}}e_{1}\oplus{\mathbb{C}}e_{2})\oplus{\mathbb{C}}e\Big{)}\bigoplus\Big{(}{\mathbb{C}}(e_{1}\otimes
e)\oplus{\mathbb{C}}(e_{2}\otimes e)\Big{)}\,.$

Correspondingly, the grand-partition function is given by:

$Z_{\rm bm}(z,\beta)=1+z(2e^{-\beta h}+e^{-\beta k})+2z^{2}e^{-\beta(h+k)}\,.$

We end the present section by noting that, if all the $I_{j}$ are singletons
(or empty sets), the block-monotone scheme will be reduced to the usual
monotone one. In the previous example, the comparison between the block-
monotone scheme and the monotone one does not depend on the order that we fix
on the first subset of eigenvectors of $H$, leading to:

$Z_{\rm m}(z,\beta)=1+z(2e^{-\beta h}+e^{-\beta k})+z^{2}\big{(}e^{-2\beta
h}+2e^{-\beta(h+k)}\big{)}\,.$

In the forthcoming analysis, we study, in some generality, a non-trivial
situation of this kind.

Conversely, the block-monotone scheme is never comparable with the weakly
monotone one. Indeed, the weakly monotone version of the last example provides
five additional elements on the basis of the two-particle subspace plus non-
trivial contributions involving all the $n$-particle subspaces. Indeed,

$\displaystyle Z_{\rm wm}(z,\beta)=$ $\displaystyle 1+z\big{(}2e^{-\beta
h}+e^{-\beta k}\big{)}+z^{2}\big{(}4e^{-2\beta h}+2e^{-\beta(h+k)}+e^{-2\beta
k}\big{)}$ $\displaystyle+$
$\displaystyle\sum_{n=3}^{+\infty}z^{n}a_{n}(h,k,\beta)\,.$

##  4\. The Grand-Partition Function for Monotone Particles

Since the explicit computation of the monotone grand-partition function is not
available for most infinite dimensional cases, we reduce the matter to the
simple case of the one-dimensional quantum harmonic oscillator briefly
described in Section 2. The spectrum of the involved Hamiltonian (7) is formed
of multiplicity-one eigenvectors. Therefore, in such a case, the block-
monotone model described in Section 3 is reduced to the usual monotone one.

Denoting such a monotone grand-partition function relative to the quantum
harmonic oscillator as $Z_{\rm m}$, we obtain the following:

######  Proposition 1.

For the grand-partition function $Z_{\rm m}$, we have

$Z_{\rm
m}=1+\sum_{n=1}^{+\infty}z^{n}e^{\frac{n\beta\hbar\omega}{2}}\prod_{k=1}^{n}\frac{1}{\big{(}e^{k\beta\hbar\omega}-1\big{)}}\,.$

In addition, $0\leq Z_{\rm m}\leq Z_{0}$, where $Z_{0}$ is the Boltzmann
grand-partition function given in (6), and thus $Z_{\rm m}$ converges for all
$z\geq 0$ and $\beta>0$.

###### Proof.

We start the second half by noting that:

$e^{k\beta\hbar\omega}-1=\big{(}e^{\beta\hbar\omega}-1\big{)}\sum_{l=0}^{k-1}e^{l\beta\hbar\omega}\geq
k\big{(}e^{\beta\hbar\omega}-1\big{)}\,,$

and, thus,

$0\leq\prod_{k=1}^{n}\frac{1}{\big{(}e^{k\beta\hbar\omega}-1\big{)}}\leq\frac{1}{n!\big{(}e^{\beta\hbar\omega}-1\big{)}^{n}}\,.$

Therefore, $Z_{\rm m}\leq Z_{0}<+\infty$.

Concerning the first half, taking into account the exclusion rule arising from
the monotone assumption, it is straightforward to verify that, for the
contribution relative to the $n$-particle subspace, $n\geq 1$,

$\displaystyle\textrm{Tr}\Big{(}$ $\displaystyle P_{\rm
m}e^{-\beta\text{d}\Gamma(H)}P_{\rm
m}\lceil_{\underbrace{{\mathscr{H}}\otimes\cdots\otimes{\mathscr{H}}}_{\textrm{n-times}}}\Big{)}=e^{-\frac{n\beta\hbar\omega}{2}}\sum_{k_{1}=0}^{+\infty}e^{k_{1}\beta\hbar\omega}\sum_{k_{2}=k_{1}+1}^{+\infty}e^{k_{2}\beta\hbar\omega}$
$\displaystyle\cdots\cdots\sum_{k_{n}=k_{n-1}+1}^{+\infty}e^{k_{n}\beta\hbar\omega}=e^{\frac{n\beta\hbar\omega}{2}}\prod_{k=1}^{n}\frac{1}{\big{(}e^{k\beta\hbar\omega}-1\big{)}}\,.$

∎

Now, we can compare both grand-partition functions $Z_{\rm m}$ and $Z_{0}$.
Indeed, after defining

$Z_{\\#}=\sum_{n=0}^{+\infty}a^{(\\#)}_{n}(\beta)z^{n}\,,$

$\\#$ standing for “0” and “monotone”, we can address two physically
interesting, cases, $\beta\downarrow 0$ (high-energy/low-density regime) and
$\beta\uparrow+\infty$ (low-energy/high-density regime).

For the high energy case,

(8) $a^{({\rm
m})}_{n}(\beta)=\frac{1}{(\hbar\omega)^{n}}\Big{(}\frac{1}{\beta}\Big{)}\Big{(}\frac{1}{2\beta}\Big{)}\cdots\Big{(}\frac{1}{n\beta}\Big{)}+o\Big{(}\frac{1}{\beta^{n}}\Big{)}\equiv\frac{1}{n!(\hbar\omega\beta)^{n}}+o\Big{(}\frac{1}{\beta^{n}}\Big{)}\,,$

which leads to

(9) $a^{(0)}_{n}(\beta)=a^{({\rm
m})}_{n}(\beta)+o\Big{(}\frac{1}{\beta^{n}}\Big{)},\,\,\text{for}\,\,\beta\downarrow
0\,.$

######  Remark 2.

Equations (8) and (9) explain that, in some sense, at least in this simple
case of the harmonic oscillator, in the limit of high energies (i.e.,
$\beta\downarrow 0$), the monotone grand-partition function $Z_{\rm m}$
behaves, term-by-term, in the same way as $Z_{0}$, corresponding to classical
particles. In the next section, we provide a more refined analysis concerning
this fact.

For the appearance (i.e., (8)) of the Gibbs correction term $n!$, we can
immediately argue that the monotone Fock space will naturally take into
account the indistinguishability of particles.

Now, we move on to the limit of low energies (typically when dealing with the
so-called ground states) $\beta\uparrow+\infty$. Indeed,

$\prod_{k=1}^{n}\frac{1}{\big{(}e^{k\beta\hbar\omega}-1\big{)}}\approx\frac{1}{e^{\beta\hbar\omega\sum_{k=1}^{n}k}}=e^{-\beta\hbar\omega\frac{n(n+1)}{2}}\,,$

and, thus,

$Z_{\rm
m}\approx\sum_{n=0}^{+\infty}z^{n}e^{-\frac{n^{2}\beta\hbar\omega}{2}}\,.$

Relative to the Boltzmann partition function, according to the reasoning
above, we obtain:

$Z_{0}\approx\sum_{n=0}^{+\infty}\frac{z^{n}}{n!}e^{-\frac{n\beta\hbar\omega}{2}},\,\,\text{for}\,\,\beta\uparrow+\infty\,.$

Now, using the Stirling formula, when $n\uparrow+\infty$ and
$\beta\uparrow+\infty$, retaining only the leading terms, we obtain:

(10) $a^{(0)}_{n}(\beta)\approx e^{-n(\beta\hbar\omega/2+\ln
n)},\,\,\text{whereas}\,\,a^{({\rm m})}_{n}(\beta)\approx
e^{-n^{2}\beta\hbar\omega/2}\,.$

######  Remark 3.

In the high-density regime described by $\beta,n\uparrow+\infty$, that is,
when the state of the matter is very condensed, the monotone particles obey a
kind of exclusion principle analogous to the Pauli exclusion principle for
fermions. The heuristic Formula (10) seems to confirm the existence of such a
principle.

##  5\. The Low-Density Regime

We discuss the low-density regime corresponding to $\beta,z\approx 0$ by
showing that, in such a limit,

$Z_{\rm m}(\beta,z)\approx Z_{0}(z,\beta),\,\,\text{for}\,\,\beta,z\to 0\,.$

######  Proposition 4.

For the monotone and Boltzmann grand-partition functions, we have:

(11) $(1-f(\beta,z))Z_{0}(z,\beta)\leq Z_{\rm m}(\beta,z)\leq
Z_{0}(z,\beta)\,,$

where
$f(\beta,z):=\frac{z^{2}e^{\beta\hbar\omega}}{4(e^{\beta\hbar\omega}-1)}$.

###### Proof.

(sketch). With $x:=e^{\beta\hbar\omega}$ and

(12)
$\Delta_{n}(x):=1-\frac{n!}{(1+x)(1+x+x^{2})\cdots\Big{(}\sum_{k=0}^{n-1}x^{k}\Big{)}}\,,\quad
n=1,2,\dots\,,$

noting that $\beta\downarrow 0\iff x\downarrow 1$, for $n\geq 2$ (for $n=0,1$,
both coincide), we obtain:

(13) $a_{n}^{(0)}-a_{n}^{({\rm
m})}=\frac{1}{n!(x-1)^{n}}\bigg{(}\Delta_{n}(1)+\Delta^{\prime}_{n}(1)(x-1)+\frac{\Delta^{\prime\prime}_{n}(\xi_{n})}{2}(x-1)^{2}\bigg{)}\,.$

Here, we apply the Taylor formula with second-order Lagrange remainder; thus,
$\xi_{n}$ is a certain number (obviously, depending on $n$) in the interval
$(1,x)$.

Since $\Delta_{n}(1)=0$, $\Delta^{\prime}_{n}(1)=\frac{n(n-1)}{4}$ and,
finally, we obtain the evidence that $\Delta^{\prime\prime}_{n}(\xi_{n})<0$
whenever $x>1$ (see the Appendix A), collecting these elements together, we
have:

$a_{n}^{(0)}-a_{n}^{({\rm
m})}\leq\frac{\sqrt{x}}{n!(x-1)^{n}}\Big{(}\frac{n(n-1)}{4}(x-1)\Big{)}\equiv\frac{n(n-1)}{4}(x-1)a_{n}^{(0)}\,.$

After defining $y:=\frac{z\sqrt{x}}{x-1}$, we can sum up, obtaining:

(14) $\begin{split}Z_{0}-Z_{\rm
m}\leq&\frac{(x-1)}{4}\sum_{n=0}^{+\infty}n(n-1)\frac{y^{n}}{n!}=\frac{(x-1)}{4}y^{2}\frac{\text{d}^{2}Z_{0}}{\text{d}y^{2}}\\\
\equiv&\frac{z^{2}x}{4(x-1)}e^{y}\equiv\frac{z^{2}x}{4(x-1)}Z_{0}\,.\end{split}$

Now, (14) easily leads to:

$Z_{0}\bigg{(}1-\frac{z^{2}e^{\beta\hbar\omega}}{4(e^{\beta\hbar\omega}-1)}\bigg{)}\leq
Z_{\rm m}\,,$

and the proof follows, according to Proposition 1. ∎

In order to progress to the investigation of the low-density regime, (11)
reads:

(15) $(1-f(\beta,z))\leq\frac{Z_{\rm m}(\beta,z)}{Z_{0}(\beta,z)}\leq 1\,,$

and (15) provides a useful condition, if and only if $(1-f(\beta,z))>0$.

It is now convenient to define $t:=z^{2}/4$ with the limitations $0\leq t<1$,
obtaining:

$1-\frac{xt}{x-1}>0\iff x>\frac{1}{1-t}\,.$

By passing to the logarithm and restoring the variables $z$ and $\beta$, we
obtain:

$\beta\hbar\omega>-\ln\big{(}1-(z/2)^{2}\big{)}\,.$

On the other hand, if for some $0<\gamma<1$,

(16)
$\beta\hbar\omega\geq-\ln\big{(}1-(z/2)^{2\gamma}\big{)}>-\ln\big{(}1-z^{2}/4\big{)}\,,$

again, in terms of $x$ and $t$, we have:

${x\geq\frac{1}{1-t^{\gamma}}}\iff{\frac{xt^{\gamma}}{x-1}\leq{1}}\iff{\frac{xt}{x-1}\leq
t^{1-\gamma}\to 0}$

whenever $t\downarrow 0$ or, equivalently, $z\downarrow 0$.

Restoring the original variables, the above computation simply means that:

$0\leq
f(\beta,z)\equiv\frac{z^{2}e^{\beta\hbar\omega}}{4(e^{\beta\hbar\omega}-1)}\leq(z^{2}/4)^{1-\gamma}\to
0\,,$

when $z$ and, necessarily, also $\beta$ in the chosen region change to $0$.

Thus, we prove the following:

######  Proposition 5.

For each fixed $0<\gamma<1$ and $(z,\beta)$ in the region $R$ delimited by
$0\leq z<2$ and by the condition

$\beta\hbar\omega>-\ln\big{(}1-(z/2)^{2\gamma}\big{)}\,,$

we obtain

$\lim_{\stackrel{{\scriptstyle(\beta,z)\to(0,0)}}{{(\beta,z)\in
R}}}\frac{Z_{\rm m}(\beta,z)}{Z_{0}(\beta,z)}=1\,.$

Concerning the low-density regime, Proposition 5 has the following meaning.
Indeed, for $\beta,z\approx 0$ in the region defined by (16),

(17) $Z_{\rm m}(\beta,z)\approx(1-f(\beta,z))Z_{0}(\beta,z)\,,$

where the minus sign is explained in Remark 3 as an analogue of the Pauli
exclusion principle for monotone particles. Such a monotone exclusion
principle tends to become very relevant in the high-density regime
($\beta\uparrow+\infty$, see Remark 3), whereas it tends to vanish in the low-
density regime (Proposition 5). In the latter case, the function
$1-f(\beta,z)$ provides the correction for the thermodynamic potentials
relative to those of the monotone case, compared with the analogous ones
relative to Boltzmann case.

Moreover, using (17), we can determine that the average number of particles is

$\displaystyle N_{\rm m}(\beta,z)=$ $\displaystyle z\frac{\partial\ln Z_{\rm
m}(\beta,z)}{\partial z}\approx z\frac{\partial\ln(1-f(\beta,z))}{\partial
z}+N_{0}(\beta,z)$ $\displaystyle=$ $\displaystyle
N_{0}(\beta,z)-\frac{z^{2}e^{\beta\hbar\omega}}{2\big{(}(e^{\beta\hbar\omega}-1)-(z/2)^{2}e^{\beta\hbar\omega}\big{)}}\,,$

where the second addendum in the last member is, indeed, negative because of
(16).

We conclude by noting that Proposition 5 allows one to compute the asymptotics
of Connes’ spectral action (cf. [1]), associated with the average number of
monotone particles, as described in [19] for $q$-particles, but with the
condition described in (16). We postpone the investigation of this aspect for
a forthcoming analysis.

##  6\. Conclusions

The previous analysis concerning monotone models suggests that their potential
applications to physics appear to be meaningful and fruitful, even if the
explicit computation of the thermodynamic quantities seems to be rather
complicated. Therefore, it is natural to systematically address the
investigation of the statistical properties of monotone systems, even if the
involved grand-partition function is not completely computable for most
infinite dimensional systems. We thus list some natural questions which could
be addressed in future investigations below.

The first one concerns the models in which the involved energy levels are
degenerate. A simple example to address is the case when the degeneracy of the
energy levels is uniform. A concrete but more complicated example is the
isotropic harmonic oscillator in a $d$-dimensional environment, $d>1$. Such a
degeneracy is not uniform but can easily be computed, obtaining:

$\begin{split}\varepsilon=\hbar\omega\sum_{i=1}^{d}(n_{i}+1/2)\,,&\quad
n_{i}\in{\mathbb{N}},\,\,i=1,2,\dots,d\,,\\\
g(\varepsilon)=\binom{d+n-1}{n}\,,&\quad n\in{\mathbb{N}}\,.\end{split}$

The second natural question to be addressed is the investigation of the free
gas formed of monotone particles. This is connected with the degeneracy and
can be addressed either by considering the free particles confined in a box
and then “passing to the continuum”, as in Sections 4 and 5 of [19], or by
removing the harmonic potential (i.e., performing an appropriate limit as the
Hook constant changes to $0$). For $d=1$, the second approach seems to be
directly applicable because, as pointed out above, the energy levels are non-
degenerate, whereas the first approach already involves a non-trivial
degeneracy in the case $d=1$.

The third but not least significant question is the systematic investigation
of the statistical properties of monotone systems, with particular attention
given to low- and high-density regimes. As explained above, the detailed
investigation of the low-density regime also involves the explicit computation
of the asymptotics for $\beta\downarrow 0$, which is connected to Connes’
spectral action (e.g., [1]) for monotone systems and, thus, to noncommutative
geometry.

The study of the high-density regime, particularly at zero temperature (i.e.,
in the limit $\beta\uparrow\infty$), could explain the quantitative effect of
the decimation induced by the monotone prescription. This is simply the effect
that we previously called the “monotone exclusion principle”, that is, the
analogue of the Pauli exclusion principle occurring in the case of fermions at
zero temperature.

For example, we can argue that the statistical properties of the (block-)
monotone systems might have reasonable applications to complex systems which
are absorbing (or emitting) quanta of energy. The system that we have in mind
is an atom which is capturing a photon and then passing into an excited state
or even emitting, again, a photon reaching a more stable state. Another
example concerns the nuclei of fissile material (such as uranium
U${}^{92}_{235}$) in a nuclear plant which are capturing thermal neutrons and
undergoing nuclear fission. In both cases, the relevant subject might not be
the absorbed particles (bosons in the former case and fermions in the latter
case) but the complex system which is absorbing (or emitting) the energy,
according to the order of the eigenvalues of its Hamiltonian. These
considerations might suggest that the block-monotone prescription has some
role in the investigation of such complex systems from a statistical point of
view.

We conclude by pointing out that such a monotone exclusion principle should
not allow for the occurrence of the condensation phenomena of monotone
particles in the fundamental state. It would interesting to provide a rigorous
proof of this conjecture for the free gas of monotone particles in a
$d$-dimensional (or, more concretely, in the euclidean 3-dimensional) space.

F.C. and F.F. acknowledge the MIUR Excellence Department Project awarded to
the Department of Mathematics, University of Rome Tor Vergata, CUP
E83C18000100006, and INDAM-GNAMPA. F.C. acknowledges the ERC Advanced Grant
669240 QUEST, “Quantum Algebraic Structures and Models”.

## Appendix A

The main aim of the present work is to investigate whether particles obeying
the monotone prescription have potential physical applications. For this
reason, we did not pursue the proof of Proposition 4 in much detail in this
stage. In particular, in the sketch of the proof, we used the negativity
condition of the second derivative of the $\Delta_{n}(x)$ for $n=2,3,\dots$
and for all $x>1$. Even though this property is reasonable and intuitive, it
is complex to provide analytic proof of it. For this reason, we numerically
computed such a second derivative for different values of $n$, reporting their
plots in Figure 1.

On the other hand, while it is easy to see that
$\Delta^{\prime}_{n}(1)=\frac{n(n-1)}{4}$, in order to estimate the last
addendum in (13), we fitted $\frac{\Delta^{\prime\prime}_{n}(1)}{2}$ with a
polynomial of degree 4 in the form

$p(n)=b_{4}n^{4}+b_{3}n^{3}+b_{2}n^{2}+b_{1}n+b_{0}\,,$

where

$b_{0}\approx 0,\,\,b_{1}\approx
0.09028,\,\,b_{2}\approx-0.13542,\,\,b_{3}\approx
0.07639,\,\,b_{4}\approx-0.03125\,,$

with negligible errors (see Figure 2).

Figure 1. Second derivatives of $\Delta_{n}(x)$, $x>1$. Figure 2. Fit of
$\frac{\Delta^{\prime\prime}_{n}(1)}{2}$.

## References

  * [1] Chamseddine, A.H.; Connes, A. The spectral action principle. _Commun. Math. Phys._ 1997, _186_ , 731–750.
  * [2] Bratteli, O.; Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics 2; Springer: Berlin/Heidelberg, Germany, 1997.
  * [3] Voiculescu, D.V.; Dykema, K.J.; Nica, A. Free Random Variables; AMS: New York, NY, USA, 1992; p. 70.
  * [4] Bożejko, M.; Kümmerer, B.; Speicher R. q-Gaussian processes: Non-commutative and classical aspects Commun. Math. Phys. 1997, 185, 129–154.
  * [5] Huang, K. Statistical Mechanics; John Wiley & Sons: Hoboken, NJ, USA, 1987.
  * [6] Accardi, L.; Fidaleo, F. Condensation of Bose and q-particles in equilibrium and non-equilibrium thermodynamics. _Rep. Math. Phys._ 2016, 77, 153–182.
  * [7] Bożejko, M.; Speicher R. Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. _Math. Ann._ 1994, _300_ , 97–120.
  * [8] von Waldenfels, W. An approach to the theory of pressure broadening of spectral lines. In Probability and Information Theory II; Lecture Notes in Mathematics; Behara, M., Krickeberg, K., Wolfowitz, J., Eds.; Springer: Berlin/Heidelberg, Germany; New York, NY, USA. 1973; Volume 29, pp. 19–69.
  * [9] Lu, Y.G. An interacting free Fock space and the arcsine law. _Probab. Math. Statist._ 1997, 17, 149-166.
  * [10] Muraki, N. Noncommutative Brownian motion in monotone Fock space. _Commun. Math. Phys._ 1997, 183, 557–570.
  * [11] Ciolli, F.; Fidaleo, F. On the thermodynamics of the $q$-particles. _Entropy_ 2022, 24, 159.
  * [12] Fidaleo, F.; Viaggiu, S. A proposal for the thermodynamics of certain open systems. _Phys. A_ 2017, 468, 677–690.
  * [13] Werner, R.F. The free quon gas suffers Gibbs’ paradox. _Phys. Rev. D_ 1993, 48, 2929.
  * [14] Dykema, K.; Fidaleo, F. Unique mixing of the shift on the $C^{*}$-algebras generated by the q-canonical commutation relations. _Houston J. Math._ 2010, 36, 275–281.
  * [15] Fidaleo, F. A note on Boolean stochastic processes. _Open Sys. Inf. Dyn._ 2015, 22, 1550004.
  * [16] Bożejko, M. Deformed Fock spaces, Hecke operators and monotone Fock space of Muraki. _Dem. Math._ 2012, XLV, 399-413.
  * [17] Bożejko, M.; Dołęga, M.; Ejsmont, W.; Gal, Ś.R. Reflection length with two parameters in the asymptotic representation theory of type B/C and application. _J. Funct. Anal._ 2023, 284, 109797.
  * [18] Fidaleo, F. On the uniform convergence of ergodic averages for $C^{*}$-dynamical systems. _Mediterr. J. Math._ 2020, _17_ , 125.
  * [19] Ciolli, F.; Fidaleo, F. Spectral actions for $q$-particles and their asymptotics. _J. Phys. A_ 2022, 55, 424001.