# Dynamics of collective action to conserve a large common-pool resource

David Andersson12, Sigrid Bratsberg2, Andrew K. Ringsmuth35467, Astrid S. de
Wijn 21
Corresponding author: Astrid de Wijn<EMAIL_ADDRESS>1Chemical Physics
Division, Department of Physics, Stockholm University, Sweden
2Department of Mechanical and Industrial Engineering, Norwegian University of
Science and Technology, Norway
3Complexity Science Hub Vienna, Vienna, Austria
4Section for the Science of Complex Systems, Medical University of Vienna,
Vienna, Austria
5Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
6Current Address: Wegener Center for Climate and Global Change, University of
Graz, Graz, Austria
7Current Address: Complexity Science Hub Vienna, Vienna, Austria

###### Abstract

A pressing challenge for coming decades is sustainable and just management of
large-scale common-pool resources including the atmosphere, biodiversity and
public services. This poses a difficult collective action problem because such
resources may not show signs that usage restraint is needed until tragedy is
almost inevitable. To solve this problem, a sufficient level of cooperation
with a pro-conservation behavioural norm must be achieved, within the
prevailing sociopolitical environment, in time for the action taken to be
effective. Here we investigate the transient dynamics of behavioural change in
an agent-based model on structured networks that are also exposed to a global
external influence. We find that polarisation emerges naturally, even without
bounded confidence, but that for rationally motivated agents, it is temporary.
The speed of convergence to a final consensus is controlled by the rate at
which the polarised clusters are dissolved. This depends strongly on the
combination of external influences and the network topology. Both high
connectivity and a favourable environment are needed to rapidly obtain final
consensus.

##  I INTRODUCTION

Common-pool resources (CPRs) are ‘non-excludable’ (open-access) and
‘subtractable’ (one individual’s consumption can preclude another’s).
Harvesting a CPR for short-term benefit without restraint can make it
unavailable through depletion – a so-called ‘tragedy of the commons’ – unless
harvesters can organise to restrain their harvesting to a sustainable level.

CPR management has typically been studied in systems where harvesters can
perceive the resource level’s linear response to their harvesting, allowing
them to adapt efforts accordingly (e.g. pastures, groundwater, fisheries). In
models, this appears as explicit interdependence between harvesting strategy
evolution and resource level [1, 2, 3, 4, 5, 6]. However, a CPR’s linear
response may be imperceptible to harvesters if, for example, the resource pool
is very large when compared with the harvesting rate. Such a resource may show
no obvious reason for restraint, even after extensive harvesting. However, the
state of the encompassing social-ecological system may depend nonlinearly on
the resource level, so unrestrained harvesting may push the system to an
unexpected critical transition into a new and potentially undesirable state
[7]. Three prominent examples of such CPRs are unpolluted atmosphere, global
biodiversity and public healthcare systems during a pandemic.

Conservation of a large CPR can be considered as a large-scale (potentially
global) collective action dilemma [8, 9]. In such a dilemma, group members
must choose whether to contribute to a collective action such as adherence to
a pro-conservation behavioural norm which, if successful, will benefit the
whole group. Below some threshold rate of cooperation, each individual
benefits more from defecting than cooperating, but all individuals are better
off if the threshold is reached [10].

The lack of obvious feedback from a large CPR amplifies the collective action
problem because the need for restraint may not be obvious to enough people
until tragedy is (almost) inevitable. Solving this problem requires achieving
a sufficient level of cooperation with a pro-conservation behavioural norm
fast enough for the collective action to successfully conserve the resource.
Precisely what this level and rate are depends on the details of a given CPR
and how it responds to changes in harvesting behaviour. In all cases, however,
the behavioural change process will depend on agents being affected by
information received from other agents and from the broader sociopolitical
environment. In general, the faster and more complete this process is, the
more likely it is that the conservation action will be successful.

Models of opinion dynamics on social networks have been extensively studied
[12, 11]. Many are agent-based and assume ‘bounded confidence’: an agent’s
interaction with another may change its opinion but only if they engage
meaningfully. Due to the psychosocial phenomenon of homophily, this is
unlikely to happen between people who have very different views. Bounded
confidence models [13, 14] therefore assume that agents engage only if their
opinions differ by less than some specified threshold. In contrast to this
psychology-based approach, models of CPR management typically assume that
agents respond rationally to available information in a manner described by
evolutionary game theory. The two types of model can behave quite differently.

Large-scale CPR conservation efforts face psychosocial constraints and are
also rationally motivated. They thus combine features of the above two model
types. Agents will face material consequences in the long term and at least
some people are aware that it is in their long-term self interest to try to
mitigate these consequences through some short-term sacrifices. They might
also realise that the collective action problem may be solved more quickly if
they are willing to discuss with others. This means actively overcoming
bounded confidence to engage with people around them even if they have very
different views.

While long-term limiting (equilibrium) behaviour in both CPR problems and
opinion dynamics has been widely studied [15, 16], the transient dynamics have
received far less attention. Successful conservation, however, may depend not
only on the long-time group behaviour but also on the rate of behavioural
change once the problem is first recognised. A small number of opinion
dynamics studies have specifically modelled transient dynamics [17, 18] and
provided evidence that this is important for understanding realistic systems
of practical interest, which are often in such transient states.

In this work, we study the transient dynamics of collective action through
behavioural change mediated by social interactions between agents who are also
influenced by a broader sociopolitical environment. Being rationally motivated
by the need for action, neighbours of different opinions overcome bounded
confidence and are just as likely to interact with each other as neighbours of
similar opinions. We also expose the agents en masse to a global field that
represents the combined effect of influences in the broader sociopolitical
environment. We study the competition between these local and global
influences, and how the speed of consensus formation emerges from different
features of the model. Finally, we discuss implications of our findings for
real-world management of large CPRs.

We study the spread of cooperation with a pro-conservation behavioural norm
using a model that is similar to those used for opinion dynamics without
bounded confidence [19]. Each agent’s level of cooperation is represented by a
real number from -1 to 1, which respectively represent total defection and
total cooperation, and we refer to agents at these extremes as defectors and
cooperators accordingly. We adopt this nomenclature from evolutionary game
theory even though our model more closely resembles an opinion dynamics model,
to reflect the mixture of rationally motivated, strategic behaviour and
psychosocial constraints in the norm-spreading dynamics that we aim to
capture. Since we are concerned with building collective action from a state
of inaction, our simulations begin with network states in which most agents
defect.

We study dynamics on clustered scale-free networks (CSF). Real social
acquaintance networks are typically clustered scale-free networks [20, 21]. In
such networks, topological path lengths between agents are typically short
(even in the case of large and highly connected online social networks [22]),
giving rise to the name ‘small-world network’[23]. Although scale-free
networks have been extensively studied [24], frameworks for constructing and
analysing clustered scale-free networks were developed only recently. In
addition to their small-world properties, such networks have high connectivity
between proximate nodes (i.e. topological clustering). However, we caution the
reader that, except where referring to CSF networks directly, in our analysis
we use ‘cluster’ not in this topological sense but rather to refer to a group
of agents who share a common cooperativity (i.e. strongly correlated states).
We study collective dynamics across many simultaneously evolving network
realizations and average the resulting network properties over the many
realizations.

##  II Results

Figure 7(g) shows a typical network evolution, averaged over 500 network
realizations. Parameters are standard (see Methods and Supplementary Material)
except $\phi=0.015$ for the external field. In this and similar plots, we
present the average cooperativity and standard deviation. The various
parameters are defined in the Methods section and summarised in Table I. The
standard deviation here is calculated by first calculating the standard
deviation of the cooperativity in each individual network realization, and
then averaging these.

(a) A Typical network evolution, here $\phi=0.015$.

(b) Grid network evolution with snapshots below.

(c) t = 0

(d) t = 0.7

(e) t = 1.7

(f) t = 3.5

(g) t = 5.2

(h) t = 7.0

(i) t = 13.9

(j) t = 27.8

Figure 1: (a) A typical average time evolution of the cooperativity for
conditions slightly favouring cooperation on a CSF network. At first, the
short range but strong social pressure of the initially abundant defectors
dominate the evolution. However, after some time, the long-range (global) but
weak external field drives the networks slowly into cooperation. The standard
deviation initially increases during cluster formation, but later decays away
as the network homogenizes. (b) An example of how a grid network of small size
evolves. Comparing (b) with grid snapshots (c-j) helps to understand how the
clustering process evolves; the dashed lines in (b) indicate the times
corresponding to (c-j). Moreover, comparing (b) with (a) shows the similarity
of dynamics in the grid and CSF network cases. (c-j) A realization of a small
grid network with periodic boundary conditions and an initial defector
majority, but a weak field favouring cooperation. Blue indicates defective
behaviour, red cooperative. We see that, initially, agents with strong
opinions efficiently convert undecided agents in their vicinity to their
cause. This is the cluster-formation regime of the dynamics. In this
realization, one can observe a strongly cooperating (red) agent in the upper
right, which appears at $t=1.7$. This agent them forms a large cooperative
cluster around it. In contrast, most of the rest of the network gradually
becomes more defective due to the initial abundance of defectors. At $t=13.9$,
the network is almost completely polarised, and by $t=27.8$, the cooperative
cluster has begun to overcome the defective cluster by gradually dissolving
its perimeter.

###  II-A Transient polarisation

The dynamics show two characteristic time scales. Initially, we see a rapid
decline in the average cooperativity and increase in its standard deviation.
However, the dynamics soon become dominated by a second, slower process that
gradually drives the network to a fully cooperative state. We can understand
this better from simulations on smaller 13x13 grids with the same degree and
periodic boundaries, shown in figure 1, along with the evolution of the
average cooperativity and standard deviation. Initially, neighbouring agents
are uncorrelated. Agents that are initially strongly cooperative or defective
spread their behaviours to their surroundings and strengthen their own
behaviours. Correlated clusters emerge around the original agent and any
interaction within the cluster will serve only to reinforce the behaviours,
leading locally to an ‘echo chamber’ and globally to a strongly clustered,
polarised network. Once the network has become completely polarised,
meaningful interactions take place only on the boundaries separating opposing
clusters.

Comparing the snapshots in fig. 1 to the time evolution shows that this
cluster formation regime corresponds to the first stage in figure 7(g). Since
the network begins with a small majority of defectors, they on average spread
their behaviours more effectively than cooperators. Hence we see strong
initial decrease in the average cooperativity, averaged over many networks.
Notably, such a polarised state is often the equilibrium to which bounded-
confidence models with a sufficiently restrictive threshold converge. In our
system, however, it is only transient. Once polarisation is complete and no or
few neutral agents remain, a ‘tug-of-war’ takes place between clusters, at
their edges. In this competition, a sufficiently strong pro-conservation
external field can tip the balance, making it more likely that one-on-one
interactions will result in defectors converting to cooperators, rather than
the converse. We see this in the second dynamical stage, in which the
cooperativity slowly rises as defective clusters shrink.

Figure 7(g) shows that the point of maximum polarisation, where the standard
deviation is maximal, does not coincide exactly with the point of minimal
cooperation. In figures 1 c-j, we see that, at the cooperation minimum, there
are regions that are not yet polarised but are responding to the pro-
conservation external field. The dynamical crossover from rapid polarisation
to slow cluster growth/shrinkage thus happens somewhat after the minimum. A
consequence of this is that the average cooperativity is not bound to
monotonically increase after the apparently minimal average cooperativity is
reached. Rather, at the end of the cluster formation stage, in some cases,
even though the average cooperativity has already started to increase
slightly, the cooperators nevertheless are not abundant enough and the system
converges to a fully defective state (Fig. 2(b)).

We can understand the above results in more detail by considering the
strengths of the two competing interactions. More-or-less neutral agents are
more susceptible than polarised agents to persuasion by others. Since neutral
agents are more abundant in the initial network configuration, social
interactions more strongly affect its early dynamics, which quickly converge
to a polarised state. Once the system is strongly polarised, persuasion is
possible only between agents of extreme, opposite levels of cooperativity at
the cluster boundaries. In this situation, even a small contribution from the
external field can tip the probability in one direction over the other. Thus,
the defector is slightly more likely to become a cooperator than the
cooperator is to become a defector. When this is the dominant dynamics, the
network evolves more slowly, but the influence of the external field becomes
apparent.

(a)

(b)

Figure 2: Long-term evolution of a system close to a tipping point
($\phi=0.0103$). (a) The average and standard deviations of the cooperativity
converge to a value near zero, meaning that in roughly half of the
realizations, the networks became completely cooperative, and in the other
half, completely defective. This evolution is very sensitive to changes in
parameter values around this point, as is a hallmark of tipping points. (b) A
heatmap corresponding to all the data for which the averages are shown in (a).
The intensity corresponds to the density of networks having a specific average
cooperativity at a specific time. Most networks converge to full cooperation
or defection within the simulation time, reinforcing that this is a tipping
point.

In some regions of the parameter space, we observe so-called social tipping
points, around which the network’s long-time behaviour depends sensitively on
the parameters. Figure 2 shows a network evolution around such a tipping
point. In panel (a), both the average and standard deviation of the
cooperativity converge to roughly zero, implying that the individual network
realizations converged to fully cooperative and fully defective states with
roughly equal frequencies. Panel (b) shows a 2D histogram, binning the
cooperativities of the individual network realizations that are averaged in
(a). The colour scale is capped at 100 counts to enhance contrast. We see a
clear division of the network realizations between the two attracting terminal
states. The final outcome for each realization depends on its precise details.
The initial conditions were chosen such that fully cooperative and fully
defective outcomes are both likely. The external field here was $\phi=0.0103$,
and all other parameters standard.

###  II-B Convergence rates

Figure 3 gives more insight into the process of convergence to a fully
cooperative network by plotting the same transient dynamics as figure 7(g) on
a logarithmic scale. We see that the second, convergent dynamical stage in
7(g) is itself separated into two stages. The first is an approximately
exponential stage of convergence, as expected. In this second, slower, stage,
it is likely the finite size of the remaining clusters that plays a role, in
combination with fluctuations.

Figure 3: Logarithmic plot for a long range simulation of the average
cooperativity in a standard clustered scale free network. We see that there is
a critical slowing down at extremely homogeneous systems.

In real-world collective action problems, achieving the threshold needed for
effective action is far more important than converting the last few hold-outs.
Therefore it is the exponential stage of the convergence regime that interests
us most. Since the length of this stage varies, we choose a simple and
consistent way to evaluate the rate: taking the time derivative of the average
cooperativity when the average cooperativity is close to 0, where the curve is
certain to be exponential.

####  II-B1 Network connectivity

We investigate how the transient dynamics depend on network topology by
comparing the convergence for different networks. Figure 4(a) shows the
average cooperativity evolution for clustered scale-free networks with
different degrees and otherwise identical, standard parameters. We see that
network connectivity is a crucial determinant of the time scale at which
clusters are dissolved. In figure 4(b), we show the convergence rate as a
function of the average degree (obtained by changing the growth degree,
$k_{0}$, to 4, 8, 16, 31, 61 respectively, see methods section). Once the
network is sufficiently connected, the convergence rate saturates. This is
most likely due to the finite size of the network. We can calculate the
average number of next neighbours as the square of the average degree. Since
our system contains around 1000 agents, if the average degree is 31 or 61,
every agent is likely to be connected to a quarter of, or all (respectively),
other agents, through its nearest neighbours.

(a)

(b)

(c)

(d)

Figure 4: The effect of the connectivity and field on the convergence to the
final state. (a) and (b) the average cooperativity as a function of time for
different network connectivities and corresponding rates. (c) and (d) the same
for field strength. Increasing the average connectivity in the network
increases the convergence speeds, because each individual agent on average
will have more ways of receiving the dominant message. As expected from the
model formulation and the analysis thereof, the strength of the external field
is paramount to both the convergence speed as well as the fate of the
networks. We see that initially, improving the conditions of the networks
yields a significant change in convergence rate. Whereas for already
favourable conditions, there is a diminishing return on the gain in
convergence rate.

####  II-B2 External field

We explore how the external field affects the convergence rate by first
letting the network polarise (reach its minimum average cooperativity), and
then changing the field. In this way we can discount any effects of the field
on the initial polarisation stage as significant determinants of the
convergence dynamics.

Figure 4(c) shows the results. Apart from the delayed change to the external
field, the parameters and simulation protocols were identical to earlier
simulations. Each curve corresponds to 500 network realizations. As shown in
figure 4(d), the external field strength strongly affects the average
cooperativity’s convergence rate, though the response diminishes as field
strength increases.

####  II-B3 Influencers

Over recent years, it has become clear that so-called _influencers_ can affect
the spread of behaviours through social networks, particularly on social
media. We represent such an influencer in our model as an agent who has both
exceptionally high connectivity and strong, inflexible behaviour. Again, we
first let the networks polarise and then begin our analysis by identifying the
agent with the highest connectivity in the network and fixing this agent as an
unyielding cooperator. To verify that the subsequent dynamics reflect both
this agent’s behaviour and connectivity, we compare this with the addition of
an influencer to a grid network in which all agents are equally well
connected. The results are shown in figure 5, again with standard parameters
and 500 network realizations. Clearly, the presence of an influencer drives
substantially faster convergence to the final state.

Figure 5: The average cooperativity as a function of time for different grid
types with and without an _influencer_ in the network. An influencer is a
highly connected (in this case the most connected) node with an unwavering
behaviour. We can see that the key property of an influencer is the first of
these attributes, as their effect in a grid network is almost negligible.

Figure 6 compares the effects of our various interventions on the convergence
rate. For brevity, we include only the strongest case observed for each
intervention. Under the given assumptions, the external field most strongly
affects the convergence rate. However, we emphasise that this result should be
interpreted with caution, since we do not account for factors affecting the
practical feasibility of each intervention in any real system. We also leave
detailed exploration of interactions between interventions for future work.

Figure 6: Summarizing cooperativity as a function of time for the various
interventions suggested in this paper to increase the rate of convergence
towards consensus. Data collected from previously presented figures. The
yellow and red curves are overlaid because their impacts are almost identical.

##  III Discussion

In this study, we used an agent-based model to investigate the transient
dynamics of collective action to conserve a large common-pool resource. This
problem combines rationally motivated behaviour with the many unconscious
processes intrinsic to social psychology. We used an opinion dynamics-type
model, without bounded confidence, to capture this dual nature and studied the
effects of network topology, an applied external field, and the presence of
‘influencers’ on the rate at which the network population converged to a state
of full cooperation with a pro-conservation behavioural norm.

Different processes dominate the model on different time scales. Initially,
formation of polarised clusters dominates but this gives way to a slower
process of cluster dissolution due to social interactions at their perimeters,
which depends strongly on cluster geometry. In larger networks, there is a
possibility of larger clusters. Since the number of agents inside clusters
will grow faster with the cluster size than the number of agents at their
perimeters, the convergence will likely be slower.

We found that the convergence rate is strongly affected by both network
connectivity and external field strength, and that some parameter regions show
tipping points in which the long-time behaviour depends sensitively on the
parameters. A pro-conservation field gives cooperators the edge to slowly
convert defectors at the cluster perimeters. We also found that a highly-
connected and zealous influencer, who can convince others but is itself
inflexible, can greatly speed up the convergence.

In the presence of any of these interventions, achieving collective action is
essentially a process of dissolving defective clusters, which is fundamentally
limited by network topology at the cluster perimeters. This result strongly
agrees with recent work by Fotouhi et al [25], who found that deliberately and
carefully increasing connectivity between polarised clusters is a powerful
mechanism for promoting cooperation. In our view, this speaks particularly to
the power of now-ubiquitous social media echo chambers as inhibitors of
adequate collective action on the extremely urgent global challenges of
climate change and biodiversity loss. Our results suggest that replacing the
mechanisms that generate and maintain such polarisation with others that
instead promote constructive engagement between disagreeing groups could be a
powerful promoter of large-scale collective action. This is an important
subject for further modelling, including additional features of real social
networks such as link rewiring, which can have a significant impact on
formation and therefore also dissolution mechanisms for polarised social
clusters [26, 2].

Previous investigations have extensively and separately discussed social
tipping points [7], clustering [14, 13], the role of connectivity [25], as
well as the impact of external parameters [27], all of which are represented
in our model. Our results suggest that real-world efforts to promote
collective action on large-scale problems can benefit from both improving
network connectivity _and_ providing a sociopolitical climate which favours
cooperation. In this sense, we combine previous results into an overarching
framework. We hope that this novel perspective will open new avenues into
investigating the social dynamics that are critically important for addressing
21st-century global challenges.

Our results also suggest that convergence rate, rather than just long-time
dynamics, are crucially important for real-world collective action. Much
effort has been invested in trying to understand which parameters ensure a
favorable final state of a social network. However, for urgent large scale
common-pool resource problems, simply concluding that there exists a stable
favorable solution that will be reached _eventually_ is in practice not
sufficient to ensure a favourable outcome, since we also have to ensure that
we will arrive there before it is too late. Otherwise the commons may have
already sustained significant, or even catastrophic and irreversible, damage.
We would therefore argue that the complexity of large-scale common-pool
resource management is such that considering a single intervention is
insufficient for advising mitigation strategies. Ensuring favourable network
connectivity would be moot if a favourable sociopolitical climate is lacking,
and vice versa.

Finally a word on the scope of this research. Our model differs from the well-
established bounded-confidence models that are conventionally used to model
opinion dynamics, and which typically produce final states with clustering.
This is founded on the fact that widespread consensus is generally not reached
on the vast majority of questions about which people have opinions, such as
which TV-show is best or which political party to vote for. However, there are
several large-scale societal issues on which opinion and/or behavioural
homogenization is occurring, such as gender equality, democratization, and
possibly the need for substantial climate change mitigation. All of these
issues are characterized by the threat of long-term system-wide material
consequences for inaction, giving agents a rational incentive towards
cooperation. An interesting avenue for future research would be to introduce a
parameter for the level of confidence bounding to our model and investigate
the transition between the clustered final state typically seen in regular
bounded confidence models and the final full consensus observed in the case of
unbounded confidence analysed here. To put things into perspective, it should
be noted that bounded confidence models with a large confidence interval can
also produce a consensus end state. However, tuning which state becomes the
end state is generally not a natural component of such models.

##  IV Methods

We study the spread of cooperation with a pro-conservation behavioural norm
using a formalism that is similar to other models of opinion dynamics without
bounded confidence [11]. Each agent’s level of cooperation ($b_{i}$) is
represented by a real number from -1 to 1, which respectively represent total
defection and total cooperation, $b_{i}\in[-1,1]$. Since we are concerned with
building collective action from a state of inaction, our simulations begin
with network states in which most agents defect.

The network evolution is simulated iteratively in 4 steps: 1. Randomly select
an agent ($i$); 2. Randomly select a neighbour ($j$) of $i$; 3. Evaluate their
interaction as described below. 4. Update $i$’s behaviour according to the
outcome. This process is iterated until the network converges to a steady
state. Updating $i$’s behaviour, rather than $j$’s, ensures that agents with
more connections are more influential.

The change $\Delta b_{i}$ in $i$’s behaviour resulting from an interaction is
given by111These equations differ slightly from the published version of the
paper. One misprint was corrected and readability was improved.

$\Delta b_{i}=|b_{i}-b_{j}|[p^{+}(1-b_{i})-p^{-}(1+b_{i})],$ (1)

where $p^{+}$ and $p^{-}$ describe the weights of opinion changes in the
positive and negative directions respectively. Here $p^{+}=f(x_{ij},r,\xi)$
and $p^{-}=1-p^{+}$ with

$f(x,r,\xi)=\begin{cases}0&\text{if }x+\xi<-r\\\
\frac{1}{2r}(x+r-\xi)&\text{if }-r\leq x+\xi\leq r\\\ 1&\text{if
}x+\xi>r,\end{cases}$ (2)

where $\xi$ is a uniform random noise term, $r$ is the width of the noise, and
furthermore

$x_{ij}=w_{i}b_{i}+w_{ij}b_{j}+\phi.$ (3)

We can simplify equation 1 to

$\Delta b_{i}=|b_{i}-b_{j}|[2f(x_{ij},r,\xi)-b_{i}-1].$ (4)

These equations are motivated as follows. The first is a continuous
generalization of a discrete opinion exchange, as reviewed in [19]. In such
models, one would weigh the probability of an agent flipping its state when
evaluating the agent interaction, this is the function of $w$ in equation 1.
Additionally, in this interaction we have added a prefactor that limits the
degree to which two already agreeing agents can influence each other and,
conversely, ensuring that an agent has to be presented with a new opinion in
order to change its opinion.

Equation 3 describes the magnitude of the agent-to-agent interaction. The
function $f(x_{ij},r,\xi)$ captures the response of the affected agent,
including the random environmental influences, which are in $r$. Finally,
equation 4 is just a more compact version of equation 1 that is more useful
for computations.

We now define every parameter in the model (summarised in Table I). The
parameter, $\phi$ represents the influence of the sociopolitical environment,
which we assume to affect all agents equally. Media influence has been
variously studied in opinion dynamics models (e.g. [27, 28]), usually
implemented to represent a specific mechanism or scenario. Since our focus is
more general, $\phi$ is intended as a simple representation of the net
influence of many factors including media messaging and regulations imposed by
institutions. A positive (negative) $\phi$ corresponds to a net positive
(negative) influence on the agents’ behaviour, making them more cooperative
(defective). We use a small but positive external field, which represents a
liberal society that is promoting cooperation on collective issues.

The agent’s resistance to change is represented by $w_{i}$. Experimentation
showed that our results do not qualitatively depend on $w_{i}$’s distribution
across agents, so we assume them to be the same for all agents. The
interaction strength between agents $i$ and $j$ is $w_{ij}$, which represents
a combination of psycho-social factors such as relatedness, charisma, and
cultural differences. We take this parameter to be normally distributed, since
it reflects a combination of a large number of random influences. To
accommodate the many highly variable instantaneous influences on social
interactions, such as mood, location and medium of communication, we screen
the interactions with a uniformly distributed stochastic noise term,
$-r\leq\xi\leq r$, where $r$ is the width of the distribution.

The factor, $|b_{i}-b_{j}|$ in eq. (4) is introduced to ensure that the final
opinion of an agent after an interaction lies inside or at least near the
range of opinions of the interacting agents. This reflects the fact that
agents can only change to other opinions that are actually presented to them.
Consequently, meaningful changes in agents’ opinions primarily occur in
exchanges with agents not already sharing their opinion. Conversely, any
interaction between two already agreeing agents will only work preserve the
agents’ present opinions.

Since we are concerned with building collective action from a state of
inaction, our simulations begin with network states in which most agents
defect. Except where specified otherwise, they begin from an uncorrelated
Gaussian distribution of agent behaviours with mean $-0.25$ and standard
deviation $=0.15$.

We use the Holme-Kim algorithm [29] to construct our clustered scale-free
networks. We also analyse simple grid networks for pedagogical purposes.
Unless otherwise stated, all of our networks consist of $33^{2}=1089$ nodes,
with average degree $\langle k\rangle=16$. The average degree is obtained as a
combination of the degree of the growth nodes added by the Holme-Kim algorithm
($k_{0}=8$), as well as the accompanying clustering parameter ($k_{t}=3.5$).
We further use external field strength $\phi=0.05$, agent self weight
$w_{i}=0.6$ and an inter-agent weight normal distribution with mean $0.5$ and
standard deviation $0.15$. Furthermore, the uniform interaction noise term
$\xi$ is bounded by $r=0.1$. Under these parameters, which we call standard,
the network always converges to a fully cooperative state, allowing us to
focus our investigation on the dynamics of this process. All of the above
parameters are summarized in table I.

The stubbornness and friendship parameter values were loosely chosen in the
mid range because with these parameters the model behaves in a way that is
qualitatively similar to what is observed in the real world: people can
influence each other but are generally reluctant to change their opinions, and
this varies widely due to many parameters [30]. We propose that refining this
aspect of the model is an interesting avenue for future work, based on data
from ongoing empirical work on opinion change.

Parameter/Variable | Represents | Value
---|---|---
Agent behaviour ($b$) | The strategy (cooperative or defecting) | -
| that an agent operates |
External field ($\phi$) | Global factors: | 0.05
| Policy regulation, mass media, morals, etc |
Self weight ($w_{i}$) | Resistance to influence: | 0.6
| Stubbornness, ideological commitment, etc |
Agent-agent weight ($w_{ij}$) | Susceptibility | 0.5
| Friendship, animosity, charisma, etc |
Agent-agent weight STD | Susceptibility variance | 0.15
Initial cooperation ratio | Initial average cooperativity | $-0.25$
Initial cooperation STD | Variance in initial cooperativity | 0.15
Randomness ($r$) | Noise intensity in agent interaction | 0.1
Growth degree | Degree of nodes added during network growth | 8
| as introduced by the Holme-Kim algorithm |
Clustering | Relative level of clustering of networks | 0.5
| as introduced by the Holme-Kim algorithm |
TABLE I: Summarizing all model parameters and variables to provide an overview
of the model’s components. Here the reference value of each parameter is
given, which is used throughout this paper unless otherwise explicitly stated.

We study the collective dynamics across multiple simultaneously evolving
networks. By obtaining data about the average cooperativity and standard
deviation of cooperativty of agents in the networks, we can learn about what
factors impact the network evolution and how. Throughout this paper, whenever
we refer to the _standard deviation_ , we are referring to the average of
standard deviations across the individual networks. The average state contains
quantitative information about the evolution of the system, such as in which
direction it is evolving, and at which rate. Combining this with the standard
deviation, and prior knowledge about the network topology, we can draw
qualitative conclusions about how the network is evolving and what mechanisms
drive this evolution.

All simulations were run using `Python 3.7.2` with the `Networkx 2.3` (for
network support) and `Community 1.0.0b1` (for the Louvain algorithm [31])
packages. Native parallelization routines were provided by the
`multiprocessing` package.

Wherever we refer to the convergence rate in quantitative terms, we use the
derivative at average cooperativity zero as an indication of the this rate.
The derivative itself was found by linear regression around the data point
closest to the root of the curve, we used a 10 point regression to safely
eliminate numerical instabilities.

For a more thorough explanation of the model parameter values, as well as the
model’s dependence on these, we refer to the Supplementary Material.

## ACKNOWLEDGMENTS

This work was supported by Swedish Research Council (Vetenskapsrådet) grant
2015-04962, Austrian Science Fund (FWF) grant P29252, Austrian Research Agency
(FFG) grant 882184 and European Research Council (ERC) grant 682472-MUSES. We
acknowledge Signe Savén for ideas that inspired this research, and Tuan Pham
for enlightening discussions. We also extend our gratitude to Effective
Altruism for promoting collective action on important issues.

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Supplementary material: Dynamics of collective action to conserve a large
common-pool resource

##  V Parameter dependence

In order to ensure the fidelity of the model across the parameter space, we
here present sweeps over the model parameters. In summary, when the parameters
are varied, the model behaves as expected based on the findings presented in
our main text. We comment below on the parameter dependence in the order of
their appearance in figure 7. Here we used 200 network realizations per
simulation.

###  V-A External field ($\phi$), fig 7a

Here we expand the interval over which the external field is varied. We see
how an extremely strong positive external field (0.25) eliminates the initial
decrease of the average state. This demonstrates that the external field is so
strong that every single interaction increases the cooperativity. Beyond this
point, increasing the external field further produces no qualitative changes.
However, we see an even faster convergence to absolute cooperation.

At the other extreme, of low external field (0.01), we observe that the
average state barely increases with time. The previous initial dip now
resembles a drop in the average state, to a persistently low value. Decreasing
the external field even further, we see a rapid drop to complete defection,
from which the system never recovers. Its monotonic decline closely mirrors
the increase we see for extremely positive external fields.

###  V-B Stubbornness ($w_{i}$), fig 7b

Whereas one might expect a monotonic dependence of the cooperativity on the
stubbornness, we instead see an optimal value at or near 0.6. We understand
this intuitively as follows. At high stubbornness, the entire population is
resistant to change and this slows the convergence. At low stubbornness,
agents are extremely susceptible to the opinion of any agent they interact
with and the convergence is dominated by the random selection of interaction
partners, rather than by the dynamics of the interaction, resulting in a very
small net increase.

###  V-C Initial state, fig 7c

The total time to reach a certain proximity near the final state depends on
the initial state, because for lower initial cooperativity, there are more
defectors to convince. For unfavourable enough conditions the abundance of
defectors will be so large that they completely overtake the cooperators, and
the networks will on average go fully defecting. The red line for -0.5 levels
off at a value not equal to 1 or -1. This is because we are plotting an
average over many realisations of the network, and in this case the final
result (1 or -1) is so sensitive to the initial conditions and realisation of
the interaction that some networks become fully cooperative and some become
fully defective. This is a finite-size effect.

###  V-D Initial state standard deviation, fig 7d

We see that a larger variance in the initial state distribution corresponds to
faster convergence. This can be understood from the grid figures in the main
text. Some slightly cooperative agents must be present for cooperative
clusters to form. The agent interaction (see Methods) is constructed such that
agents on average can switch opinions only to other opinions that are
presented to them. So, if no cooperators are initially present, none can
appear subsequently either.

###  V-E Friendship ($w_{ij}$), fig 7e

As friendship is implemented in the agent interaction almost identically to
stubbornness, it is not surprising that the effects of changing the two are
qualitatively similar. Friendship acts conversely to stubbornness: weak
friendship makes it easier for an agent to retain their state and strong
friendship makes the agent susceptible to changing its state.

###  V-F Friendship standard deviation, fig 7f

Since we have chosen a favourable friendship value, giving it a large standard
deviation introduces more nodes that either have very strong or weak
friendships. This slows down the convergence, as described in the previous
subsection.

###  V-G Randomness, fig 7g

The randomness parameter in the interaction does not strongly influence the
convergence rate. Each individual realisation can vary more when the
randomness is large, but the average convergence does not change.

(a) External field

(b) Stubbornness.

(c) Initial cooperativity.

(d) Initial cooperativity standard deviation.

(e) Friendship.

(f) Friendship standard deviation.

(g) Randomness.

Figure 7: Summarizing parameter dependence of the model, c.f. table I in the
methods section of the main text.


CORRELATED COHERENT STATES - QUANTUM ANALOGUE OF THERMAL STATES
(ON THE PROBLEM OF INCORPORATING THERMODYNAMICS INTO QUANTUM THEORY  )

A.D.Sukhanov1, O.N. Golubjeva2, and V.G.Bar’yakhtar3

1Theoretical Physics Laboratory, Joint Institute for Nuclear Research, Dubna,
Russia<EMAIL_ADDRESS>
2People’s Friendship University of Russia, Moscow, Russia<EMAIL_ADDRESS>
3 Institute of Magnetism, National Academy of Sciences of Ukraine, Kiev,
Ukraine<EMAIL_ADDRESS>
###### Abstract

In this paper, we show that correlated coherent states (CCSs) are the most
adequate candidates for the role of quantum analogues of the thermal states.
The main result of our study reduces to the fact that quantum thermal effects
under conditions of the equilibrium of an object with the stochastic
environment at any temperatures can be explained consistently only on the
basis of the set of CCSs.

## 1\. Wave functions of correlated coherent states of an arbitrary vacuum

In our paper [1], we proposed an original approach to the incorporation of
stochastic thermodynamics into quantum theory. It is based on the concept of
consistent inclusion of the holistic stochastic environmental action
consisting of the thermal environmental action in addition to the quantum one.
Bogoliubov arbitrary vacua were used as a model of the generalized thermostat.
In this case, it becomes possible to bring the set of quantum states described
by the complex wave function in correspondence with equilibrium thermal
states. In the next paper [2], it was shown that squeezed correlated states
can adequately describe only the cases of the contact with the cold vacuum,
which exists at the zero Kelvin temperature, which is not significant for
thermodynamics.

Now we consider a system (a quantum oscillator) under the conditions of the
contact with the thermostat modeled by an arbitrary vacuum. Under these
conditions, this system can be described by the complex wave function that is
dependent on the parameter $\alpha\neq\frac{\pi}{2}$. 111The states
corresponding to $\alpha=0$ were considered in our paper [2] and were
qualified as squeezed correlated states (SCSs), which are inadequate to the
thermal states. In the coordinate representation, in the general case, it has
the form

$\psi_{\alpha}(q)=[2\pi(\Delta
q_{0})^{2}\frac{1}{\cos\alpha}]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{0})^{2}}e^{i\alpha}\right\\}.$ (1)

Here,

$(\Delta q_{0})^{2}=\frac{\hbar}{2\gamma},\;\;\;\;\;\;$ (2)

where the coefficient $\gamma>0$ and $\hbar$ is the Planck constant. The
coordinate variance $\overline{(\Delta q_{\alpha})^{2}}\equiv(\Delta
q_{\alpha})^{2}$ in the arbitrary-vacuum state $|\psi_{\alpha}\rangle$
calculated using wave function (1) has the form

$(\Delta
q_{\alpha})^{2}=\int_{-\infty}^{+\infty}\psi^{*}_{\alpha}q^{2}\psi_{\alpha}dq=\frac{\hbar}{2\gamma}\cdot\frac{1}{\cos\alpha}=(\Delta
q_{0})^{2}\cdot\frac{1}{\cos\alpha}.$ (3)

Similar calculations for the momentum variance $\overline{(\Delta
p_{\alpha})^{2}}\equiv(\Delta p_{\alpha})^{2}$ in this state
$|\psi_{\alpha}\rangle$ lead to the result

$(\Delta
p_{\alpha})^{2}=\frac{\hbar\gamma}{2}\cdot\frac{1}{\cos\alpha}=(\Delta
p_{0})^{2}\cdot\frac{1}{\cos\alpha},$ (4)

where $\dfrac{\hbar\gamma}{2}\equiv(\Delta p_{0})^{2}.$

We show that wave function (1) is completely identical to the state of the
contact with the arbitrary vacuum $\psi_{\tau,\phi}$, which was found in [3]
using the Bogoliubov $(u,v)$-transformations from the equations

$\hat{b}\psi_{\tau,\varphi}(q,\omega)=0,$ (5)

where the operator $\hat{b}$ is the quasiparticle annihilation operator for
the arbitrary vacuum, or

$\frac{d\psi_{\tau,\varphi}}{dq}+\frac{u-v}{u+v}\cdot\frac{\omega}{\hbar}\;q\;\psi_{\tau,\varphi}=0$
(6)

In the general case [4], the complex functions $u$ and $v$ containing in (6)
are determined in terms of the free parameters $(\tau,\theta,\varphi)$ as
follows:

$u=\mathop{\rm ch}\tau\cdot e^{i\varphi};\;\;\;\;v=\mathop{\rm sh}\tau\cdot
e^{-i\varphi}.$ (7)

In this case, $(\tau,\theta,\varphi)$ can be interpreted as Euler angles,
which are used to parametrize the group of rotations $O$(3). Without loss of
generality, we assume that $\theta=0$ in this case.

The solution of Eq. (6) for arbitrary $\tau$ and $\varphi$ has the form of the
complex Gaussoid

$\psi_{\tau,\varphi}(q,\omega)=C\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{0})^{2}}\cdot\frac{u-v}{u+v}\right\\},$ (8)

where $(\delta q_{0})^{2}\equiv\dfrac{\hbar}{2\omega}$.

If formulas (7) and the normalization conditions are taken into account,
expression (8) becomes

$\psi_{\tau,\varphi}(q,\omega)=\left[2\pi(\ \Delta
q_{\tau,\varphi})^{2}\right]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{\tau,\varphi})^{2}}(1-i\beta_{\tau,\varphi})\right\\}.$ (9)

Here,

$(\Delta q_{\tau,\varphi})^{2}=(\Delta q_{0})^{2}(\mathop{\rm
ch}2\tau-\mathop{\rm sh}2\tau\cdot\cos 2\varphi)$ (10)

$\beta_{\tau,\varphi}=\mathop{\rm sh}2\tau\cdot\sin 2\varphi$ (11)

For the convenience of comparison of different representations (1) and (9) of
the wave function, we endow them with the same forms. Writing the exponent
$e^{i\alpha}$ in (1) in the trigonometric form, we obtain the result

$\psi_{\alpha}(q)=[2\pi(\Delta
q_{0})^{2}\frac{1}{\cos\alpha}]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{0})^{2}}\cos\alpha(1+i\tan\alpha)\right\\}.$ (12)

To make expressions (1) and (12) more similar, we assume

$\beta_{\alpha}\equiv\tan\alpha$ (13)

and use the value of $(\Delta q_{\alpha})^{2}$ in accordance with (3).
Finally, expression (1) can be written in the form

$\psi_{\alpha}=\left[2\pi(\ \Delta
q_{\alpha})^{2}\right]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{\alpha})^{2}}(1-i\beta_{\alpha})\right\\}.$ (14)

It is natural to assume that formulas (1) and (14) obtained using different
initial preconditions and, accordingly, expressed in terms of different
parameters are nevertheless related to identical states. Then the condition
for the coincidence between $(\Delta q_{\alpha})^{2}$ (3) and $(\Delta
q_{\tau,\varphi})^{2}$ (10) can be obtained if it is required that the
following conditions be satisfied at the same time:

$(\mathop{\rm ch}2\tau-\mathop{\rm sh}2\tau\cdot\cos
2\varphi)\Leftrightarrow\frac{1}{\cos\alpha}$ (15)

$\mathop{\rm sh}2\tau\cdot\sin 2\varphi\Leftrightarrow\tan\alpha.$ (16)

This turns out to be possible if in (15) $\cos 2\varphi=0$ and in (16) $\sin
2\varphi=1$, which agrees with the well-known assertion that the correlated
coherent states (CCS) in the $(u,v)$-transformations are fixed by the
parameter $\varphi=\frac{\pi}{4}.$ Thus, we demonstrated that the states
$\psi_{\alpha}(q)$ (1) and
$\psi_{\tau,\varphi}(q)\Big{|}_{\varphi=\frac{\pi}{4}}$ (14) under condition
$\tan\alpha=\sinh 2\tau$ agree well with each other.

## 2\. Correlated coherent states as thermal ones

To study the possibility of endowing the states $|\psi_{\alpha}\rangle$ with
the meaning of thermal states, it is necessary to bring the parameter $\alpha$
in correspondence with the temperature that has no pre-image in quantum
mechanics. To do this, we consider the expression for the Planck energy

$\mathcal{E}_{Pl}=\frac{\hbar\omega}{2}\coth\frac{\hbar\omega}{k_{B}T}.$ (17)

We note that this formula is strictly equilibrium and corresponds to the
Kelvin temperature $T$. Therefore, based on formula (17), we can bring the
parameter $\alpha\neq\frac{\pi}{2},0$ in correspondence with the temperature
explicitly. In what follows, we use the fact that the average values of the
kinetic $\overline{K}$ and potential $\overline{U}$ energies are equal for a
quantum oscillator in the thermal-equilibrium state, so that from formula
(17), we obtain

$\overline{K}=\overline{U}=\frac{\mathcal{E}_{Pl}}{2}=\frac{\hbar\omega}{4}\coth\frac{\hbar\omega}{k_{B}T}.$
(18)

Taking into account that $\overline{K}=\cfrac{1}{2m}(\overline{\Delta p)^{2}}$
and $\overline{U}=\cfrac{m\omega^{2}}{2}(\overline{\Delta q)^{2}}$ and
assuming that $m=1$, from formula (18), we obtain the coordinate and momentum
variances, letting the subscript $T$ denote their relation with the Planck
distribution:

$(\Delta q_{\scriptscriptstyle T})^{2}=\frac{\hbar}{2\omega}\mathop{\rm
coth}\frac{\hbar\omega}{k_{B}T}=(\Delta q_{0})^{2}\mathop{\rm
coth}\frac{\hbar\omega}{k_{B}T}.$ (19)

$(\Delta p_{\scriptscriptstyle T})^{2}=\frac{\hbar\omega}{2}\mathop{\rm
coth}\frac{\hbar\omega}{k_{B}T}.$ (20)

As follows from formulas (19) and (20), as $T$ increases, the coordinate and
momentum variances increase synchronously for thermal states (unlike the
SCSs), so that their product $(\mathcal{U}\mathcal{P})_{pq}$ also increases in
this case.

The qualitative distinction between the thermal-like SCSs ($\alpha=0$) and the
thermal states (CCS) can be demonstrated obviously by giving them the
geometric interpretation in the phase plane. For convenience, we put $\hbar=1$
and choose the dimensionless variables $\hat{\mathcal{P}}=\dfrac{\scriptstyle
1}{i}\dfrac{d}{d\mathcal{Q}};\hat{\mathcal{Q}}$. Then, for cold-vacuum states
(SCSs of the $|\psi_{0}\rangle$ type), the quantities $\Delta\mathcal{P}_{0}$
and $\Delta\mathcal{Q}_{0}$ (in view of their equality in accordance with
formulas (3) and (4)) have the meaning of the sides of a single square with
area of $1/4$. At the same time, the thermal CCSs (of the
$|\psi_{\scriptscriptstyle T}\rangle$ type) in this plane correspond to
different squares whose sides and areas increase consistently with
temperature.

We now compare the coordinate $(\Delta q_{\alpha})^{2}$ (3) and momentum
$(\Delta p_{\alpha})^{2}$ (4) variances calculated using wave function (1)
with their corresponding values $(\Delta q_{\scriptscriptstyle T})^{2}$ (19)
and $(\Delta p_{\scriptscriptstyle T})^{2}$ (20) obtained using the Planck
energy. They turn out to be completely identical if we set

$\gamma=\omega$ (21)

and

$\frac{1}{\cos\alpha}=\mathop{\rm coth}\frac{\hbar\omega}{k_{B}T}.$ (22)

Thus, in accordance with relation (22), the parameter $\alpha$ in formula (1)
fixes the states corresponding to the equilibrium at the temperature $T$.

Expression (22) allows representing the exponent $e^{i\alpha}$ in formula (1)
in the trigonometric form, explicitly indicating its relation with the
temperature $T$ in this case. Calculating
$\sin\alpha=\sqrt{1-\tanh^{2}\frac{\hbar\omega}{k_{B}T}}=\dfrac{1}{\cosh\frac{\hbar\omega}{k_{B}T}}$
in advance, we obtain

$e^{i\alpha}=\cos\alpha+i\sin\alpha=\tanh(\frac{\hbar\omega}{k_{B}T})+i\frac{1}{\cosh(\frac{\hbar\omega}{k_{B}T})}=\tanh(\frac{\hbar\omega}{k_{B}T})\left[1+i\frac{1}{\sinh\frac{\hbar\omega}{k_{B}T}}\right]$
(23)

Returning to wave function (1), we can demonstrate the explicit temperature
dependence of its amplitude and phase by labeling it with the subscript $T$

$\psi_{\scriptscriptstyle T}(q)=[2\pi(\Delta q_{0})^{2}\mathop{\rm
coth}\frac{\hbar\omega}{k_{B}T}]^{-1/4}\exp\left\\{-\frac{q^{2}}{4(\Delta
q_{0})^{2}}\tanh(\frac{\hbar\omega}{k_{B}T})[1+i\frac{1}{\sinh\frac{\hbar\omega}{k_{B}T}}]\right\\}.$
(24)

## 3\. Saturation of the Schrödinger UR in the correlated coherent states

It is interesting to analyze certain CCS peculiarities. As is well known, the
most general _Schrödinger uncertainties relation_ (SUR) for the coordinate and
momentum has the form

$\Delta p\cdot\Delta
q\geqslant\big{|}\langle\psi|\delta{\hat{p}}\cdot\delta{\hat{q}}|\psi\rangle\big{|}.$
(25)

Here, the left-hand side of the inequality contains the product of the
momentum and coordinate uncertainties in the state $|\psi\rangle$ calculated
using the definition

$(\Delta
p)^{2}\equiv\langle\psi|(\delta{\hat{p}})^{2}|\psi\rangle;\;\;\;\;(\Delta
q)^{2}\equiv\langle\psi|(\delta{\hat{q}})^{2}|\psi\rangle.$ (26)

The expression in the right-hand side

$\Big{|}\langle\psi|\delta{\hat{p}}\cdot\delta{\hat{q}}|\psi\rangle\Big{|}=\Big{|}\langle\delta
p|\delta q\rangle\Big{|}$ (27)

has the meaning of the correlator of object momentum and coordinate
fluctuations in the same state, which is expressed in terms of the fluctuation
operators $\delta\hat{p}$ and $\delta\hat{q}$. We recall that the left- and
right-hand sides of relation (22) must be calculated independently.

In the arbitrary-vacuum state, the equality for the means

$(\Delta p_{\alpha})^{2}=\gamma^{2}(\Delta q_{\alpha})^{2}$ (28)

is valid [5] so that the _left-hand_ side of SUR (25) becomes equal to

$\Delta p_{\alpha}\cdot\Delta q_{\alpha}=\gamma\Delta q_{\alpha}^{2}.$ (29)

Calculating the correlator in the _right-hand_ side of SUR (25) with respect
to the state $|\psi_{\alpha}\rangle$, we obtain

$\Big{|}\langle\psi_{\alpha}|\hat{p}\cdot\hat{q}|\psi_{\alpha}\rangle\Big{|}=\sqrt{\frac{\hbar^{2}}{4}\tan^{2}\alpha+\frac{\hbar^{2}}{4}}=\frac{\hbar}{2}\cdot\frac{1}{\cos\alpha}.$
(30)

We note that the correlator here is expressed in terms of the $\alpha$ \-
phase of the wave function. Taking into account that in accordance with (3),
$\dfrac{1}{\cos\alpha}=\dfrac{2}{\hbar}\cdot\gamma\Delta q_{\alpha}^{2},$ we
reduce correlator (30) to the final form

$\Big{|}\langle\psi_{\alpha}|\hat{p}\cdot\hat{q}|\psi_{\alpha}\rangle\Big{|}=\gamma\Delta
q_{\alpha}^{2},$ (31)

which completely coincides with product (29) of uncertainties, i.e., with the
left-hand side of the SUR (25).

Thus, we see that the correlated state $|\psi_{\alpha}\rangle$ is outlined by
the fact that the SUR in it indeed acquires the form of the  _equality_ ,
i.e., becomes saturated

$\Delta p_{\alpha}\cdot\Delta
q_{\alpha}=\Big{|}\langle\psi_{\alpha}|\hat{p}\cdot\hat{q}|\psi_{\alpha}\rangle\Big{|}.$
(32)

The interpretation of the radicand in (30) can be related to that of a similar
expression for the cold-vacuum state $\psi_{0}$. For $\alpha=0$, saturated SUR
(30) transforms into the saturated Heisenberg UR

$\Delta p_{0}\cdot\Delta
q_{0}=\big{|}\langle\psi_{0}|\frac{1}{2}[\hat{p},\hat{q}]|\psi_{0}\rangle\big{|}=\frac{\hbar}{2}.$
(33)

Here, as is known, $\dfrac{\hbar}{2}$ is the measure of a purely quantum
environmental action occurring in the cold vacuum. Thus, for $\alpha=0$, the
correlator
$\big{|}\langle\psi_{\alpha}|\hat{p}\cdot\hat{q}|\psi_{\alpha}\rangle\big{|}$
has a value that is as minimum as possible. As was expected, the state
$|\psi_{0}\rangle$ indeed has the meaning of the state of equilibrium with the
cold vacuum because it corresponds to the minimum value of the vacuum energy
$\dfrac{\hbar\omega}{2}$.

It is adopted to regard the Heisenberg relation as a fundamental equality
reflecting the presence of unavoidable purely quantum effects in the Nature.
We assume that in comparison with (33), the origin of the additional term in
the radicand of correlator (30) in the form
$\dfrac{\hbar^{2}}{4}\tan^{2}\alpha$ is related to precisely the inclusion of
the thermal influence of the arbitrary vacuum, which is manifested in the
complex character of the wave function. The fact that the macroparameter of
the effective quantum-thermostat action

$\mathbb{J}_{\alpha}\equiv\sqrt{\frac{\hbar^{2}}{4}\tan^{2}\alpha+\mathbb{J}_{0}^{2}}=\frac{\hbar}{2}\frac{1}{\cos\alpha},$
(34)

(here, $\mathbb{J}_{0}^{2}=\dfrac{\hbar}{2}$ relates to the purely quantum
influence) which was introduced previously in the framework of stochastic
thermodynamics [6], coincides with the right-hand side of formula (30) is
evidence in favor of this assertion. This fact gives grounds to assume from
now on that for the arbitrary vacuum, saturated SUR (32) also corresponds to
the more general equilibrium state $|\psi_{\alpha}\rangle$ occurring at the
simultaneous presence of the quantum and thermal actions. Thus, among the
functions providing the saturation of SUR (32), there exist the functions
$\psi_{\alpha}(q)\Big{|}_{\alpha\neq 0,\frac{\pi}{2}}$, which allow taking
into account extra thermal effects in addition to the quantum ones in a
certain temperature range. Thus, the CCSs can be regarded as quantum analogues
of thermal states in a sufficiently substantiated way.

References

[1] A.D. Sukhanov and O.N. Golubjeva. Toward a quantum generalization of
equilibrium statistical thermodynamics: $(\hbar,k)$\- dynamics. Theoretical
and Mathematical Physics, 160(2): 1177 (2009)

[2] A.D. Sukhanov, O.N. Golubjeva, V.G. Bar’yakhtar. arXiv:1211.3017v1 [quant-
ph] 13 Nov 2012

[3] A.D. Sukhanov and O.N. Golubjeva. Arbitrary vacuum as a model of
stochastic influence of environment: on the problem of incorporating
thermodynamics into quantum theory. Physics of Particles and Nuclei Letters,
Vol. 9, No. 3, p. 303. Pleiades Publishing, Ltd., 2012.

[4] V.V. Dodonov and V.I.Man’ko. Trudy Fiz. Inst. Lebedev, 1987. 183, 71

[5] J.von Neumann. Mathematical Foundations of Quantum Mechanics. Princeton
University Press,1996

[6] Thermodynamics. Ed. Tadashi Mizutani, 2011. InTech. P.P. 73-98.