###### Abstract

Using mobile phone records and information theory measures, our daily lives
have been recently shown to follow strict statistical regularities, and our
movement patterns are, to a large extent, predictable. Here, we apply entropy
and predictability measures to two datasets of the behavioral actions and the
mobility of a large number of players in the virtual universe of a massive
multiplayer online game. We find that movements in virtual human lives follow
the same high levels of predictability as offline mobility, where future
movements can, to some extent, be predicted well if the temporal correlations
of visited places are accounted for. Time series of behavioral actions show
similar high levels of predictability, even when temporal correlations are
neglected. Entropy conditional on specific behavioral actions reveals that in
terms of predictability, negative behavior has a wider variety than positive
actions. The actions that contain the information to best predict an
individual’s subsequent action are negative, such as attacks or enemy
markings, while the positive actions of friendship marking, trade and
communication contain the least amount of predictive information. These
observations show that predicting behavioral actions requires less information
than predicting the mobility patterns of humans for which the additional
knowledge of past visited locations is crucial and that the type and sign of a
social relation has an essential impact on the ability to determine future
behavior.

###### keywords:

human behavior; mobility; computational social science; online games; time-
series analysis; social dynamics

10.3390/e16010543 16 Received: 01 December 2013; in revised form: 16 December
2013 / Accepted: 30 December 2013 /
Published: 16 January 2014 Entropy and the Predictability of Online Life
Roberta Sinatra 1 and Michael Szell 2,* E-Mail<EMAIL_ADDRESS>Tel.:
+1-617-324-4474; Fax: +1-617-258-8081.

##  1 Introduction

Capturing the regularities of our daily lives and the occasional deviations
from the steady diurnal patterns has traditionally eluded an all-encompassing
approach, due to tremendous efforts in monitoring detailed human activities
over long times and the bias in behavior caused by obtrusive methods of
observation Rosenthal (1991). However, the recent ability to address questions
in social science by using huge datasets that have emerged over the past
decades as a result of digitalization has opened previously unimaginable ways
of conducting research in the field Lazer et al. (2009).

On the one hand, these new datasets give a highly detailed protocol of our
ordinary lives, for example, in the form of mobile phone data, which enables a
deeper understanding of the regularities in our mobility patterns González et
al. (2008); Schneider et al. (2013), and how the regularities in human
behavior are reflected in the geographic regions that emerge from our
interactions Sobolevsky et al. (2013). On the other hand, from a previous
point of view, “extraordinary” new forms of human behavior can now be observed
online, where the full set of all actions performed in the system is typically
available for study, spanning an even deeper level of detail. Online social
networking services, such as Twitter or Facebook, or discussion forums allow
new insights into the rhythms of social actions and interactions, as expressed
online Golder and Macy (2011); Golder et al. (2007); Mitrović and Tadić
(2010); Tadić et al. (2013), and how these interactions relate to the
underlying offline events Szell et al. (2013). An even richer insight can be
gained into human-led lives that unfold _entirely_ in artificial online
environments, such as in persistent, massive multiplayer online games, where
human-controlled characters spend their whole virtual lives within an online
world interacting with other characters Bainbridge (2007). The playing of
online games is one of the most wide-spread forms of collective human behavior
in the world; the “massive multiplayer” aspect allows one to not only study
single individuals, but also collective behavioral phenomena that typically
emerge in complex social systems Ball (2003). Here, data can be available of
all actions, decisions and interactions between many thousands of individuals
over long time spans Szell and Thurner (2012), allowing understanding of the
structure and evolution of socio-economic networks Szell et al. (2010); Szell
and Thurner (2010); Klimek and Thurner (2013), mobility Szell et al. (2012) or
the emergence of good conduct Thurner et al. (2012) and elite structures
Corominas-Murtra et al. (2013) in large social systems.

Going hand in hand with the new availability of large-scale, longitudinal
behavioral datasets of various kinds, well-known methods from the mathematical
and physical sciences, especially statistical physics and information theory
Castellano et al. (2009); Sinatra et al. (2010); Gallotti et al. (2012), have
been extended and/or re-applied successfully in this context. In particular,
principal component analysis and the concept of “eigenbehavior” has been used
to quantify behavioral regularities and to predict future activities in the
daily lives of a group of 100 subjects Eagle and Pentland (2009). Similarly,
information theory measures provide an adequate quantification between uniform
distributions (maximal entropy) and maximally uneven distributions of states
(minimal entropy), which, in the case of human behavior, can inform us about
the extent of uniformity and, thus, predictability in our activity patterns.
The concept of entropy has been applied specifically to assess the
predictability of mobility patterns Song et al. (2010); Gallotti et al.
(2013), of economic behavior Krumme et al. (2010), the order of human-built
structures, such as urban street networks Gudmundsson and Mohajeri (2013), or
the complexity of online chatting behavior Takaguchi et al. (2011); Wang and
Huberman (2012); Tadić et al. (2013). Further, a theoretical framework for
non-extensive entropies has been recently developed that might be well
applicable tocomplex systems Hanel and Thurner (2011); Hanel et al. (2012).

##  2 Behavior and Mobility Data of Human Players in the Online World, Pardus

Here, our goal is to apply classical entropy measures to study the patterns of
various kinds of behavior in a single, closed socio-economic system, as
generated by thousands of users in the online game “Pardus” Szell and Thurner
(2012), to provide an insight into the regularity of life in online worlds
and, eventually, to draw possible conclusions on how humans lead their offline
lives.

###  2.1 The Online World, Pardus

The online world Pardus, www.pardus.at, is a browser-based, massive
multiplayer online game open to the public for over nine years. Over 400,000
users have registered to play so far. The game features three independent,
persistent game universes, which had a defined starting time, but no scheduled
end. There are no predefined goals in the game: many aspects of social life
within Pardus are self-organized, for example, the emergence of social groups
(alliances) and the politics between them. Players are engaged in a multitude
of social activities, i.e., chatting, cultivating friendships, building up
alliances, but also negative interactions, such as destructive attacks, and
economic activities, such as producing commodities in factories and selling
them to other players. We focus on the “Artemis” game universe, in which we
recorded player actions over the first 1,238 consecutive days of the
universe’s existence. Communication between any two players can take place
directly, by using a one-to-one, e-mail-like private messaging system. We
focus on one-to-one interactions between players only and discard indirect
interactions, such as, e.g., participation in chats or forums. There are
global interactions, i.e., interactions that can be performed independently of
the spatial position of players in the game universe, which are communication,
setting and removing friendship or enemy links, or placing a bounty on another
player. The actions of trade or attack, however, need players to meet in
space. All data used in this study are fully anonymized.

###  2.2 Mobility Time Series

For studying regularities in mobility patterns, we use the same dataset of
Pardus player movements that has been used in Szell et al. (2012). A universe
in Pardus can be represented as a network with 400 nodes, called sectors, and
1,160 links. Each sector is like a city, where players can have social
relations or entertain economic activities. Typically, sectors adjacent on the
universe map, as well as a few far-apart sectors, are interconnected by links
that allow players to move from sector to sector. At any point in time, each
sector is usually attended by a large number of players. The universe network
has a large diameter of 27, which means that, on average, players have to move
through a non-negligible number of sectors to traverse the universe. Due to a
limited pool of actions that players can spend on movement, traveling large
distances can take a player several days. Using this dataset, we previously
studied the statistical movement patterns of players and found that locations
are visited in a specific order, leading to strong long-term memory effects
Szell et al. (2012). In detail, we extract player mobility data from day 200
to day 1,200 of the universe’s existence. We discard the first 200 days,
because social networks between players of Pardus have shown aging effects in
the beginning of the universe Szell and Thurner (2010). To make sure we only
consider active players, we select all who exist in the game between the days
200 and 1,200, yielding 1,458 players active over a time-period of 1,000 days.
The sector IDs of these players, i.e., their positions on the universe
network’s nodes, are logged every day at 05:35 GMT.

###  2.3 Behavioral Action Time Series

For studying regularities in behavioral time series, we use the same dataset
of Pardus player actions that has been used in Thurner et al. (2012). Players
can express their sympathy (distrust) toward other players by establishing so-
called friendship (enmity) links. These links are only seen by the player
marking another as a friend (enemy) and the respective recipient of that link.
For more details on the game, see Szell and Thurner (2010). We consider eight
different actions every player can execute at any time. These are
communication (C),trade (T), setting a friendship link (F), removing an enemy
link (forgiving) (X), attack (A), placing a bounty on another player
(punishment) (B), removing a friendship link (D) and setting an enemy link
(E). While C, T, F and X can be associated with positive actions, A, B, D and
E are hostile or negative actions. We classify communication as positive,
because only a negligible part of communication takes place between enemies
Szell and Thurner (2010). Following a previous formalism Szell and Thurner
(2010), we say that positive actions have a positive sign, and negative
actions have a negative sign. The alphabet, $\mathcal{X}$, of all possible
dyadic actions happening in each player’s life therefore spans 16 letters:
eight possible performed actions (four negative, four positive) and eight
possible received actions (four negative, four positive). We denote received
actions with the suffix, $r$, e.g., $A_{r}$ for a received attack. Due to the
heterogeneous activity patterns of players, we operate in action-time rather
than in actual time; for example, indices of $t$ and $t-1$ denote that two
actions were subsequent, regardless of whether the actual time difference was
seconds or weeks Thurner et al. (2012). From all sequences of all 34,055
Artemis players who performed or received an action at least once within 1,238
days, we removed players with a life history of less than 1,000 actions,
leading to the set of the most active 1,758 players that are considered
throughout this work.

##  3 Entropy and Predictability Measures

To study the regularity and predictability of behavior from the discrete time
series, we use three entropy measures. Following Song et al. (2010), we call
the binary logarithm of the number of distinct states, $N_{i}$, of a player,
$i$, the _random entropy_ :

$S^{\mathrm{rand}}_{i}=\log_{2}N_{i}$ (1)

In the case of mobility, “states” refer to the 400 possible sectors in the
universe visitable at a given point in time by a player. The maximal possible
random entropy is $S^{\mathrm{rand}}=\log_{2}400\approx 8.6$, reached when all
sectors are visited at least once. In the case of behavioral actions, a state
can be one of the 16 possible action or received action types; here, the
maximum possible random entropy is $S^{\mathrm{rand}}=\log_{2}16=4$.

The Shannon entropy, $S^{\mathrm{unc}}_{i}$, of a player, $i$, is defined as:

$S^{\mathrm{unc}}_{i}=-\sum_{x\in\mathcal{X}_{i}}p_{i}(x)\log_{2}p_{i}(x)$ (2)

where $p_{i}(x)$ is the measured probability over the respective time span
that player $i$ has occupied a state, $x$, and $\mathcal{X}_{i}$ is the
ensemble of the $N_{i}$ distinct states. In this context, we call the Shannon
entropy the _temporal-uncorrelated entropy_ , because it captures the entropy
when the temporal order of states is ignored Song et al. (2010). The random
and temporal-uncorrelated entropies are equal,
$S^{\mathrm{rand}}_{i}=S^{\mathrm{unc}}_{i}$, if all of the $N_{i}$ distinct
states, $x$, were occupied with uniform probability $p_{i}(x)=1/N_{i}$ by the
player, $i$. For mobility, the occupation of a single sector over the whole
time span of 1,000 days would result in the smallest possible random and
temporal-uncorrelated entropy of $S^{\mathrm{rand}}=S^{\mathrm{unc}}=0$.

Finally, we make use of the conditional entropy $S^{\mathrm{cond}}_{i}$ of a
player, $i$, capturing the entropy conditional on temporal short-term
correlations over one previous state in the time series,

$S^{\mathrm{cond}}_{i}=-\sum_{x_{t}\in\mathcal{X}_{i}}\sum_{x_{t-1}\in\mathcal{X}_{i}}p_{i}(x_{t-1},x_{t})\log_{2}p_{i}(x_{t}|x_{t-1})$
(3)

with $p_{i}(x_{t-1},x_{t})$ being the probability of occurrence of the pair of
subsequent states, $x_{t-1}$ and $x_{t}$,
$p_{i}(x_{t}|x_{t-1})=p_{i}(x_{t-1},x_{t})/p(x_{t-1})$, the probability of the
state, $x_{t}$, at time $t$ given a preceding state, $x_{t-1}$. The
conditional and temporal-uncorrelated entropies are equal,
$S^{\mathrm{cond}}_{i}=S^{\mathrm{unc}}_{i}$, if there are no temporal
correlations.

It is easy to show that we have $S^{\mathrm{cond}}\leq S^{\mathrm{unc}}\leq
S^{\mathrm{rand}}$ for each user Cover and Thomas (2006). The differences in
these two inequalities quantify the effects of short-term temporal
correlations and the uniformity of the occupation distribution, respectively.
To assess the predictability of specific states or of classes of states, we
also define the conditional entropy for the set of states, $\mathcal{Z}$,

$S^{\mathrm{cond}}_{i}(\mathcal{Z})=-\sum_{x_{t}\in\mathcal{X}_{i}}\sum_{x_{t-1}\in\mathcal{X}_{i}\cap\mathcal{Z}}p_{i}(x_{t-1},x_{t})\log_{2}p_{i}(x_{t}|x_{t-1})$
(4)

which is the conditional entropy given that the previous state belonged to
$\mathcal{Z}$, where $\mathcal{Z}$ can be fixed as any subset of all the
possible states, $\mathcal{X}$. Notice that $S^{\mathrm{cond}}_{i}\equiv
S^{\mathrm{cond}}_{i}(\mathcal{Z})+S^{\mathrm{cond}}_{i}(\mathcal{X}_{i}\setminus\mathcal{Z})$.

Complementary to entropy measures of information content or _unpredictability_
are measures of _predictability_ that denote in a percent value how likely an
appropriate predictive algorithm could foresee an individual’s future behavior
Song et al. (2010). The predictability, $\Pi_{i}^{\bullet}$, of an individual
$i$ is bounded above by:

$S_{i}^{\bullet}=H(\Pi_{i}^{\bullet})+(1-\Pi_{i}^{\bullet})\log_{2}(N_{i}-1)$
(5)

with the binary entropy function:

$H(\Pi_{i}^{\bullet})=\Pi_{i}^{\bullet}\log_{2}(\Pi_{i}^{\bullet})-(1-\Pi_{i}^{\bullet})\log_{2}(1-\Pi_{i}^{\bullet})$
(6)

where $\bullet$ is a placeholder for any of the types, $\mathrm{rand}$,
$\mathrm{unc}$ or $\mathrm{cond}$. Unlike the measure of entropy, which is
well established, the application of this predictability measure to practical
problems is relatively recent. It is based on the idea that predictability is
related to the error probability in guessing the outcome of a discrete random
variable Feder and Merhav (1994). The upper bound given in Equation (5) comes
from Fano’s inequality Fano (1961); Cover and Thomas (2006). For a detailed
discussion on this bound and on possible lower bounds, see Feder and Merhav
(1994); Song et al. (2010).

For being able to study in more detail the effects of memory in the system
Sinatra et al. (2011); Chierichetti et al. (2012), we generalize the
conditional entropy:

$S^{\mathrm{cond},k}_{i}=-\sum_{x_{t}\in\mathcal{X}_{i}}\cdots\\!\sum_{x_{t-k}\in\mathcal{X}_{i}}p_{i}(x_{t-k},\ldots,x_{t})\log_{2}p_{i}(x_{t}|x_{t-k},\ldots,x_{t-1})$
(7)

where $k$ is an integer denoting the memory window. Note that
$S^{\mathrm{\mathrm{cond},1}}_{i}\equiv S^{\mathrm{cond}}_{i}$ and that we can
identify $S^{\mathrm{\mathrm{cond},0}}_{i}$ with $S^{\mathrm{unc}}_{i}$. It
follows from Fano’s inequality Fano (1961) that
$S^{\mathrm{\mathrm{cond},1}}_{i}\geq S^{\mathrm{\mathrm{cond},2}}_{i}\geq
S^{\mathrm{\mathrm{cond},3}}_{i}\geq\cdots$. The differences between
subsequent values in this chain inform us about the gain of predictability
when we increase the memory window one by one. If such a difference starts
becoming negligible from a particular level, $k$ to $k+1$, it means that the
system does not exhibit relevant memory effects beyond a window of $k$ steps.
If this level is at $k=0$, the events are uncorrelated; if at $k=1$, the
system is Markovian, otherwise, it is non-Markovian.

##  4 Results and Discussion

###  4.1 Predictability in Mobility

We applied all entropy and predictability measures to the mobility time
series, Figure 1a,b, respectively. Results show almost identical
predictability behavior for humans in our online world as for the mobility of
humans in geographic space Song et al. (2010); Gallotti et al. (2012). The
distributions for $S^{\mathrm{unc}}$ and $S^{\mathrm{rand}}$ are both
qualitatively and quantitatively matching, showing that also online, movements
of human avatars have the same highly predictable patterns when temporal
correlations are accounted for, but are mostly unpredictable when the order of
visitations is ignored. In particular, also here, $S^{\mathrm{rand}}$ peaks
around six, indicating that an individual who chooses her next location
randomly could be found, on average, in any of $2^{S^{\mathrm{rand}}}\approx
64$ locations, which is a substantial part of the 400 possible sectors. The
contrasting peak of $S^{\mathrm{cond}}$ below two shows that the actual
uncertainty of a typical player’s location is not 64, but rather, less than
$2^{2}=4$ sectors. The conditional entropy, $S^{\mathrm{cond}}$, is not
directly comparable to the actual entropy, $S$, in Song et al. (2010), but
shows the same tendency in that temporal correlations are substantial, even if
just having a memory of one. However, for re-creating the statistical features
of mobility thoroughly, longer memory is needed Szell et al. (2012). The peak
of $\Pi^{\mathrm{cond}}$ around $0.9$ means that only in around $10\%$ of
cases does a player choose her location in a manner that appears to be random,
but in $90\%$ of the cases, we can hope to predict her whereabouts with an
appropriate predictive algorithm. This high predictability stands in contrast
to the moderately predictive case given by $\Pi^{\mathrm{unc}}$ peaking around
$0.5$ and the highly unpredictive case of $\Pi^{\mathrm{rand}}$ peaking
narrowly and close to zero.

Figure 1: The distribution of (a) entropy and (b) the predictability measures
of the mobility of the Pardus players. Both are almost identical to the
mobility of humans in geographic space Song et al. (2010): Each considered
entropy measure improves predictability substantially, from considering the
uniformity of occupation to additionally short-term temporal correlations.

###  4.2 Predictability in Behavioral Actions

A similar picture to mobility arises for behavioral actions. Figure 2a,b,
respectively, report the entropy and predictability distributions of all 16
types of actions and received actions. Here, $S^{\mathrm{rand}}$ is peaked at
four, showing that most players are making full use of their behavioral
possibilities of $16=2^{4}$ action and received action types in the course of
their online lives. However, the sharp drop to the distribution of
$S^{\mathrm{unc}}$, which peaks around two, shows that, in practice, most of
these actions and received actions are focused on around $2^{2}=4$ action or
received action types only. The even narrower curve of $S^{\mathrm{cond}}$,
which peaks around $1.5$, with a corresponding peak of $\Pi^{\mathrm{cond}}$
at $0.8$, demonstrates that the conditional information allows us to predict
$80\%$ of actions. This is only slightly more than the $73\%$ prediction rate
peak from $\Pi^{\mathrm{unc}}$; however, $\Pi^{\mathrm{cond}}$ is distributed
more widely. In conclusion, the predictability gained from considering the
uniformity of occupation is much larger than the predictability gained from
also considering Markovian temporal correlations, as opposed to the case of
mobility where temporal correlations add substantial predictive value.

Figure 2: Distribution of (a) entropy and (b) predictability measures of the
behavioral actions of the Pardus players. As in the case of mobility,
behavioral actions are highly regular and predictable. However, the
predictability gained from considering the uniformity of occupation is much
larger than the predictability gained from also considering temporal
correlations.

One previous key observation on Pardus players is the fundamental structural
and dynamic difference between positive and negative action types and their
interaction networks Szell et al. (2010); Szell and Thurner (2012, 2010);
Thurner et al. (2012). To see if this difference is also apparent in the
extent of predictability, we plotted the distribution of the conditional
entropy of the players given that the previous action or received action was
positive/negative (Figure 3a), i.e., the set $\mathcal{Z}$ in Equation (4)
corresponds to $\mathcal{Z}=\\{C,T,F,X,C_{r},T_{r},F_{r},X_{r}\\}$ or to
$\mathcal{Z}=\\{A,B,D,E,A_{r},B_{r},D_{r},E_{r}\\}$, respectively. We aim to
understand whether the actions that follow positive actions are more
predictable than those that follow negative actions. If the distributions were
identical, the sign of an action would cause no difference in the
predictability of the subsequent action. In fact, although both distributions
peak around $0.55$, showing that there is a moderate amount of predictive
value gained from the information of an action’s sign, the positive
distribution is much more narrow than the negative one, implicating that there
is a much wider range of negative behavior in terms of predictability than
positive behavior. This result suggests that “good” people are much alike, but
“bad” persons behave badly in more various and, sometimes, more unpredictable
ways.

Figure 3: The distribution of the conditional entropy measures of the
behavioral actions of the Pardus players, given that the previous action
belonged to a certain category. (a) Entropy given that the previous action or
received action was positive/negative. The positive and negative distributions
have their maxima both around $0.55$, but the former is much more narrow than
the latter one, showing that there is a much wider range of negative behavior
in terms of predictability than positive behavior. (b) Entropy given that the
previous action was performed/received. Both distributions peak very close to
one, showing that the information of whether an action was performed or
received does, in general, not have a high predictive value. The peak for the
received actions is slightly closer to one than for the performed actions.

The conditional entropy for performed or received actions, i.e.,
$\mathcal{Z}=\\{C,T,F,X,A,B,D,E\\}$ or
$\mathcal{Z}=\\{C_{r},T_{r},F_{r},X_{r},A_{r},B_{r},D_{r},E_{r}\\}$ in
Equation (4), respectively, is peaked very narrowly and close to one for both
cases and slightly more so for received actions; Figure 3b. This observation
shows that the directionality of actions contains much less predictive
information than the sign of an action.

We can further refine the conditional entropy measure by considering single
actions as the condition, i.e., where $\mathcal{Z}$ in Equation (4) is a
singleton, to assess how much each action or received action type allows one
to predict the subsequent action that is about to happen in a player’s life.
The conditional entropy of trade peaks around $1.3$; the distribution of
communication is more wide, peaking around six bits; Figure 4a. Distributions
of received trades and communications are almost identical, only received
communication is slightly more right-skewed than performed actions of
communication. The reason why communication is associated with higher
unpredictability might have to do with the game’s action point system Szell
and Thurner (2010): every action, except the action of communication, costs an
amount of so-called action points for which every player has only a limited
pool. Therefore, players are not limited in their communication behavior, but
are so for trade, friendship markings, etc. The entropy distribution of
friendship marking, $F$, Figure 4b, peaks around one bit and is, therefore,
much less unpredictable. The entropy of enemy marking $E$ peaks even closer to
zero (Figure 4d); all of the actions related to enemy markings, $E$, $E_{r}$,
$X$ and $X_{r}$, show a bimodal distribution with an extra peak at zero, but
this is clearly not the case for friendship markings $F$ or $F_{r}$. This
bimodality could hint towards two different kinds of effects that arise from
enemy marking, where, for example, either the person who makes or removes the
marking immediately predictably sends a message to the recipient in a fraction
of cases, or in the remaining fraction, this does not happen. Finally, the
conditional entropy of received attacks, $A_{r}$, peaks around one, and
performed attacks, $A$, are more wide peaking at a smaller value; Figure 4c.
In all the distributions that deal with friendship or enemy markings, $F$,
$D$, $E$ and $X$, we observe a right-shift of peaks for received actions,
meaning that a player’s next action is more predictable given that a
friend/enemy event happened to her, as opposed to when she performed such an
action towards somebody else. For attacks, however, we see the opposite. It is
unclear what causes this phenomenon or how relevant it is: we can only
speculate that a received attack could have a possibly stronger emotional
impact on a player and, therefore, a more adverse effect on the predictability
on her next action, while this is vice versa for friendship/markings. Further,
it is interesting to note that the removal of a friendship link has a similar
pattern to the addition of an enmity link, suggesting that these two actions
might be closely related, since they have a similar impact on future behavior.
In general, however, the removal of a positive/negative tie cannot always be
put on the same level as the addition of a negative/positive tie, as the
reversed case of friendship addition and enemy removal shows.

Figure 4: The distribution of conditional entropy measures of the behavioral
actions of the Pardus players, given that the previous action was of a certain
type. (a) The distributions for performed and received communication events
($C$ and $C_{r}$) and for performed and received trade events ($T$ and
$T_{r}$). Communication peaks around six bits, trade around $1.3$ bits.
Performed and received actions do not show substantial deviations here. (b)
The distributions for performed and received friendship marking events ($F$
and $F_{r}$) and for performed and received friendship removals ($D$ and
$D_{r}$). The curves peak around one or lower. (c) The distributions for
performed and received attacks ($A$ and $A_{r}$). The former curve peaks below
one; the latter peaks around one and is narrower. (d) The distributions for
performed and received enemy marking events ($E$ and $E_{r}$) and for
performed and received enemy removals ($X$ and $X_{r}$). All the curves peak
once around $0.6$ and another time close to zero.

Finally, we are interested in assessing the memory dependence of the
behavioral actions in the system Chierichetti et al. (2012), i.e., the gain of
predictability from conditional entropies with longer time windows, using the
measures, $S^{\mathrm{cond},k}$, for increasing $k$. Unfortunately, in
practice, these rely on the empirical probabilities,
$p_{i}(x_{t-k},\ldots,x_{t})$, of all possible substrings
$x_{t-k},\ldots,x_{t}$ (see Equation (7)), which would lead to combinatorial
explosion with our alphabet size of 16. For example, $k=3$ would mean
$16^{3}=4,096$ possible substrings of a length of three, many of which do not
exist at all or are statistically not reliable to assess from a dataset of
1,758 players, each having performed up to a few thousand actions. Therefore,
in the following, we used the simplified alphabet of a size of two of negative
or positive actions, allowing feasible calculation of $S^{\mathrm{cond},k}$ up
to $k=5$. The distributions of these entropies are shown in Figure 5. The
distributions converge quickly, showing only a small difference between
$S^{\mathrm{cond},1}$ and $S^{\mathrm{cond},2}$ and almost no difference
between higher order distributions. We quantify these differences via the
Kullback–Leibler divergence between the distributions of the conditional
entropy of subsequent memory levels, $S^{\mathrm{cond},k-1}$ and
$S^{\mathrm{cond},k}$,

$D(k)=D(S^{\mathrm{cond},k}||S^{\mathrm{cond},k-1})=\sum_{j}S^{\mathrm{cond},k}(j)\log\frac{S^{\mathrm{cond},k}(j)}{S^{\mathrm{cond},k-1}(j)}$
(8)

which provides the information gain for going from a memory of a length of
$k-1$ to $k$ Cover and Thomas (2006); Sinatra et al. (2011). A divergence of
zero means that two distributions are identical. The first values from $D(2)$
to $D(5)$ read $0.0097$, $0.0020$, $0.0006$ and $0.0005$. For comparison, the
Kullback–Leibler divergence between $S^{\mathrm{unc}}$ and
$S^{\mathrm{cond}}$, $D(1)$, yields the much higher value of $0.38$, showing
that the system is, to a large part, Markovian and that the predictability
gained from higher-order correlations is negligible.

Figure 5: The convergence of the conditional entropy of the positive and
negative behavioral actions of Pardus players with an increasing memory
window. The difference between $S^{\mathrm{cond},1}$ and $S^{\mathrm{cond},2}$
is small, $D(2)=0.0097$, showing that the system is almost Markovian. For
higher memory windows, we have $D(3)=0.0020$, $D(4)=0.0006$ and $D(5)=0.0005$,
indicating almost identical distributions, which implies that there are
practically no long-term correlations in the signs of behavioral actions.

##  5 Conclusions

We applied three measures of entropy to two sets of time series of the
behavioral actions and the movements of a large number of players in a virtual
universe of a massive multiplayer online game. We found that movements in
virtual human lives follow identical levels of predictability as offline
mobility. This result reasserts previous observations on the similarities
between the online and offline movements of humans Szell et al. (2012) and is
especially striking considering that in online worlds, individuals are not
performing physical movements, but rather, navigate a virtual avatar.

Extending the approach to behavioral time series, also, here, we were able to
provide evidence for high predictability. However, in this case, we found that
due to weaker temporal correlations, there is hope to more easily predict
behavioral actions than the temporally correlated mobility patterns of humans
for which information about previously visited locations is required. Findings
using entropy measures conditional on positive and negative actions suggest
that “good” people are much alike, but “bad” persons behave badly in more
various and, sometimes, more unpredictable ways. Actions containing the
highest predictive information for an individual’s next behavior are negative,
such as attacks or enemy markings, while the positive actions of friendship
marking, trade and communication contain the least amount of predictive
information. However, we show that the system is, to a large part, Markovian
and almost devoid of any higher order correlations when taking into account
the sign of the action, showing that positive or negative behavior is not more
predictable when a longer history of previous actions is accounted for.

The distributions of entropies and predictability found here is strikingly
similar to distributions found for datasets of offline mobility Song et al.
(2010), economic transactions Krumme et al. (2010), online conversations and
online location check-ins Wang and Huberman (2012), therefore suggesting a
possible universality in the limitations of human behavior and its
independence of the concrete medium or context. However, contrary to our
result of little high-order correlations in behavior, a recent study has shown
that the behavior of browsing web pages is, to a large extent, non-Markovian
Chierichetti et al. (2012). Non-extensive entropies have been recently
developed that might be well applied for non-Markovian settings in complex
social systems Hanel and Thurner (2011); Hanel et al. (2012).

Our observations also provide additional evidence for the fundamental
differences in positive and negative behavior that were previously found on
dynamic Thurner et al. (2012) and structural Szell et al. (2010); Szell and
Thurner (2012, 2010) levels. Although previously large-scale evidence has
confirmed in online human behavior a number of known or hypothesized
behavioral phenomena of offline behavior, it is not immediately clear how
asymmetries between positive and negative behavior in our, to some extent,
artificial, online world can be translated to the offline world. Future
research should aim to analyze positive and negative relationships and
behaviors that happen in real-life societies and organizations Labianca and
Brass (2006), especially considering the multi-relational aspect of social
organization Szell et al. (2010); Kivelä et al. (2013). Fine-grained datasets
of socio-economic behavior, such as the one presented, offer the further
possibility of going beyond observations and measurements, to study the
mechanisms and origins of behavior in the view of collective phenomena Tadić
et al. (2013).

##  6 Notes added in proof

During the redaction of this paper, we were made aware of a relevant study
that applied the conditional entropy of signed messages to model growth of
entropy in emotionally charged online dialogues Sienkiewicz et al. (2013).

###### Acknowledgements.

Acknowledgments Roberta Sinatra is supported by the James S. McDonnell
Foundation. Michael Szell thanks the National Science Foundation, the
Singapore-Massachusetts Institute of Technology Alliance for Research and
Technology (SMART) program, the Center for Complex Engineering Systems (CCES)
at King Abdulaziz City for Science and Technology (KACST) and Massachusetts
Institute of Technology (MIT), Audi Volkswagen, Banco Bilbao Vizcaya
Argentaria (BBVA), The Coca Cola Company, Ericsson, Expo 2015, Ferrovial and
all the members of the MIT Senseable City Lab Consortium for supporting the
research. Both authors also thank the Santa Fe Institute for the opportunities
offered during the Complex Systems Summer School 2010, where some ideas for
this project originated. Conflicts of Interest Michael Szell is an associate
of the company, Bayer & Szell OG, which is developing and maintaining the
online game, Pardus, from which the data was collected.

## References

  * Rosenthal (1991) Rosenthal, R.  Meta-Analytic Procedures for Social Research; Sage, Newbury Park, USA: 1991; Volume 6.
  * Lazer et al. (2009) Lazer, D.; Pentland, A.; Adamic, L.; Aral, S.; Barabási, A.L.; Brewer, D.; Christakis, N.; Contractor, N.; Fowler, J.; Gutmann, M.; et al. Computational social science.  Science 2009,  323, 721–723.
  * González et al. (2008) González, M.; Hidalgo, C.; Barabási, A.L.  Understanding individual human mobility patterns.  Nature 2008,  453, 779–782.
  * Schneider et al. (2013) Schneider, C.M.; Belik, V.; Couronné, T.; Smoreda, Z.; González, M.C.  Unravelling daily human mobility motifs.  J. R. Soc. Interface 2013,  10, doi:10.1098/rsif.2013.0246.
  * Sobolevsky et al. (2013) Sobolevsky, S.; Szell, M.; Campari, R.; Couronné, T.; Smoreda, Z.; Ratti, C.  Delineating geographical regions with networks of human interactions in an extensive set of countries.  PLoS One 2013,  8, e81707.
  * Golder and Macy (2011) Golder, S.A.; Macy, M.W.  Diurnal and seasonal mood vary with work, sleep, and daylength across diverse cultures.  Science 2011,  333, 1878–1881.
  * Golder et al. (2007) Golder, S.; Wilkinson, D.; Huberman, B.  Rhythms of Social Interaction: Messaging within a Massive Online network. In Communities and Technologies 2007; Steinfield, C., Pentland, B., Ackerman, M., Contractor, N., Eds.; Springer, London, UK: 2007;  pp. 41–66.
  * Mitrović and Tadić (2010) Mitrović, M.; Tadić, B.  Bloggers behavior and emergent communities in blog space.

Eur. Phys. J. B

2010,  73, 293–301.

  * Tadić et al. (2013) Tadić, B.; Gligorijević, V.; Mitrović, M.; Suvakov, M.  Co-evolutionary mechanisms of emotional bursts in online social dynamics and networks.  Entropy 2013,  15, 5084–5120.
  * Szell et al. (2013) Szell, M.; Grauwin, S.; Ratti, C.  Contraction of Online Response to Major Events. 2013, arXiv preprint arXiv:1308.5190. arXiv.org e-Print archive. Available online: http://arxiv.org/abs/1308.5190 (accessed on 14 January 2014).
  * Bainbridge (2007) Bainbridge, W.  The scientific research potential of virtual worlds.  Science 2007,  317, 472–476.
  * Ball (2003) Ball, P.  The physical modelling of human social systems.  Complexus 2003,  1, 190–206.
  * Szell and Thurner (2012) Szell, M.; Thurner, S.  Social dynamics in a large-scale online game.  Adv. Complex Syst. 2012,  15, 1250064.
  * Szell et al. (2010) Szell, M.; Lambiotte, R.; Thurner, S.  Multirelational organization of large-scale social networks in an online world.  Proc. Natl. Acad. Sci. USA 2010,  107, 13636–13641.
  * Szell and Thurner (2010) Szell, M.; Thurner, S.  Measuring social dynamics in a massive multiplayer online game.  Soc. Netw. 2010,  32, 313–329.
  * Klimek and Thurner (2013) Klimek, P.; Thurner, S.  Triadic closure dynamics drives scaling laws in social multiplex networks.  New J. Phys. 2013,  15, 063008.
  * Szell et al. (2012) Szell, M.; Sinatra, R.; Petri, G.; Thurner, S.; Latora, V.  Understanding mobility in a social petri dish.  Sci. Rep. 2012,  2, doi:10.1038/srep00457.
  * Thurner et al. (2012) Thurner, S.; Szell, M.; Sinatra, R.  Emergence of good conduct, scaling and zipf laws in human behavioral sequences in an online world.  PLoS One 2012,  7, e29796.
  * Corominas-Murtra et al. (2013) Corominas-Murtra, B.; Fuchs, B.; Thurner, S.  Detection of the Elite Structure in a Virtual Multiplex Social System by Means of a Generalized $K$-core. 2013, arXiv preprint arXiv:1309.6740. arXiv.org e-Print archive. Available online: http://arxiv.org/abs/1309.6740 (accessed on 14 January 2014).
  * Castellano et al. (2009) Castellano, C.; Fortunato, S.; Loreto, V.  Statistical physics of social dynamics.  Rev. Mod. Phy. 2009,  81, 591–646.
  * Sinatra et al. (2010) Sinatra, R.; Condorelli, D.; Latora, V.

Networks of motifs from sequences of symbols. Phys. Rev. Lett.

2010,  105, 178702.

  * Gallotti et al. (2012) Gallotti, R.; Bazzani, A.; Rambaldi, S.  Towards a statistical physics of human mobility.  Int. J. Mod. Phys. C 2012,  23, 1250061.
  * Eagle and Pentland (2009) Eagle, N.; Pentland, A.  Eigenbehaviors: Identifying structure in routine.  Behav. Ecol. Sociobiol. 2009,  63, 1057–1066.
  * Song et al. (2010) Song, C.; Qu, Z.; Blumm, N.; Barabási, A.L.  Limits of predictability in human mobility.  Science 2010,  327, 1018–1021.
  * Gallotti et al. (2013) Gallotti, R.; Bazzani, A.; Esposti, M.D.; Rambaldi, S.  Entropic Measures of Individual Mobility Patterns. 2013, arXiv preprint arXiv:1305.1836. arXiv.org e-Print archive. Available online: http://arxiv.org/abs/1305.1836 (accessed on 14 January 2014).
  * Krumme et al. (2010) Krumme, C.; Cebrian, M.; Pentland, A.  Patterns of Individual Shopping Behavior. 2010, arxiv preprint arXiv:1008.2556. arXiv.org e-Print archive. Available online: http://arxiv.org/abs/1008.2556 (accessed on 14 January 2014).
  * Gudmundsson and Mohajeri (2013) Gudmundsson, A.; Mohajeri, N.

Entropy and order in urban street networks.

Sci. Rep.

2013,  3, 3324.

  * Takaguchi et al. (2011) Takaguchi, T.; Nakamura, M.; Sato, N.; Yano, K.; Masuda, N.  Predictability of conversation partners.  Phys. Rev. X 2011,  1, 011008.
  * Wang and Huberman (2012) Wang, C.; Huberman, B.A.  How random are online social interactions?  Sci. Rep. 2012,  2, 633.
  * Hanel and Thurner (2011) Hanel, R.; Thurner, S.  A comprehensive classification of complex statistical systems and an axiomatic derivation of their entropy and distribution functions.  Europhys. Lett. 2011,  93, 20006.
  * Hanel et al. (2012) Hanel, R.; Thurner, S.; Gell-Mann, M.  Generalized entropies and logarithms and their duality relations.  Proc. Natl. Acad. Sci. USA 2012,  109, 19151–19154.
  * Cover and Thomas (2006) Cover, T.; Thomas, J.

Elements of Information Theory 2nd Edition (Wiley Series in Telecommunications

and Signal Processing), 2nd ed.; Wiley-Interscience, New York, USA:  2006\.

  * Feder and Merhav (1994) Feder, M.; Merhav, N.  Relations between entropy and error probability.  IEEE Trans. Inf. Theory 1994,  40, 259–266.
  * Fano (1961) Fano, R.  Transmission of Information; MIT Press, Cambridge, USA:  1961\.
  * Sinatra et al. (2011) Sinatra, R.; Gómez-Gardeñes, J.; Lambiotte, R.; Nicosia, V.; Latora, V.  Maximal-entropy random walks in complex networks with limited information.  Phys. Rev. E 2011,  83, 030103.
  * Chierichetti et al. (2012) Chierichetti, F.; Kumar, R.; Raghavan, P.; Sarlós, T.  Are web users really markovian?  In Proceedings of the 21st International Conference on World Wide Web, Lyon, France, 16–20 April 2012; ACM, New York, USA: 2012;  pp. 609–618.
  * Labianca and Brass (2006) Labianca, G.; Brass, D.J.  Exploring the social ledger: Negative relationships and negative asymmetry in social networks in organizations.  Acad. Manag. Rev. 2006,  31, 596–614.
  * Kivelä et al. (2013) Kivelä, M.; Arenas, A.; Barthelemy, M.; Gleeson, J.P.; Moreno, Y.; Porter, M.A.  Multilayer Networks. 2013, arXiv preprint arXiv:1309.7233. arXiv.org e-Print archive. Available online: http://arxiv.org/abs/1309.7233 (accessed on 14 January 2014).
  * Sienkiewicz et al. (2013) Sienkiewicz, J.; Skowron, M.; Paltoglou, G.; Hołyst, J.  Entropy-growth-based model of emotionally charged online dialogues.  Adv. Complex Syst. 2013,  16, 1350026.


# Scrambling and the black hole atmosphere

Ted Jacobson1 and Phuc Nguyen2,3 1 Maryland Center for Fundamental Physics
University of Maryland
College Park, MD 20742

2 Department of Physics and Astronomy
Lehman College, City University of New York
Bronx NY 10468, USA

3 Department of Mathematics and
Haifa Research Center for Theoretical Physics and Astrophysics
University of Haifa, Haifa 3498838, Israel

###### Abstract

We argue that the scrambling time is the same, up to a numerical factor in
three or more spacetime dimensions, as the time for the atmosphere to fall
across the horizon or escape, to be replaced by new atmosphere. We propose
that these times agree because the physical scrambling process is part and
parcel of the atmosphere refreshment process. We provide some support for this
relation also in two dimensions, but the atmosphere is not as localized, so
the argument is less justified.

##  1 Introduction

The notion that black holes are fast scramblers of quantum information was
motivated by the assumption of unitarity of the black hole S-matrix [1].
Several distinct time scales that might be associated with black hole
scrambling converged on the same value,

$t_{*}\sim\frac{\beta}{2\pi}\log S$ (1)

where $\beta$ is the inverse temperature and $S$ the entropy of the black
hole. These include apparent spreading of charge that falls into a black hole,
a no-cloning constraint, and a chaos timescale probed holographically with
Planckian shockwave collisions [1, 2, 3, 4]. A notion similar to the last of
these was introduced some three decades earlier [5], as the time after which
“a pure state description of a black hole will be very difficult”. In the
shockwave analysis of [4] and some later work, the timescale depends on the
energy input, and is longest when that energy is minimized. Taking the minimal
energy disturbance to be one thermal unit at the Hawking temperature yields
again the result (1). For an asymptically Anti-de Sitter black hole, according
to AdS/CFT duality, black hole scrambling is dual to thermalizing interactions
of the dual, strongly coupled conformal quantum field theory, and the
scrambling time in one thermal cell of the CFT is again of the form (1), but
now with $S\sim N^{2}$, the entropy per thermal cell. The nature of scrambling
in various quantum systems, including holographic duals of black holes, is
fairly clear, and has been studied in a number of models. However, the nature
of scrambling in terms of black hole physics remains mysterious.

At the classical level, spacetime outside a large black hole is quiescent and
gently curved, and presents no reason to suspect any kind of “scrambling”
might be taking place when a disturbance falls in across the horizon. However,
the black hole is a deformed quantum field vacuum, so there is in fact much
quantum activity around a black hole. Indeed, the tidal acceleration acting on
these vacuum fluctuations produces the Hawking radiation. Previous authors
have suggested that collisions of infalling quanta with outgoing quantum field
fluctuations are responsible for scrambling. This mechanism was first
discussed in [5], and the idea was further explored in the setting of AdS/CFT
[4]. It is an elusive notion, since the outgoing fluctuations are in their
ground state. That is, infalling quanta encounter nothing but vacuum outside
the black hole, so there seems to be nothing with which they might interact.
Here we propose a picture of the nature of black hole scrambling that involves
the outgoing quantum field fluctuations in a different way.

The near horizon quantum vacuum is a thermal “atmosphere”. Most of the entropy
in this atmosphere comes from outgoing modes with high transverse momentum,
which quickly reflect from the effective potential and fall into the black
hole, while modes of sufficiently high frequency and low angular momentum
escape. The thermal atmosphere of a black hole must therefore be continually
resupplied. On a fixed classical spacetime background, with Lorentz invariant
quantum fields, there would be no UV cutoff, and the atmosphere would be
resupplied from a transPlanckian reservoir of modes just outside the horizon.
But that picture neglects the gravitational effect of the vacuum fluctuations,
and must be far from the truth. In particular, the finiteness of black hole
entropy implies that no such reservoir exists. In the local vacuum state the
outside field modes are entangled with partners inside the horizon, so a
transPlanckian reservoir would carry an infinite entanglement entropy,
exceeding the finite Bekenstein-Hawking entropy $S_{\rm BH}$ of the black
hole. Evidently quantum gravity somehow cuts off this would-be divergent
entropy contribution,111For discussions of the possible role of quantum
horizon fluctuations in this cutoff see [6, 7, 8]. and it follows that the
resupply mechanism for the atmosphere must also involve quantum gravity.

Our proposal is that scrambling is part of the atmosphere replacement
mechanism. In particular, we provide evidence that the scrambling time and the
atmosphere refresh time agree up to a numerical factor, if the cutoff is
chosen so that the atmosphere entropy matches the black hole entropy minus the
extremal value for the same charge. The refresh time shares aspects of some
other definitions of the scrambling time, but it is conceptually distinct.

This interpretation of scrambling provides some insight into the causality
puzzle posed by the rapidity of the scrambling [1]. Hayden and Preskill noted
that, when viewed as thermalization on the “stretched horizon”, the scrambling
process is superluminal in the transverse dimensions. If the stretched horizon
were an ordinary thermal system, no local causal dynamics could scramble that
quickly. But it is not an ordinary thermal system: the atmosphere continually
falls into the black hole or escapes from the near horizon region, and is
replaced by “fresh” atmosphere. That replacement is itself a mysterious
process. How do outgoing modes emerge from the near horizon region, in the
absence of a transPlanckian reservoir to supply them [9]? It seems that the
process must be nonlocal, because locally the horizon is an unremarkable place
in spacetime. Together with the agreement of the refresh time and scrambling
times, this inherent nonlocality strongly suggests that the mechanism of
scrambling is intimately related to the atmosphere resupply mechanism.

##  2 Atmosphere model

To estimate the refresh time, we model the atmosphere as a gas of massless
field modes, and use the ray optics approximation to evaluate the mode
propagation. This crude model captures the consequences of radial causality,
which we presume are all that really matters. While most of the entropy
resides in outgoing modes that fall back across the horizon more quickly, the
longest lived modes are those of the smallest angular momentum and highest
frequency, some of which escape from the near horizon region. Those are also
the modes that are the most relevant for the Hawking radiation. We define the
refresh time as the time for those modes to escape from the “Rindlersphere”
i.e., the region where the Rindler (flat spacetime) approximation is valid.
More precisely, for static, spherical black holes, we define the Rindlersphere
as the region where the norm of the Killing vector is approximately
proportional to the proper radial distance (normal to the Killing vector) from
the horizon.

While we lack a sharp reason for using the top of the Rindlersphere to
demarcate the “tropopause” of the black hole atmosphere, it is a natural
choice because, for an evaporating black hole, the Rindlersphere is the region
that is in equilibrium. As for the “bottom” of the atmosphere, as discussed
above, quantum gravity presumably imposes a cutoff on modes close to the
horizon. If the atmosphere entropy contributed by a single species is to be of
order the Bekenstein-Hawking entropy $S_{\rm BH}=A/4l_{P}^{2}$, we should
exclude modes localized closer to the horizon than the Planck length $l_{P}$,
as measured on a spatial hypersurface orthogonal to the horizon generating
Killing vector [10, 5]. (We consider here only spherical black holes.)

For charged black holes we set the lower cutoff so as to match only the
portion of the black hole entropy above the extremal value for the same
charge. We make this choice in order to match the scrambling time found by
shockwave analytics in [11], but it is not entirely unmotivated. The extremal
portion of the entropy resides in a ground state degeneracy. It is a different
kind of entropy, and is not carried by an atmosphere that is continually
refreshed. (See below for more discussion of this point.)

##  3 Atmosphere refresh time

We define the refresh time as the Killing time for a radial null geodesic to
climb from the bottom to the top of the Rindlersphere. This is well-defined in
spherical symmetry if the Killing time coordinate is chosen so its level sets
are orthogonal to the Killing vector.222Alternatively, we could define the
refresh time using a round trip that reflects at a centrifugal barrier (or an
imaginary mirror) at the top of the Rindlersphere, and falls back to the
initial Killing orbit. The elapsed Killing time along that orbit does not
depend on which Killing time coordinate is employed, and is just twice the
time defined above. The spherically symmetric, static line element then takes
the form

$ds^{2}=-N^{2}dt^{2}+dy^{2}+r^{2}d\Omega^{2},$ (2)

where $N=N(y)$ and $r=r(y)$. The coordinate $y$ is the proper radial distance
orthogonal to the Killing vector $\partial_{t}$, whose norm is $N$. For a
black hole we set $y=0$ at the horizon, so in the Rindlersphere we have
$N(y)\approx\kappa y$, where $\kappa$ is the surface gravity.

A radial light ray satisfies $dt=dy/N$, so as it propagates from the bottom to
the top of the Rindlersphere the elapsed Killing time is

$\Delta
t=\int_{y_{b}}^{y_{t}}\frac{dy}{N}\approx\kappa^{-1}\log(y_{t}/y_{b}),$ (3)

where $y_{b}$ and $y_{t}$ are the radial distances from the horizon to the
bottom and top of the Rindlersphere. For a Schwarzschild black hole in four
spacetime dimensions, the lower cutoff is the Planck length, $y_{b}\sim
l_{P}$, and we find $y_{t}\sim r_{+}$, so the log in (3) is
$\sim\log(r_{+}/l_{P})\sim{\textstyle{\frac{1}{2}}}\log S_{\rm BH}$. The
refresh time (3) therefore agrees with (1), up to the prefactor
${\textstyle{\frac{1}{2}}}$, since $\kappa=2\pi T_{\rm H}/\hbar=2\pi/\beta$.

To locate the top of the Rindlersphere we consider the Taylor expansion of the
lapse $N(y)=\kappa y+\dots$. The Rindlersphere ends where the higher order
terms compete with the linear one. The even terms in $y$ vanish for the static
metrics we are considering, because regularity at the horizon implies that
$N^{2}$ must be even under $y\rightarrow-y$ reflection. Assuming that
$(N_{yyy})_{+}$ ($\equiv d^{3}N/dy^{3}|_{y=0}$) is nonzero, the top of the
Rindlersphere lies where $\kappa y\sim|N_{yyy}|_{+}y^{3}$, i.e. at
$y_{t}\sim[{\kappa}/{|N_{yyy}|_{+}}]^{1/2}$. The metrics we consider all
satisfy $N=dr/dy$. A straightforward computation using this relation shows
that $(N_{yyy})_{+}={\textstyle{\frac{1}{2}}}\kappa[(N^{2})_{rr}]_{+}$, so a
general expression for the location of the top of the Rindlersphere is

$y_{t}\sim|(N^{2})_{rr}|_{+}^{-1/2}.$ (4)

For Schwarzschild black holes this yields $y_{t}\sim
r_{+}[(D-2)(D-3)]^{-1/2}$. For AdS-Schwarzschild black holes that are much
larger than the AdS length $L$, and for the planar case, we find $y_{t}\sim
L[(D-1)(D-4)]^{-1/2}$. For Reissner-Nordstrom (charged) black holes in $D=4$,
we find $y_{t}\sim r_{+}|1-2r_{-}/r_{+}|^{-1/2}$, which yields $y_{t}\sim
r_{+}$ as long as $r_{-}$ is not equal (or very close) to $r_{+}/2$. In
exceptional cases where $(N_{yyy})_{+}$ vanishes, we find instead
$y_{t}\sim|\kappa(N^{2})_{rrr}|_{+}^{-1/4}$. This happens in $D=4$ for for
planar Schwarzschild-AdS black holes and for Reissner-Nordstrom black holes
with $r_{-}=r_{+}/2$. When using this higher order relation, we find that the
dependence of $y_{t}$ on the metric parameters is the same as in the generic
cases.

To locate the bottom of the atmosphere, $y_{b}$, we require that the
atmosphere entropy accounts for the portion of the black hole entropy above
the extremal value, as discussed above. The thermal entropy can be estimated
as $S\sim A/y_{b}^{D-2}$ [10, 5]. Setting this equal to the entropy above the
extremal value, $(A-A_{0})/l_{\rm P}^{D-2}$, where $A_{0}$ is the extremal
area for the given charge, yields

$y_{b}=l_{P}[A/(A-A_{0})]^{1/(D-2)}.$ (5)

Close to extremality, this can differ significantly from $l_{P}$, and it can
even become greater than $y_{t}$, the top of the Rindlersphere. However, the
thermal approximation we are employing should not be trusted too close to
extremality [12]. For a Reissner-Nordstrom black hole in $D=4$ spacetime
dimensions, (5) yields $y_{b}=l_{P}[r_{+}/(r_{+}-r_{-})]^{1/2}$. The condition
required for validity of the equilibrium treatment is $r_{+}-r_{-}\gtrsim
l_{P}^{2}/r_{+}$, and as long as this holds, we have $y_{b}\lesssim r_{+}\sim
y_{t}$. That is, the bottom of the atmosphere lies below the top of the
Rindler region.

##  4 Examples

We are now ready to evaluate the refresh time (3) for several examples.

###  4.1 Schwarzschild in $D$ dimensions

For a Schwarzschild black hole in any dimension $D=O(1)>2$ we have
$y_{t}/y_{b}\sim r_{+}/l_{P}\sim S^{1/(D-2)}$, so (3) yields $\Delta
t\sim[\kappa(D-2)]^{-1}\log S$. Apart from the prefactor $(D-2)^{-1}$, this
agrees with (1).

It is interesting to consider also the very large $D\gg 1$ limit, if we
presume that there exists a consistent UV completion of large $D$ quantum
gravity in which black holes have finite entropy.333See also [13] for an
investigation of scrambling for large $D$ black holes. Keeping track of just
the leading order $D$ dependence, we have $y_{t}\sim r_{+}/D$. As for $y_{b}$,
to apply our criterion that the atmosphere entropy matches the black hole
entropy, we need to consider the large $D$ limit of entropy density of
massless fields in a thermal state. The temperature dependence is $T^{D-1}$,
and for each polarization there is a leading order $D^{D/2}$ numerical factor
(coming from the large phase space contributing to the high energy tail of the
Planck distribution) [14]. In addition, there are $D(D-3)/2$ graviton
polarizations, but this does not change the leading order $D$ dependence, so
the leading $D$-dependence of the entropy density is $s\sim D^{D/2}T^{(D-1)}$.
Integrating this with the local temperature $T(y)\sim 1/y$ down to $y_{b}$ we
obtain an atmosphere entropy $S_{\rm atm}\sim D^{D/2}A/y_{b}^{D-2}$, and
equating this to the black hole entropy $S_{\rm BH}\sim A/l_{P}^{D-2}$ yields
$(y_{b}/l_{P})^{D-2}\sim D^{D/2}$, i.e. $y_{b}/l_{P}\sim D^{1/2}$. Putting the
results together we thus have $y_{t}/y_{b}\sim D^{-3/2}r_{+}/l_{P}$. If this
is not greater than unity, the thermal atmosphere lies entirely above the
tropopause of the Rindlersphere, which means the calculation is inconsistent.
However, also in this case the Hawking evaporation timescale is shorter than
the Planck time, so the black hole cannot be treated semiclassically [13]. If
we assume that the black hole is large enough that $r_{+}\gg D^{3/2}l_{P}$,
then the atmosphere refresh time (3) is
$\sim\kappa^{-1}\log(D^{-3/2}r_{+}/l_{P})\sim(\kappa D)^{-1}\log S$. This
indicates a $1/D$ suppression of the refresh time compared to (1), no
different from what we found for $D\sim O(1)$.

###  4.2 Reissner-Nordstrom in $D=4$

For the Reissner-Nordstrom case in $D=4$ we have from the above results
$y_{t}\sim r_{+}$ and $y_{b}/l_{P}\sim[S/(S-S_{0})]^{1/2}$ (5). Since also
$r_{+}/l_{P}\sim S^{1/2}$, this yields $y_{t}/y_{b}\sim(S-S_{0})^{1/2}$, so
the refresh time is $\Delta
t\sim{\textstyle{\frac{1}{2}}}\kappa^{-1}\log(S-S_{0})$. Up to the factor 1/2,
this agrees with the scrambling time found in Ref. [11] via shockwave
analytics with the injected energy of order the temperature. This result
suggests that the scrambling does not involve the degrees of freedom counted
by the extremal entropy. This makes some sense, since the extremal entropy
does not correspond to a thermal atmosphere that falls into the black hole, so
does not need to be refreshed. Perhaps when a charged black hole relaxes to
the extremal state, unitarity may not require the scrambling of all the
degrees of freedom.

A different interpretation of the shortened scrambling for near-extremal
charged black holes was proposed in Ref. [15]. The viewpoint adopted there was
that the infalling energy behaves like a fundamental string, and thus exhibits
an effective transverse spreading as its momentum relative to the local static
frame increases during its fall through the long throat region between the
Newtonian exosphere and the Rindlersphere. That paper argued that this earlier
spreading decreases the time needed to complete scrambling at the stretched
horizon, and reproduces the result of [11] while maintaining the hypothesis
that all of the degrees of freedom counted by the black hole entropy are
involved in the scrambling. Note that this argument assumes that the
excitation falls freely into the black hole, which produces the effective
transverse spread. But a particle need not fall freely. If it is lowered
toward the horizon, and only dropped in from the top of the Rindlersphere,
then the long throat should play no role.

###  4.3 Very large Schwarzschild-AdS in $D=O(1)>2$ dimensions

For an AdS-Schwarzschild black hole with $r_{+}\gg L$, in any dimension
$D=O(1)>2$, we have $y_{t}/y_{b}\sim L/l_{P}$, so Eq. (3) yields $\Delta
t\sim\kappa^{-1}\log(L/l_{P})\sim[\kappa(D-2)]^{-1}\log N^{2}$, where
$N^{2}\sim(L/l_{P})^{(D-2)}$ is the entropy per thermal cell of the dual
$SU(N)$ gauge theory [16]. Apart from the $1/(D-2)$ prefactor, this agrees
with the time estimated from charge spreading in [2]. This time can also be
inferred from the shockwave analytics of Ref. [4].444We thank Douglas Stanford
for an explanation of this point. The time found there is (1), with $S$ the
black hole entropy, if the (homogeneously) injected energy corresponds to one
unit of entropy. To express this in terms of quantities associated with a
single thermal cell, note that $S=N^{2}/(N^{2}/S)$. The numerator is the
number of degrees of freedom in a thermal cell, while the denominator is the
reciprocal of the number of thermal cells, which is equal to the entropy
injected per cell, given that just one unit of entropy was injected globally.
If one unit of entropy were injected per thermal cell, the shockwave
scrambling time would have been $\sim\kappa^{-1}\log N^{2}$, in agreement with
our defnition of the atmosphere refresh time.

###  4.4 de Sitter spacetime

It was noticed in Ref. [3] that the charge spreading timescale implies that de
Sitter space, too, is a fast scrambler, whose scrambling time is
$t_{*}\sim\kappa^{-1}\log(L/l_{P})$, where $L$ is the de Sitter radius.555In
[3] the string length was used rather than the Planck length, but this makes
little difference to the log. Also, the prefactor ${\cal R}$ in Eq. (2.3) of
that paper is a typographical error. Moreover, recent work [17, 18] has
studied scrambling in de Sitter space via shockwave computations of chaos
timescales. A de Sitter horizon also has a thermal atmosphere [19], to which
we may apply our definition of the refresh time. The top of the Rindlersphere
is or order $\sim L$, so we obtain $y_{t}/y_{b}\sim L/l_{P}$, and thus $\Delta
t\sim\kappa^{-1}\log(L/l_{P})\sim(\kappa(D-2))^{-1}\log S$.

Despite the similarities with black hole horizons, the atmosphere refresh
mechanism for de Sitter horizons may differ, since it must take place within
each static patch, whereas for a black hole it could be connected to infinity.
While the refresh time according to our criterion is the same as for black
hole horizons, perhaps that criterion is not correct for de Sitter horizons.
It was suggested recently in Ref. [20] that the difference with black hole
horizons is more profound, and leads to a “hyperfast” scrambling time
$\sim\kappa^{-1}$, with no dependence on the horizon entropy.

###  4.5 Black holes in $D=2$ dimensions

In two spacetime dimensions our previous considerations for the black hole
geometries and our estimate of the thermal entropy of the atmosphere do not
apply, yet various dilaton gravity theories in two spacetime dimensions have
black hole solutions [21]. A horizon cross section for such black holes is
just a single point, but they have an entropy determined by the value of the
dilaton at the horizon, and an associated scrambling time if the theory
supports black hole dynamics and Hawking radiation. In this subsection we
briefly consider whether the close relation between the scrambling time and
the atmosphere refresh time may hold in such theories, despite the differences
from the higher dimensional cases. Our conclusion will be “perhaps”.

The atmosphere in two spacetime dimensions is qualitatively different from
that in higher dimensions. The entanglement entropy of a half-line in a gapped
theory with correlation length $\xi$ and a UV fixed point with central charge
$c$ is $\frac{c}{6}\log(\xi/\epsilon)$, where $\epsilon$ provides a UV cutoff
[22]. Thus, unlike in higher dimensions, the entropy depends on the log of the
cutoff rather than a power. Also, if the entangled field is gapless, there is
an IR divergence. Our criterion for locating the bottom of the atmosphere by
setting the entanglement entropy equal to the black hole entropy thus appears
not to be well-justified. If we ignore this and postulate that $\epsilon$
should be identified with the cutoff $y_{b}$ at the bottom of the atmosphere,
and if we identify the IR cutoff with the top of the atmosphere $y_{t}$, the
entanglement entropy would be $\frac{c}{6}\log(y_{t}/y_{b})$. If this is set
equal to the black hole entropy $S$,666For a near-extremal black hole one
would set it equal to the entropy $S-S_{0}$ that exceeds the extremal value
$S_{0}$ one obtains $\log(y_{t}/y_{b})=\frac{6}{c}S$, so the atmosphere
refresh time (3) becomes $\Delta t=\frac{6}{c}\kappa^{-1}S$. This strongly
disagrees with the usual scrambling time $\kappa^{-1}\log S$. However, since
this calculation is not well-justified, we do not regard the disagreement as
necessarily indicating a failure of the atmosphere refresh time interpretation
of scrambling. It may rather indicate that in two dimensions the nature of the
atmosphere and its entanglement entropy in an underlying fundamental theory
with finite black hole entropy is qualitatively different from that in higher
dimensions.

While it is beyond the scope of this paper to attempt to characterize the
black hole atmosphere in 2d theories, we shall just note here one possible
alternative to the estimate given above, making use of the UV cutoff that is
implied by the matrix model dual to two-dimensional string theory [23, 24,
25]. According to the analysis of Ref. [25], the entropy of an interval of
length $L$ in the emergent linear dilaton vacuum, in a region where the string
coupling $g_{s}=e^{\Phi}$ is weak, is $S=\frac{1}{3}\log(L\exp(-\Phi_{\rm
tw}-\Phi_{1}/2-\Phi_{2}/2))$, where $\Phi$ is the dilaton, which appears in
the semiclassical action in combination with the Ricci scalar as
$e^{-2\Phi}R$, and the string length is set to unity. The subscripts “tw,1,2”
label the values at the “tachyon wall” and the two endpoints of the interval,
and it is assumed here that the interval is located in a region where
$\Phi_{1,2}<\Phi_{w}\ll-1$. This formula indicates that the effective cutoff
at the interval endpoint 1 scales with $\Phi_{1}$ as $\epsilon\sim
e^{\Phi_{1}/2}$, which is exponentially smaller than the string length. If we
set $y_{b}$ equal to this cutoff, $y_{b}\sim e^{\Phi_{\rm hor}/2}$, then the
refresh time (3) becomes $\Delta t\sim\kappa^{-1}(-\frac{1}{2}\Phi_{\rm
hor}+\log y_{t})$. If black holes existed in the emergent spacetime of the
matrix model, their entropy would be determined by the coefficient
$e^{-2\Phi_{\rm hor}}$ of the Ricci scalar in the action, in which case this
refresh time would be $\kappa^{-1}(\frac{1}{4}\log S_{BH}+\log y_{t})$. And if
the entropy term were to dominate over the $y_{t}$ term, then up to the factor
$\frac{1}{4}$ this would agree with the scrambling time. However, this
identification of $y_{b}$ with the cutoff inferred from the matrix model
entanglement entropy is admittedly only vaguely motivated. Moreover, that
theory does not even support black hole formation [26]. We are only pointing
out that perhaps a UV cutoff set by stringy effects could be essential to the
log rather than linear dependence of the refresh time on the black hole
entropy in two dimensions.

A recently much studied alternative to the matrix model for emergent 2d string
theory that does contain black holes is the SYK model, which is dual to JT
gravity. In Ref. [27] the chaos time scale in the SYK model was studied via
the Lyapunov exponent in correlation functions. It was observed there that,
although “the bulk theory dual to SYK has a tower of light fields roughly
similar to a string spectrum” with string length of order the AdS radius, the
contribution of the duals of those fields to the Lyapunov exponent (and hence
to the scrambling time) in the SYK model is suppressed relative to the
“gravity” contribution for near-extremal black holes. Perhaps the nature of
the black hole atmosphere can be understood in this model, allowing its
refresh time to be found. Stringy effects would presumably impose the UV
cutoff of the atmosphere.

##  5 Conclusion

We have shown that the black hole scrambling time is the same, up to a
numerical factor in three or more spacetime dimensions, as our estimate of the
time for the thermal atmosphere to be refreshed. We propose that these times
agree because the physical scrambling process is part and parcel of the
atmosphere refreshment process. The discrepancy of the numerical factor is
worrisome, but it may not point to a fundamental flaw in the proposal, since
our crude model of the atmosphere and kinematic criterion for its refresh time
may just be too crude to accurately capture this factor. We provide some
evidence supporting also a relation in two spacetime dimensions between the
refresh time and the scrambling time, but the picture is not as clear because
the atmosphere is apparently not as localized near the horizon.

The atmosphere refreshment process must be nonlocal, since locally nothing
distinguishes the horizon. The picture of atmosphere refreshment thus goes
some way toward illuminating why the scrambling is nonlocal in spacetime, and
therefore how it can be so fast. We have said nothing, however, that
elucidates the dynamics of atmosphere replacement. The origin of the outgoing
black hole modes remains as obscure as ever [9]. We may hope that, by having
linked it to scrambling, perhaps some light may be shined on it.

## Acknowledgements

We thank Douglas Stanford and Brian Swingle for helpful discussions. This work
was supported in part by the National Science Foundation grants PHY-1708139
and PHY-2012139 at UMD, PHY-1748958 at the KITP, and PHY-1820734 at CUNY. PN
acknowledges support from Israel Science Foundation grant 447/17 for the
Israel portion and from U.S. National Science Foundation grant PHY-1820734 at
CUNY for the U.S. portion of the work.

## References

  * [1] P. Hayden and J. Preskill, _Black holes as mirrors: Quantum information in random subsystems_ , _JHEP_ 09 (2007) 120 [0708.4025].
  * [2] Y. Sekino and L. Susskind, _Fast Scramblers_ , _JHEP_ 10 (2008) 065 [0808.2096].
  * [3] L. Susskind, _Addendum to fast scramblers_ , _arXiv:1101.6048_ (2011) .
  * [4] S. H. Shenker and D. Stanford, _Black holes and the butterfly effect_ , _JHEP_ 03 (2014) 067 [1306.0622].
  * [5] G. ’t Hooft, _On the Quantum Structure of a Black Hole_ , _Nucl. Phys._ B256 (1985) 727.
  * [6] J. W. York, Jr., _Dynamical Origin of Black Hole Radiance_ , _Phys. Rev. D_ 28 (1983) 2929.
  * [7] V. P. Frolov and I. Novikov, _Dynamical origin of the entropy of a black hole_ , _Phys. Rev. D_ 48 (1993) 4545 [gr-qc/9309001].
  * [8] T. Jacobson, _Gravitation and vacuum entanglement entropy_ , _Int. J. Mod. Phys. D_ 21 (2012) 1242006 [1204.6349].
  * [9] T. Jacobson, _On the origin of the outgoing black hole modes_ , _Phys. Rev._ D53 (1996) 7082 [hep-th/9601064].
  * [10] R. D. Sorkin, _1983 paper on entanglement entropy: ”On the Entropy of the Vacuum outside a Horizon”_ , in _Proceedings, 10th International Conference on General Relativity and Gravitation: Padua, Italy, July 4-9, 1983_ , vol. 2, pp. 734–736, 1984, 1402.3589.
  * [11] S. Leichenauer, _Thermal Corrections to Entanglement Entropy from Holography_ , _JHEP_ 09 (2015) 014 [1502.07348].
  * [12] J. Preskill, P. Schwarz, A. D. Shapere, S. Trivedi and F. Wilczek, _Limitations on the statistical description of black holes_ , _Mod. Phys. Lett. A_ 6 (1991) 2353.
  * [13] F. Holdt-Sørensen, D. A. McGady and N. Wintergerst, _Black hole evaporation and semiclassicality at large $D$_, _Phys. Rev. D_ 102 (2020) 026016 [1908.08083].
  * [14] S. AL-Jaber, _Planck’s spectral distribution law in n dimensions_ , _International Journal of Theoretical Physics_ 42 (2003) 111.
  * [15] A. R. Brown, H. Gharibyan, A. Streicher, L. Susskind, L. Thorlacius and Y. Zhao, _Falling Toward Charged Black Holes_ , _Phys. Rev. D_ 98 (2018) 126016 [1804.04156].
  * [16] J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * [17] L. Aalsma and G. Shiu, _Chaos and complementarity in de Sitter space_ , _JHEP_ 05 (2020) 152 [2002.01326].
  * [18] H. Geng, _Non-local entanglement and fast scrambling in de-Sitter holography_ , _Annals Phys._ 426 (2021) 168402 [2005.00021].
  * [19] G. Gibbons and S. Hawking, _Cosmological Event Horizons, Thermodynamics, and Particle Creation_ , _Phys. Rev. D_ 15 (1977) 2738.
  * [20] L. Susskind, _Entanglement and Chaos in De Sitter Holography: An SYK Example_ , 2109.14104.
  * [21] D. Grumiller, W. Kummer and D. V. Vassilevich, _Dilaton gravity in two-dimensions_ , _Phys. Rept._ 369 (2002) 327 [hep-th/0204253].
  * [22] M. Headrick, _Lectures on entanglement entropy in field theory and holography_ , 1907.08126.
  * [23] S. R. Das, _Geometric entropy of nonrelativistic fermions and two-dimensional strings_ , _Phys. Rev. D_ 51 (1995) 6901 [hep-th/9501090].
  * [24] S. R. Das, _Degrees of freedom in two-dimensional string theory_ , _Nucl. Phys. B Proc. Suppl._ 45BC (1996) 224 [hep-th/9511214].
  * [25] S. A. Hartnoll and E. Mazenc, _Entanglement entropy in two dimensional string theory_ , _Phys. Rev. Lett._ 115 (2015) 121602 [1504.07985].
  * [26] J. L. Karczmarek, J. M. Maldacena and A. Strominger, _Black hole non-formation in the matrix model_ , _JHEP_ 01 (2006) 039 [hep-th/0411174].
  * [27] J. Maldacena and D. Stanford, _Remarks on the Sachdev-Ye-Kitaev model_ , _Phys. Rev. D_ 94 (2016) 106002 [1604.07818].