Birth, survival and death of languages by Monte Carlo simulation
C. Schulze1, D. Stauffer1, S. Wichmann2
1 Institute for Theoretical Physics, Cologne University, D-50923 Köln, Euroland
2 Department of Linguistics, Max Planck Institute for Evolutionary Anthropology
Deutscher Platz 6, D-04103 Leipzig, Germany
Abstract: Simulations of physicists for the competition between adult
languages since 2003 are reviewed. How many languages are spoken by how
many people? How many languages are contained in various language fam-
ilies? How do language similarities decay with geographical distance, and
what effects do natural boundaries have? New simulations of bilinguality are
given in an appendix.
1 Introduction
While the emergence and learning of human languages has been simulated
since decades on computers [1], and while a later economics Nobel laureate
also contributed to linguistics long ago [2], the competition between existing
languages of adults is a more recent research trend, where physicists have
tried to play a major role. It follows the principle of survival of the fittest,
as known from Darwinian evolution in biology, and indeed many of the tech-
niques have been borrowed from simulational biology [3]. This emphasis from
physics on the competition of existing languages for adult humans started
with Abrams and Strogatz[4] and was then followed by at least six groups
independently [5, 6, 7, 8, 9, 10]. More recently, of course, reviews[11, 3] and
conferences brought them together, and others followed them [12, 13, 14].
Today about 7000 different languages (as defined by linguists) are spoken,
and about every ten days one of them dies out. On the other hand, the split
of Latin into different languages spoken from Portugal to Romania is well
documented. In statistical physics, we can describe and explain the pressure
which air molecules of a known density and temperature exert on the walls.
But we cannot predict were one given molecule will be one second from
now. Similarly, the application of statistical physics tools to linguistics may
describe the ensemble of the seven thousand presently existing languages,
http://arxiv.org/abs/0704.0691v1
 1000
 1  10  100  1000  10000  100000  1e+06  1e+07  1e+08  1e+09
size s
Distribution of language sizes from Grimes, Ethnologue, and  550 exp[-0.05{ln(size/7000)}**2]
Figure 1: Empirical variation of the number Ns of languages spoken by
s people each. For better presentation, the language sizes s are binned in
powers of two. Data from Ethnologue [16], as plotted in [17]. The parabola
corresponds to a log-normal distribution; we see deviations from it for the
smallest sizes [18].
but not the extinction of one given language in one given region on Earth.
Figure 1 shows how many languages exist today, as a function of the number
of speakers of that language. A statistical theory of language competition
thus first of all should try to reproduce such results, in order to validate the
model. If if fails to describe this fact, why should one trust it at all? Or as
stated by linguist Yang on page 216 of [15]: It is time for the ancient field of
linguistics to join the quantitative world of modern science.
This review starts with our own model for numerous languages in section
2, followed by a review of the alternative model of Viviane de Oliveira and
coworkers [10]. Then we review more shortly the many other models which
at present do not allow the simulation of thousands of different languages.
2 Schulze Model
2.1 Definition
Our own simulations, also called the Schulze model, characterise each lan-
guage (or grammar) by F independent features each of which can take one
of Q different values; the binary case Q = 2 allows the storage in bit-strings.
Three basic mechanisms connected with probabilities p, q and r are common
to all variants:
i) With probability p at each iteration and for each feature, this feature
is changed (or mutated in biological language). This change is random or
not, depending on process ii).
ii) With probability q the mutation/change under i) is not random but
instead transfers the value of this feature from another person in the popu-
lation. This transfer is called diffusion by linguists. With probability 1− q,
the change is random.
iii) With probability (1 − x)2r (also (1 − x2)r has been used instead)
somebody discards the mother language and takes over the whole language
(all F features) from another person in the population. Here x is the fraction
of people speaking the old language. This flight is called shift by linguists.
Several variants are possible: One can use one joint population where
everybody can meet everybody for transfer and shift; or we put people onto
a square L× L lattice or more complicated network, where diffusion and/or
shift are possible only from a randomly selected neighbour. People may
migrate on this lattice, which physicists would call diffusion. The population
can be fixed, meaning that at every iteration all adults are replaced by their
children. Or it can grow by a suitable birth and death process; in this case the
shifting probability can include also a factor proportional to the population.
If one dislikes to have three free parameters p, q, r one may set q = r = 1
without much loss in results.
For the number F of features, from 8 to 64 were used in simulations. Real
languages contain many thousands of words for everyday use, and thus one
should identify one feature rather with an independent grammatical element
(like the order of subject, object and verb in a sentence) than with a word. F
for real languages was estimated as about 30 [19] or about 40 to 50 [15] such
choices, and the Word Atlas of Language Structures [20] lists 138 features
with up to Q = 9 values. These grammar sizes thus correspond roughly to
what has been simulated. According to [21] the rate of change in normal
linguistic typological features, i.e., excluding a few extraordinarily unstable
ones, is 16 % per 1000 years.
2.2 Results
If we start with everybody speaking one language (or with just one Eve), then
at low p this language still dominates and is spoken by more than half of the
total population, with the remaining people speaking a minor and short-lived
variant of this dominating language. At high p, on the other hand, the whole
population soon fragments into many languages, roughly such that everybody
selects nearly randomly one of the QF possible languages. This corresponds
to the biblical story of the Tower of Babel. We thus have dominance for
small p and fragmentation for large p, with a first-order phase transition or
jump at some threshold value which depends on the other parameters and
details of the model.
If instead we start with everybody speaking a randomly selected lan-
guage, then for high p this situation remains. For low p, however, after some
time one language by random accidents happens to grow to a sufficiently
large size such that it then grows rapidly to be spoken by more than half of
the population. Thus a transition from initial fragmentation to final dom-
inance happens in Fig.2. The threshold value is different from the one for
the opposite direction from dominance to fragmentation: we have hysteresis
as is common for first-order transitions, Fig.3. Empirically, this transition
to one dominating language was observed on the American continent; within
the last five centuries, two thirds of the native Brazilian languages have died
out. And in the last half century we observed the rise of English in physics
research publications. While 85 years ago, physicist Bose sent from India
his paper to Einstein in order to have it translated from English to German
(which lead to Bose-Einstein condensation), after World War II physics re-
search was usually published in English, first in Japan, since the 1960’s in
(West) Germany, a decade later in France, since the 1990’s in Russia; finally,
China has witnessed a surge in physics papers written in English since 2000.
The time needed to go from fragmentation to dominance increases roughly
logarithmically with population size, at least in the binary case Q = 2 with-
out lattice. Thus a mathematical solution for an infinite population might
never get this transition. In other words, proper models should be agent-
based [22], with individuals acting on their own; one should not average over
10000
100000
100 M
0 100 200 300 400 500 600
Size of largest language for 100M, 10M, 1 M, 0.1 M, 10 K, 1 K people, 0.48 mutations per person, q=0.94
Figure 2: Variation with time of the number of people speaking the most
widespread language, for various population sizes. The larger the population
is, the longer is the time until the transition from fragmentation to dominance
takes place. Q = 2, F = 8, p = 0.06, q = 0.94; from [11].
the whole population, using differential equation for the concentrations. Such
simulations have been standard in computational physics for half a century
(Monte Carlo and Molecular Dynamics), while mean field approximations
average over many individuals and can give somewhat or completely wrong
results. (The transition from fragmentation to dominance may require a shift
probability (1− x)2r instead of (1− x2)r.)
The language size distribution to be compared with Fig.1 is shown in
Fig.4a. To get it, we looked at non-equilibrium results and introduced ran-
dom multiplicative noise, since otherwise the language sizes were too small
and their distribution too irregular. Fig.4b avoids these tricks and instead
places the people on a directed scale-free network, discussed below.
No lattice or other spatial structure was employed in the above simula-
tions. On a lattice one can look at language geography [23, 24, 25]. North
and South of the Alps, different languages are spoken, and the same sepa-
ration is made by the English Channel. Genetic and linguistic boundaries
in Europe mostly coincide, and about two thirds of them agree with natural
boundaries like a mountain chain or sea [26]. We simulated this effect on a
0.01 0.1 1 10
population in millions
Critical mutation rate, no transfer, initially one (top) or initially many random languages (bottom)
Figure 3: Dependence of the mutation threshold for the phase transition
on the population size; upper data from dominance to fragmentation, lower
data from fragmentation to dominance. Above the curves we arrive at frag-
mentation, below at dominance. From [11].
lattice [27] with contact only between nearest neighbours and a horizontal
barrier separating the upper from the lower half. The shift from a small
to a large language happens across the barrier only with a small crossing
probability c. For c = 0 one thus has two completely separated halves of the
lattice, and trivially the languages which evolve as dominating are different
on both sides of the border. With c = 1 the border has no effect, and only
one language dominates. Fig.5 shows how often for small but finite c two
separate dominating languages may coexist; already quite small c suffice,
particularly for large lattices, to unify the two regions into only one with the
same language dominating on both sides.
Fig.4b employed a directed Barabási-Albert scale-free network. These
networks are grown from a small fully connected core such that each new
network member selects m already existing members as teachers. The more
people have selected a certain teacher before, the higher is the probability
that this teacher will again be selected. Information only flows from the
teacher to the person who selected this teacher, not in the opposite direction
[31].
 1000
 10000
 100000
 10 M
 1  10  100  1000  10000  100000
number of speakers
Two runs of 200 million; fragmented start, mutation rate 0.0032 per bit-string, t < 5000 or 6000
 10 K
 100K
 1  10  100  1 K  10 K  100K  1 M  10 M
number of speakers
N = 1 million (+), 3 M (x), 10 M (*), 30 M (sq.)
Figure 4: Language size distribution Ns. Top: without lattice, with random
multiplicative noise, not in equilibrium [17]. Bottom: On scale-free network
in equilibrium [30].
 0.0001  0.001  0.01  0.1
crossing rate
100 samples, L = 50 (right line), 100 (+,x), 200 (left line)
Figure 5: Fraction of cases when a semi-permeable barrier allowed two
different languages to dominate on its two sides in the Schulze model.
3 Viviane Model
3.1 Definition
The model of Viviane de Oliveira et al [10] has become known as the Viviane
model (following the Brazilian tradition of how to call people). It simulates
the colonisation of an uninhabited continent by people. Each site j of an L×L
square lattice can later be populated by cj people; this carrying capacity cj
is selected randomly between 1 and some m ∼ 102. Initially only one site i
is occupied by ci people.
Then at each time step, one randomly selected empty lattice neighbour j
of occupied sites becomes occupied with probability cj/m by cj people. Thus
after some time the whole lattice becomes occupied and the simulation stops.
In contrast to the Schulze model, the Viviane model is a growth process and
not one eventually fluctuating about some equilibrium.
Languages have no internal structure and are simply numbered 1, 2, 3,,,
with 1 being the number of the language spoken on the originally occupied
site. All people within one lattice site speak the same language. First, if a
new site has been colonized the language spoken there is taken from one of the
occupied neighbours k, proportional to the fitness Fk of that neighbour site k.
This fitness is the total number of people speaking anywhere in the lattice the
language of k, except that it is bounded from above by a maximum fitness
Mk fixed randomly between 1 and some Mmax ∼ 10
3. Second, mutations
(language change) are made with probability α/Fj on the freshly occupied
site j only, from the selected language of neighbour k. A mutation means
that a new language is created which gets a new number not used previously.
In this way, the flight (shift) from small language and the mutations,
which were two separate processes in the Schulze model, are combined into
one process; and this process also is a transfer (diffusion) process which in
the Schulze model was dealt with separately. Thus we have here only one
free parameter α, the mutation coefficient, instead of three parameters p, q, r
in the Schulze model.
Variants allow mutations also later, after a site is occupied. Or a language
is characterised by a string of F bits (Q = 2 in the Schulze notation) and only
different bit-strings count as different languages [28]. Or the capacities ck are
not homogeneously distributed between 1 and m but more often small than
large, with a frequency proportional to 1/cj, as long as it is not larger than
the maximum m. Lot of computer time then is saved if after the occupation
of the new sites one selects two of its occupied neighbours and takes the
language from the one with the bigger capacity; if only one neighbour is
occupied then its language is taken over.
3.2 Results
In contrast to the Schulze model, the Viviane model gives languages spoken
by 109 people if a sufficiently large lattice is used. The language size distri-
bution Ns has a maximum at moderately small language sizes s. However,
instead of a round parabola as in Fig.1, the log-log plot of Ns versus s gives
two straight lines meeting at the maximum, that means one power law for
small s (where Ns increases with s) and another power law for large s (where
Ns decreases with s). So, not everything is solved yet.
Crucial progress was made by Paulo Murilo de Oliveira (not the same
family as Viviane de Oliveira), who introduced the above-mentioned modifi-
cations [28]: Languages (grammars) are characterized by bit-strings (Q = 2)
of length F ≃ 13 and count as different only when their bit-strings differ; the
carrying capacities c are selected with a probability ∝ 1/c, and the newly
colonized site gets the fitter language of two previously occupied neighbours.
 100K
 1  10  100  1 K  10K 100K 1 M  10M 100M 1 G
number of speakers
L = 20,000; 7500 languages; 5940 million people
Figure 6: Variation of the number Ns of languages spoken by s people each
in the Viviane model, as modified and published in [28].
Now the distribution is roughly log-normal, Fig.6, with enhancement for very
small sizes; the total population and the total number of languages can be
made close to the present reality, the maximum of the parabola in a double-
logarithmic plot (with binning by factors of two in s) is near s ≃ 104, while
the largest language is spoken by 109 people, similar to Mandarin Chinese.
Fig.7 shows for both the modified Viviane model and the Schulze lattice
model that languages in general are less similar to each other if they are
widely separated geographically, in agreement with reality [24, 25]. Note the
difference in scales: One lattice constant (distance between nearest neigh-
bours) corresponds to about one kilometer in the Viviane model and 1000
kilometers in the Schulze model, if Fig.7 is compared to reality [24].
Also the classification of different languages into one family, like the Indo-
European languages, has been simulated with moderate success. Following
the history of the mutations during the colonisation, a language tree like can
be constructed in the unmodified version (Fig.11.15 in [11]). One can imagine
that this is Latin, splitting up into Romanian, Italian, Spanish and French,
with Spanish then splitting into Castellan, Galego and Catalan, and Catalan
mutating further into Mallorquin. (Many small branches were omitted for
clarity.) More quantitative information is obtained from the modified bit-
 0  2  4  6  8  10  12  14  16  18  20
geographical distance in 1000
Modified Viviane model, line = random differences
 1  2  3  4  5  6  7  8  9  10
geographical distance
Schulze model, Q=5, F=9, p=0.5, q=r=0.9, s=0.5
Figure 7: Language differences (in arbitrary units [24]) as a function of
geographical distance in the Viviane model (top) and the Schulze model
(bottom). The horizontal line corresponds to completely uncorrelated lan-
guages. In the bottom part, + and x correspond to start with fragmentation
(+) and with dominance (x). From [24, 30].
string version [28]. The mutated language on a newly occupied site starts a
new family if it differs in two or more bits from the bit-string characterising
the historically first language of the old family. The size distribution of
language families in Fig.8 agrees in its central part with the empirically
observed [29] exponent –0.525 and is independent of the length F = 8, 16,
32 or 64 of the bit-strings for the Viviane model and independent of the
population size for the Schulze model.
 1000
 10000
 100000
 1  10  100  1000
number of languages
100 * L=10,000; 8(+), 16(x), 32(*), 64(sq.) bits
 10 K
 100K
 1  10  100  1000
number of families
N = 1 million (+), 3 M (x), 10 M(*), 30 M(sq.)
Figure 8: Number of families as a function of the number of different lan-
guages in this family [30]. Top: Modified Viviane model for various lengths
of the bit-strings. Bottom: Schulze model with Q = 5, F = 8 on directed
Barabási-Albert scale-free network, p = 0.5, q = 0.59, r = 0.9 for various
population sizes.
4 Other Models
Years before physicists invaded en masse the field of linguistics, Nettle [33]
already wrote down a differential equation for the number L of languages,
dL/dt = 70/t− L/20 ,
where time t is measured in millennia. For long times, only one language
(mathematically: zero languages) will remain; however that time lies far in
the future. A more detailed splitting mechanism was introduced by Novotny
and Drozd [34] for the emergence of new languages from one mother lan-
guage, and gave a log-normal distribution of language sizes, in agreement
with reality except presumably at the smallest sizes [18]. In the same spirit
of looking at languages as a whole, ignoring the individuals, are the very re-
cent models of Tuncay [14], who coupled a splitting mechanism with random
multiplicative noise in the size of the growing population, plus an extinction
probability, and found a power-law decay or a log-normal size distribution
for the simulated languages, depending on parameters. He also checked the
lifetimes of the simulated languages. An “early” attempt to apply the Ising
model of statistical physics to linguistics [36] had little success.
Numerous coupled differential equations were studied by scientists coming
from theoretical chemistry, mathematics and computer science [35] for the
purpose of language learning by children. They have been applied [3] also
to the competition of up to 8000 languages of adults, but since the original
authors have to our knowledge not followed this re-interpretation of their
learning model we now refer to [35, 3] for details and results.
It was the population dynamics of Abrams and Strogatz [4] which started
the avalanche of physics papers on language competition. They assume two
languages X and Y, spoken by the fractions x and y = 1 − x of a fixed
population with a time dependence
dx/dt = yxas− xya(1− s) ,
with a status or prestige variable S which is close to one if X has a high
prestige and close to zero for low prestige of X. The neutral case is s = 1/2.
The exponent a = 1.31 was fitted to some empirical data of how minority
languages decay in size. If a is replaced by unity we arrive at the logistic
equation of Verhulst from the 19th century, which was applied to languages
by Shen in 1997.
0 1 2 3 4 5 6 7 8 9 10
Abrams-Strogatz model for a = 1.31 and s = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 (bottom to top); unsymmetric
Figure 9: Abrams-Strogatz model: Fraction of people speaking language
X, versus time, for an initial concentration of only ten percent and various
prestige values s of this language. For s ≥ 0.7 this language finally wins over
the other language Y. From [3].
Fig.9 shows the resulting x(t) if X is spoken initially by a minority of
ten percent only. Then for low, neutral or slightly higher status s of its
language X, the fraction decays further towards zero, but for a higher status
like x = 0.7 it finally wins over and is spoken by everybody (not shown).
This may correspond to the influence of a colonial power; indeed in France
today most people speak French as a result of the Roman conquest of more
than two millennia ago, and in Brazil many of the native languages have
become extinct in the last five centuries since the Portuguese arrived there.
This Abrams-Strogatz approach was soon generalized to a lattice by Pa-
triarca and Lepännen [4], and later to populations with bilingual speakers
[5, 38], coexistence of the two languages [8] as well simulations based on
individuals [13].
Such agent-based simulations were also made by Kosmidis et al [6] who
gave each person a string of 20 bits. The first 10 belonged to one language,
the last 10 to another language. In this way they were able to simulate
people speaking, more of less correctly, one or two languages. One could
also interpret their model as one for English which is a mixture of German
(Anglo-Saxon) and French words, due to the conquest of England by the
Normans in 1066.
Finally, Schwämmle [9] also used bit-strings, but to describe biological
ageing through the Penna model. The child can learn the languasge from
the mother, the father, or both, thus also allowing for bilinguals. This model
allowed to simulate that languages are learned easier in youth than at old age.
In this way it builds a bridge between language competition and language
learning [35].
5 How physics may inform linguistics:
prospects for future research
As the research described above has progressed a larger design has become
apparent, which consists in an empirical side looking for quantitative distribu-
tions involving languages [29, 24, 37] and a development of models simulating
similar quantitive distributions. The hope is that as more and more quantifi-
able relations in and among languages are discovered and simulation models
are developed which can adequately replicate these distributions, the simu-
lation models will of necessity become more and more adequate as models
of actual languages, and could therefore be employed for purposes beyond
the ones for which they were designed. For instance, the revised Viviane
model, which was designed to capture the distribution of speaker popula-
tions and the population of languages within families [30], could potentially
be employed for investigating absolute rates of language change, an issue
with which linguists are very much concerned [39], inasmuch as knowledge
of how fast languages change could provide us with a way to date prehis-
toric events involving people speaking given reconstructed languages. Thus,
a strand of research where linguists and physicists can and will continue to
cooperate is the search for quantifiable distributions on the one hand and the
fine-tuning of models which can adequately simulate an increasing range of
such distributions.
Apart from some exceptions [5, 6, 38], most work on language compe-
tition has assumed monolingual speakers. Since most of the world’s pop-
ulation is bi- or multilingual this is clearly not adequate. Language shift
will normally involve transitional bilingualism, or bilingualism may persist
for centuries without the majority language necessarily replacing minority
languages. Diglossia, i.e. the use of different languages for different pur-
poses may help sustain bilingualism. Current models can be extended to
investigate under which conditions bilingualism may persist or get reduced
to monolingualism. Different kinds of situations can be modelled, such as
the replacement of certain, but not all, languages within the domain of the
Roman empire, the development of so-called linguistic areas, where several
languages share a number of features (e.g., the Balkans, India, Mesoamer-
ica), multilingualism caused by linguistic exogamy (the northwest Amazon
region), the shift from one to another lingua franca with retention of minority
languages (Mayan immigrants in urban United States shifting from Spanish
to English but retaining Mayan languages), etc., and may be applied to sit-
uations where prehistoric interaction has left linguistic traces but where the
nature of the interaction is unknown (e.g., the sharing of linguistic features
around the entire coast of the Pacific Ocean). In the Appendix we provide a
preview of the extension of the Schulze model to bilingualism.
An area where physicists may wish to try out their hands more is that of
language change. Simulations may help linguists come to terms with realities
that are accessible through empirical research only in small fragments. Lan-
guages develop and change through the interaction of multitudes of agents
using large lexical inventories and complex grammars. The kinds of regu-
larities that linguist can identify, such as the regularity of sound changes
or directed paths of grammaticalization (roughly, the process whereby sep-
arate words become part of the morphology), are mostly accessible only to
a retrospective view, trough the comparison of language stages dozens or
hundreds of years apart. What lies between is a flux whose behavior is not
easy to understand. 19th century historical linguists, with their focus on
regular sound changes, lived in a universe of clean equations such as Latin
p = English f (as in pater = father). The advent of 20th century soci-
olinguistics, with its focus on the social mechanisms behind sound changes,
complicated the picture, much like the picture gets complicated when one
moves from clean Newtonian physics to modern statistical physics, which
tries to model the actual behavior and interaction of entities. Nevertheless,
unlike physicists, linguists working on the way that languages change have
taken little recourse to simulations that might help them understand the
complexity of how language change or shift percolates within a community.
For instance, a leading sociolinguist has argued that ”networks constituted
chiefly of strong ties function as a mechanism to support minority languages,
resisting institutional pressures to language shift, but when these networks
weaken, language shift is likely to take place” (p. 558 in [40]). This hypoth-
esis is based on just a few case studies, and such case studies are, one the
one hand, extremely costly and, on the other, cannot even begin to cover
the multitute of different situations that actually obtains. In addition to the
strength of network ties, other important parameters are presumably the size
of the group speaking the minority language, geography, prestige of one as
opposed to the other language, economic gain involved in shifting language,
age- and gender-determined mobility, and maybe more. The behaviors of
such parameters can be investigated in simulations (e.g., for geography see
[11, 32]).
Finally, more works needs to be done towards the integration of the mod-
elling of language competition by physicists reviewed here and the modelling
of language evolution by computational linguists [41, 42, 43, 44, 45]. While
physicists have been adept in modelling the interaction among agents but
have operated with languages represented only by numbers or bit-strings,
computational linguists offer elaborate grammar models. With more com-
plex models of the interior structure of languages carried by agents, research
need not be limited to a focus on language competition, but could be ex-
tended to issues of language structure itself.
6 Appendix: Bilingualism
Several authors studied the possibility that people speak more than one lan-
guage [5, 6, 38], and we do here the same for the Schulze model on the square
lattice, with F = 8 features of Q = 3 different values, using only interactions
to nearest neighbours [27]. For this purpose we modify the switch process.
Before, languages spoken by a fraction x of the four neighbours were
dropped in favour of the language of a randomly selected neighbour with
probability (1 − x)2r (r = 0.9). Now we do this at lattice site i only if none
of the four neighbours of i speaks the mother language of site i: x = 0;
then with probability r we replace the mother language of i by the mother
language of a randomly selected neighbour. Otherwise, for x > 0, with the
above probability (1− x)2r, site i learns as an additional “foreign” language
a randomly selected (foreign or mother) language of a randomly selected
neighbour. If in the latter case x > 0, site i has already learned a foreign
language before, then this old foreign language is replaced by the new foreign
language.
 .0001
 0.001
 0.01
 1  10  100  1000  10000
Leading mother(+,x) and foreign(*,sq.) languages
 1e-04
 0.001
 0.01
 1  10  100  1000  10000
Leaders with (+,x) and without (*,sq.) bilinguals
Figure 10: Largest and second-largest languages in the Schulze lattice at
p = 0.01, q = r = 0.9 in a 8000 × 8000 lattice without migration. Part
a includes bilinguals, part b compares only the mother languages with and
without bilinguals.
 .0001
 0.001
 0.01
 1  10  100  1000  10000
As 10a but with forgetting and migration d=0.01
 10  100  1000
100 x 100 migration: d = 1 (left) to 0.0001 (right)
Figure 11: Part a: As Fig 10a, but including a migration probability d of 1 %
and a site-dependent forgetting probability between zero and 5 %, on a 6000×
6000 lattice. Part b shows the drastic speed-up of dominance if the migration
probability d is enhanced: d = 1, 0.5, 0.2, 0.1, . . . , 0.0005, 0.0002, 0.0001.
These are the learning and replacement events if everybody can speak at
most two languages. If instead the number of languages for each person is
restricted by an overall upper limit, then for x > 0 the last-learned foreign
language is replaced by the newly selected neighbour language. If this upper
limit is set equal to one, we go back to the model of monolingual speakers.
In all cases, x is the fraction of neighbours of i speaking as their mother
tongue the mother language of site i. For language diffusion we took q = 0.9
throughout, and for language change mostly p = 0.01. Thus one iteration
may correspond to about one human generation.
We start with a fragmented distribution of mother languages and no
foreign languageas, except that one particular language is spoken initially
by ten percent of the people, randomly spread over the lattice. Then we
check if this “lingua franca” finally (after at most 105 iterations) is spoken
by about everybody: transition from fragmentation to dominance of initially
favoured language. (If another language dominated we count this case as
fragmentation.)
For ten 50× 50 lattices, this transition happened up to p = 0.04 if bilin-
guality is allowed, while for monolinguals it happens up to the larger chang-
ing rate p = 0.09: Bilinguality makes dominance of one language less stable
against continuous changes; see also [38]. For Q = 5 instead of 3 this limit
shifts from 0.04 to 0.05, while for Q = 3, F = 16 it is about 0.03. Fig.10
shows for 8000× 8000 lattices the time dependence of the fraction of people
speaking the largest and the second-largest language, separately for mother
tongue and foreign languages; the comparison for the case without bilinguals
is restricted to the mother language and shows that dominance is reached
faster without bilinguals.
In these simulations, after a short time everybody speaks two languages,
and if up to ten languages are allowed, then again after a short time every-
body speaks ten languages. This is nice but unrealistic. In order to take
into account that people forget again foreign languages which were learned
but not used, or give up learning a foreign language when the need for it
dissipates, we assume that at every time step each speaker (more precisely,
each lattice site) may give up the last-learned foreign language if none of the
neighbours at that time speaks this language. This forgetting happens with
a probability between zero and five percent, fixed for each site randomly and
independently at the beginning.
In addition we included migration via exchanging locations: A speaker
or family exchanges residence with a randomly selected neighbour, and both
carry their languages with them. This happens at each iteration with a
probability d; physicists call d the diffusion constant. Fig.11 shows that ap-
preciable migration can drastically speed up the growth of the lingua franca
from having an initial advantage of being spoken by ten percent of the pop-
ulation to being dominant.
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	Introduction
	Schulze Model
	Definition
	Results
	Viviane Model
	Definition
	Results
	Other Models
	How physics may inform linguistics:  prospects for future research
	Appendix: Bilingualism

  Simulations of physicists for the competition between adult languages since
2003 are reviewed. How many languages are spoken by how many people? How many
languages are contained in various language families? How do language
similarities decay with geographical distance, and what effects do natural
boundaries have? New simulations of bilinguality are given in an appendix.

Introduction
While the emergence and learning of human languages has been simulated
since decades on computers [1], and while a later economics Nobel laureate
also contributed to linguistics long ago [2], the competition between existing
languages of adults is a more recent research trend, where physicists have
tried to play a major role. It follows the principle of survival of the fittest,
as known from Darwinian evolution in biology, and indeed many of the tech-
niques have been borrowed from simulational biology [3]. This emphasis from
physics on the competition of existing languages for adult humans started
with Abrams and Strogatz[4] and was then followed by at least six groups
independently [5, 6, 7, 8, 9, 10]. More recently, of course, reviews[11, 3] and
conferences brought them together, and others followed them [12, 13, 14].
Today about 7000 different languages (as defined by linguists) are spoken,
and about every ten days one of them dies out. On the other hand, the split
of Latin into different languages spoken from Portugal to Romania is well
documented. In statistical physics, we can describe and explain the pressure
which air molecules of a known density and temperature exert on the walls.
But we cannot predict were one given molecule will be one second from
now. Similarly, the application of statistical physics tools to linguistics may
describe the ensemble of the seven thousand presently existing languages,
http://arxiv.org/abs/0704.0691v1
 1000
 1  10  100  1000  10000  100000  1e+06  1e+07  1e+08  1e+09
size s
Distribution of language sizes from Grimes, Ethnologue, and  550 exp[-0.05{ln(size/7000)}**2]
Figure 1: Empirical variation of the number Ns of languages spoken by
s people each. For better presentation, the language sizes s are binned in
powers of two. Data from Ethnologue [16], as plotted in [17]. The parabola
corresponds to a log-normal distribution; we see deviations from it for the
smallest sizes [18].
but not the extinction of one given language in one given region on Earth.
Figure 1 shows how many languages exist today, as a function of the number
of speakers of that language. A statistical theory of language competition
thus first of all should try to reproduce such results, in order to validate the
model. If if fails to describe this fact, why should one trust it at all? Or as
stated by linguist Yang on page 216 of [15]: It is time for the ancient field of
linguistics to join the quantitative world of modern science.
This review starts with our own model for numerous languages in section
2, followed by a review of the alternative model of Viviane de Oliveira and
coworkers [10]. Then we review more shortly the many other models which
at present do not allow the simulation of thousands of different languages.
2 Schulze Model
2.1 Definition
Our own simulations, also called the Schulze model, characterise each lan-
guage (or grammar) by F independent features each of which can take one
of Q different values; the binary case Q = 2 allows the storage in bit-strings.
Three basic mechanisms connected with probabilities p, q and r are common
to all variants:
i) With probability p at each iteration and for each feature, this feature
is changed (or mutated in biological language). This change is random or
not, depending on process ii).
ii) With probability q the mutation/change under i) is not random but
instead transfers the value of this feature from another person in the popu-
lation. This transfer is called diffusion by linguists. With probability 1− q,
the change is random.
iii) With probability (1 − x)2r (also (1 − x2)r has been used instead)
somebody discards the mother language and takes over the whole language
(all F features) from another person in the population. Here x is the fraction
of people speaking the old language. This flight is called shift by linguists.
Several variants are possible: One can use one joint population where
everybody can meet everybody for transfer and shift; or we put people onto
a square L× L lattice or more complicated network, where diffusion and/or
shift are possible only from a randomly selected neighbour. People may
migrate on this lattice, which physicists would call diffusion. The population
can be fixed, meaning that at every iteration all adults are replaced by their
children. Or it can grow by a suitable birth and death process; in this case the
shifting probability can include also a factor proportional to the population.
If one dislikes to have three free parameters p, q, r one may set q = r = 1
without much loss in results.
For the number F of features, from 8 to 64 were used in simulations. Real
languages contain many thousands of words for everyday use, and thus one
should identify one feature rather with an independent grammatical element
(like the order of subject, object and verb in a sentence) than with a word. F
for real languages was estimated as about 30 [19] or about 40 to 50 [15] such
choices, and the Word Atlas of Language Structures [20] lists 138 features
with up to Q = 9 values. These grammar sizes thus correspond roughly to
what has been simulated. According to [21] the rate of change in normal
linguistic typological features, i.e., excluding a few extraordinarily unstable
ones, is 16 % per 1000 years.
2.2 Results
If we start with everybody speaking one language (or with just one Eve), then
at low p this language still dominates and is spoken by more than half of the
total population, with the remaining people speaking a minor and short-lived
variant of this dominating language. At high p, on the other hand, the whole
population soon fragments into many languages, roughly such that everybody
selects nearly randomly one of the QF possible languages. This corresponds
to the biblical story of the Tower of Babel. We thus have dominance for
small p and fragmentation for large p, with a first-order phase transition or
jump at some threshold value which depends on the other parameters and
details of the model.
If instead we start with everybody speaking a randomly selected lan-
guage, then for high p this situation remains. For low p, however, after some
time one language by random accidents happens to grow to a sufficiently
large size such that it then grows rapidly to be spoken by more than half of
the population. Thus a transition from initial fragmentation to final dom-
inance happens in Fig.2. The threshold value is different from the one for
the opposite direction from dominance to fragmentation: we have hysteresis
as is common for first-order transitions, Fig.3. Empirically, this transition
to one dominating language was observed on the American continent; within
the last five centuries, two thirds of the native Brazilian languages have died
out. And in the last half century we observed the rise of English in physics
research publications. While 85 years ago, physicist Bose sent from India
his paper to Einstein in order to have it translated from English to German
(which lead to Bose-Einstein condensation), after World War II physics re-
search was usually published in English, first in Japan, since the 1960’s in
(West) Germany, a decade later in France, since the 1990’s in Russia; finally,
China has witnessed a surge in physics papers written in English since 2000.
The time needed to go from fragmentation to dominance increases roughly
logarithmically with population size, at least in the binary case Q = 2 with-
out lattice. Thus a mathematical solution for an infinite population might
never get this transition. In other words, proper models should be agent-
based [22], with individuals acting on their own; one should not average over
10000
100000
100 M
0 100 200 300 400 500 600
Size of largest language for 100M, 10M, 1 M, 0.1 M, 10 K, 1 K people, 0.48 mutations per person, q=0.94
Figure 2: Variation with time of the number of people speaking the most
widespread language, for various population sizes. The larger the population
is, the longer is the time until the transition from fragmentation to dominance
takes place. Q = 2, F = 8, p = 0.06, q = 0.94; from [11].
the whole population, using differential equation for the concentrations. Such
simulations have been standard in computational physics for half a century
(Monte Carlo and Molecular Dynamics), while mean field approximations
average over many individuals and can give somewhat or completely wrong
results. (The transition from fragmentation to dominance may require a shift
probability (1− x)2r instead of (1− x2)r.)
The language size distribution to be compared with Fig.1 is shown in
Fig.4a. To get it, we looked at non-equilibrium results and introduced ran-
dom multiplicative noise, since otherwise the language sizes were too small
and their distribution too irregular. Fig.4b avoids these tricks and instead
places the people on a directed scale-free network, discussed below.
No lattice or other spatial structure was employed in the above simula-
tions. On a lattice one can look at language geography [23, 24, 25]. North
and South of the Alps, different languages are spoken, and the same sepa-
ration is made by the English Channel. Genetic and linguistic boundaries
in Europe mostly coincide, and about two thirds of them agree with natural
boundaries like a mountain chain or sea [26]. We simulated this effect on a
0.01 0.1 1 10
population in millions
Critical mutation rate, no transfer, initially one (top) or initially many random languages (bottom)
Figure 3: Dependence of the mutation threshold for the phase transition
on the population size; upper data from dominance to fragmentation, lower
data from fragmentation to dominance. Above the curves we arrive at frag-
mentation, below at dominance. From [11].
lattice [27] with contact only between nearest neighbours and a horizontal
barrier separating the upper from the lower half. The shift from a small
to a large language happens across the barrier only with a small crossing
probability c. For c = 0 one thus has two completely separated halves of the
lattice, and trivially the languages which evolve as dominating are different
on both sides of the border. With c = 1 the border has no effect, and only
one language dominates. Fig.5 shows how often for small but finite c two
separate dominating languages may coexist; already quite small c suffice,
particularly for large lattices, to unify the two regions into only one with the
same language dominating on both sides.
Fig.4b employed a directed Barabási-Albert scale-free network. These
networks are grown from a small fully connected core such that each new
network member selects m already existing members as teachers. The more
people have selected a certain teacher before, the higher is the probability
that this teacher will again be selected. Information only flows from the
teacher to the person who selected this teacher, not in the opposite direction
[31].
 1000
 10000
 100000
 10 M
 1  10  100  1000  10000  100000
number of speakers
Two runs of 200 million; fragmented start, mutation rate 0.0032 per bit-string, t < 5000 or 6000
 10 K
 100K
 1  10  100  1 K  10 K  100K  1 M  10 M
number of speakers
N = 1 million (+), 3 M (x), 10 M (*), 30 M (sq.)
Figure 4: Language size distribution Ns. Top: without lattice, with random
multiplicative noise, not in equilibrium [17]. Bottom: On scale-free network
in equilibrium [30].
 0.0001  0.001  0.01  0.1
crossing rate
100 samples, L = 50 (right line), 100 (+,x), 200 (left line)
Figure 5: Fraction of cases when a semi-permeable barrier allowed two
different languages to dominate on its two sides in the Schulze model.
3 Viviane Model
3.1 Definition
The model of Viviane de Oliveira et al [10] has become known as the Viviane
model (following the Brazilian tradition of how to call people). It simulates
the colonisation of an uninhabited continent by people. Each site j of an L×L
square lattice can later be populated by cj people; this carrying capacity cj
is selected randomly between 1 and some m ∼ 102. Initially only one site i
is occupied by ci people.
Then at each time step, one randomly selected empty lattice neighbour j
of occupied sites becomes occupied with probability cj/m by cj people. Thus
after some time the whole lattice becomes occupied and the simulation stops.
In contrast to the Schulze model, the Viviane model is a growth process and
not one eventually fluctuating about some equilibrium.
Languages have no internal structure and are simply numbered 1, 2, 3,,,
with 1 being the number of the language spoken on the originally occupied
site. All people within one lattice site speak the same language. First, if a
new site has been colonized the language spoken there is taken from one of the
occupied neighbours k, proportional to the fitness Fk of that neighbour site k.
This fitness is the total number of people speaking anywhere in the lattice the
language of k, except that it is bounded from above by a maximum fitness
Mk fixed randomly between 1 and some Mmax ∼ 10
3. Second, mutations
(language change) are made with probability α/Fj on the freshly occupied
site j only, from the selected language of neighbour k. A mutation means
that a new language is created which gets a new number not used previously.
In this way, the flight (shift) from small language and the mutations,
which were two separate processes in the Schulze model, are combined into
one process; and this process also is a transfer (diffusion) process which in
the Schulze model was dealt with separately. Thus we have here only one
free parameter α, the mutation coefficient, instead of three parameters p, q, r
in the Schulze model.
Variants allow mutations also later, after a site is occupied. Or a language
is characterised by a string of F bits (Q = 2 in the Schulze notation) and only
different bit-strings count as different languages [28]. Or the capacities ck are
not homogeneously distributed between 1 and m but more often small than
large, with a frequency proportional to 1/cj, as long as it is not larger than
the maximum m. Lot of computer time then is saved if after the occupation
of the new sites one selects two of its occupied neighbours and takes the
language from the one with the bigger capacity; if only one neighbour is
occupied then its language is taken over.
3.2 Results
In contrast to the Schulze model, the Viviane model gives languages spoken
by 109 people if a sufficiently large lattice is used. The language size distri-
bution Ns has a maximum at moderately small language sizes s. However,
instead of a round parabola as in Fig.1, the log-log plot of Ns versus s gives
two straight lines meeting at the maximum, that means one power law for
small s (where Ns increases with s) and another power law for large s (where
Ns decreases with s). So, not everything is solved yet.
Crucial progress was made by Paulo Murilo de Oliveira (not the same
family as Viviane de Oliveira), who introduced the above-mentioned modifi-
cations [28]: Languages (grammars) are characterized by bit-strings (Q = 2)
of length F ≃ 13 and count as different only when their bit-strings differ; the
carrying capacities c are selected with a probability ∝ 1/c, and the newly
colonized site gets the fitter language of two previously occupied neighbours.
 100K
 1  10  100  1 K  10K 100K 1 M  10M 100M 1 G
number of speakers
L = 20,000; 7500 languages; 5940 million people
Figure 6: Variation of the number Ns of languages spoken by s people each
in the Viviane model, as modified and published in [28].
Now the distribution is roughly log-normal, Fig.6, with enhancement for very
small sizes; the total population and the total number of languages can be
made close to the present reality, the maximum of the parabola in a double-
logarithmic plot (with binning by factors of two in s) is near s ≃ 104, while
the largest language is spoken by 109 people, similar to Mandarin Chinese.
Fig.7 shows for both the modified Viviane model and the Schulze lattice
model that languages in general are less similar to each other if they are
widely separated geographically, in agreement with reality [24, 25]. Note the
difference in scales: One lattice constant (distance between nearest neigh-
bours) corresponds to about one kilometer in the Viviane model and 1000
kilometers in the Schulze model, if Fig.7 is compared to reality [24].
Also the classification of different languages into one family, like the Indo-
European languages, has been simulated with moderate success. Following
the history of the mutations during the colonisation, a language tree like can
be constructed in the unmodified version (Fig.11.15 in [11]). One can imagine
that this is Latin, splitting up into Romanian, Italian, Spanish and French,
with Spanish then splitting into Castellan, Galego and Catalan, and Catalan
mutating further into Mallorquin. (Many small branches were omitted for
clarity.) More quantitative information is obtained from the modified bit-
 0  2  4  6  8  10  12  14  16  18  20
geographical distance in 1000
Modified Viviane model, line = random differences
 1  2  3  4  5  6  7  8  9  10
geographical distance
Schulze model, Q=5, F=9, p=0.5, q=r=0.9, s=0.5
Figure 7: Language differences (in arbitrary units [24]) as a function of
geographical distance in the Viviane model (top) and the Schulze model
(bottom). The horizontal line corresponds to completely uncorrelated lan-
guages. In the bottom part, + and x correspond to start with fragmentation
(+) and with dominance (x). From [24, 30].
string version [28]. The mutated language on a newly occupied site starts a
new family if it differs in two or more bits from the bit-string characterising
the historically first language of the old family. The size distribution of
language families in Fig.8 agrees in its central part with the empirically
observed [29] exponent –0.525 and is independent of the length F = 8, 16,
32 or 64 of the bit-strings for the Viviane model and independent of the
population size for the Schulze model.
 1000
 10000
 100000
 1  10  100  1000
number of languages
100 * L=10,000; 8(+), 16(x), 32(*), 64(sq.) bits
 10 K
 100K
 1  10  100  1000
number of families
N = 1 million (+), 3 M (x), 10 M(*), 30 M(sq.)
Figure 8: Number of families as a function of the number of different lan-
guages in this family [30]. Top: Modified Viviane model for various lengths
of the bit-strings. Bottom: Schulze model with Q = 5, F = 8 on directed
Barabási-Albert scale-free network, p = 0.5, q = 0.59, r = 0.9 for various
population sizes.
4 Other Models
Years before physicists invaded en masse the field of linguistics, Nettle [33]
already wrote down a differential equation for the number L of languages,
dL/dt = 70/t− L/20 ,
where time t is measured in millennia. For long times, only one language
(mathematically: zero languages) will remain; however that time lies far in
the future. A more detailed splitting mechanism was introduced by Novotny
and Drozd [34] for the emergence of new languages from one mother lan-
guage, and gave a log-normal distribution of language sizes, in agreement
with reality except presumably at the smallest sizes [18]. In the same spirit
of looking at languages as a whole, ignoring the individuals, are the very re-
cent models of Tuncay [14], who coupled a splitting mechanism with random
multiplicative noise in the size of the growing population, plus an extinction
probability, and found a power-law decay or a log-normal size distribution
for the simulated languages, depending on parameters. He also checked the
lifetimes of the simulated languages. An “early” attempt to apply the Ising
model of statistical physics to linguistics [36] had little success.
Numerous coupled differential equations were studied by scientists coming
from theoretical chemistry, mathematics and computer science [35] for the
purpose of language learning by children. They have been applied [3] also
to the competition of up to 8000 languages of adults, but since the original
authors have to our knowledge not followed this re-interpretation of their
learning model we now refer to [35, 3] for details and results.
It was the population dynamics of Abrams and Strogatz [4] which started
the avalanche of physics papers on language competition. They assume two
languages X and Y, spoken by the fractions x and y = 1 − x of a fixed
population with a time dependence
dx/dt = yxas− xya(1− s) ,
with a status or prestige variable S which is close to one if X has a high
prestige and close to zero for low prestige of X. The neutral case is s = 1/2.
The exponent a = 1.31 was fitted to some empirical data of how minority
languages decay in size. If a is replaced by unity we arrive at the logistic
equation of Verhulst from the 19th century, which was applied to languages
by Shen in 1997.
0 1 2 3 4 5 6 7 8 9 10
Abrams-Strogatz model for a = 1.31 and s = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 (bottom to top); unsymmetric
Figure 9: Abrams-Strogatz model: Fraction of people speaking language
X, versus time, for an initial concentration of only ten percent and various
prestige values s of this language. For s ≥ 0.7 this language finally wins over
the other language Y. From [3].
Fig.9 shows the resulting x(t) if X is spoken initially by a minority of
ten percent only. Then for low, neutral or slightly higher status s of its
language X, the fraction decays further towards zero, but for a higher status
like x = 0.7 it finally wins over and is spoken by everybody (not shown).
This may correspond to the influence of a colonial power; indeed in France
today most people speak French as a result of the Roman conquest of more
than two millennia ago, and in Brazil many of the native languages have
become extinct in the last five centuries since the Portuguese arrived there.
This Abrams-Strogatz approach was soon generalized to a lattice by Pa-
triarca and Lepännen [4], and later to populations with bilingual speakers
[5, 38], coexistence of the two languages [8] as well simulations based on
individuals [13].
Such agent-based simulations were also made by Kosmidis et al [6] who
gave each person a string of 20 bits. The first 10 belonged to one language,
the last 10 to another language. In this way they were able to simulate
people speaking, more of less correctly, one or two languages. One could
also interpret their model as one for English which is a mixture of German
(Anglo-Saxon) and French words, due to the conquest of England by the
Normans in 1066.
Finally, Schwämmle [9] also used bit-strings, but to describe biological
ageing through the Penna model. The child can learn the languasge from
the mother, the father, or both, thus also allowing for bilinguals. This model
allowed to simulate that languages are learned easier in youth than at old age.
In this way it builds a bridge between language competition and language
learning [35].
5 How physics may inform linguistics:
prospects for future research
As the research described above has progressed a larger design has become
apparent, which consists in an empirical side looking for quantitative distribu-
tions involving languages [29, 24, 37] and a development of models simulating
similar quantitive distributions. The hope is that as more and more quantifi-
able relations in and among languages are discovered and simulation models
are developed which can adequately replicate these distributions, the simu-
lation models will of necessity become more and more adequate as models
of actual languages, and could therefore be employed for purposes beyond
the ones for which they were designed. For instance, the revised Viviane
model, which was designed to capture the distribution of speaker popula-
tions and the population of languages within families [30], could potentially
be employed for investigating absolute rates of language change, an issue
with which linguists are very much concerned [39], inasmuch as knowledge
of how fast languages change could provide us with a way to date prehis-
toric events involving people speaking given reconstructed languages. Thus,
a strand of research where linguists and physicists can and will continue to
cooperate is the search for quantifiable distributions on the one hand and the
fine-tuning of models which can adequately simulate an increasing range of
such distributions.
Apart from some exceptions [5, 6, 38], most work on language compe-
tition has assumed monolingual speakers. Since most of the world’s pop-
ulation is bi- or multilingual this is clearly not adequate. Language shift
will normally involve transitional bilingualism, or bilingualism may persist
for centuries without the majority language necessarily replacing minority
languages. Diglossia, i.e. the use of different languages for different pur-
poses may help sustain bilingualism. Current models can be extended to
investigate under which conditions bilingualism may persist or get reduced
to monolingualism. Different kinds of situations can be modelled, such as
the replacement of certain, but not all, languages within the domain of the
Roman empire, the development of so-called linguistic areas, where several
languages share a number of features (e.g., the Balkans, India, Mesoamer-
ica), multilingualism caused by linguistic exogamy (the northwest Amazon
region), the shift from one to another lingua franca with retention of minority
languages (Mayan immigrants in urban United States shifting from Spanish
to English but retaining Mayan languages), etc., and may be applied to sit-
uations where prehistoric interaction has left linguistic traces but where the
nature of the interaction is unknown (e.g., the sharing of linguistic features
around the entire coast of the Pacific Ocean). In the Appendix we provide a
preview of the extension of the Schulze model to bilingualism.
An area where physicists may wish to try out their hands more is that of
language change. Simulations may help linguists come to terms with realities
that are accessible through empirical research only in small fragments. Lan-
guages develop and change through the interaction of multitudes of agents
using large lexical inventories and complex grammars. The kinds of regu-
larities that linguist can identify, such as the regularity of sound changes
or directed paths of grammaticalization (roughly, the process whereby sep-
arate words become part of the morphology), are mostly accessible only to
a retrospective view, trough the comparison of language stages dozens or
hundreds of years apart. What lies between is a flux whose behavior is not
easy to understand. 19th century historical linguists, with their focus on
regular sound changes, lived in a universe of clean equations such as Latin
p = English f (as in pater = father). The advent of 20th century soci-
olinguistics, with its focus on the social mechanisms behind sound changes,
complicated the picture, much like the picture gets complicated when one
moves from clean Newtonian physics to modern statistical physics, which
tries to model the actual behavior and interaction of entities. Nevertheless,
unlike physicists, linguists working on the way that languages change have
taken little recourse to simulations that might help them understand the
complexity of how language change or shift percolates within a community.
For instance, a leading sociolinguist has argued that ”networks constituted
chiefly of strong ties function as a mechanism to support minority languages,
resisting institutional pressures to language shift, but when these networks
weaken, language shift is likely to take place” (p. 558 in [40]). This hypoth-
esis is based on just a few case studies, and such case studies are, one the
one hand, extremely costly and, on the other, cannot even begin to cover
the multitute of different situations that actually obtains. In addition to the
strength of network ties, other important parameters are presumably the size
of the group speaking the minority language, geography, prestige of one as
opposed to the other language, economic gain involved in shifting language,
age- and gender-determined mobility, and maybe more. The behaviors of
such parameters can be investigated in simulations (e.g., for geography see
[11, 32]).
Finally, more works needs to be done towards the integration of the mod-
elling of language competition by physicists reviewed here and the modelling
of language evolution by computational linguists [41, 42, 43, 44, 45]. While
physicists have been adept in modelling the interaction among agents but
have operated with languages represented only by numbers or bit-strings,
computational linguists offer elaborate grammar models. With more com-
plex models of the interior structure of languages carried by agents, research
need not be limited to a focus on language competition, but could be ex-
tended to issues of language structure itself.
6 Appendix: Bilingualism
Several authors studied the possibility that people speak more than one lan-
guage [5, 6, 38], and we do here the same for the Schulze model on the square
lattice, with F = 8 features of Q = 3 different values, using only interactions
to nearest neighbours [27]. For this purpose we modify the switch process.
Before, languages spoken by a fraction x of the four neighbours were
dropped in favour of the language of a randomly selected neighbour with
probability (1 − x)2r (r = 0.9). Now we do this at lattice site i only if none
of the four neighbours of i speaks the mother language of site i: x = 0;
then with probability r we replace the mother language of i by the mother
language of a randomly selected neighbour. Otherwise, for x > 0, with the
above probability (1− x)2r, site i learns as an additional “foreign” language
a randomly selected (foreign or mother) language of a randomly selected
neighbour. If in the latter case x > 0, site i has already learned a foreign
language before, then this old foreign language is replaced by the new foreign
language.
 .0001
 0.001
 0.01
 1  10  100  1000  10000
Leading mother(+,x) and foreign(*,sq.) languages
 1e-04
 0.001
 0.01
 1  10  100  1000  10000
Leaders with (+,x) and without (*,sq.) bilinguals
Figure 10: Largest and second-largest languages in the Schulze lattice at
p = 0.01, q = r = 0.9 in a 8000 × 8000 lattice without migration. Part
a includes bilinguals, part b compares only the mother languages with and
without bilinguals.
 .0001
 0.001
 0.01
 1  10  100  1000  10000
As 10a but with forgetting and migration d=0.01
 10  100  1000
100 x 100 migration: d = 1 (left) to 0.0001 (right)
Figure 11: Part a: As Fig 10a, but including a migration probability d of 1 %
and a site-dependent forgetting probability between zero and 5 %, on a 6000×
6000 lattice. Part b shows the drastic speed-up of dominance if the migration
probability d is enhanced: d = 1, 0.5, 0.2, 0.1, . . . , 0.0005, 0.0002, 0.0001.
These are the learning and replacement events if everybody can speak at
most two languages. If instead the number of languages for each person is
restricted by an overall upper limit, then for x > 0 the last-learned foreign
language is replaced by the newly selected neighbour language. If this upper
limit is set equal to one, we go back to the model of monolingual speakers.
In all cases, x is the fraction of neighbours of i speaking as their mother
tongue the mother language of site i. For language diffusion we took q = 0.9
throughout, and for language change mostly p = 0.01. Thus one iteration
may correspond to about one human generation.
We start with a fragmented distribution of mother languages and no
foreign languageas, except that one particular language is spoken initially
by ten percent of the people, randomly spread over the lattice. Then we
check if this “lingua franca” finally (after at most 105 iterations) is spoken
by about everybody: transition from fragmentation to dominance of initially
favoured language. (If another language dominated we count this case as
fragmentation.)
For ten 50× 50 lattices, this transition happened up to p = 0.04 if bilin-
guality is allowed, while for monolinguals it happens up to the larger chang-
ing rate p = 0.09: Bilinguality makes dominance of one language less stable
against continuous changes; see also [38]. For Q = 5 instead of 3 this limit
shifts from 0.04 to 0.05, while for Q = 3, F = 16 it is about 0.03. Fig.10
shows for 8000× 8000 lattices the time dependence of the fraction of people
speaking the largest and the second-largest language, separately for mother
tongue and foreign languages; the comparison for the case without bilinguals
is restricted to the mother language and shows that dominance is reached
faster without bilinguals.
In these simulations, after a short time everybody speaks two languages,
and if up to ten languages are allowed, then again after a short time every-
body speaks ten languages. This is nice but unrealistic. In order to take
into account that people forget again foreign languages which were learned
but not used, or give up learning a foreign language when the need for it
dissipates, we assume that at every time step each speaker (more precisely,
each lattice site) may give up the last-learned foreign language if none of the
neighbours at that time speaks this language. This forgetting happens with
a probability between zero and five percent, fixed for each site randomly and
independently at the beginning.
In addition we included migration via exchanging locations: A speaker
or family exchanges residence with a randomly selected neighbour, and both
carry their languages with them. This happens at each iteration with a
probability d; physicists call d the diffusion constant. Fig.11 shows that ap-
preciable migration can drastically speed up the growth of the lingua franca
from having an initial advantage of being spoken by ten percent of the pop-
ulation to being dominant.
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	Introduction
	Schulze Model
	Definition
	Results
	Viviane Model
	Definition
	Results
	Other Models
	How physics may inform linguistics:  prospects for future research
	Appendix: Bilingualism
