Fernando Peruani

Gustavo J Sibona

Dynamics and steady states in excitable mobile agent systems

Keywords: numbers: 87.23.Cc, 87.19.Xx, 02.50.-r, 87.18.Bb

We study the spreading of excitations in 2D systems of mobile agents where the excitation is transmitted when a quiescent agent keeps contact with an excited one during a non-vanishing time. We show that the steady states strongly depend on the spatial agent dynamics. Moreover, the coupling between exposition time (ω) and agent-agent contact rate (CR) becomes crucial to understand the excitation dynamics, which exhibits three regimes with CR: no excitation for low CR, an excited regime in which the number of quiescent agents (S) is inversely proportional to CR, and for high CR, a novel third regime, model dependent, where S scales with an exponent ξ -1, with ξ being the scaling exponent of ω with CR.

The understanding of information propagation through a system of moving agents is crucial for many applications, ranging from chemical reactions to epidemic spreading [1]. A particular example of such a process is the propagation of an excitation through a system of mobile agents. In an excitable system, agents typically have three states, quiescent, excited, and refractory, and pass through them in such a way that once excited become first refractory, and then quiescent again. A classical example of an excitable system is the forest-fire [2]. In this case, the immobility of the agents is well justified. Another example of excitable system is a disease in which agents undergo a cycle by which an infected agent becomes first recovered and then susceptible (SIRS dynamics). Though in the spreading of diseases the mobility of the agent has been for a long time ignored [3,[START_REF] Bailey | The mathematical theory of infectious diseases and its applications[END_REF], its importance has been recently recognized [START_REF] Hufnagel | Proc. Natl. Acad. Sci[END_REF][START_REF] Cross | [END_REF]7,8,9]. The first attempt to incorporate interactions among individuals was through the study of disease spreading on complex networks which typically represent social structures [10,11,12,[START_REF] Mickens | Proceedings of WiSE'05[END_REF][START_REF] Van Ballegooijen | Proc. Natl. Acad. Sci[END_REF][START_REF] Joo | [END_REF]16,17]. The next step was to incorporate the actual motion of individuals across subpopulation structures where nodes represent containers of individuals such as urban areas or just communities [START_REF] Hufnagel | Proc. Natl. Acad. Sci[END_REF][START_REF] Cross | [END_REF]7,8,9]. Despite the fact that complex networks can describe realistic social interactions or traffic networks, these models constitute frozen pictures where edges do not exhibit any dynamics. Gonzalez et al. in [18] showed that the degree distribution and clustering coefficient of empirical social networks can be described by a systems of off-lattice mobile agents, provided the agent density is appropriately chosen. Interestingly, in this model agents are constantly moving and the underlying agent-agent contact network is highly dynamical.

While agent motion may be central to information/excitation spreading, however, it poses questions that are not easy to answer. How does the spatial agent dynamics affect the excitation spreading? In a SIRSdynamics, is the agent-agent contact rate (CR) the key parameter that controls the the epidemic dynamics?

In classical epidemiological models [3,[START_REF] Bailey | The mathematical theory of infectious diseases and its applications[END_REF] as well as in previous mobile agent models [10,11,12,[START_REF] Mickens | Proceedings of WiSE'05[END_REF], it is assumed that the mean number of neighbors κ (directly associated to CR) and the effective infection probability λ are independent. Consequently, for either a SIRS or SIS dynamics there are only two regimes with CR, resp. κ . For low CR the system remains in the epidemic extinction regime, while for CR above a critical value the disease propagates such that the number of susceptible agents is inversely proportional to CR.

In this paper we study for the first time an excitable mobile agent system where the excitation is transmitted by keeping physical contact for a finite time (see Fig. 1). We derive expressions for the threshold and steady states of the excitation dynamics, which highly depend on the spatial agent dynamics, and show that the coupling between exposition time (which directly affects λ) and CR (resp. κ ) is crucial to understand the evolution of the excitation. In particular, we show that due to this coupling the excitation dynamics exhibits three regimes with the CR. These results applied to epidemic spreading suggests that, contrary to what classical epidemiological models state, the epidemic dynamics exhibits more than two regimes with CR.

Spatial agent dynamics.-We assume agents are selfpropelled disks which, in absence of interactions, move at constant speed in a box with periodic boundary condition and change their direction of motion at Poissonian distributed times. The equation of motion of the i-th agent can be expressed in the following way:

ẋi (t) = F i ζ + 1 ζ j =i ∇U (x i (t), x j (t)) (1)
where F i /ζ is the active velocity of the agent and ζ denotes a friction coefficient. As mentioned above, the active direction of motion follows a Poisson process by which an agent changes the direction of F i with a rate α while keeping constant the active speed υ

= |F i | /ζ [19].
As example of agent-agent interaction we have chosen a repulsive soft-core two-body potential that penalizes agent overlapping:

U (x, x ′ ) =      γ |x -x ′ | -β -(2r) -β if |x -x ′ | < 2r 0 if |x -x ′ | ≥ 2r ( 2
)
where r is the radius of the agents, β is a constant, and γ is a function of υ such that the maximum overlapping area between two agents is independent of υ [START_REF]Simulation parameters[END_REF].

The collision between two or more agents is a relatively slow process in which agents keep physical contact for a non-vanishing time. Notice that while the implemented dynamics is suitable to describe migration of some microorganisms [START_REF] Szabó | [END_REF], it becomes unrealistic as a model for human mobility. Excitation dynamics.-Without lack of generality we use as excitation dynamics a SIRS dynamics. A susceptible agent gets the disease by keeping physical contact with an infected one for certain time. The transmission of the infection (=excitation) is modeled by a Poissonian process whose characteristic time is τ T . Once infected, an agent carries the infection for a certain period of time, after which gets recovered and remains immune to the disease for another period of time, to finally become susceptible of being infected. The transitions from infected to recovered, and from recovered to susceptible are modeled by Poisson processes characterized by the characteristic times τ I and τ R , respectively.

Low density.-At low density the agents do not form large clusters and we can reduce the dynamics to a much FIG. 2: Populations of agents at the steady state vs. linear system size L. Simulations performed with the parameters given in [START_REF]Simulation parameters[END_REF] and υ = 0.1. Full black lines corresponds to Eqs. ( 6), (7), and ( 8 simpler one in which we consider only binary collisions. In this gas-like phase we describe the time evolution of the populations of susceptible, infected, and recovered agents by the following mean-field equations:

Ṡ = R τ R -1 -e -ω/τT υσ 0 ρIS (3)
İ = 1 -e -ω/τT υσ 0 ρIS -

I τ I (4)
Ṙ = I τ I - R τ R (5)
where S, I and R are defined as S = N S /N , I = N I /N and R = N R /N , with N being the total number of agents in the system, and N S , N I , and N R being the number of susceptible, infected and recovered agents, respectively.

Here, σ 0 represents the scattering cross section of the agents, defined as σ 0 = 4r, ρ is the density (ρ = N/V ), and ω denotes the mean duration of the collision event.

The exact dependency of ω with υ depends on the particular agent-agent interaction rule. Since in general, for υ → ∞, ω has to vanish, while for υ → 0, ω → ∞, and considering to the Buckingham π theorem [21], we conclude that ω takes the functional form ω (υ) = K (r/υ) ξ , where ξ is a positive constant and K a dimensional function of the interaction rule parameters. We choose values of K and ξ that best fit simulation data of ω vs. υ [START_REF]Simulation parameters[END_REF].

In consequence, 1 -e -ω/τT represents the probability per contact event (of mean duration ω) of transmitting the disease, while υσ 0 ρ in Eqs. ( 3) and ( 4) represents the agent-agent contact rate (CR). From Eqs. (3)-( 5), we obtain the two steady states of the system: i) S = 1, I = 0, and R = 0, corresponding to the epidemic extinction, and ii)

S st = R -1 0 (6) I st = (1 -S st ) τ I τ I + τ R (7) R st = (1 -S st ) τ R τ I + τ R , (8)
corresponding to the endemic state, where R 0 = 1 -e -ω/τT υσ 0 ρτ I denotes the basic reproductive number. Recall that R 0 > 1 corresponds to the endemic state, while R 0 < 1 to epidemic extinction. It is instructive to represent R 0 as R 0 = λ κ , where κ denotes the mean number of individuals in contact with an infected agent per time unit, i.e., κ = υσ 0 ρτ I , while λ refers to the probability for a susceptible individual to contract the disease during a contact event with an infected one, i.e., λ = 1 -e -ω/τT . Notice that though λ and κ are typically independent model parameters, for the current mobile agent-based model, they are coupled. This coupling has proved to be essential to understand the disease spreading dynamics.

Fig. 2 shows a comparison of Eqs. ( 6)-( 8) and agentbased simulations. For large values of L, there is a good agreement between simulations and theory. For small values of L, spatial correlations become important and multiple collisions and shielding effects take place. However, since the spatial dynamics is faster than the excitation dynamics, the mean-field approach still approximates the simulation data as observed in Fig. 2. Clearly, the smaller L, the worse the approximation becomes. Notice that the critical system size L c that separates the endemic from the epidemic extinction regime takes the form: L c = N 1 -e -ω/τT υσ 0 τ I 1/2 (dashed vertical line in Fig. 2). High density.-At high enough densities, agents form percolating clusters and agent mobility is strongly reduced, though exchange of neighbors still takes place. The maximum conceivable density, ρ M , corresponds to the one associated to the maximum packing fraction of disks, η M = ρ M a, where a is the area of a single agent. η M is 0.91 and corresponds to an hexagonal lattice. At densities close to this limit, the system can be reduced to a regular lattice of agents through which a SIRS dynamics evolves. Since we assume that locally there is still some mixing of particles, we neglect correlations between nodes, and express the temporal evolution of agent-populations in terms of a simple mean-field which yields the following endemic states:

S HD st = τ T / (τ I κ), I HD st = (1 -S st ) (τ I / (τ I + τ R )), and R HD st = (1 -S st ) (τ R / (τ I + τ R ))
, where κ denotes the mean number of neighbors. These values are shown in Fig. 2 as open symbols and represent a rough estimate for the system behavior at densities close to ρ M . Better estimates can be obtained by considering spacial correlations through the two-point probability function as FIG. 3: Populations of agents at the steady state vs. agent active speed υ. Simulations performed with the parameters given in [START_REF]Simulation parameters[END_REF] and L = 150. Full black lines corresponds to Eqs. ( 6)-( 8)). The vertical lines correspond to υm and υc, while dashed, dotted, and dash-dotted curves correspond to different values of ξ, see text.

indicated in [START_REF] Joo | [END_REF]. Notice that κ is constrained to the structure of the lattice and that S HD st > 0. The role of the active speed.-The active speed of the agents υ is a control parameter which directly regulates the degree of interaction among agents. Particularly, CR is directly proportional to υ. The values of the epidemic steady state, as indicated by Eqs. ( 6)-( 8) and confirmed by agent-based simulations, exhibit a non-trivial behavior with υ (see Fig. 3). For low enough density, we expect to observe the epidemic extinction state in the limit of υ going to zero. This implies that to observe the endemic state, υ has to be larger than a threshold value υ m . From Eq. ( 6) we can estimate the minimum speed required by the disease to survive. This is set by the condition S st = 1 from which we get the following transcendental equation for the minimum speed υ m :

e -ω(υm) τ T = 1 - 1 υ m σ 0 ρτ I . (9)
By solving numerically Eq. ( 9) for the parameters used in Fig. 3, we obtain υ m = 0.027 (dashed vertical line). Note that the expression for R 0 predicts a crossover at ω(υ c ) = τ T (dash-dotted vertical line).

As shown in Fig. 3, there are three regimes with the agent active speed υ. For υ < υ m , the mean time between subsequent collisions is larger than the infection period τ I , and so the disease can not spread and gets extinguished. On the contrary, for υ m < υ < υ c , the mean time between subsequent collisions is smaller than τ I and the system ends up in an endemic state. In this range of υ, the mean collision time is such that ω(υ) ≫ τ T , and thus the probability of transmitting the disease in a collision event is very high. As a consequence, the epi-demic size increases with the CR leading to S st ∼ υ -1 . Notice that in the limiting case of τ T → 0, this regime is valid for all υ > υ m . However, any realistic disease has some non-vanishing characteristic transmission time, and consequently, a third regime, ignored so far, emerges for υ > υ c , i.e., for larger values of CR. In this case, ω(υ) < τ T and the mean collision time is not enough to assure the disease transmission in each collision event. The interplay between CR and the transmission probability per collision event determines the third regime which, contrary to the second one, is model dependent and such that S st ∼ υ ξ-1 . In consequence, there are three possible behaviors depending on the interaction rule, resp. ξ. If ξ = 1, in the limit of υ → ∞ the epidemic size reaches an asymptotic value S ∞ = τ T / [Krσ 0 τ I ρ] > 0. For ξ > 1, the disease spreading exhibits a non-monotonic behavior with υ and in the limit of υ → ∞ the systems reaches the epidemic extinction regime. Thus, an increase in υ, resp. CR, leads to a reduction of the epidemic size. Finally, for 0 < ξ < 1 and in the limit of υ → ∞, S st goes to zero, which indicates that the epidemic size at low density and for large enough values of υ can become larger that the bounded epidemic size corresponding to S HD st . Note that simulations performed with the specific agent-agent interaction rule given by Eq. ( 2) fall into this case, with ξ = 0.91.

Concluding remarks.-We have shown that in mobile agent systems the coupling between mean exposition time and CR is crucial to understand the excitation dynamics, which generically exhibits three regimes with CR: excitation extinction for low CR, an excited regime where the excitation is such that the number of quiescent agents is inversely proportional to CR, and a third regime, for high CR, where the number of quiescent agents scales with an exponent ξ -1, with ξ being the scaling exponent of the mean exposition time with CR. The novel third regime is clearly model dependent and opens, in the context of epidemic spreading, the counterintuitive possibility of "curing" by increasing CR, which can be achieved in simulations by modifying the interaction rule among agents such that ξ > 1.

At the experimental level, the introduced mobile agent model might help to understand spatial distribution (and levels) of gene expression induced by cell-cell signaling as observed in some bacterial colonies [START_REF]Myxobacteria II[END_REF]. It also might shed some light on disease spreading on bacteria [START_REF] Weinbauer | [END_REF]24] and also on competing strain bacteria experiments [25]. Clearly, the current agent model is not suitable to describe disease spreading on structured populations, i.e., when the disease moves across metapopulations, as in humans. Further improvements to account for structured communities might include preferred interacting partners, which can be modeled by a long-range potential acting among members of a community, as well as by dividing the space into several connected compartments.

FIG. 1 :

 1 FIG. 1: (a) simulation snapshot. Number of agents N = 1024. Colors indicate agent states: quiescent/susceptible (green), excited/infected (red) and refractory/recovered (blue). A short segment indicates the active direction of motion. (b) amplification of the area indicated in (a). (c) temporal evolution of the populations.

  FIG.2:Populations of agents at the steady state vs. linear system size L. Simulations performed with the parameters given in[START_REF]Simulation parameters[END_REF] and υ = 0.1. Full black lines corresponds to Eqs. (6),(7), and (8) while the open symbols to S HD st , I HD st and R HD st . The vertical line corresponds to Lc.

We thank L.G. Morelli for fruitful discussions. FP thanks M. Bär and A. Deutsch for support. We acknowledge funding from SECyT-DAAD (Proalar DA0603).


Fernando Peruani
email: peruani@pks.mpg.de

Luis G Morelli

Self-propelled particles with fluctuating speed and direction of motion

Keywords: numbers: 05, 40, Jc,87, 17, Jj

We study general aspects of active motion with fluctuations in the speed and the direction of motion in two dimensions. We consider the case in which fluctuations in the speed are not correlated to fluctuations in the direction of motion, and assume that both processes can be described by independent characteristic time-scales. We show the occurrence of a complex transient that can exhibit a series of alternating regimes of motion, for two different angular dynamics which correspond to persistent and directed random walks. We also show additive corrections to the diffusion coefficient. The characteristic time-scales are also exposed in the velocity autocorrelation, which is a sum of exponential forms.

The study of cell movement on surfaces can shed light on the processes that underly cell motility [START_REF] Bray | Cell movements: from molecules to motility[END_REF]. In vitro experiments that characterize cell movement include wound closure assays and individual cell tracking to determine cell trajectories [START_REF] Czirók | [END_REF]3,4,5]. To interpret and understand such experiments it is necessary to rely on a phenomenological description of the motion, providing expressions that allow to fit the experimental observations and compute motility indexes [4,6].

Persistent motion subject to fluctuations has been described by a class of stochastic process known as persistent random walk [7,8,[START_REF] Mikhailov | Stochastic Dynamics[END_REF][START_REF] Schweitzer | Brownian Agents and Active Particles[END_REF][START_REF] Ebeling | Statistical thermodynamics and stochastic theory of non-equilibrium systems[END_REF]. In such processes the direction of motion fluctuates, but on short timescales a persistence to move in the current direction is observed. Formally, the velocity autocorrelation function v(t) • v(0) exhibits a finite decay time, giving rise to a ballistic regime x 2 (t) ∼ t 2 for short times and a crossover to a diffusive regime x 2 (t) ∼ t for long times [7,8,[START_REF] Mikhailov | Stochastic Dynamics[END_REF].

A related problem is that of the directed motion of selfpropelled particles driven by an external field. Single cells can be directed by external signals in the form of molecule gradients, as in the case of fibroblasts [15] or the amoebae D. discoideum [16]. Directed motion with fluctuations can be described by another broad class of stochastic process known as the directed random walk [8,[START_REF] Schweitzer | Brownian Agents and Active Particles[END_REF], which displays a diffusive regime for short times followed by a ballistic regime for long times [15,17].

In previous works either fluctuations in the speed and direction of motion were considered to occur simultaneously, or fluctuations of the speed were simply neglected [7,8,[START_REF] Mikhailov | Stochastic Dynamics[END_REF][START_REF] Schweitzer | Brownian Agents and Active Particles[END_REF][START_REF] Ebeling | Statistical thermodynamics and stochastic theory of non-equilibrium systems[END_REF][START_REF] Tojo | [END_REF]13,14]. In this paper, we study two dimensional stochastic motion with uncorrelated fluctuations of the speed and the direction of motion, assuming that both processes can be described by independent characteristic time-scales. We investigate persistent and directed random walks, and derive exact expressions for the mean squared displacement and the asymptotic diffusion coefficient for arbitrary speed and angular stationary distributions.

General aspects. We consider self-propelled particles that move in two dimensions. The velocity v(t) at time t is represented by an angle θ(t), and a modulus -the speed-v(t). The dynamics of the velocity v(t) is given by a stochastic process which for the moment we do not specify. We introduce d(θ, t), the total distance covered by the particle moving along the direction θ since the beginning of the trajectory at t = 0. Given a particular trajectory characterized by θ(t) and ṽ(t), we can express this distance as

d(θ, t) = t 0 dt ′ ∞ 0 dv v δ(v -ṽ(t ′ ))δ(θ -θ(t ′ )).
The ensemble average of this distance is

d(θ, t) = t 0 dt ′ ∞ 0 dv v p(θ, v, t ′ ) , (1)
where we have introduced the probability density to find the particle moving in the direction θ with speed v at time t, p(θ, v, t) = δ(v -ṽ(t))δ(θ -θ(t)) [START_REF] Risken | The Fokker-Planck equation[END_REF]. Here we denote ensemble averages by . . . . The correlations d(θ ′ , t)d(θ ′′ , t) between the distances can be written in terms of the joint probability distribution. We first express these correlations in terms of the ensemble average of delta distributions, and use

p(θ ′ , v ′ , t ′ ; θ ′′ , v ′′ , t ′′ ) = δ(v -ṽ(t ′ ))δ(v -ṽ(t ′′ ))δ(θ -θ(t ′ ))δ(θ -θ(t ′′ )) to obtain d(θ ′ , t)d(θ ′′ , t) = t 0 dt ′ t 0 dt ′′ (2) ∞ 0 dv ′ ∞ 0 dv ′′ v ′ v ′′ p(θ ′ , v ′ , t ′ ; θ ′′ , v ′′ , t ′′ ) .
We can use the distance d(θ, t) to express the position of the particle at time t as x(t) = π -π d(θ, t)ř(θ)dθ, where ř(θ) = cos(θ)x + sin(θ)y is the unit vector along the direction θ. Then the mean value of the position is

x(t) = π -π d(θ, t) ř(θ)dθ , (3)
and the mean square displacement

x 2 (t) = π -π dθ ′ dθ ′′ d(θ ′ , t)d(θ ′′ , t) ř(θ ′ ) • ř(θ ′′ ). (4)
Eqs. ( 1) to (4) provide a general way to calculate the mean value of the position and the mean square displacement, which so far does not involve any assumptions. In the following, we consider the special case in which the fluctuations in the speed v are not correlated with the fluctuations in the direction θ. As a consequence, p(θ, v, t) = p(θ, t)p(v, t) and p(θ

′ , v ′ , t ′ ; θ ′′ , v ′′ , t ′′ ) = p(θ ′ , t ′ ; θ ′′ , t ′′ )p(v ′ , t ′ ; v ′′ , t ′′
). Such a situation could naturally arise if fluctuations of the speed are endogenous and produced by an irregular engine, while fluctuations of the direction of motion are produced by random changes in the environment. In the following, we further assume that the speed fluctuations are in the stationary state with an arbitrary speed distribution p(v, t) = ρ(v), and the joint probability is given by

p(v ′ , t ′ ; v ′′ , t ′′ ) = ρ(v ′′ )δ(v ′ -v ′′ )e -β|t ′ -t ′′ | (5) + ρ(v ′′ )ρ(v ′ ) 1 -e -β|t ′ -t ′′ | .
This expression for the joint probability distribution implies that the speed correlations decay exponentially as

v(t)v(0) -v 2 = ( v 2 -v 2 )e -βt .
Eq. ( 5) describes particles that keep on moving with roughly the same speed for a characteristic time β -1 , while for larger times the values of the speed become uncorrelated. An example of a stochastic process which generates such statistics is given by v(t) = η(t), where the value of η(t) is taken from a distribution ρ(v) with waiting times given by a Poisson process of rate β. The evolution Eq. for the probability density p(v, t) can be expressed as

∂ t p(v, t) = -βp(v, t) + βρ(v)
, leading to the conditional probability given by Eq. ( 5). Under these assumptions Eqs. ( 1) and ( 2) can now be simplified performing the integrals on the speed:

d(θ, t) = v t 0 dt ′ p(θ, t ′ ) , (6)
d(θ ′ , t)d(θ ′′ , t) = v 2 t 0 dt ′ dt ′′ p(θ ′ , t ′ ; θ ′′ , t ′′ ) (7) +( v 2 -v 2 ) t 0 dt ′ dt ′′ p(θ ′ , t ′ ; θ ′′ , t ′′ )e -β|t ′ -t ′′ | .
Persistent random walk. As a first application, we consider the case of a persistent random walk. We study a problem in which the angular probability distribution function obeys a diffusion equation characterized by the diffusion constant κ. An example of a stochastic process described by such an equation is given by θ(t) = η(t), where η(t) is an uncorrelated white noise.

We assume that particles start moving from the origin in all possible directions with equal probability, so p(θ, t) = 1/2π for all times and x(t) = 0. However a given particle starts moving along a particular direction and smoothly explores other directions, so a characteristic time must elapse before we can find this particle pointing with equal probability in any direction. This is described by the conditional probability distribution p(θ ′ , t ′ |θ ′′ , t ′′ ), which obeys the diffusion equation

∂ t ′ p(θ ′ , t ′ |θ ′′ , t ′′ ) = κ∂ θ ′ θ ′ p(θ ′ , t ′ |θ ′′ , t ′′ ), with the initial condition p(θ ′ , t ′ |θ ′′ , t ′ ) = δ(θ ′ -θ ′′ ).
To warrant the conservation of the probability we impose the periodic boundary condition p(π, t ′ |θ ′′ , t ′′ ) = p(-π, t ′ |θ ′′ , t ′′ ) and

∂ θ ′ p(π, t ′ |θ ′′ , t ′′ ) = ∂ θ ′ p(-π, t ′ |θ ′′ , t ′′ ).
The solution for the conditional probability is

p(θ ′ , t ′ |θ ′′ , t ′′ ) = 1 2π + 1 π ∞ m=1 cos [m(θ ′ -θ ′′ )] e -m 2 κ|t ′ -t ′′ | .
(8) As |t ′ -t ′′ | → ∞ the information about the direction of motion at time t ′′ is completely lost, and the conditional probability approaches the asymptotic value 1/2π. The slowest mode m = 1 sets the characteristic time-scale κ -1 that describes the duration of the transient.

Recalling that p(θ ′ , t ′ ; θ ′′ , t ′′ ) = p(θ ′ , t ′ |θ ′′ , t ′′ )p(θ ′′ , t ′′ ) and using Eq. ( 8) in Eq. ( 7), we obtain from Eq. ( 4) that

x 2 (t) = 2 v 2 κ 2 κt -1 + e -κt (9)
+ 2 v 2 -v 2 (κ + β) 2 (κ + β)t -1 + e -(κ+β)t .
In the absence of speed fluctuations, the speed variance σ 2 = v 2 -v 2 vanishes and Eq. ( 9) reduces to the well known result for persistent Brownian particles [7], which exhibits a single crossover at t ∼ κ -1 , see thin solid red line in Fig. 1. When fluctuations become relevant, a previous crossover from a quadratic to a linear regime occurs at t ∼ (κ + β) -1 . If the separation of time-scales allows it another crossover can be observed between these two, when the linear regime of the second term turns into the quadratic regime of the first one, see the thick solid black line in Fig. 1.

In order to unveil the different regimes that Eq. ( 9) permits, we introduce non-dimensional variables ξ = xκ/ v and τ = κt, and parameters µ = σ/ v and γ = β/κ. For γ ≪ 1 there is a single crossover at τ ∼ 1, see dashed green line in Fig. 1. For larger values of γ, solutions lie between the dashed green line and thin solid red line. For τ ≪ (1 + γ) -1 we observe ξ 2 ≃ (1 + µ 2 )τ 2 . A first crossover occurs at τ 1 ∼ (1+γ) -1 . For larger times, if the separation of time-scales is such that (1 + γ) -1 ≪ τ ≪ 1 then ξ 2 ≃ τ 2 + 2µ 2 τ /(1 + γ). Provided that µ 2 is sufficiently large, a second crossover occurs at τ 2 ∼ 2µ 2 /(1 + γ) separating a transient linear regime from a second quadratic regime. Finally, for τ ≫ 1 the asymptotic diffusive regime emerges with ξ 2 ≃ 2(1+µ 2 /(1+γ))τ , after the third crossover at τ 3 ∼ 1. Such asymptotic regime can be described in terms of an effective diffusion coefficient, defined as D = lim t→∞ ( x 2 (t)x(t) 2 )/2t. From Eq. ( 9) we obtain

D = v 2 κ -1 + v 2 -v 2 (κ + β) -1 .
Speed fluctuations introduce an additive correction to the well known diffusion coefficient for constant speed [START_REF] Mikhailov | Stochastic Dynamics[END_REF], and can lead up to four consecutive regimes of motion separated by three crossovers, see Fig. 1. In the absence of speed fluctuations only one crossover is found [8,[START_REF] Mikhailov | Stochastic Dynamics[END_REF]. In Fig. 1 we use a large speed variance and plot x 2 (t) over a huge range to illustrate all the features of Eq. ( 9). Below we discuss experimental constraints in the observation of the phenomena described here.

Directed persistent random walk. As a second example we consider the directed random walk, in which the particles have some preferred direction of motion. This could be the case for particles moving in a symmetrybreaking field or gradient. We assume that angular fluctuations are in the stationary state p(θ, t) = ρ(θ). We describe the presence of an external field by assuming that π/2 -π/2 ρ(θ)dθ > 1/2 together with the symmetry requirement ρ(-θ) = ρ(θ), setting a preferred direction of motion along θ = 0. Time correlations decay exponentially with a characteristic time α -1

p(θ ′ , t ′ ; θ ′′ , t ′′ ) = ρ(θ ′′ )δ(θ ′ -θ ′′ )e -α|t ′ -t ′′ | (10) + ρ(θ ′′ )ρ(θ ′ ) 1 -e -α|t ′ -t ′′ | .
A realization of such stochastic process is θ(t) = η(t), where the value of the noise η(t) is taken from a distribution ρ(θ) at times given by a Poisson process of rate α. Using this expressions for the angular probability distributions together with Eqs. ( 6) and ( 7) in Eqs. ( 3) and ( 4) we arrive at the following expressions for the mean value of the position, x(t) = √ c v t x, and the mean square displacement

x 2 (t) = v 2 ct 2 + 2(1 -c)ϕ α (t) (11)
+ 2 v 2 -v 2 [(1 -c)ϕ α+β (t) + cϕ β (t)]
where c = cos θ 2 and ϕ α

(t) = α -2 [αt -(1 -e -αt )].
The effective diffusion coefficient D as defined above also exhibits an additive correction,

D = (1 -c) v 2 α -1 + ( v 2 -v 2 )[(1 -c)(α + β) -1 + cβ -1 ]
. Speed fluctuations introduce a new time-scale which together with the independent time-scale of angular fluctuations can lead up to five alternating regimes of motion separated by four crossovers, see Fig. 2. In the absence of speed fluctuations the second line in Eq. ( 11) vanishes and the mean square displacement exhibits only two regimes [17].

Concluding remarks. The interplay of speed and angular fluctuations gives rise to a sequence of regimes, revealing a complex transient not observed when speed fluctuations are absent. The occurrence of such a complex transient can wreck the interpretation of experimental observations, due to the constrains imposed by resolution and finite size limitations. Particle size -or a fluctuating cell shape-sets the smallest accessible length-scale, while the field of the experimental setup sets the largest. The temporal window is similarly bounded. If the window of observation is limited to a part of the complex transient, anomalous diffusion could be wrongly interpreted. Superdiffusion has been repeatedly reported from experimental data [21,22]. However, distinguishing true superdiffusion from a persistent or directed random walk is a subtle task [23]. Our results suggest that in some cases the observed anomalous behavior could be related to one or more of the reported crossovers. In Fig. 3 (a) we display D(τ ) = ( ξ 2 (τ ) -ξ(τ ) 2 )/2τ for a persistent random walk, using time and space ranges which are reasonable for current experimental setups [START_REF] Czirók | [END_REF]4]. Furthermore, 12) for rescaled variables. In both panels, the solid red line corresponds to µ = 0. For the other curves µ = 1.31, with γ = 10 -1 (dashed green line) and γ = 10 (dotted blue line). Dots correspond to numerical simulations as described in the text.

we choose µ = 1.31 according to data reported in [4], and values of γ within experimental ranges. The solid red line is the result without speed fluctuations. Dots correspond to numerical simulations performed with µ = 1.31 and γ = 10. In the simulations speeds are chosen at a rate

β = 4.0 h -1 from a speed distribution ρ(v) ∼ v -3/2 for v ∈ [1, v c
] and zero otherwise, with v c such that µ = 1.31.

Angles are chosen at a rate 9.6 h -1 from a uniform distribution of width 1 rad centered around the direction of motion, and so yielding κ = 0.4 h -1 . Error bars are the standard deviation from the mean value obtained for 100 realizations with 100 particles each. This means that a particular 100 particles experiment should fall within the range of such error bars. The two characteristic time-scales of the system are also exposed in the velocity autocorrelation function, which is given by a sum of two exponentials v(t) • v(0) = v 2 e -κt + ( v 2 -v 2 ) e -(β+κ)t . (12) Autocorrelations of similar functional form have been observed in cell motility experiments, but the microscopic origin of the two time-scales has not been established [4]. Here we show that independent fluctuations in speed and direction could produce such autocorrelations, see Fig. 3 (b). Fast intracellular processes could give rise to such fluctuations in speed with small characteristic time-scales [START_REF] Joanny | [END_REF]20]. The simulations suggest that fluctuations in speed as the ones observed in experiments might be enough to cause visible deviations from the classical result [7]. In the case of directed motion, the presence of an external field decouples the time-scale of speed fluctuations and the velocity autocorrelation results in a sum of three exponentials: v(t) • v(0) = ( v 2 + σ 2 e -βt )[c + (1 -c)e -αt ]. In this case we are not aware of experiments showing such autocorrelations. While here we have considered the case in which speed and angular fluctuations are not correlated, the case in which they are is also of much interest and deserves future attention.

2 FIG. 1 :

 21 FIG.1: Scaled mean squared position as a function of scaled time for persistent random walks. The lines correspond to Eq. (9) for rescaled variables. The thin solid red line corresponds to µ = 0, and µ = 100 for the other curves, with γ = 10 -3 (dashed green line), γ = 10 3 (dotted blue line) and γ = 10 6 (thick solid black line).

FIG. 2 :

 2 FIG.2: Scaled mean squared position as a function of scaled time for directed random walks. The lines correspond to Eq. (11) for the rescaled variables, with c = 10 -4 . The thin solid red line corresponds to µ = 0. The other curves correspond to µ = 100, with γ = 10 -3 (dashed green line), γ = 10 3 (dotted blue line) and γ = 10 6 (thick solid black line).

FIG. 3 :

 3 FIG. 3: (a) Re-scaled mean squared displacement and (b) velocity autocorrelation as a function of scaled time for persistent random walks. The lines in (b) correspond to Eq. (12) for rescaled variables. In both panels, the solid red line corresponds to µ = 0. For the other curves µ = 1.31, with γ = 10 -1 (dashed green line) and γ = 10 (dotted blue line). Dots correspond to numerical simulations as described in the text.

We thank S. F. Nørrelykke and B. Lindner for insightful comments and valuable suggestions on the manuscript. FP thanks M. Bär and A. Deutsch for support and acknowledges funding from the Deutsche Forschungsgemeinschaft through Grant No. DE842/2.


Candice Delaite
email: candice.delaite@atilf.fr

Alain Polguère
email: alain.polguere@univ-lorraine.fr

Sex-Based Nominal Pairs in the French Lexical Network: It's Not What You Think

Keywords: Lexical relation, lexical function, Explanatory Combinatorial Lexicology, grammatical gender, lexical database, French Lexical Network (fr-FLN)

We examine French nominal pairs of the type BERGER 'shepherd' ~BERGÈRE 'shepherdess' and CHIEN 'dog' ~CHIENNE 'bitch dog'. These pairs are considered from the viewpoint of their lexicographic modeling in an Explanatory Combinatorial lexical database: the French Lexical Network (fr-FLN). We first present, in section 1, the linguistic problem we are dealing with; we identify two main types of semantic relations involved in these lexical pairs: sex-based quasi-synonymy-that is analyzed in detail in section 2-and sex-based contrastive opposition-dealt with in section 3. As conclusion, section 4, we look at the implications of our modeling of such relations on the graph structure of the fr-FLN.

3. N fem carries the feminine grammatical gender and denotes an animate being whose sex is, necessarily or by default, female;

4. N masc and N fem are generally, but not necessarily, morphologically related.

Here are a few examples of some N masc ~ N fem pairs: BERGER 'shepherd' ~BERGÈRE 'shepherdess', HÔTE 'host' ~HÔTESSE 'hostess', NAGEUR '(male) swimmer' ~NAGEUSE 'female swimmer', CHIEN 'dog' ~CHIENNE 'bitch dog', ÉTALON 'stallion' ~JUMENT 'mare', FIANCÉ ~ FIANCÉE. 1 We examine these lexical pairs in the context of the RELIEF lexicographic project, that targets the building of the French Lexical Network or fr-FLN [START_REF] Lux-Pogodalla | Construction of a French Lexical Network: Methodological Issues[END_REF][START_REF] Gader | Hand-Crafting a Lexical Network With a Knowledge-Based Graph Editor[END_REF]. The fr-FLN is a model of the French lexicon that is being built according to Explanatory Combinatorial Lexicology's principles [START_REF] Mel'čuk | Introduction à la lexicologie explicative et combinatoire[END_REF]; its main characteristic is to be formally structured as a multidimensional lexical graph, instead of being a simple "lexicographic text." The bulk of the fr-FLN structuring results from the weaving of Meaning-Text lexical function relations [START_REF] Mel'čuk | Lexical Functions: A Tool for the Description of Lexical Relations in the Lexicon[END_REF].

Our perspective being lexicographic in essence, it is useful to begin with a brief summary of how "classical" dictionaries handle N masc ~ N fem pairs, focusing on pairs that are morphologically related.

The three main general public dictionaries of French-Petit Robert 2 , Petit Larousse 3 and Trésor de la Langue Française 4 -offer heterogeneous descriptions of morphologically related N masc ~ N fem pairs. In some cases, there exist individual entries for each pair member-for instance, MARQUIS ~MARQUISE or POÈTE 'poet' ~POÉTESSE 'poetess'-, and in other cases, both lexical units are grouped under a unique entry-for instance, COIFFEUR ~COIFFEUSE, that are being described under a common entry named "coiffeur, -euse." The above-mentioned dictionaries seem to converge in their descriptive choices, for which it is normally possible to find a logical explanation. Thus, for the few cases that we have cited:

• the choice of a separate entry for MARQUISE is almost imposed to dictionary makers by the polysemy that developed around the feminine noun-marquise (d'une gare) 'canopy (of a railway station)', marquise (au chocolat) 'chocolate cake'-, with no corresponding sense within the masculine vocable;

• the derivational suffix -esse possesses a special status in French morphology that makes POÉTESSE appear as morphologically more "remote" from POÈTE than COIFFEUSE appears to be from COIFFEUR. 5 It is obvious that the lexicographic treatment of sex-based semantic derivations will depend on the grammatical characteristics of each language. Because there is no gender inflection in English, the morphological nature of the relation holding between the source and target of such derivation poses no problem. Furthermore, there are relatively few derivations of that kind-based on the use of the suffix -ess (borrowed from French). For these reasons, English dictionaries always create two separate entries, one for the noun that denotes a male animate being and one for the noun that denotes the corresponding female being-see, for instance, 1

We do not provide glosses for French lexical units that exist with identical meaning in English as borrowings. The pairs listed above form a rather heterogeneous set from a semantic and morphological viewpoint. However, we believe that they have to be considered and dealt with together, as will be shown shortly. Spanish is a more interesting case here, as Spanish nouns possess a grammatical gender, and Spanish grammar contains gender inflection for adjectives, participials and articles. As a side effect, we find many N masc ~ N fem pairs in Spanish, as in French. Similarly to French dictionaries, the Diccionario de la lengua español (Real Academia Española)9 and the Diccionario de uso del español [START_REF] Moliner | Diccionario de uso del español, primera reimpresión, edición abreviada por la Editorial Gredo[END_REF] propose a heterogeneous treatment of N masc ~ N fem pairs: two separate entries for MODISTO 'couturier, (male) fashion designer' MODISTA 'couturière, female fashion designer', but one single entry for PASTOR 'shepherd' and PASTORA 'shepherdess' named "pastor, ra" in the first dictionary and "pastor, -a" in the second. The case of Spanish is however very different from French because Spanish common nouns, in standard cases, possess a suffix that carries grammatical gender-cf. Mel'čuk (to appear) for an analysis of the morphological status of Spanish gender nominal suffixes.

In the fr-FLN, we treat morphologically related N masc ~ N fem pairs in a systematic fashion, based on the derivational (and not inflectional) nature of the morphological link that unites them. Let us examine this point briefly.

Derivation and not inflection

It is obvious that, from a strictly formal viewpoint, pairs of wordforms such as berger 'shepherd' ~bergère 'shepherdess' display inflection-like characteristics: the signifier bergère looks like a "feminine of" berger. However, as shown by [START_REF] Mel'čuk | Un FOU/une FOLLE: un lexème ou deux? Bulag[END_REF], there are at least five reasons why one should consider that no inflectional mechanism is involved here.

1. A description based on inflection would force us to postulate an inflectional category of nominal gender in French, that would apply only to nouns denoting sexed animate beings, whereas typical inflection applies globally to all lexical units of given parts of speech (see grammatical gender for French adjectives).

2. This hypothetical inflection would be very irregular-cf. professeur '(male) teacher' professeure 'female teacher', but coiffeur ~coiffeuse-and non-systematic-cf. [un] poisson masc '[a] fish', sex-neutral masculine noun that does not have a feminine counterpart such as *[une] poisson fém or *poissonne.

3. For these reasons, it is not uncommon that even native speakers hesitate on the proper choice of term, which is rather strange if we were to be in presence of inflection. Additionally, diatopic variations, that are so common for N masc ~ N fem pairs, are clearly perceived as lexical rather than grammatical in essence-cf. écrivaine 'female writer', that is almost systematically used in Québec when talking about a woman writer, vs. the feminine noun [une] écrivain fem , that is much more commonly used in France.

4. As mentioned in section 1.1, each element of a N masc ~ N fem pair tends to develop its own individual polysemy; there are therefore in French many pairs of vocables that are autonomously structured, which contradicts an inflectional modeling of the phenomenon we are examining.

5. Finally, a derivational rather than inflectional description of French N masc ~ N fem pairs can be done in a very simple and direct manner. The only justification one can find for the approach commonly adopted by standard dictionaries, that tend to model these pairs as if they were inflectionally related (section 1.1 above), is the significant economy in printed pages it entails for printed versions of the dictionaries.

These observations logically lead us to consider that morphologically related N masc ~ N fem French pairs are cases of pure derivations, where a nominal masculine lexical units denoting a male animate being is used as a source from which a feminine lexical units denoting the corresponding female being is derived. Dictionaries and lexical databases, such as the fr-FLN, should explicitly and systematically reflect this fact in their macro-and microstructure.

Hypothesis: two types of N masc ~ N fem pairs

Now that it is clearly established that pairs of nominal lexical units such as BOULANGER '(male) baker' ~BOULANGÈRE 'female baker', FILS 'son' ~FILLE 'daughter', CHAT '(male) cat' CHATTE 'female cat', etc., are indeed displaying cases of morphological derivations and that no inflection mechanism is involved, we will identify the different types of such derivations and indicate how each type has to be modeled in the fr-FLN.

Let us recall that we are interested in semantic derivations that are not necessarily expressed by morphological means and that we are considering together morphologically related pairs such as those examined in section 1.2 above as well as pairs such as FRÈRE 'brother' ~SOEUR 'sister' or ÉTALON 'stallion' ~ JUMENT 'mare'.

We adopt the lexicographic perspective of the construction of the fr-FLN and of the weaving of its graph structure by means of lexical function links. In this context, it is of paramount importance for all N masc ~ N fem lexical pairs to be described in a homogeneous way, whether they display or not a morphological connection.

We believe there exist two relations that can hold between a masculine nominal lexical unit denoting a male animate being and a feminine nominal unit denoting the corresponding female being. These relations are illustrated by means of the following two lexical pairs:

1. the pair AVOCAT '(male) lawyer, advocate' ~AVOCATE 'female lawyer', that displays, according to us, a synonymy-like lexical relation;

2. the pair ÉTALON 'stallion' ~JUMENT 'mare', that displays a lexical relation that is contrastive, rather than quasi-synonymic in essence.

We are now proceeding with the presentation of these two types of relations: sex-based quasisynonymy in section 2 and sex-based contrastive opposition in section 3.

2 Sex-based quasi-synonymy

Characterization

In lexical pairs of the AVOCAT '(male) lawyer' ~AVOCATE 'female lawyer' type, one of the two lexical units-the masculine or the feminine noun (section 2.2 below)-functions as generic. Thus, avocat in (1a) below necessarily denotes a male individual, while the sex can be unspecified in (1b).

(1) a. Le jeune avocat s'est avancé résolument vers les journalistes. 'The young lawyer walked resolutely towards the journalists' b. Il a demandé la présence d'un avocat.

'He asked for a lawyer'

The plural form avocats in (2a) can designate a group of both men and women, and (2b) as well as (2c) can be used by a woman-i.e. she can either use the masculine or feminine noun.

(2) a. Tous les avocats du barreau parisien ont protesté. 'All the lawyers of the Parisian Bar have protested' b. Je veux devenir avocat.

'I want to become a lawyer' c. Je veux devenir avocate.

'I want to become a female-lawyer'

All this shows that there is some form of semantic unevenness between AVOCAT and AVOCATE:

(3) a. avocat = '(male) individual whose profession consists in [...] or corresponding profession' b. avocate = 'female individual whose profession consists in [...] or corresponding profession'

In other words, these lexical units are both semantically ambivalent 10 , denoting either an individual or a profession, but not in the same fashion. Whether it is used to denote an individual or the profession she exercises, AVOCATE implies the female sex, which is not the case for AVOCAT regarding the male sex. The 'male' semanteme that is embedded in the definition of this latter lexeme must be considered as being a weak semantic component (Mel'čuk et al., 1995: 95); it is this semantically weak nature that is being signaled by the parentheses in gloss (3a) above.

The complex semantic ratio holding between lexical units such as AVOCAT and AVOCATE entails that lexical units that form pairs of this type are not perceived as being in opposition vis-à-vis the denotation of sex. Elements of the pair AVOCAT ~AVOCATE are not contrasting on the ground that one would denote a "male" and the other the corresponding "female." Because of this, we consider such lexical units as quasi-synonyms and we call this type of relation sexbased quasi-synonymy.

It we take into consideration the semantic unevenness identified above, we need to distinguish the following two cases.

First, a speaker who says (4a) instead of the prototypical sentence (4b) chooses to be more specific with regards to the designation of the sex.

(4) a. Second, a speaker who says (4b), rather that (4a), chooses to be less specific with regards to the designation of the sex. We therefore encode the semantic derivation link AVOCATE ➔ AVOCAT by means of the Syn⊂ sex lexical function, that stands for 'potentially less 10 On semantic ambivalence and its lexicographic treatment, see [START_REF] Milićević | Ambivalence sémantique des noms de communication langagière du français[END_REF]. Note that because this particular semantic ambivalence is systematic in French-all nouns denoting an individual who exercises a given profession can also denote the profession itself-, [START_REF] Milićević | Ambivalence sémantique des noms de communication langagière du français[END_REF] propose to not make the ambivalence explicit in the structure of the definition and to simply formulate the genus of this type of lexical unit using the general pattern 'individual who has a given profession'. 11 The exponent sex in Syn⊃ sex is a new standardized element we introduce in the formal language of lexical therefore, it stands for Lat. sexus and not for Eng. sex.

rich synonym with regards to the sex'. We are considering here a synonymy that is potentially less rich because the speaker can of course also use avocat in order to specifically designate a male individual.

Notice that the two lexical functions that have just been introduced correspond to two reciprocal relations:

Syn⊃ sex ( L 1 ) = L 2 entails that Syn⊂ sex ( L 2 ) = L 1 , and vice versa.

There exists clearly a considerable amount of lexical pairs that are based on these quasisynonymy relations. Let us mention, among others: We do not mean to imply that all nominal pairs that fit into the above categories are in Syn⊃ sex vs. Syn⊂ sex relation. In many instances, such as for instance the pair of animal names ÉTALON 'stallion' ~JUMENT 'mare', we are in presence of another type of relation, contrastive in essence, that will be dealt with in section 3.

•

Masculine or feminine generic

It is important to stress that the semantic ratio between two lexical units that are linked by sex-based quasi-synonymy as well as the very existence of this type of quasi-synonym for a masculine vs. feminine noun are not at all systematic in French, as demonstrated by the following two cases.

1. It can be the feminine noun that functions as generic, while denoting primarily a female animate being. For instance, OIE 'goose' denotes a female animal or a species; its Syn⊃ sex is JARS 'gander', which cannot be used to denote a species.

2. A feminine noun denoting an animal can be completely neutral with regards to the denotation of the sex; therefore, it can have no corresponding masculine noun in French. Such is the case of GIRAFE 'giraffe'. To specifically denote a male or female giraffe, one will have to say girafe mâle ~ girafe femelle.

It is however still possible to make generalizations. For instance, we believe that in French the generic is always the masculine noun for nouns of N masc ~ N fem pairs denoting human beings. We have not found any exception to this rule. It applies even for a pair such as INFIRMIER '(male) nurse' ~INFIRMIÈRE 'female nurse', about which one may easily think that the feminine noun is the generic. Rather than being a potential generic, the feminine noun is rather the default value for extra-linguistic reasons, bearing to the fact that this profession has for a long time been exclusively exercised by women. But infirmière denotes specifically a female individual and cannot be used as generic; (5) below cannot be used in cases where one wants to refer to a mixed group of women and men.

(5) Un groupe d'infirmières se tenait à l'entrée de l'hôpital. 'A group of female nurses was standing at the entrance of the hospital'

Contrary to what one may first believe, it is indeed the masculine noun that is used in contemporary French as generic. For instance, vêtements d'infirmiers 'nurse cloths' in ( 6) below (found on the Web) can refer to both men and women cloths.

(6) L'association a ainsi acheminé du matériel médical: 140 lits médicalisés manuels ou électriques, un bloc opératoire, 4 750 paires de lunettes de vue et des vêtements d'infirmiers.

'The association has thus shipped medical equipment: 140 manual or automatic medicalized beds, one operating room, 4,750 spectacles and nurse cloth'

Other generalizations can probably be drawn about sex-based quasi-synonymy, but we do not have yet enough data in the fr-FLN to be able to extract them. We now proceed with the second type of sex-based semantic derivation.

Sex-based contrastive opposition

As mentioned above, some N masc ~ N fem pairs do not fall in the scope of quasi-synonymy. For instance, it would be odd to state that ÉTALON and JUMENT are quasi-synonyms as they seem to be linked by a semantic opposition: a stallion is a male horse, in contrast with a mare, that is a female horse. Which lexical function should be used to account for the lexical relation connecting lexical elements of such pairs, given that the use of Syn is ruled out?

There exists indeed the Contr, "contrastive," lexical function, that can be characterized as follows.

Contr( L 1 ) = L 2 if there exist phraseological expressions that feature both lexical units L 1 and L 2 in contrastive opposition. For instance, Contr( noir 'black' ) = blanc 'white' because there exist in French expressions such as (photo) en noir et blanc 'black and white (picture)', Tout n'est pas tout blanc ou tout noir 'Everything is not always black and white', etc.

This characterization, which is not based on the identification of an intrinsic semanticosyntactic content of Contr, entails that Contr is not an "ordinary" lexical function. A lexical unit that is the value of Contr for another lexical unit is not, strictly speaking, a semantic derivative of this unit. All this shows that pairs such as ÉTALON ~JUMENT, that display an obvious semantic contrast, are not in a Contr relation. Indeed, elements of these pairs are related through a semantic opposition that directly results from their lexical definition, rather than through the presence in the language of phraseological expressions that put them in contrast with each other.

Though not Contr, the semantic relation we consider here is much more related to contrastivity than it is, for instance, to antonymy-cf. the Anti lexical function. It is not based on the negation of a definitional semantic component lexical units it connects: compare true antonyms such as PRÉSENT (Adj) 'that is here' ~ABSENT (Adj) 'that is not here', with non-antonymic contrastive pairs such as ÉTALON ~JUMENT. The non-antonymic nature of the relation holding between ÉTALON and JUMENT clearly appears if we draft the definitions of the two semantemes that distinguish them-'female [X]' included in the meaning of JUMENT and 'male [X]' included in the meaning of ÉTALON:12

• 'female [X]' = '[X] whose sex enables her/it to give birth';

• 'male [X]' = '[X] whose sex enables him/it to participate in the fact that a female individual of the same species gives birth'.

The relation that concerns us being close to, but distinct from Contr, it should in theory be lexicographically modeled by means of two non-standard lexical functions:

1. Corresponding female animate being( étalon ) = jument;

2. Corresponding male animate being( jument ) = étalon.

these two non-standard lexical functions seem to meet most of the requirements for standardness [START_REF] Polguère | Selected Lexical and Grammatical Issues in the Meaning-Text Theory[END_REF]: they correspond to very regular relations in French (and probably in all natural languages), they are applicable to a very large and varied set of arguments, the returned values are many and varied, and finally they are often expressed by morphological means. For these reasons, we have decided to standardize those two symmetrical relations and make use of two "normalized" encodings in the fr-FLN: Fem for lat. 'feminus' and Masc for lat. 'masculus'.

Lexical function Fem and Masc have already been used in the DiCo project [START_REF] Mel'čuk | Dérivations sémantiques et collocations dans le DiCo/LAF[END_REF], following proposals made by A.-L. Jousse in order to normalize the encoding of some recurrent non-standard lexical functions (Jousse, 2010: 139-140). Technically, however, those were not the exact same Fem and Masc that are being used in the fr-FLN because the DiCo used them indiscriminately for both families of semantic derivations that we have identified-synonymic and contrastive.

Lexical pairs that are connected by Fem vs. Masc relations can very well interact in the lexicon with a third lexical unit that connects to the first two as quasi-synonym. Such is the case for CHEVAL 'horse', with regards to its relation with the ÉTALON ~JUMENT pair. The lexical definition of CHEVAL is similar in structure to those of lexical units such as AVOCAT 'lawyer' examined in section 2. The masculine noun CHEVAL denotes a certain type of animal which is, by default but not necessarily, a male. Figure 1 below visualizes the complete system of lexical relations connecting CHEVAL, ÉTALON and JUMENT. To conclude this section, we give below two characteristics of sex-based contrastive oppositions in French that we have induced from the lexicographic work done on the fr-FLN.

1. Fem and Masc relations are much less present in the lexicon than Syn⊃ sex and Syn⊂ sex . This is easily explained by the fact that approximate synonymy is precisely the most common relation in the graph of any natural language lexicon.

2. Syn⊃ sex vs. Syn⊂ sex is often realized morphologically, whereas it is hardly the case for Fem vs. Masc. In fact, we are yet to find a clear-cut case of a Fem that would be morphologically built out of its corresponding Masc, or vice versa.

Those are only preliminary observations, that ought to be systematically checked on a larger set of data.

Implications on the structure of the fr-FLN

Our analysis of sex-based semantic derivatives influences in two ways the development of the the fr-FLN's structure.

Firstly, the fr-FLN, unlike standard French dictionaries, systematically possesses distinct lexicographic entries for N masc ~ N fem pairs. Because polysemy tends to develop independently for both vocables of these pairs, one should never try to unite them under a single entry, even when they make use of the same signifiers-cf. LÂCHE (N, masc) '[a] coward' ~LÂCHE (N, fem) 'female coward'. The saving of printed paper is not a parameter in the context of the fr-FLN and this method allows us to remain closer to what we believe is the actual structuring of the lexicon.

Secondly, paradigmatic links that connect lexical units involved in the system of N masc ~ N fem oppositions are systematically encoded by fr-FLN lexicographers using the appropriate lexical functions: Let us mention an interesting problem, that we will not develop for lack of space. There exist degrees in the synonymic vs. contrastive nature of links connecting N masc ~ N fem lexical units. For instance, the FILS 'son' ~FILLE 'daughter' pair pertains to sex-based contrastive opposition because none of the two lexical units can be used as generic denoting someone's child without specification of the sex. However, the contrastive nature of the link appears to be weak for at least two reasons:

1. what dominates in the meaning of both lexical units is their relational nature '[X is] son/daughter of Y' and not, strictly speaking, the denotation of a certain type of individual;

2. the relation, central to the meaning, is identical in both cases.

To conclude, we would like to draw the reader's attention to [START_REF] Steffens | Homme/femme, mode d'emploi. Existe-t-il des "antonymes de genre[END_REF], a paper that we happened to come across right when the final version of the present text was being submitted. It takes a non-lexicographic approach to the question of N masc ~ N fem pairs while reaching conclusions that are astonishingly (and reassuringly) similar to ours.

Figure 1 :

 1 Figure 1: Lexical relations connecting CHEVAL 'horse', ÉTALON 'stallion' and JUMENT 'mare'

  ~LIONESS in the American Heritage, 6 Longman Dictionary of Contemporary English 7 or Cobuild.8

2 CD-ROM edition, 2011. 3 Online consultation: http://www.larousse.com/fr/dictionnaires/francais-monolingue. 4 Online consultation: http://atilf.atilf.fr. 5

The suffix -esse is archaic and rather rare in modern French; it is almost no longer used to produce neologisms. See the etymological notice of the Trésor de la Langue Française, in its entry -ESSE 2 . In comparison, the derivation based on -eur ➔ -euse is extremely productive. LION

  -Je veux une avocate ! 'I want a female lawyer' b. -Je veux un avocat ! [man or woman] 'I want a lawyer' Consequently, we encode the semantic derivation link AVOCAT ➔ AVOCATE by means of the Syn⊃ sex lexical function, that stands for 'richer synonym with regards to the sex'. 11

  nouns denoting individuals who hold an official title or function-cf. ROI 'king' REINE 'queen', MARQUIS ~MARQUISE 'marquess', PRÉSIDENT 'president' ~PRÉSIDENTE 'female president' or PAPE 'pope' ~ PAPESSE 'female pope';

• nouns denoting individuals who have a given citizenship, philosophy, religion, etc.cf. FRANÇAIS 'French person' ~FRANÇAISE 'female French person', JUIF 'Jew' ~JUIVE 'female Jew' or BOUDDHISTE (N, masc) 'Buddhist' ~ BOUDDHISTE (N, fem) 'female Buddhist'; • nouns denoting individuals who experience a given emotion or possess a given behavioral characteristic-cf. AMOUREUX (N) 'person in love' ~AMOUREUSE (N) 'female person in love' or LÂCHE (N, masc) '[a] coward' ~ LÂCHE (N, fem) 'female coward'; • nouns denoting animals, such as CHIEN 'dog' ~CHIENNE 'bitch dog' or ÉLÉPHANT 'elephant' ~ ÉLÉPHANTE 'female elephant'.

  Syn⊃ sex , Syn⊂ sex , Masc or Fem. At the time of writing, the fr-FLN contains 18,470 lexical units (senses) connected by a total of 26,959 lexical links; among those, 1,618 lexical function links encode lexical relations geared to N masc ~ N fem pairs: 1,592 links for Syn⊃ sex vs. Syn⊂ sex relations and 116 links only for Fem vs. Masc relations. These statistics confirm what was said earlier: sex-based synonymy is significantly more present in the lexicon than the sexbased contrastive opposition.

On-line consultation: http://www.ahdictionary.com.

On-line consultation: http://www.ldoceonline.com.

On-line consultation: http://dictionary.reverso.net/english-cobuild/teach.

20 th edition, on-line cconsultation: http://www.rae.es/rae.html.

For the sake of simplicity, we define two English semantemes though, strictly speaking, the corresponding French semantemes 'femelle' and 'mâle' should be analyzed. The following definitions are loosely based on the semantic treatment of lexical units denoting sexed beings that is proposed byWierzbicka (1972: 34-56).

Acknowledgements

All our thanks to Igor Mel'čuk, Dorota Sikora and two anonymous MTT'13 reviewers for their comments on an earlier version of this paper.


Fernando Peruani

Andreas Deutsch

Markus Bär

Non-equilibrium clustering of self-propelled rods

Keywords: numbers: 87.18.Bb, 05.70.Ln, 87.18.Ed, 87.18.Hf

Motivated by aggregation phenomena in gliding bacteria, we study collective motion in a twodimensional model of active, self-propelled rods interacting through volume exclusion. In simulations with individual particles, we find that particle clustering is facilitated by a sufficiently large packing fraction η or length-to-width ratio κ. The transition to clustering in simulations is well captured by a mean-field model for the cluster size distribution, which predicts that the transition values κc of the aspect ratio for a fixed packing fraction η is given by κc = C/η -1 where C is a constant.

Introduction.-Emergent large-scale patterns of interacting self-driven motile elements are observed in a wide range of biological systems of different complexity: from human crowds, herds, bird flocks, and fish schools [1] to multicellular aggregates, e.g. of bacteria and amoebae [2] as well as sperms [3]. A recurrent question is how these entities coordinate their behavior to form groups which move collectively. At a theoretical level, several qualitative approaches have been made to incorporate the diverse collective behaviors of such different systems in a common framework [1,4,5]. More specific models for bacteria like E. coli as well as for amoebae like D. discoideum [2], have been based on chemotaxis, a long-range cell interaction mechanism according to which individual cells move in response to chemical signals produced by all other cells. However, in some bacteria there is no evidence for chemotactic cues and cells coordinate their movement by cell-to-cell signalling mechanisms in which physical contact between bacteria is needed [6]. Consequently, one may ask how such bacteria aggregate in order to communicate.

Another relevant aspect is the influence of the shape of the bacteria. The shape has been shown to be essential for individual motion of swimming bacteria [START_REF] Dusenbery | Proc. Natl. Acad. Sci U.S.A[END_REF]. In contrast, the role of the cell shape for collective motion has remained mostly unexplored. It has been demonstrated experimentally [START_REF] Kemkemer | [END_REF] that migrating elongated amoeboid cells exhibit alignment effects similar to those reported in liquid crystals [9]. A prominent example for collective behavior with no apparent long range interactions are the striking patterns observed during the life-cycle of gliding myxobacteria, see e.g. [6,10]. Earlier modeling work has reproduced many of these patterns in three dimensions assuming either perfect alignment [11] or a phenomenological alignment force [12]. These models have all considered patterns resulting from exchange of chemical signals, that are absent in an early stage of the myxobacterial life cycle. Nevertheless, a trend from initial independent motion towards formation of larger clusters of aligned bacteria is often observed ( Fig. 1). Here, we study a model of self-propelled rods that have only repulsive excluded volume interactions in two dimensions. We find that the interplay of rod geometry, self-propulsion and repulsive short-range interaction is sufficient to facilitate aggregation into clusters.

In simulations of an individual based model (IBM), clustering of self propelled particles (SPP) is observed for large enough packing fraction η resp. aspect ratio κ of the rods (see Fig. 2). We define the onset of clustering by the transition from a unimodal to a bimodal cluster size distribution. A mean-field approximation (MFA) for the cluster size distribution is derived and reproduces the change from a unimodal to a bimodal shape upon increase of either η or κ. The MFA yields a simple equation κ c = C/η -1 for the critical rod aspect ratio, κ c , at the onset of clustering in line with the IBM simulation results. fitted with C = 1.46. If diffusion is added to the active motion (active Brownian rods), the clustering transition is shifted to higher values of κ, whereas clustering is absent for pure diffusive motion (Brownian rods) as well as for isotropic particles with κ = 1. Hence, clustering of particles with excluded volume interaction requires both active motion, i. e. a non-equilibrium system, and elongated particles (= rods).

Individual-based model (IBM).-We consider N rodlike particles moving on a plane. Each particle is equipped with a self-propelling force acting along the long axis of the particle. We assume that particles are submerged in a viscous medium. Velocity and angular velocity are proportional to the force and torque, correspondingly. The rod-shape of the particles requires three different friction coefficients which correspond to the resistance exerted by the medium when particles either rotate or move along their long and short axes. Inertial terms are neglected (overdamped motion). Consequently the movement of the i-th rod is governed by the following equations for the velocity of its center of mass and angular velocity:

(v (i) , v (i) ⊥ ) = 1 ζ (F - ∂U (i) ∂x ), - 1 ζ ⊥ ∂U (i) ∂x ⊥ θ(i) = - 1 ζ θ ∂U (i) ∂θ (1)
where

v (i) , v (i)
⊥ refer to the velocities along the long and short axis of the rods, respectively, ζ i indicates the corresponding friction coefficients (ζ θ is related to the friction torque), U (i) refers to the energy of the interaction of the i-th rod with all other rods, and F is the magnitude of the self-propelling force. The motion of the center of mass ẋ(i) = (v

(i) x , v (i)
y ) of the i-th rod is given by

v (i) x = v (i) cos θ (i) + v (i) ⊥ sin θ (i) v (i) y = v (i) sin θ (i) -v (i) ⊥ cos θ (i) (2)
Particles interact by ,,soft volume exclusion, i. e. by a potential that penalizes particle overlaps in the following way:

U (i) (x (i) , θ (i) , x (j) , θ (j) ) = φ N j=1,j =i (γ -a o (x (i) , θ (i) , x (j) , θ (j) )) -β -γ -β (3)
where a o (x (i) , θ (i) , x (j) , θ (j) ) is the area overlap of the rods i and j, γ is a parameter which can be associated to the maximum compressibility, β controls the stiffness of the particle, and φ is the interaction strength. The simulations were performed placing N identical particles initially at random inside a box of area A with periodic boundary conditions. The values of the parameters are given in [START_REF][END_REF].

There are three key parameters which control the dynamics: i) persistence of particle motion, regulated by F , ii) the packing fraction η, i.e., the area occupied by rods divided by the total area (η = N a/A, where N is the number of particles in the system, a is the area of a single particle, and A is the total area of the box), and iii) the length-to-width aspect ratio κ (κ = L/W , where L is the length and W is the width of the rods). Simulations yield an increase of cluster formation with increasing κ or η, see Fig. 2. Individual clusters are defined by connected particles that have non-zero overlap area. Simulations can be characterized by the mean maximum cluster size, M , and the weighted cluster size distribution, p(m), which indicates the probability of finding a given particle inside a cluster of mass m. Fig. 3a shows that for a given η, M seems to saturate after the critical κ c which is defined as the value of κ for which the shape of p(m) changes from unimodal to bimodal. In Fig. 3b typical shapes of p(m) are shown: before clustering and corresponding to low values of κ (circles), and after clustering and corresponding to large values of κ (crosses). We define the onset of clustering by the emergence of a second peak in p(m). We have also tested the robustness of the model against fluctuations by inserting additive noise terms R i /ζ i in Eqs. (1), which correspond to a switch from active to active Brownian particles [5]. We found that clustering is still present in rods of the latter kind, albeit the transition is moved to larger values of κ and η. Clustering was absent in all simulations with purely Brownian rods (F = 0).

Mean field approximation (MFA).-We have studied the clustering effects described above through a MFA by deriving kinetic equations for the number n j of clusters of a given size j. The equations for n j contain terms for cluster fusion and fission. For the fusion terms we have adopted kinetic equations originally derived for coagulation of colloids [14], while the fission terms are empirically defined from the typical behavior seen in the above simulations. The numbers n j change in time -we have {n j (t)} ∞ j=1 , where n j (t) is the number of clusters of mass j at time t. This description neglects the geometry of clusters as well as spatial fluctuations. This allows us to consider a single rate constant for all possible collision processes between clusters of mass i and j, as well as a unique disintegration constant for any cluster of mass i. In addition we make four crucial assumptions: i) The total number of particles in the system, N = N j=1 jn j (t), is conserved. ii) Only binary cluster collisions are considered. Collisions between any two clusters are allowed whenever the sum of the cluster masses is less or equal to N . iii) Clusters suffer spontaneous fission only by losing individual particles at the boundary one by one, i. e. a cluster can only decay by a process by which a j-cluster split into a single particle plus a (j -1)-cluster. This is motivated by observations in the above simulations. iv) All clusters move at constant speed, v ≈ F/ζ , which implies that rods in a cluster have high orientational order and interact only very weakly with their neighbors. Under all these assumptions the evolution of the n j 's is given by the following N equations:

ṅ1 = 2B 2 n 2 + N k=3 B k n k - N -1 k=1 A k,1 n k n 1 ṅj = B j+1 n j+1 -B j n j - N -j k=1 A k,j n k n j + 1 2 j-1 k=1 A k,j-k n k n j-k for j = 2, ....., N -1 ṅN = -B N n N + 1 2 N -1 k=1 A k,N -k n k n N -k (4)
where the dot denotes time derivative, B j represents the fission rate of a cluster of mass j, defined by B j = ( v/R) √ j, and A j,k is the collision rate between clusters of mass j and k, defined by A j,k = ( vσ 0 /A) √ j + √ k . σ 0 is the scattering cross section of a single rod. R is the only free parameter and indicates the characteristic length a rod at the boundary of a cluster moves before it is leaving the cluster in a typical fission event. We assume R = αL taken into account that longer rods will stay attached to cluster for a longer time.

Since σ 0 can be approximated by

σ 0 ≈ L + W = √ a √ κ + 1 √ κ
, the MFA depends only on the parameters κ, a, A, v and α. If one integrates Eqs. ( 4) with parameters used in IBM simulations and an initial condition n j (t = 0) = N δ 1,j , their solution yields steady state values n 0 j for t → ∞. From these values, we obtain a MFA for the weighted cluster size distribution p(m) = n 0 m m/N for given values of the free parameters R resp. α. The best agreement between the MFA and the IBM simulations is found for a choice of α = 1.0 ± 0.05 (see Fig. 3b). Hence, we will use R = L in the following. To understand the relation between the parameters of the model and clustering effects, we can rescale Eqs. (4) by introducing a new time variable: τ = t v/ √ aκ. The resulting equations [15] depend only on a dimensionless parameter P = (κ + 1)a/A. Note that v = 0 is scaled and does not affect the qualitative dynamics of the system. In the dimensionless model the parameter P stands for ratio between fusion and fission processes and therefore triggers the transition from a unimodal to a bimodal cluster size distribution. We can easily establish a transition criterion, and by using a bisection method, we can accurately determine the critical transition parameter P c . Given the system area A, the rod area a and the number of rods N , this method provides a way to calculate κ c :

κ c = P c (N ) A a -1 (5)
At this point it is crucial that P c depends on N , which is formally the number of equations in the MFA. By numerically solving the MFA equations for different particle numbers N up to N = 1024, we find P c (N ) ∝ N -1.026±0.023 . This result indicates that in the MFA the critical parameter value κ c for the clustering transition does not depend on the number of particles. This does not imply that the weigthed distribution p(m) is independent of N ; in fact, we find that the probability for a rod to be in a large cluster increases with N . We proceed by assuming that P c is inversely proportional with N , we can express κ c as a simple function of the packing fraction:

κ c = C/η -1 (6)
where the constant is found to be C ≈ 1.46. The κ-η phase diagram (Fig. 4) shows a reasonable agreement of the transition line given by Eq. ( 6) and the IBM simulation results. So, for the range of parameters used in the IBM, we retrieve in the MFA the unimodal shape of the weighted cluster size distribution for small values of κ and η, and the bimodal shape for large values of the two parameters. Fig. 3b gives a comparison of the cluster size distribution in the IBM and MFA.

In summary, we have found non-equilibrium clustering for interacting self-propelled rod shaped-particles with sufficient packing density η and aspect ratio κ in simulations. The rods interact via strong short range repulsive interactions that approximate excluded volume interactions. The onset of clustering has been defined by a transition from a unimodal to bimodal cluster size distribution. This transition is reproduced by a meanfield description of the cluster size distribution, which yielded a simple criterion, κ = C/η -1, for the onset of clustering. This functional form with C ≈ 1.46 provides a good fit to the results of the simulations. The high density inside the cluster leads also to alignment of rods and coordinated motion of all particles in the cluster. The transition to clustering defined here is practically independent of the system size resp. the number of particles. It is instructive to compare our result rewritten in the form κη + η ≈ 1.46 with the formula for the isotropic-nematic transition κη ≈ 4.7 found in the twodimensional version [16] of Onsagers mean-field theory for Brownian rods [9]. This shows that actively moving rods can achieve alignment at much lower densities than Brownian rods resp. particles in equilibrium systems. The clustering phenomenon is absent in simulations with isotropic self-propelled particles as well as with Brownian rods. Our model provides also an alternative explanation for collective behavior of rod-shaped objects -previous swarming models have achieved aggregation and clustering by assuming attractive long-range interactions [4,5]. With respect to biology, our observation offers a simple physical explanation for the formation of clusters in many gliding rod-shaped bacteria, that often precedes the formation of biofilms and the appearance of more complex patterns. Acknowledgement: We acknowledge financial support of Deutsche Forschungsgemeinschaft (DFG) through grant DE842/2 and fruitful discussions with L. Morelli, L. Brusch, J. Starruss and L. Sogaard-Andersen.

FIG. 1 :

 1 FIG. 1: Example for clustering of myxobacteria (M. xanthus) in the early stage of the life cycle. (a) Immediately after maturation of spores. (b) Afterwards, during the vegetative phase. Snapshots are taken from a movie (Ref. [10](b)), the frame size is 40 x 30 µm 2 . Similar phenomena were seen in other bacterial species (cf. Ref. [6](a)).

FIG. 2 :

 2 FIG. 2: Simulation snapshots of the steady states for different particle anisotropy κ and the same packing fraction η (a-c), and the same κ and different η (d-f). Fixing η = 0.24: (a) before the transition, κ = 1; (b) almost at the transition, κ = 5; (c) after the transition, κ = 8. Fixing κ = 6: (d) before the transition, η = 0.18; (e) just crossing the transition, η = 0.24; (f) after the transition, η = 0.34. In all cases, particles N = 100 and particle area a = 0.2. The arrows indicate the direction of motion of some of the clusters.

FIG. 3 :

 3 FIG. 3: (a) The mean maximum cluster size M vs κ for IBM simulations (N = 50). (b) p(m) as function of the cluster size m for η = 0.34. Symbols show the average over eight IBM simulations for active particles with N = 50 and κ = 1 (circles) and κ = 8 (crosses), errors bases give distributions of individual runs. The lines correspond to the mean field theory for κ = 1 (solid) and κ = 8 (dashed)(color online).

FIG. 4 :

 4 FIG. 4: κ-η phase diagram. The solid line corresponds to the transition curve predicted by equation 6. The symbols indicate IBM simulations (N = 100). Crosses refer to unimodal p(m) and circles to bimodal p(m) of active particles. Stars refer to unimodal p(m) and hexagrams to bimodal p(m) of active-Brownian particles (color online).

  jk)n k n j-k -N-j k=1 ( √ j + √ k)n k nj ] for j = 2, ..., N -1.[16] R. F. Kayser and J. Raveche, Phys. Rev. A 17, 2067 (1978).