J M Guadarrama-Cetina

A Mongruel

M.-G Medici

E Baquero

A R Parker

I Milimouk-Melnytchuk

W González-Viñas

D Beysens

Dew condensation on desert beetle skin

Keywords: 68, 03, Fg Evaporation and condensation of liquids -87, 19, -j Properties of higher organisms

Some tenebrionind beetles inhabiting the Namib desert are known for using their body to collect water droplets from wind-blown fogs. We aim to determine whether dew water collection is also possible for desert insects. For this purpose, we investigated the infra-red emissivity, and the wetting and structural properties, of the surface of the elytra of a preserved specimen of Physasterna cribripes (Tenebrionidae) beetle, where the macro-structure appears as a series of "bumps", with "valleys" between them. Dew formation experiments were carried out in a condensation chamber. The surface properties (infra-red emissivity, wetting properties) were dominated by the wax at the elytra surface and, to a lower extent, its micro-structure. We performed scanning electron microscope on histological sections and determined the infra-red emissivity using a scanning pyrometer. The emissivity measured (0.95± 0.07 between 8-14 µm) was close to the black body value. Dew formation occurred on the insect's elytra, which can be explained by these surface properties. From the surface coverage of the condensed drops it was found that dew forms primarily in the valleys between the bumps. The difference in droplet nucleation rate between bumps and valleys can be attributed to the hexagonal microstructure on the surface of the valleys, whereas the surface of the bumps is smooth. The drops can slide when they reach a critical size, and be collected at the insect's mouth.

Introduction

Harvesting water in arid or semi-arid regions is a challenge for life. Some insects such as beetles living in the Namib desert have developed a strategy to collect water drops [1,2]. When air is humid enough at night such that fog (and/or dew) can form, the beetle tilts its body forwards into the wind to collect the fog water in a manner that is well-known within some Tenebrionidae, and termed fog-basking [1]. Droplets form on the upper surface of the fused fore-wings (elytra) and roll down the beetles surface to its mouth parts. The elytrae of these water-collecting insects show surfaces that are either smooth or with grooves or bumps (Fig. 1). However, in addition to fog droplets striking the elytra, the other possible mechanism by which water is extracted from the air to form large a Present address: Facultad de Ciencias, Universidad Nacional Autónoma de México b email: daniel.beysens@espci.fr droplets, is dew droplets nucleation and growth on the elytrae. Laboratory studies have focused so far on the collection of fog droplets, either by spraying water or producing fog [3][4][5]. In a seminal paper [3], Parker and Lawrence highlighted the possible role of hydrophilic and hydrophobic microstructures to improve water drop collection. The droplets observed on Physasterna cribripes elytrae have a diameter 1 -40 µm and the water collection is assumed to be due to the hydrophilic property of the bumps and the hydrophobic characteristics of the valleys in between.

Water accumulates on the top of the bumps and when its volume is sufficiently large, the droplets roll down along the hydrophobic surface towards the insect's mouth. Their study was the starting point for further investigation for nanomaterials with mixed wetting properties for improved water collection efficiency (see e.g. [4,6,7]).

However, further study on living beetles [5] have shown that tenebrionids that exhibit a fog-basking behaviour rather possess a uniform hydrophobic and smooth sur-face of their elytrae. Moreover, fog collecting efficiency on elytrae (of preserved beetles) showing different surface structures (smooth or with bumps), is not significantly influenced by these differences in hydrophobicity. Thus, the microstructure of the elytrae may not be the primary parameter for fog collecting but rather is secondary to the posture of the insect. Here, the beetle assumes a constant angle with the horizontal (23 • ). This angle is necessary for the fog drops to strike the surface and also to collect dew or fog water by gravity. In all the above studies, there is some confusion between fog and dew phenomena [4]. We note that fog is formed of liquid water droplets with a diameter in the range of 10 µm. In contrast, dew involves the condensation of water vapour on a surface following a process of heterogeneous nucleation, which depends on the vapour supersaturation and on the wetting properties and the chemical and geometrical heterogeneities of the substrate, and further growth. One can make, however, the assumption that the collection of water drops is the same for both captured fog drops and condensed dew drops. It happens often that dew and fog form during the same night. It is a matter of fact that during dewy nights, dew starts to form first, with fog appearing only later, in the morning, when the atmosphere is at its coolest. The dew point temperature is reached when air is saturated at 100% relative humidity. Therefore it is reasonable to question whether dew can also form on an insect's back, since insects have no temperature regulation and the elytrae are disconnected from the insect's body by a layer of air that favours some thermal insulation. The climate where such insect lives (Gobabeb, Namibia from Ref. [5]) is characterised for year 2013 [8] by clear sky (only were recorded six rainy days, in March, September and December) and two seasons of high (60-80%, September to March) and small (45-56%, April to August) mean nocturnal humidity. The mean nocturnal temperature is about 18 • C when air is humid and 15 • C when it is dry. The number of days where dew can form corresponds typically to a temperature difference between air and dew point that does not exceed 10 K, corresponding to relative humidity larger than 55% [9]. It should correspond for year 2013 to more than 60% of dewy nights. During condensation, the substrate temperature is typically between 1 K and 7 K below the dew point temperature [10].

It is the object of this study to determine whether dew can form on the elytrae of Physasterna cribripes. The elytrae of these water-collecting insects show surfaces that are either smooth or with grooves or bumps 0.5-1.5 mm apart, each about 0.5 mm in diameter and around 0.4 mm high (Fig. 1). In particular, we address the question of whether the particular infra-red emissivity, and the wetting and geometrical properties of the elytrae of Physasterna cribripes, (i.e. its very characteristic morphology), are of particular interest for dew harvesting when compared to regular, smooth surfaces.

Experimental investigations

An elytra from a preserved specimen of Physasterna cribripes collected from the Skeleton Coast, Namibia, was dissected to provide two sections, each about 3 mm × 7 mm. Half of one section was studied in a Zeiss DSM 940A scanning electron microscope (SEM), and the remaining sections were used as a surface for dew condensation experiments.

Elytra histology

A schematic representation of a transverse section of elytra is shown in Fig. 2. For water condensation, the outer surface of the elytra (valleys and bumps) is most important. From the electron micrographs, the surface of the valleys only reveals a hexagonal periodic patterning with approximately 6 µm period, and both the valleys and the bumps show some microsetae (Fig. 3). The periodic structure is similar to the structure already reported by Parker and Lawrence [3]. The surface is coated with wax, however it is difficult to estimate whether the layer thickness of the wax varies from valleys to bumps, which would alter the surface energy and consequently the wetting properties. The wax on the exoskeleton of insects serves a protective function and reduces desiccation. The amount of wax is larger in animals which live in dry areas. The wettability properties depends on both the chemical properties of the wax and the geometrical (roughness) properties of the surface. The thickness of the layer of wax in Physasterna cribripes is approximately 15 nm.

Infra-red emissivity

In nature, surface cooling is insured by infra-red (IR) radiative cooling. The IR emissivity of the elytra was determined in the wavelength domain between 8-14 µm (the atmospheric window [START_REF] Houghton | The Physics of Atmospheres[END_REF]) with a scanning pyrometer with In this first set of experiments, the pyrometer did not detect any temperature difference between the elytra and the glass, thus indicating that the elytra has an emissivity close to that of the glass. In the further experiments, type (2) and (2'), a section of aluminium foil with low emissivity (0.25) was used instead of glass. Here, the elytra was well detected by the pyrometer, in agreement with an emissivity much larger than the aluminium foil (Fig. 4).

The analysis of experiments type (1) and (2) in terms of emissivity gave the following values for the different surfaces investigated. The uncertainty was ± 0.07. Glass: 1; aluminium film: 0.28; cardboard: 0.76; elytra: 0.95. The good agreement between the measured and expected emissivities for glass, cardboard and aluminium film confirms the accuracy of the measurements. The high value of the elytra emissivity conforms to that reported for Schistocerca gregaria, a desert insect called the bird grasshopper [START_REF] Hunt | [END_REF].

Condensation

The condensation experiments have to reproduce conditions typical of the desert conditions. However, for practical reasons, the experiment time has to be reduced, which corresponds to somewhat larger degree of supersaturation than found in an actual environment. The laboratory experiments will thus correspond to very favourable conditions and will somewhat emphasise the effects.

The second section of the elytra was attached with heat conductive grease onto the copper disk and placed inside the condensation chamber, which is cylindrical (diameter 2L =10 cm, height: 1 cm). A flux of air saturated with water (35 ml/min and 200 ml/min) at room temperature (23 • C ± 0.3 • C), formed by bubbling air through ultra-pure water, was sent into the chamber through two holes ("nozzles") with 4.5 mm diameter. They are set at a chamber perimeter and perfectly aligned on a chamber diameter right under the level of the plane surface. The latter is slightly hollowed out like a channel over a length of 1 cm. This considerably slows down the flow in the ratio hole/chamber cross-section (≃1.6•10 -2 ) even before both flows meet at the expected stagnation point. Two other holes connected to the room are set on the chamber perimeter on a diameter perpendicular to the previous one. The maximum velocity is u m =0.2 m/s, corresponding to a nozzle output for the maximum flow rate (200 mL/min.). The corresponding Reynolds number Re = u m L/ν = 750 (with ν =1.4•10 -5 m 2 /s the air kinematic viscosity), a value much lower than the critical Reynolds number (5×10 5 ) where turbulence should occur. Flow even at the nozzles output for the largest flow rate is thus in the laminar regime. The elytra was positioned in this central region of the chamber. The copper disk temperature can be adjusted between 4 • C and 14 • C to provide water vapour condensation. The temperature measured on the elytra with a very thin thermocouple was the same than the copper disk temperature within at most 1 • C.

Note that here the cooling process (conduction) is not the same as the radiative process that occurs during dew formation. However, the temperature is also nearly homogeneous on the surface of the elytra in both cases.

The development of the condensed pattern was observed by optical microscopy (magnification of 20X, resolution of 2 µm) using a CCD camera. The images were recorded on a computer for further analysis.

Observations and analyses

The development of water vapour condensation on the elytra was observed from the time the water vapour flux was sent into the chamber. We chose an area of observation where the two surface geometries (valley and bump) could be focused at the same time (Fig. 5). Applying the flux of water vapour, we found that water always condensed firstly on the most depressed parts of the elytra (valleys). This can be due to two reasons: (i) a thermal effect, the valleys being slightly colder due to their close vicinity with the copper disk and exhibiting a lower thermal exchange with the surrounding atmosphere that is at higher temperature; (ii) a difference in nucleation barrier (e.g. due to a different wettability and/or different nucleation sites and roughness); nucleation in the valleys can thus be promoted greater than on the bumps. The shapes of water drops in the valleys appear more irregular than on the bumps, corresponding to a smaller contact angle and/or a larger contact angle hysteresis [13]. Thermal and contact angle effects are investigated in the following experiments.

Growth, contact angle and roughness

The growth of water droplets on a plane surface obeys different regimes [14,15]. Nucleation of microdroplets proceeds first. The energy barrier for nucleation corresponds to the cost of formation of a vapour-liquid interface. It thus depends on the water contact angle θ on the surface. The barrier is maximum for θ = 180 • (very hydrophobic surface) and minimum (zero) for θ = 0 • (purely hydrophilic surface). In addition, geometric defects, which lower the vapour-liquid interface, also favour nucleation. Then nucleation proceeds preferentially on these nucleation sites that correspond to surface chemical and/or geometric irregularities. After having nucleated, droplets grow with radius ∼ t 1/3 (t is time) and the drop surface coverage ε 2 increases on the surface. The surface coverage is defined as the ratio of surface occupied by condensed droplets to the total substrate surface. Droplets coalescence and then growth is accelerated. As the coalescence process lowers the surface coverage, a balance between the increase (by droplet growth) and decrease (by coalescence events) follows, where the drop surface coverage becomes constant (ε 2 = ε 2 ∞ ), and droplet growth becomes self-similar (statistically scale invariant) with mean droplet radius ∼ t. The constant ε 2 ∞ depends on the wetting properties (water contact angle and surface roughness) of the substrate [13]. These growth laws were indeed observed on both valleys and bumps, for the two flow rates as reported in Figs. 6ab. On a smooth surface with small contact angle hysteresis, a mean contact angle (deg.) can then be obtained from the measurement of ε 2 in the self-similar, coalescencelimited stage of growth [13]:

ε 2 ∞ ≃ 1 -(θ/200). (1)
The value of the contact angle should be between the advancing contact angle (the drops are growing) and the receding angle (drops that have coalesced). However, as there are far fewer drops in the latter case, this angle becomes close to the advancing contact angle. The analysis was performed on the valley and bump areas of two different elytrae exposed to the same room (23 • C) and surface (10 • C) temperatures, but to two different flow rates (set 1: 200 ml/min; set 2: 35 ml/min). The results for set 1 are shown in Fig. 5. The measurements of surface coverage ε 2 in the valley and bump areas are reported in Fig. 6. The data can be fitted to an empirical exponential function:

ε 2 = ε 2 ∞ 1 -exp t -t 0 τ . (2)
Here ε 2 ∞ is the surface coverage limiting value in the self-similar growth regime, t 0 is the nucleation time and τ is the typical time to reach the self-similar growth regime. t 0 is contact angle and supersaturation dependent; τ depends on supersaturation and on water vapour flow rate.

The following values were found for the valleys: The time t 0 is smaller for valleys than for bumps (∼ 100 s) in both sets 1 and 2, regardless of the flow rate. Thus, the nucleation time for valleys is taken as the origin of time. This result indicates that nucleation is faster on the valleys than on the bumps. The typical times τ are comparable for both bumps and valleys (set 1: ∼ 80 s; set 2: ∼ 400 s) but are larger in set 2 than in set 1 because of the lower flow rate.

Set ε 2 ∞ t 0 (s) τ (
The surface coverage limiting values were found to be larger in the valleys than on the bumps, for the two sets of data. Taking the mean values of ε 2 ∞ between set 1 and set 2, i.e. 0.72 ± 0.04 for the valleys and 0.38 ± 0.02 for the bumps, respectively, one obtains from Eq. 1: θ ≃ (64 ± 8) • in the valleys and θ ≃ (124 ± 4) • on the bumps. The surface of the valleys thus appears more hydrophilic than the bumps. However, this difference between valley and bump contact angles cannot be explained by a difference in surface coating as the valleys and bumps are both covered with wax. The only difference is that the surface of the bump is smooth while the valley surface exhibits a 6 µm period hexagonal patterning (Fig. 3). The patterning (i) favours nucleation, (ii) pins the droplets contact line, resulting in an increase of the contact angle hysteresis and an irregular droplet shape. The effects of patterning at this spatial scale are thus similar to the effects of lowering the water contact angle on a smooth surface.

Thermal effects

In order to test for differential thermal effects between valleys and bumps, we inverted the gradient of temperatures between the lowermost valley region and the highest (peak) part of the bump. To achieve this, we decreased the substrate temperature to 1 • C while streaming dry air (23 • C) to prevent water condensation. Then we rapidly increased the substrate temperature to ≈ 11 • C while simultaneously streaming humid air at the same temperature. The surface temperature of the valley was then larger than the bump temperature. However, as shown in Fig. 7, water vapour still began to condense in the valleys. This was in accordance with the largest contact angle at the bump location, where nucleation requires further supersaturation than in the valleys.

Conclusions

This study reveals specific properties of the beetle skin concerning dew water condensation. The high IR emissivity of the wax-coated elytra provides efficient radiative cooling. Higher droplet nucleation rate is found in valleys, where dew forms primarily. This is due to the hexagonal microstructure on the surface of the elytra. The latter provides more nucleation sites than on the bumps whose surface is smooth at the micron level. The role of these bumps, which are found on the backs of many other insects (but not in the fog basking species Onymacris unguicularis and Onymacris bicolor ) remains thus unclear. They could serve as wind shields for the valleys, increas-ing there the dew yields. They can also be used to deter possible predators.

The dew collection properties of Physasterna cribripes is basically due to the high infra-red emissivity of wax. As insects are wax coated, this property should be shared by all of them, in particular those which, like the Onymacris unguicularis and Onymacris bicolor species, are known to harvest fog. Evidence that dew can be collected in addition to fog water on the backs of these beetles has not been examined yet in field experiments, where traditionally fog and dew phenomena have not been distinguished. This study reveals that dew collection is indeed possible on insect elytrae when the external conditions are favourable: clear sky and relative humidity larger than 55%. These climatic conditions can be found in many coastal arid areas in the world.

Fig. 1 .

 1 Fig. 1. Physasterna cribripes (Tenebrionidae), female. The scale bar is 4 mm.

Fig. 2 .

 2 Fig. 2. Sketch of the elytra structure. (a) General cross-sectional view. (b) Details of exocuticle and epicuticle (from (a)).

Fig. 3 .

 3 Fig. 3. SEM photos of elytra regions of Physasterna cribripes. (a) A bump and surrounding valleys. At the top of the bump, there is a small depression with a microseta. Another microseta is observed in the valley. The surface of the top of the bump is smooth; the surface of the valley contains an hexagonal structure (see (c)). (b) Transverse section through a bump. (c) Structured surface of the valley. (d) The smooth layer observed on the surface of the bump, which corresponds to the reticulated layer of cement plus wax, above the outer epicuticle.

Fig. 4 .

 4 Fig. 4. Scanning pyrometer study on a section of elytra approximately 3 mm ×7 mm attached to aluminium film (see text).

Fig. 5 .

 5 Fig. 5. Measurement of drop surface coverage (see text) in a valley (V) and a bump (B) of the elytra. (a) t = 5 s, (b) t = 297 s (image is focused on region V) and (c) t = 309 s (image is focused on region B).

Fig. 6 .

 6 Fig. 6. Development of drop radius and surface coverage in a valley (white triangles) and on a bump (dark triangles) for two different elytrae and flow rates. (a): set 1 (200 ml/min.); (b): set 2 (35 ml/min.).

Fig. 7 .

 7 Fig. 7. A valley and a bump where the gradient of temperatures have been reversed; the water vapour still condensed first in the valley.

Acknowledgements

The authors thank S. Zeine for help in the IR measurements. This work was partly supported by the Spanish MEC (Grant n. FIS2011-24642). JGC acknowledges financial support from the "Asociación de Amigos de la Universidad de Navarra".


Hugues Berry

Thomas Lepoutre
email: thomas.lepoutre@inria.fr

†1

Mateos González
email: alvaro.mateosgonzalez@ens-lyon.fr

Quantitative convergence towards a self similar profile in an age-structured renewal equation for subdiffusion

Keywords:

Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the subdiffusive continuous-time random walk using age-structured partial differential equations with age renewal upon each walker jump, where the age of a walker is the time elapsed since its last jump. In the spatially-homogeneous (zero-dimensional) case, we follow the evolution in time of the age distribution. An approach inspired by relative entropy techniques allows us to obtain quantitative explicit rates for the convergence of the age distribution to a self-similar profile, which corresponds to convergence to a stationnary profile for the rescaled variables. An important difficulty arises from the fact that the equation in self-similar variables is not autonomous and we do not have a specific analyitcal solution. Therefore, in order to quantify the latter convergence, we estimate attraction to a time-dependent "pseudo-equilibrium", which in turn converges to the stationnary profile.

Introduction 1.Brief model description

Recent methodological advances in cell biology allowed the measurements of the displacement of single molecules (or assemblies thereof) in living single cells. Those investigations have consistently reported that the random displacement of molecules inside cells often deviates from Brownian motion, with the mean squared displacement r 2 (t) that does not scale linearly with time, as in Brownian motion, but sublinearly, with a power-law behavior: r 2 (t) ∝ t µ [START_REF] Golding | Physical Nature of Bacterial Cytoplasm[END_REF][START_REF] Bronstein | Transient Anomalous Diffusion of Telomeres in the Nucleus of Mammalian Cells[END_REF][START_REF] Parry | The bacterial cytoplasm has glass-like properties and is fluidized by metabolic activity[END_REF][START_REF] Di Rienzo | Probing short-range protein brownian motion in the cytoplasm of living cells[END_REF]. This behavior is usually referred to as "anomalous" diffusion or "subdiffusion", since µ < 1 usually, for non-active transport (for a review see e.g. [START_REF] Höfling | Anomalous transport in the crowded world of biological cells[END_REF]).

Continuous-time random walks (CTRW) are one of the main mechanisms that are recurrently evoked to explain the emergence of subdiffusion in cells. CTRW were introduced fifty years ago by Montroll and Weiss as a generalisation of random walks [START_REF] Montroll | Random walks on lattices[END_REF], where the residence time (the time between two consecutive jumps) is a random variable τ with probability distribution φ(τ ) (see [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF] for a review). If the expectation of τ is defined, for instance when τ is dirac-distributed or decays exponentially fast, one recovers the "normal" Brownian motion. However, when the expectation of τ diverges, for instance when φ(τ ) is heavy-tailed, φ(τ ) ∝ τ -(1+µ) with 0 < µ < 1, the CTRW describes a subdiffusive behavior, with r 2 (t) ∝ t µ .

One great achievement of CTRW is that they can readily be used to derive mean-field equations for the spatio-temporal dynamics of the random walkers. Indeed, starting from φ(τ ), combinations of Laplace and Fourier transforms lead to a "subdiffusion" equation for the density of random walkers located at position x at time t: ∂ t ρ(x, t) = D µ D 1-µ t ∇ 2 ρ(x, t) where D µ is a generalised diffusion coefficient and D 1-µ t is the Riemann-Liouville fractional derivative operator [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF][START_REF] Mendez | Reaction-Transport Systems[END_REF]. Such a fractional dynamics formulation is very attractive for modelling in biology, in particular because of its apparent similarity with the classical diffusion equation. However, contrarily to the diffusion equation, the Rieman-Liouville operator is non-Markovian. This non-Markovian property becomes a serious obstacle when one wants to couple subdiffusion with chemical reaction [START_REF] Henry | Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations[END_REF][START_REF] Bravo Yuste | Anomalous Transport, chapter Subdiffusion-Limited Reactions[END_REF][START_REF] Fedotov | Nonlinear degradation-enhanced transport of morphogens performing subdiffusion[END_REF].

Here, we take an alternative approach to CTRW that maintains the Markovian property of the transport equation at the price of a supplementary independent variable. We associate each random walker with an age a, that is reset when the random walker jumps. In one dimension of space, we note n(t, x, a) the density probability distribution of walkers at time t that have been residing at location x during the last span of time a. The dynamics of the CTRW is then described with an age-renewal equation with spatial jumps that reinitialise the age:

    
∂ t n(t, x, a) + ∂ a n(t, x, a) + β(a)n(t, x, a) = 0 , t ≥ 0, a > 0 , x ∈ R n(t, x, a = 0) = ∞ 0 R β(a )ω(x -x )n(t, x , a )dxda n(t = 0, x, a) = n 0 (x, a).

(1)

The kernel ω describes the spatial distribution of jump destinations (typically a Gaussian distribution centred at the origin position), and the function β(a) gives the jump rate. Since we are mostly interested here in the subdiffusive case (where the expectation of the residence time diverges), we will focus throughout this article on the case: aβ(a) ---→ a→∞ µ ∈ (0, 1).

(

) 2
The precise meaning of the limit will be given later on. The limit µ in eq.( 1) is the subdiffusion exponent: for µ > 1, eq.( 1) describes a diffusive process, whereas for 0 < µ < 1 the mean time a particle has to wait between two consecutive renewals diverges and the mean squared displacement exhibits subdiffusion with exponent µ. The distribution of residence time φ(τ ) evoked above is related to the jump rate as: φ(τ ) = β(τ ) exp -τ 0 β(s) ds . Note that this agestructured approach is not uncommon in the CTRW literature [START_REF] Mendez | Reaction-Transport Systems[END_REF][START_REF] Fedotov | Subdiffusive master equation with space-dependent anomalous exponent and structural instability[END_REF]. Our main contribution here is to use it in conjunction with approaches borrowed from the study of partial differential equations.

In the present article, we restrict our attention to the temporal evolution of the age distribution of the walkers. To this end, we simplify the problem by considering its spatially-homogenous version, namely:

     ∂ t n(t, a) + ∂ a n(t, a) + β(a)n(t, a) = 0 , t ≥ 0, a > 0 n(t, a = 0) = ∞ 0 β(a )n(t, a )da n(t = 0, a) = n 0 (a).
(3)

Self-similar solutions

The only steady state solution of eq.( 3) in L 1 is 0, which doesn't allow us to describe the dynamics of the system in a satisfactory way. Hence the search for self-similar solutions. An educated guess is that they should be of the following form, with A(t) to be determined:

n(t, a) = 1 A(t) w ln(1 + t) τ , a/A(t) b .
Let us consider, for the sake of simplicity, an initial condition supported on [0, 1). By injecting the previous expression into eq.( 3), we find that the natural choice A(t) = 1 + t preserves the initial condition n(0, b): = n 0 (b) = w(0, b): = w 0 (b), and yields : (4) Note that for an initial condition supported on [0, A + ), A(t) = A + + t is a better choice and leads to a similar analysis. If the initial condition is not compactly supported, the tail of the age distribution can influence the convergence rate we give below.

     ∂ τ w + ∂ b ((1 -b)w) + e τ β(
It is important to note that the previous system is not autonomous, for the term e τ β(e τ b) depends on b. This rescaling does not lead to a classical steady state, and we could not find a particular solution of the previous equation. However, we may look for a stationary state satisfying formally the following equation, since we consider here β(a) ∼ µ a .

∂ b ((1 -b)W ∞ ) + µ b W ∞ = 0
where the boundary condition cannot be stated as an equality since W ∞ is expected to blow up at 0, but can be understood as an equivalence as ε tends to 0 of W ∞ (ε) and

∞ ε µ b W ∞ (b)
db. This leads us to define the self-similar equilibrium as:

W ∞ (b) = c ∞ b µ (1 -b) 1-µ (5)
which is called the arcsine distribution, or Dynkin-Lamperti distribution. c ∞ is defined such that W ∞ 1 = 1. Under some conditions, we can expect that w(τ, b) will converge to eq.( 5) when τ → ∞.

A similar result in probability theory appears in Feller's book [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF] tome II, chapter XI, especially in section 5 and onwards, where the renewal problem is tackled by considering the waiting time before the n th renewal. For an introduction to renewal theory, see the eponymous chapter (8.6) in [START_REF] Bingham | Regular Variation[END_REF]. However, no convergence rate is given for our infinite mean waiting time problem in any of these books, and we have been unable to locate such a convergence rate in the subsequent literature. Recent developments in Ergodic Theory for mildly related problems (see chapter 8.11 of [START_REF] Bingham | Regular Variation[END_REF] for an introduction to Darling-Kac theory), have yielded convergence rates, that are optimal in certain cases, as shown in [START_REF] Melbourne | Operator renewal theory and mixing rates for dynamical systems with infinite measure[END_REF] and [START_REF] Terhesiu | Error rates in the Darling-Kac law[END_REF].

Main results

Throughout the article, the following set of hypotheses will intervene. Hypothesis (H1) will be used in properties of convergence without a rate while hypothesis (H2) will allow convergence rate estimates.

(H1). β is a positive, bounded, and non-increasing function satisfying lim a→∞ aβ(a) = µ ∈ (0, 1).

Remark. We will always assume β to be non-increasing for the sake of simplicity (in particular in theorem 5, propositions 3 and 20, and lemmas 13 and 22). The monotonicity can be replaced by the following hypothesis:

β is a positive, bounded function satisfying lim a→∞ aβ(a) = µ ∈ (0, 1), such that β defined as follows

β(a) = inf x∈[0,a] β(x) (6)
also satisfies lim a→∞ aβ(a) = µ. This leads to minor changes in the proofs, the loss of a multiplicative constant in the affected results and replacing β by β where it corresponds. (H2). β satisfies (H1). Additionally, β(a) = µ 1+a + g(a), where g ∈ L 1 and there exist K, α > 0 such that

∞ a |g(s)|ds ≤ K (1 + a) α .
Remark. For the sake of clarity we will investigate separately the particular case g = 0, called the "reference case". Then, all our results will be extended to the general case at the expense of the convergence rates.

Due to the specific shape of W ∞ and to the boundary condition, it is difficult to investigate in a direct way the evolution of w -W ∞ 1 : the methods we describe subsequently fail to do so. However, we could recover a quantitative explicit convergence rate with respect to a "pseudo-equilibrium" W which will be proved to converge in L 1 to W ∞ . Definition 1. We define the pseudo-equilibrium W over R + × [0, 1) as follows :

W (τ, b) = C(τ ) e B(e τ b) (1 -b) 1-µ . ( 7
)
where B(a) = a 0 β(s)ds and C is defined so that W (τ,

•) L 1 = 1.
In particular in the reference case β(a) = µ 1+a , it may be written as

W (τ, b) = C(τ )e -µτ (e -τ + b) µ (1 -b) 1-µ .
Note the similarity between this expression and that for W ∞ in eq.( 5). In the following, we obtain explicit convergence rates of w(τ, b) to W , developing proofs based on Relative Entropy estimates. Importantly, we show that W converges to W ∞ at the same rate (up to multiplication by a constant) as the rate with which w converges to W . Hence, the convergence to the pseudoequilibrium W yields a very good estimate of the convergence to the self-similar equilibrium W ∞ . Finally, we carry out Monte-Carlo simulations of zero-dimensional CTRW to illustrate and question the optimality of our main analytical results. Definition 2. For the moment and for the sake of simplicity, let us define:

H(τ ) = w(τ, •) -W (τ, •) L 1 ([0,1]) .
Remark. Later on, we shall define more generally H as a relative entropy, the L 1 distance being a particular case more suited to our purposes.

In this paper, we prove the following propositions: Proposition 3. Under hypothesis (H1), we have:

H(τ ) ----→ τ →∞ 0.
Our first quantitative result is a convergence rate for the reference case of hypothesis (H2). Theorem 4. Let β(a) = µ 1+a . Then we have the following convergence rates:

     H(τ ) ≤ e -µτ H(0) 2 1+e -τ µ -8 2µ-1 2 1+e -τ µ + e -(1-µ)τ 8 2µ-1 2 1+e -τ µ if µ = 1 2 H(τ ) ≤ e -τ 2 2 1+e -τ (H(0) + 8τ ) if µ = 1 2 . (8)
A modified, yet analogous, convergence rate still holds for g = 0:

Theorem 5. Suppose hypothesis (H2) holds. If α > 1 -µ, we recover the optimal rate of convergence

H(τ ) ≤ K(e -µτ + e -(1-µ)τ ), if µ = 1/2 Kτ e -τ /2 , if µ = 1/2.
If α ≤ 1 -µ, we need to distinguish between several cases:

H(τ ) ≤                    K(e -ατ + e -µτ ), if µ = α < 1 -µ K(1 + τ )e -µτ , if α = µ < 1 -µ, K(τ e -(1-µ)τ + e -µτ ), if α = 1 -µ = 1/2, K(1 + τ 2 )e -τ /2 , if α = µ = 1 -µ = 1/2.
We finally reinterpret our results in terms of non-rescaled variables, for instance in the reference case β(a) = µ 1+a .

Corollary 6. Assume n 0 is supported in [0, 1) and β(a) = µ 1+a , then if we denote

N ∞ (t, a) = c∞ a µ (1+t-a) 1-µ , a < 1 + t, 0, a > 1 + t. Then if µ = 1/2, there exists K such that n(t, .) -N ∞ (t, .) 1 ≤ K (1 + t) µ + K (1 + t) 1-µ . If µ = 1/2, then we have n(t, .) -N ∞ (t, .) 1 ≤ K(1 + log(1 + t)) √ 1 + t .
Remark. An analogous version in non-rescaled variables can be given for theorem 5.

Outline of the paper

The paper is organized as follows. In Section 2 we set the entropic structure of the equation and the main properties of the pseudo equilibrium W . In particular we establish (non quantitatively) that lim

τ →∞ w(τ, •) -W (τ, •) 1 = 0, lim τ →∞ W ∞ (τ, •) -W (τ, •) 1 = 0, proving thereby lim τ →∞ w(τ, •) -W ∞ (τ, •) 1 = 0.
Section 3 deals with quantitative convergence rates towards the pseudo-equilibrium W , proving Theorems 4 and 5. A convergence rate for w -W ∞ 1 , some effects of initial conditions on the convergence rates, and convergence rates in non-rescaled variables are dealt with in Section 4. Finally, we show the results of some simulations in the eponymous section.

Entropic structure

Even if we are mainly estimating L 1 -norms, we see our proof as a specific case of relative entropy inequalities. Rates could be obtained following the lines of our proofs for other entropies.

L 1 contraction for compactly supported solutions

The first evidence of an attractor is the L 1 contraction of compactly supported solutions. If we consider two initial data supported in [0, 1), w 0 1 , w 0 2 , non-negative and of mass 1 and the associated solutions w 1 , w 2 , then we have the following property (we take β(a) = µ 1+a for this computation)

d dτ 1 0 |w 1 (τ, b) -w 2 (τ, b)|db ≤ D(τ )
where

D(τ ) = 1 0 µ e -τ + b (w 1 (τ, b) -w 2 (τ, b))db - 1 0 µ e -τ + b |w 1 (τ, b) -w 2 (τ, b)| db.
Since mass is conserved i.e., 1 0 w 1 -w 2 = 0, we have easily

D(τ ) = 1 0 µ e -τ + b - µ e -τ + 1 (w 1 (τ, b) -w 2 (τ, b))db - 1 0 µ e -τ + b |w 1 (τ, b) -w 2 (τ, b)| db ≤ 1 0 µ e -τ + b - µ e -τ + 1 |w 1 (τ, b) -w 2 (τ, b)|db - 1 0 µ e -τ + b |w 1 (τ, b) -w 2 (τ, b)| db.
Thereby, we obtain

d dτ 1 0 |w 1 (τ, b) -w 2 (τ, b)|db ≤ - µ 1 + e -τ 1 0 |w 1 (τ, b) -w 2 (τ, b)|db.

And this leads to

1 0 |w 1 (τ, b) -w 2 (τ, b)|db = O(e -µτ ).
In the next section we identify the attractor towards which solutions converge.

Pseudo equilibrium

We start by recalling the definition of what we call the pseudo equilibrium. We recall Definition 1:

W (τ, b) = C(τ ) (1 -b) 1-µ e B(e τ b)
where B(a) = a 0 β(s)ds and C is defined so that W (τ,

•) L 1 = 1. Remark. By definition, W (τ, b = 0) = C(τ ). (9)
Firstly we establish the fact that W (τ, •) is an approximation of W ∞ Lemma 7. Assume hypothesis (H1). Then, defining the Dynkin-Lamperti distribution as in (5):

W ∞ (b) = b -µ (1 -b) µ-1 1 0 b -µ (1 -b) µ-1 db we have lim τ =+∞ W (τ, •) -W ∞ 1 = 0.
Proof. We start with the model case β(a) = µ 1+a . In this case, we can write

W (τ, b) = (e -τ + b) -µ (1 -b) µ-1 1 0 (e -τ + b) -µ (1 -b) µ-1 db
, and the result is immediate.

For the general case, we use the following useful bound on W :

Lemma 8. Under hypothesis (H1), for any η > 0 satisfying η < min(µ, 1 -µ), there exists a constant (depending on β, but not on τ ) C η > 0 such that

W (τ, b) ≤ C η (1 -b) µ-1 (e -τ + b) -(µ+η) 1 b (1 -b ) µ-1 (e -τ + b ) -(µ+η) db
Proof. We first notice that there always exists a function g η ≥ 0, compactly supported, such that

β(a) ≤ µ + η 1 + a + g η (a).
Thereby, for all b ≥ b and all τ ≥ 0, we have

B(e τ b) -B(e τ b ) = - e τ b e τ b β(s)ds ≥ (µ + η) ln 1 + e τ b 1 + e τ b -g η 1 ,
and e B(e τ b)-B(e τ b ) ≥ e -gη 1 (e

-τ + b ) -(µ+η) (e -τ + b) µ+η (10)
Then we recall that by definition

W (τ, b) = e -B(e τ b (1 -b) µ-1 1 0 e -B(e τ b ) (1 -b ) µ-1 db ≤ (1 -b) µ-1 1 b e B(e τ b)-B(e τ b ) (1 -b ) µ-1 db .
Inserting [START_REF] Melbourne | Operator renewal theory and mixing rates for dynamical systems with infinite measure[END_REF] in the latter, we obtain the result with

C η = e gη 1 .
It is worth noticing that we can establish with the same proof

∀ε ≤ 1/2, ε 0 W ≤ C η ε (1-(µ+η))
We denote W ref for β(a) = µ 1+a and notice that in our general case

β(a) = µ 1 + a + g(a).
We denote G(a) = a 0 g and we can write

W (τ, b) = K(τ )e G(e τ )-G(e τ b) W ref (τ, b).
for some K(τ ) > 0 that insures the normalisation

1 0 W = 1.
We introduce some η > 0 as in lemma 8. We already establish in the proof of lemma 10 that for ε ≤ 1/2,

ε 0 W ≤ C η ε 1-(µ+η) . Therefore, 1 ≥ K(τ ) 1 ε e G(e τ )-G(e τ b) W ref (τ, b)db ≥ 1 -C η ε 1-(µ+η) .
Let ε > 0 be fixed. In particular, for ε > 0 fixed, e G(e τ )-G(e τ b) → 1, uniformly on (ε, 1) as τ → +∞. As a consequence, we have for all ε > 0

1 ε e G(e τ )-G(e τ b) W ref (τ, b)db - 1 ε W ref (τ, b)db → 0.
This leads, for any ε > 0, to the bounds:

lim sup +∞ K(τ ) ≤ 1 1 ε W ∞ , lim inf +∞ K(τ ) ≤ 1 -C η ε 1-(µ+η) 1 ε W ∞ . Letting ε → 0, we obtain lim +∞ K(τ ) = 1.
What we established proves that, for any ε > 0,

1 ε |W -W ref | → 0.
We can conclude using lemma 8 that we have

1 0 |W -W ref | → 0. Consequently, lim +∞ W -W ∞ 1 = 0.
This ends the proof of lemma 7.

The main property of the pseudo equilibrium is the following Proposition 9. W satisfies the following system :

∂ τ W (τ, b) + ∂ b ((1 -b)W (τ, b)) + e τ β(e τ b)W (τ, b) = W (τ, b)C(τ )δ(τ ) W (τ, 0) (1 + δ(τ )) = 1 0 e τ β(e τ b)W (τ, b)db, (11)
where δ(τ ) is defined by the equation

δ(τ ) = C (τ ) (C(τ )) 2 - µ C(τ ) . ( 12
)
Proof. By computing the partial derivatives of W with respect to b and to τ , we obtain:

∂ b ((1 -b)W (τ, b)) = -W (τ, b) [µ + (1 -b)e τ β(e τ b)]
and:

∂ τ W (τ, b) = W (τ, b) C (τ ) C(τ ) -be τ β(e τ b) .
Therefore, W satisfies :

∂ τ W (τ, b) + ∂ b ((1 -b)W (τ, b)) + e τ β(e τ b)W (τ, b) = W (τ, b) C (τ ) C(τ ) -µ .
If we take into account that ∀τ ≥ 0 W (τ, •) L 1 = 1, by integrating the previous equation over b ∈ [0, 1], we obtain the value of W (τ, 0), hence the claimed system.

The next results justify that ( 11) is close to (4). Lemma 10. Under hypothesis (H1), we have

lim τ →+∞ C(τ )δ(τ ) = 0 where C(τ )δ(τ ) = C (τ ) C(τ ) -µ = 1 0 [be τ β(e τ b) -µ] W (τ, b)db. ( 13
)
Proof. We recall first, by definition

|C(τ )δ(τ )| ≤ 1 0 |e τ bβ(e τ b) -µ|W (τ, b)db.
We can split then the integral into two parts

|C(τ )δ(τ )| ≤ e -τ /2 0 |e τ bβ(e τ b) -µ|W (τ, b)db I1(τ ) + 1 e -τ /2 |e τ bβ(e τ b) -µ|W (τ, b)db I2(τ )
.

Firstly, we have

I 2 (τ ) ≤ sup (e τ /2 ,+∞) |aβ(a) -µ| → 0, τ → +∞.
To estimate I 1 (τ ) we notice

I 1 (τ ) ≤ sup R+ |aβ(a) -µ| e -τ /2 0 W ≤ sup R+ |aβ(a) -µ| W -W ∞ 1 + e -τ /2 0 W ∞ .
We already know from lemma 7 W -W ∞ 1 → 0. Furthermore, for large τ we have

0 ≤ e -τ /2 0 W ∞ ≤ c ∞ (1 -e -τ /2 ) 1-µ e -τ (1-µ)/2 1 -µ → 0.
Therefore, we have lim +∞ I 1 = 0, which concludes the proof of the lemma.

Dissipation of entropy with respect to W

We now introduce the most important tool we will use: the relative entropy (similar to the entropy rate of a stochastic process, or the general relative entropy used in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Perthame | Transport Equations in Biology[END_REF]).

Definition 11. Let w be a solution of the equation ( 4) with support included in [0, 1). Let H be a convex, continuous function, C 1 by parts, which reaches its minimum, 0, at 1. We define the generalised relative entropy as:

H(τ ) = 1 0 H w(τ, b) W (τ, b) W (τ, b)db. ( 14
)
And for a non-negative measure ν on [0, 1) the entropy dissipation DH(u|ν) is defined by

DH(u|ν) = 1 0 H(u(b))dν(b) -H 1 0 u(b)dν(b) .
Note that DH(u|ν) ≥ 0 if ν is a probability (by Jensen's inequality).

We are now in position to establish a first important inequality on the relative entropy Proposition 12. Under (H1), the entropy H satisfies the following equality:

H (τ ) = -C(τ )DH(u|dγ τ ) + C(τ )δ(τ ) 1 0 (H(u) -uH (u)) W (τ, b)db (15)
where dγ τ (b) = e τ β(e τ b)W (τ,b)

W (τ,0)
is a non-negative measure of mass (1 + δ(τ )) and u = w/W .

Proof. The mass of dγ is immediately derived from the equation on W (τ, 0). We have then:

∂ τ w + ∂ b ((1 -b)w) + e τ β(e τ b)w = 0 ∂ τ W + ∂ b ((1 -b)W ) + e τ β(e τ b)W = W (τ, b)C(τ )δ(τ ).
Denoting u = w/W we arrive at

∂ τ u + (1 -b)∂ b u = -Cδu.
We multiply this equation by H (u) and get Remark. C(τ )δ(τ ) appears naturally as a remainder we will have to estimate in order to prove convergence-related properties for H, and also for W -W ∞ 1 .

∂ τ (H(u)) + (1 -b)∂ b (H(u)) = -CδuH (u).    W ∂ t (H(u)) + (1 -b)W ∂ b (H(u)) = -W CδuH (u) H∂ τ W + (1 -b)H∂ b W + (e τ β(e τ b) -1) HW = CδHW ∂ τ (H(u)W ) + ∂ b ((1 -b)H(u)W ) + e τ β(e τ b)H(u)W = W Cδ [H(u) -uH (u)]

L 1 convergence (without a rate) to W

In this section we prove proposition 3. We take therefore H(x) = |x -1|, we have then,

DH(u|dγ τ ) = 1 0 |u -1|dγ τ - 1 0 udγ τ -1 .
The core of the proof is the following Lemma 13. Under hypothesis (H1), we have

d dτ 1 0 |w -W |db ≤ -e τ β(e τ ) 1 0 |w -W |db + 2|C(τ )δ(τ )|. ( 16
)
Proof. By definition,

dγ τ = e τ β(e τ b)W (τ, b) W (τ, 0) ≥ e τ β(e τ ) C(τ ) W (τ, b) = K(τ )W (τ, b).
Furthermore, since

1 0 (u -1)W = 1 0 w -W = 0, we have DH(u|dγ τ ) = 1 0 |u -1|dγ τ - 1 0 udγ τ -1 = 1 0 |u -1|dγ τ - 1 0 (u -1)dγ τ -δ = 1 0 |u -1|dγ τ - 1 0 (u -1) (dγ τ -K(τ )W ) ≥0 -δ = K(τ ) 1 0 |u -1|W + 1 0 |u -1|(dγ τ -K(τ )W ) - 1 0 (u -1)(dγ τ -K(τ )W ) -|δ| ≥ K(τ ) 1 0 |u -1|W -|δ(τ )|.
Since we also have, for

H(x) = |x -1|, |H(u) -uH (u)| = | -sign(u -1)| ≤ 1,
we obtain for this case

1 0 (H(u) -uH (u))W ≤ 1.
And since C(τ )K(τ ) = e τ β(e τ ), we obtain equation [START_REF] Perthame | Transport Equations in Biology[END_REF].

Since we already have by hypothesis e τ β(e τ ) → µ > 0, standard ODE arguments yield lim sup

+∞ 1 0 |w -W |db ≤ 2 lim sup +∞ |C(τ )δ| µ .
We can conclude the proof of proposition 3 using lemma 10.

Remark. We let the reader check that, by defining β as in ( 6), we may replace the non-increasing β hypothesis by aβ(a) ---→ a→∞ µ, obtaining the following equation instead of ( 16):

d dτ 1 0 |w -W |db ≤ -e τ β(e τ ) 1 0 |w -W |db + 2|C(τ )δ(τ )|. ( 17
)
Remark. We have now finished developing a framework which allows us to deduce the behaviour of the entropy H from suitable hypotheses made on β, and showed that, under mild conditions, the entropy tends to 0. The following section will extract a convergence rate from more restrictive hypotheses.

3 Rates of convergence to the pseudo equilibrium We consider it best to start by presenting this simple case, since the following proofs contain the key innovative elements of the general case while simplifying the presentation of our results.

Here, the pseudo equilibrium W becomes, with c(τ ) = e -µτ C(τ ):

W (τ, b) = C(τ ) (1 -b) 1-µ (1 + e τ b) µ = c(τ ) (1 -b) 1-µ (e -τ + b) µ .
We can now compute δ as follows.

Lemma 14.

δ(τ ) = - e -τ 1 + e -τ . (18)
Proof.

C(τ )δ(τ ) = C (τ ) C(τ ) -µ = c (τ ) c(τ ) = 1 0 [be τ β(e τ b) -µ] W (τ, b)db = 1 0 µ e -τ +b [b -(e -τ + b)] W (τ, b)db = -e -τ 1
0 e τ β(e τ b)W (τ, b)db. By applying Proposition 9, we obtain:

C(τ )δ(τ ) = -e -τ (1 + δ(τ ))W (τ, 0) = -e -τ (1 + δ(τ ))C(τ )
resulting in the claimed equality since C doesn't vanish. Definition 15. We call c ∞ the limit at ∞ of c(τ ) when such limit exists. Here, it is easy to see c is a decreasing function and c ∞ is well defined. Lemma 16.

-4e

-(1-µ)τ ≤ C(τ )δ(τ ) ≤ - c ∞ 1 + e -τ e -(1-µ)τ ≤ 0. ( 19
)
Proof. We have c(τ ) = e -µτ C(τ ), thus C (τ ) C(τ ) -µ = c (τ ) c(τ ) , and we establish two trivial bounds on c(τ ). By definition:

1 c(τ ) = 1 0 1 (e -τ + b) µ (1 -b) 1-µ db
hence, since τ ≥ 0 and 0 ≤ µ ≤ 1, we have:

1 c(τ ) ≥ 1 0 (1 + b) -µ (1 -b) µ-1 db ≥ 1 0 1 -b 1 + b µ db ≥ 1 0 1 2 (1 -b) µ db = 1 2(1 + µ) ≥ 1 4 ;
and likewise:

1 c(τ ) ≤ 1 0 b -µ (1 -b) µ-1 db = 1 c ∞ . It follows that: 0 ≤ c ∞ ≤ c(τ ) ≤ 2(1 + µ) ≤ 4. ( 20
)
The result of Lemma 14 allows us to conclude.

Putting together this lemma and equation [START_REF] Perthame | Transport Equations in Biology[END_REF] gives us Corollary 17. The following inequality holds

H (τ ) ≤ -µ - µ 1 + e τ H(τ ) + 8e -(1-µ)τ . ( 21
)
All is ready to prove Theorem 4 by applying Gronwall's Lemma to the previous inequality.

Proof of Theorem 4. We set f (s) = -µ 1+e s , which gives τ 0 f (s)ds = -µ ln 1+e -τ

2

. Corollary 17 implies that:

d dτ exp τ 0 (µ + f (s))ds H(τ ) ≤ 8e -(1-µ)τ exp τ 0 (µ + f (s))ds .
By integrating over τ , we obtain:

H(τ ) ≤ H(0)e -τ 0 µ+f + 8 τ 0 e -(1-µ)τ e τ 0 µ+f e -τ 0 µ+f dτ ≤ e -µτ H(0)e -τ 0 f + 8e -µτ τ 0 e (2µ-1)τ e - τ f dτ ≤ e -µτ H(0) 2 1+e -τ µ + 8e -µτ τ 0 e (2µ-1)τ 1+e -τ 1+e τ µ dτ ≤ e -µτ 2 1+e -τ µ H(0) + 8
τ 0 e (2µ-1)τ dτ .

A larger class of β

We now consider β satisfying (H2).

Remark. The bound on G is not necessary to prove lemmas 18 and 19: g ∈ L 1 is a strong enough hypothesis. However, that precise bound in necessary for our convergence rate estimates and as such, we assume it holds throughout the section.

We take the following notations:

G(a) = ∞ a |g(s)|ds W ref (τ, b) = C ref (τ ) (1-b) 1-µ (1+e τ b) µ W (τ, b) = C(τ ) exp -e τ b 0 g(s)ds (1-b) 1-µ (1+e τ b) µ where C ref and C ensure W ref (τ ) 1 = W (τ ) = 1.
(We use the same notation as in the proof of lemma 7).

We have:

C(τ )δ(τ ) = 1 0 [e τ bβ(e τ b) -µ] W (τ, b)db = 1 0 e τ bg(e τ b)W db + µ 1 0 e τ b 1+e τ b -1 W (τ, b)db.
Lemma 18. Assume (H2) holds. Then there exists K > 0 such that for any τ ≥ 0 and 0 ≤ b ≤ 1:

K -1 e µτ (1 + e τ b) µ (1 -b) 1-µ ≤ W (τ, b) ≤ Ke µτ (1 + e τ b) µ (1 -b) 1-µ . ( 22
)
Proof. We have:

W (τ, b) W ref (τ, b) = C(τ ) C ref (τ ) exp - e τ b 0 g(s)ds ∈ C(τ ) C ref (τ ) e -g 1 , e g 1 .
And since

1 0 W (τ, b)db = 1 0 W ref (τ, b)db = 1, it follows that: e -g 1 ≤ C(τ ) C ref (τ ) ≤ e g 1 which gives us W W ref ∈ L ∞ in the sense given above.
This result leads, through a proof analogous to that of lemma 16, to the following Lemma 19. Under hypothesis (H2), there exists a positive M such that:

µ 1 0 e τ b 1 + e τ b -1 W (τ, b)db ≤ M e -(1-µ)τ . ( 23
)
We now give the strategy for estimating the rate of convergence. It is based on the same procedure as before. Consider a non-increasing β and equation ( 16), which measures the dissipation of entropy (or equation [START_REF] Terhesiu | Error rates in the Darling-Kac law[END_REF] under the corresponding hypothesis).

H (τ ) ≤ -e τ β(e τ )H(τ ) + 2|C(τ )δ(τ )|.
We have then

He B(e τ ) (τ ) ≤ 2|C(τ )δ(τ )|e B(e τ ) . (24)
We recall

C(τ )δ(τ ) = 1 0 (e τ bβ(e τ b) -µ)W (τ, b)db.
And also by the definition of W and lemma 18

W (τ, b) ≤ K (e -τ + b) µ (1 -b) 1-µ .
Therefore, we easily obtain

|C(τ )δ(τ )| ≤ K 1 0 e τ b|g|(e τ b) (e -τ + b) µ (1 -b) 1-µ db + µ 1 0 e τ b 1 + e τ b -1 W (τ, b)db .
Lemma 19 then gives us

|C(τ )δ(τ )| ≤ K 1 0 e τ b|g|(e τ b) (e -τ + b) µ (1 -b) 1-µ db + M e (µ-1)τ .
The constant K may change value from line to line.

We integrate equation ( 24) and get

H(τ )e B(e τ ) ≤ H(0) + K τ 0 e B(e τ ) 1 0 e τ b|g|(e τ b) (e -τ + b) µ (1 -b) 1-µ dbdτ + K τ 0 e (µ-1)τ +B(e τ ) dτ .
Using the fact the B(e τ ) -µτ is bounded from above and below, we can replace B(e τ ) by µτ with just a change of constants.

H(τ )e µτ ≤ K 1 + τ 0 1 0 e µτ e τ b|g|(e τ b) (e -τ + b) µ (1 -b) 1-µ dbdτ + τ 0 e (2µ-1)τ dτ . ( 25
)
Note that for α > (1 -µ) the rate is the same than the one for g = 0.

Proof. We need an estimate of

1 0 τ 0 1 (1 + e τ b) α e µτ (e -τ + b) µ (1 -b) 1-µ dτ db = 1 0 τ 0 e (µ-α)τ (e -τ + b) µ+α (1 -b) 1-µ dτ db
Putting all together with (25), we have

H(τ ) ≤ K e -µτ + e -µτ 1 0 τ 0 e (µ-α)τ (e -τ + b) µ+α (1 -b) 1-µ dτ db + e -µτ τ 0 e (2µ-1)τ dτ .
We essentially need to estimate the middle term.

Lemma 21. Under (H2), the following holds true:

1 0 τ 0 µ e (µ-α)τ (e -τ + b) µ+α (1 -b) 1-µ dτ db ≤              K τ 0 e (µ-α)τ dτ , if α < 1 -µ, K τ 0 e (2µ-1)τ dτ , if α > 1 -µ, K τ 0 (1 + τ )e (2µ-1)τ dτ , if α = 1 -µ.
Proof of the lemma. Case 1: α < 1 -µ.

For α < 1 -µ, we simply use the fact that

1 0 1 (e -τ + b) µ+α (1 -b) 1-µ db ≤ 1 0 1 b µ+α (1 -b) 1-µ db < +∞ Case 2: α > 1 -µ.
It is obvious that we can restrict to the case α < 1. We first need a few intermediate computations. Firstly, for γ > 0, we have

1 0 γ (e -τ + b) γ+1 (1 -b) 1-γ = 1 1 + e -τ 1 0 - d db f (b)e f (b) db.
Where

f (b) = -γ ln(e -τ + b) + γ ln(1 -b).
Therefore, we have

1 0 γ (e τ + b) γ+1 (1 -b) 1-γ = 1 1 + e -τ - (1 -b) e -τ + b γ 1 0 = e γτ 1 + e -τ . ( 26
)
Using this computation and noticing γ = µ + α -1, we can easily establish that, for α < 1,

1 0 1 (e -τ + b) µ+α (1 -b) 1-µ db = 1 0 (1 -b) 1-α (e -τ + b) γ+1 (1 -b) 1-γ db.
Applying then (26) and using (1 -b) 1-α ≤ 1, we arrive at

1 0 1 (e -τ + b) µ+α (1 -b) 1-µ db ≤ Ke γτ .
Injecting, we obtain

I α (τ ) ≤ K τ 0 e (γ+µ-α)τ dτ = K τ 0 e (2µ-1)τ dτ . Case 3: α = 1 -µ.
In this case

I α (τ ) = τ 0 1 0 e (2µ-1)τ (e -τ + b)(1 -b) 1-µ dbdτ ,
Cutting the integral on b at 1/2 for instance, it is easy to establish

I α (τ ) ≤ K τ 0 1 0 e (2µ-1)τ 1 + 1 (e-τ + b) dbdτ ≤ K τ 0 e (2µ-1)τ 1 + log(1 + e τ ) dτ ,
thereby, we have

I α (τ ) ≤ K τ 0 (2 + τ )e (2µ-1)τ dτ .
This ends the proof of lemma 21.

To end the proof of proposition 20, we essentially just need to discuss whether the integrals of type τ 0 e λτ dτ take value e λτ -1 λ or τ , and similarly for integrals of type τ 0 τ e λτ dτ .

Rates of convergence towards the equilibrium W

∞ . 4.1 Quantitative estimate of W -W ∞ 1
In what follows, we justify how the rate of convergence of w to W can be extended to quantify (up to a multiplicative constant) the rate of convergence towards W ∞ . The main remark is the following.

Lemma 22. Under hypothesis (H2), we have

W -W ∞ 1 ≤ 2 1 0 |e τ bβ(e τ b) -µ|W (τ, b)db.
Proof. We already know from lemma 7 lim +∞ W (τ, •) -

W ∞ 1 = 0. Therefore W (τ, •) -W ∞ 1 ≤ ∞ τ d dτ 1 0 |W (τ , b) -W ∞ (b)|db dτ ≤ ∞ τ 1 0 |∂ τ W (τ , b)|dbdτ .

Since we have

∂ τ W = C (τ ) C(τ ) W -e τ bβ(e τ b)W = C (τ ) C(τ ) -µ W + (µ -e τ bβ(e τ b))W, it follows that 1 0 |∂ τ W | ≤ C (τ ) C(τ ) -µ + 1 0 |e τ bβ(e τ b) -µ|W.
And since

C (τ ) C(τ ) -µ ≤ 1 0 |e τ bβ(e τ b) -µ|W,
we can conclude.

Remark. The bound on G has not been used in the proof of the previous lemma, for which g ∈ L 1 is a strong enough hypothesis.

We encounter yet again the quantity 1 0 |e τ bβ(e τ b) -µ|W , for which we have already given a time-weighted average estimate in the reasoning following equation (25). Let us now provide a pointwise estimate. In the reference case, we have

W (τ, •) -W ∞ 1 ≤ K ∞ τ e (µ-1)τ 1 + e -τ dτ ≤ Ke (µ-1)τ
In the situation described by proposition 20, with β = µ 1+a + g(a) and for some α > 0, we have,

∞ a |g| ≤ K (1 + a) α .
In this case, we can split 1 0 |e τ bβ(e τ b) -µ|W into two parts and use the previous arguments of the proof of lemma 19 to claim

1 0 |e τ bβ(e τ b) -µ| W ≤ ∞ τ 1 0 µb e -τ + b -µ 1 (e -τ + b) µ (1 -b) 1-µ dbdτ +K ∞ τ 1 0 |e τ bg(e τ b)| 1 (e -τ + b) µ (1 -b) 1-µ dbdτ
The first term is already know to be bounded by Ke (µ-1)τ by lemma 16 . The second term satisfies

∞ τ 1 0 |e τ bg(e τ b)| 1 (e -τ + b) µ (1 -b) 1-µ dbdτ ≤ ∞ τ 1 0 |e τ bg(e τ b)| 1 b µ (1 -b) 1-µ dbdτ ≤ 1 0 1 b µ (1 -b) 1-µ ∞ e τ b |g| db ≤ 1 0 1 b µ (1 -b) 1-µ 1 (1 + e τ b) α db ≤ Ke -ατ
Hence the rate of convergence W (τ, •) -W ∞ 1 ≤ K e (µ-1)τ + e -ατ .

Possible influence of the initial condition

Let us prove a lower bound on the convergence rate of ln w(τ, •) -W (τ, •) L 1 (0,1) for an initial age distribution w(0, b) = δ 0 (b).

Proposition 23. Consider the reference case

β(a) = µ 1 + a .
Suppose the initial age distribution satisfies:

w 0 (b) = δ 0 (b).
We can bound below the total variation:

W (τ, b) -w(τ, b) T V ≥ e -µτ . (27)
Proof. For φ(τ ) = 1 -e -τ ξ(τ ) = exp τ 0 e s β(e s φ(s))ds we have:

d dτ [ξ(τ )w(τ, φ(τ )] = ξ ξ ξ w + ∂ τ w + φ ∂ b w = ξ [∂ τ w + ∂ b ((1 -φ)w) + e τ β(e τ φ)w] = 0. It follows that w(τ, φ(τ )) = exp - τ 0 e s β(e s φ(s))ds w(0, φ(0))
which, after injecting the corresponding values, yields w(τ, 1 -e -τ ) = e -µτ δ with δ a Dirac mass. Therefore :

w(τ, •) = e -µτ δ 1-e -τ + w j
where w j ≥ 0 is the distribution of particles that have jumped at least once over (0, τ ].

Since W (τ, b)db is an atomless measure, any Dirac mass and W db are stranger measures, hence:

W (τ, b) -w(τ, b) T V ≥ e -µτ .
For µ < 1 2 , this lower bound agrees up to multiplication by a constant with the upper bound given in theorem 4: our convergence exponent is optimal for µ < 1 2 . Remark. It is worth noting that we have the trivial bound:

H(0) ≤ 2. ( 28
)
(H(0) can be greater than 1 if w 0 (b)db has atoms.)

Remark. Our results are proved for compactly-supported initial age distributions, and they will most likely hold for initial age distributions that decrease fast enough. However, if this is not the case, the convergence rates might be affected in a way left for future investigation. Definition 24. We set:

Convergence rates for natural variables

N (t, a) = e -τ W (τ, b) = c(ln(1 + t)) (1 + t) 2 (1 + a) µ (1 + t -a) 1-µ (29)
which leads to the following Proposition. In order to illustrate graphically the behaviour of the solution to our equations and its convergence towards the pseudo-equilibrium, Figure 2 displays, for µ = 0, 6 and an initial age distribution n 0 = w 0 = δ 0 , the time evolution of the simulation results expressed either in the original variables n(t, •) (histograms), N (t, •) (full line) on the left-hand side column or in the rescaled variables w(τ, •) (histograms), W (τ, •) (full line) on the right-hand side column. Moreover, the rescaled variables panels also show as grey dotted lines the equilibrium W ∞ , to which W converges. From visual inspection of the is figure, it is clear that n(t, a) largely flattens as t → ∞ (note the difference in the y-axis scale between the panels). The figure depicts a pointwise convergence of the simulated w to the pseudo-equilibrium W which in turn converges pointwise to W ∞ . Moreover, it illustrates how rescaling allows a better description of the self-similar behaviour, which is difficult to grasp in natural variables since n converges pointwise to 0. The next sections quantify the simulated convergence rates.

Exponential fit of w(τ, •) -W (τ, •) 1

To quantify the convergence rates in the simulations, we fit the distance w(τ, •) -W (τ, •) by the following function:

f (λ, A, B) = Ae -λτ + Be -(1-λ)τ + C. (31)
Remark (Heuristic estimate of the error term). C is a simulation error, that we evaluate to C ≈ 0, 1. This is consistent both with empirical evidence and with a simple heuristic overevaluation of C as #bins/#particles, which is roughly 0, 16.

Remark. According to the above analysis one expects λ = µ. A and B are multiplicative parameters: we expect A around 2 and |B| close to 0, since H(0) = 2 and our upper boundary is of the form H(τ ) ≤ [H(0) -(≤ 0)] e -µτ + ke -(1-µ)τ , with k small.

Remark. Another possible explanation of the predominance of e -µτ over e -(1-µ)τ in the convergence rate is linked to the fact that, for a given β, two solutions w 1 and w 2 corresponding to different, compactly supported initial conditions, satisfy, for a certain constant K (see Subsection 2.1):

w(τ, •) -w 1 (τ, •) 1 ≤ Ke -µτ .
Figure 3 presents as examples three cases that exhibit a certain diversity : µ = 0, 9, µ = 0, 5 and µ = 0, 2. We plot in red dots the evolution along τ of the simulated value of w(τ, •) -W (τ, •) 1 and use function f defined in equation (31) to fit the results (blue curves). The fit results are given in the figures, ± one standard deviation.

We first note that in the three panels of figure 3, C ≈ 0.1 expected and our estimates for λ are very close to µ. Note that in the second panel, with µ = 0.5, the values of A and B cannot be estimated independently thus the large inaccuracy/variance on their determination. Finally, the third panel shows a marked discontinuity around τ = 6. This is due to the discretisation of the age distribution: with small values of µ, the number of random walkers that have never experienced a single renewal during the simulation period becomes large. Since, according to our initial conditions, all walkers have the same initial age, many walkers will enter the last age bin simultaneously thus causing the observed discontinuity. However even in this case, we obtain a very good fit for λ by restricting the fit to the values before the discontinuity and fixing C to 0.8.

Figure 4 summarizes the values of λ determined from Monte-Carlo simulations identical to those shown in Fig. 3 (red crosses), together with the diagonal line λ = µ (blue). For all the values of µ tested, the simulations confirm that w tends to W with a sum of exponential rates given by µ and 1-µ. Therefore, taken together, those simulation results, while agreeing with our analytical estimations, suggest that our estimate of w -W 1 may not be optimal, in particular for larger values of µ. This figure shows that for for µ ≤ 0.5, W (τ, •) and W ∞ are systematically much closer to each other than to w. However, as µ increases, this trend reverses: for large enough τ , w becomes significantly closer to W than to W ∞ : the distance between w and W converges much faster. Therefore, according to those simulation results W is a much better asymptotic approximation w for µ > 0.5, thus justifying further its utility here.

Future developments

Throughout the article we have estimated L 1 norms, but we have presented the estimates in the context of an entropic structure. It is indeed possible by means analogous to ours to prove entropy inequalities for dissipations corresponding to other H functions than |1-•|. For instance, the classical H(x) = x ln xx + 1 also allows us to prove a convergence rate of the corresponding entropy to 0: it is also K(e -µτ + e (µ-1)τ ). Thanks to the Csiszár-Kullback inequality, it is also possible to prove a rate of convergence of w(τ, •) -W (τ, •) 1 to 0, albeit one worse than that obtained in theorems 4 and 5.

We may encounter inequalities such as that of proposition 12, bounding the derivative of an entropy with respect to a probability measure W db by an entropy dissipation with respect to another measure (which we can compare to the dissipation with respect to a probability measure dγ τ ). When the comparison of DH(u|W db) and DH(u|dγ) does not follow calculations as straightforward as ours, an alternative may be to rely on a precise Jensen estimate comparing the entropy dissipations with respect to two absolutely continuous probability measures.

Here, we have considered a spatially-homogeneous (zero-dimensional), age-dependent renewal probability β(a). We believe the ideas we have exposed may be used to tackle the problem with a spatial extension, for instance in a discrete space setting. One major interest of our age-structure approach of CTRW is that the dynamics remain Markovian. We believe that keeping Markovian properties will be crucially helpful when introducing the coupling between sub-diffusive CTRW and reaction, since the coupling should simply consist in the addition of the reaction and the subdiffusion terms (contrarily to the case of fractional dynamics). However, the extent to which the supplementary age variable will make this process more complex remains to be evaluated.

Appendix

The case µ = 1

It is quite interesting to notice that even if the behaviour is not really self similar, our method gives a precise asymptotic for the case µ = 1. To illustrate this, we focus on the reference case: β(a) = 1 1+a . In this case the 'pseudo equilibrium reads'

W (τ, b) = 1 (e -τ + b) log(1 + e τ )
.

This pseudo equilibrium tends to a Dirac mass but gives still quantitative information. Indeed, following the same computation than for equation ( 16) for the case µ < 1, we obtain easily

  e τ b)w = 0 w(τ, 0) = ∞ 0 e τ β(e τ b)w(τ, b)db w(0, b) = w 0 (b).

  Furthermore, since g(a) = o( 1 a ), we have sup b≥ε |G(e τ ) -G(e τ ε)| = o e τ e τ ε da a = o (ln ε) = o(1).

η.dτ 1 0H 1 0e

 11 Taking the integral over b of the previous expression yields: d (u)W db -W (τ, 0)H(u(τ, 0)) + τ β(e τ b)H(u)W db = definition of the entropy dissipation, proves the proposition.

3. 1

 1 The key situation: β(a) = µ 1+a

Figure 1 :

 1 Figure 1: w(τ, •) -W (τ, •) 1 lies under the theoretical bound (black curve) for an initial age distribution δ 0 (red dots) and U(0, 1) (blue dots).

Figure 2 :

 2 Figure 2: Evolution of n, N , w, W and W ∞ along time, for µ = 0, 6 and an initial age distribution n 0 = w 0 = δ 0 .

λλ

  = 0.899 ± 0.007 A = 1.795 ± 0.007 B = 0.050 ± 0.008 C = 0.098 ± 0.004 = 0.523 ± 0.033 A = 1.895 ± 1.240 B = -7.560 × 10 -5 ± 1.240 C = 0.110 ± 0.001.

Figure 3 :

 3 Figure 3: Fit by f defined in equation (31) (blue curves), for different µ, of the simulated w(τ, •) -W (τ, •) 1 (red dots). Initial age distribution: w 0 = δ 0 .

Figure 4 :

 4 Figure 4: Values of the exponent λ found by using function f from equation (31) to fit the simulated values of w(τ, •) -W (τ,•) 1 for µ ∈ {0, 1 ; 0, 2 ; . . . ; 0, 9}.

5. 2 Figure 5

 25 Figure5compares the distances between w and W (red dots), w and W ∞ (black dots), or W and W ∞ (green curve), for three values of µ. This figure shows that for for µ ≤ 0.5, W (τ, •) and W ∞ are systematically much closer to each other than to w. However, as µ increases, this trend reverses: for large enough τ , w becomes significantly closer to W than to W ∞ : the distance between w and W converges much faster. Therefore, according to those simulation results W is a much better asymptotic approximation w for µ > 0.5, thus justifying further its utility here.

Figure 5 :

 5 Figure 5: Influence of µ on ln w(τ, •) -W (τ, •) 1 (red dots) and ln w -W ∞ (black dots): for higher values of µ, w is significantly closer to W than to W ∞ .

1 0 e τ b 1 + e τ b - 1 W 1 0

 1111 | + 2 |C(τ )δ(τ )| .Where we also haveC(τ )δ(τ ) = (τ, b)db. )δ(τ ) = -e τ (1 + e τ ) log(1 + e τ )→ 0.And we can still claim that |w -W | → 0. We can give a (rough) estimate for a rate of convergence. Integrating, we have This term behaves as τ . Indeed, we have easily (splitting the integral at e ατ for α < 1

Acknowledgements

This work was initiated within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

We wish to thank Sergei Fedotov for many valuable discussions. This work could not have been written without the help of Vincent Calvez.

Remark. We let the reader check that the non-increasing β hypothesis may be replaced by the following condition on the function defined in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF]: aβ(a) ---→ a→∞ µ. Replacing the use of B by that of B(a) = a 0 β, we still obtain equation (25) up to multiplication by a constant, since B(e τ ) -µτ is also bounded from above and below. Now the work is focused on the estimate of the middle quantity

We integrate by parts with respect to τ and recall that d dτ G(e τ b) = -e τ b|g(e τ b)|. We have

The first term is bounded from above (since G ≥ 0) by

Finally, since e -τ e -τ +b ≤ 1,

We need a sharp estimate on the second term. We focus our efforts on the case

Proposition 20. Assume hypothesis (H2).

Proof. The e -τ appearing as the Jacobian of the change of integration variables is compensated by the e -τ in the definition of w and we get the claimed result.

Therefore, in the reference case β(a) = µ 1+a , the distribution of walkers that have age a at time t converges to N (t, a) algebraically fast, with a rate that is essentially given by t -min{µ,1-µ} : we recover corollary 6.

Monte Carlo simulations

In order to illustrate the evolution of the age distribution of the system and check the accuracy of the convergence rates to self-similar equilibrium, we have carried out Monte-Carlo simulations for our reference case β(a) = µ 1+a . In these simulations, we describe explicitly each individual walker i by associating it with an age a i and a first jumping time τ i . The initial age of each walker is chosen according to some initial distribution, for instance uniform distribution in [0, 1] or a Dirac distribution at age a = 0. The first jumping time of each random walker is sampled from the distribution φ(τ ) = µ/(1 + τ ) 1+µ , that corresponds to our reference jump rate β(a) = µ/(1 + a). The simulation then iterates the following steps: (i ) find k, the walker with the earlier jump time: k = arg min i τ i , then (ii ) make it jump, i.e. reset its age a k = 0 and finally, (iii ) pick its next jump time τ k according to φ(τ ). During the simulation, we store the distance between the dynamic equilibrium W at that time and the observed distribution of rescaled ages b i = a i /(1 + t) of all the walkers i in the simulation: w(τ, •) -W (τ, •) L 1 ([0,1]) . We also compute at each time step the L 1 norm of the difference w(τ, •) -W ∞ . Unless stated otherwise we use 20,000 random walkers in each simulation.

First, we note that in all cases, the simulated L 1 distance between w and the pseudoequilibrium W is indeed bounded above by the expression given in theorem 4 (except at very high τ , when our bound becomes lower than the numerical error of the simulation). The example given in figure 1 corresponds to µ = 0, 4 and µ = 0, 8, and an initial age distribution Dirac at 0 for the red dots, and uniform on [0, 1) for the blue dots, the black curve representing the upper bound proved in theorem 4 taken for H(0) = 2, which is an upper bound for H(0). As we see, the multiplicative constant we lose (the overestimation of K in theorem 4 corresponding to the losses throughout the inequalities used to prove our bound) is not too high.