J. Duchesne1, P. Raimbault2 and C. Fleurant1 Etude de la morphométrie des arbres par combinaison de la géométrie fractale et de la physique statistique J. Duchesne1, P. Raimbault2 and C. Fleurant1 1. UMR 105 Paysages et biodiversité 2. UMR SAGAH

Une loi universelle de la morphométrie des arbres ? Introduction Démonstration de la loi Résultats et discussion Conclusion

Une loi universelle de la morphométrie des arbres ? Introduction Démonstration of the law Résultats et discussion Conclusion

Réseaux hydrographiques Arbres Ont en commun l’invariance d’échelle Structures fractales ramifiées

2 2 3 2 2 3 2 4 2 2 2

RC : rapport de bifurcation N1 : nombre de tronçons du 1er ordre N2 : nombre de tronçons du 2ème ordre ……….. Ni: nombre de tronçons d’ordre i Rapports N1/ N2, N2/ N3, … Ni/ Ni+1 sont constants et notés RC : rapport de bifurcation

RL : rapport de longueur : longueur moyenne des tronçons d’ordre 1 : longueur moyenne des tronçons d’ordre 2 ……….. : longueur moyenne des tronçons d’ordre n Les rapports L2/ L1, L3/ L2, … Li+1/ Li sont constants et notés RL : rapport de longueur

Ces deux résultats sont la marque d’une structure fractale ramifiée

Une loi universelle de la morphométrie des arbres ? Introduction Démonstration of the law Résultats et discussion Conclusion

d’utiliser un raisonnement de physique statistique Nous proposons d’utiliser un raisonnement de physique statistique

Une loi universelle de la morphométrie des arbres ? 1. Introduction 2. Démonstration de la loi Choix de l’espace symbolique

Symbolic space of Maxwell : the space of speeds vz vy dvz dvy dvx vx

d3N is the number of molecules which the speed vector ends to the elementary volume dvx dvy dvz , among a total number of molecules N

The two hypotheses of Maxwell the independence of the 3 speed components ; the isotropy of the speed directions

The independence of the 3 speed components involves : So, the 3 variables are separated

The isotropy of the speed directions is a natural hypothesis because one can hardly imagine that some directions be privilegied The distributions f1, f2, f3 have the same form :

are sufficient conditions to determine the function F(v) The two hypotheses of Maxwell are sufficient conditions to determine the function F(v)

Analogy between thermodynamics and natural networks Maxwell approach Our approach

Notion of speed vector module Notion of hydraulic length Analogy ... thermodynamics natural networks Notion of speed vector module Notion of hydraulic length v L

Analogy … thermodynamics natural networks Maxwell approach vz vy dvz dvy dvx vx

Analogy … thermodynamics natural networks Our approach L = l1 + l2 + l3 +… + ln

Analogy … thermodynamics natural networks There are two differences between the two approaches ...

Analogy … thermodynamics natural networks first difference : In thermodynamics In natural networks there are 3 components there are n components

Analogy … thermodynamics natural networks • • •

Analogy … thermodynamics natural networks

Analogy … thermodynamics natural networks second difference : In thermodynamics In natural networks the 3 components have the same mean the n components have not the same mean

Analogy … thermodynamics natural networks • • • • • •

Analogy … thermodynamics natural networks • • •

Analogy … thermodynamics natural networks

Maxwell’s two hypotheses the independence of the 3 speed components ; the isotropy of the speed directions become the independence of the n length components ; the isotropy of the directions of the symbolic space

The Maxwell function becomes

The Maxwell results for f And the same for vx, vy, vz become become

Une loi universelle de la morphométrie des arbres ? Introduction Démonstration of the law Résultats et discussion Conclusion

… it is necessary to respect two conditions which are strongly related to the statistical physics : 1. the size of the system much be very large compared with the elementary constituent which will be taken into account 2. the local properties of the system must be homogeneous

A large number of elementary constituents Homogeneity of the population

Results with a population of trees The population : 12 apple trees 4 years old grown from the same parents

1. The number of hydraulic lengths, corresponding to apexes, cannot exceed a few thousand for a given class ; 2. Moreover, the distribution of hydraulic lengths, as well as the distribution of their n components can be more or less influenced by the environment constraints.

Results with a Cupressocyparis Order n : the maximum order observed in the tree Mean hydraulic length : the average of all the hydraulic lengths of the tree In the same way, RB and RL are calculated for all branches of the tree

Une loi universelle de la morphométrie des arbres ? Introduction Démonstration de la loi Résultats et discussion Conclusion

Réseaux sur Titan (source : ESA)

Biblio sommaire Fleurant C., Duchesne, J., Raimbault, P., 2004. An allometric model for trees. Journal of Theorical Biology, 227, 137-147. Cudennec C., Fouad Y., Sumarjo Gatot I. & Duchesne J. 2004. A geomorphological explanation of the unit hydrograph concept. Hydrological processes, 18, 603-621. Duchesne J., Raimbault P. & Fleurant C. 2002. Towards a universal law of tree morphometry by combining fractal geometry and statistical physics, in Proceeding "Emergent Nature", 7th multidisciplinary conference, M. M. Novak (ed.), World Scientific 2002, pp.93-102. Roland B. 2002. An attempt to characterise Hedgerow lattice by means of fractal geometry, , in Proceeding "Emergent Nature", 7th multidisciplinary conference, M. M. Novak (ed.), World Scientific 2002, pp. 103-112. Duchesne, J, Fleurant, C. & Capmarty-Tanguy, F., inventeurs, 2002, Procédé d’implantation de végétaux, plan d’implantation de végétaux obtenu et système informatique pour l’élaboration d’un tel plan, déposé à l’INPI le 25 juin 2002.

Merci de votre attention

This is the density of the points representing the speeds

As we have :