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Elena Agliari University of Freiburg YEP 2008 Eurandom, Eindhoven, The Netherlands, March 10-14 2008

DIFFUSIVE THERMAL DYNAMICS FOR THE ISING MODEL ON THE ERDÖS-RÉNYI RANDOM GRAPH. Elena Agliari University of Freiburg YEP 2008 Eurandom, Eindhoven, The Netherlands, March 10-14 2008. SUMMARY. DIFFUSIVE THERMAL DYNAMICS - Motivations - How it works → BRW

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Elena Agliari University of Freiburg YEP 2008 Eurandom, Eindhoven, The Netherlands, March 10-14 2008

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  1. DIFFUSIVE THERMAL DYNAMICS FOR THE ISING MODEL ON THE ERDÖS-RÉNYI RANDOM GRAPH Elena Agliari University of Freiburg YEP 2008 Eurandom, Eindhoven, The Netherlands, March 10-14 2008

  2. SUMMARY DIFFUSIVE THERMAL DYNAMICS - Motivations - How it works → BRW - Results on Regular Lattices Thermodynamics, Geometric, Diffusive Properties DIFFUSIVE THERMAL DYNAMICS ON THE ERDÖS-RÉNYI RG - Extension of previous results - Applications to Social systems Diffusive Dynamics → Strategy

  3. DIFFUSIVE THERMAL DYNAMICS Magnetic system evolves according to relaxation dynamics → asymptotically drives it to equilibrium steady state Probability given configuration occurs proportional to Boltzmann factor Relaxation dynamics (single spin-flip) Rule to select site Rule to decide whether to flip relevant spin Markov chain Physical interpretation: spin flips ascribed to coupling magnetic system & heat-bath HEAT CAN BE INJECTED INTO A SYSTEM NON-UNIFORMLY INDEED, HEAT USUALLY PROPAGATES THROUGHOUT SAMPLE IN DIFFUSIVE WAY DIFFUSIVE CHARACTER DIFFUSION MORE LIKELY TOWARDS THOSE REGIONS WHERE ENERGY VARIATIONS ARE MORE PROBABLE TO OCCUR BIAS

  4. RANDOM- WALK (RW) THOUGHT OF AS A LOCALIZED EXCITATION POSSIBLY INDUCING A SPIN-FLIP PROCESS AT EVERY SITE IT VISITES Rw NON ISOTROPIC: BIAS TOWARDS SITES WHERE SPIN-FLIP MORE LIKELY TO OCCUR - Diffusive dynamics with isotropic hopping probabilities equivalent to single-spin-dynamics with random update - Diffusive dynamics strictly local character, different from delocalized heat-bath energy exchanges • No restrictions on the geometry of the underlying structure, neither on S Degrees of freedom Each jump (zi+1)(2S+1) options Spin-flips are the result of a stochastic process featuring a competition between energetic and entropic term PBC h=0, J cost [1] P. Buonsante, R. Burioni, D. Cassi, A. Vezzani, Phys. Rev. E, 66, 36121 (2002) [2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, 46, 109 (2005)

  5. THERMODYNAMIC PROPERTIES S=½ S=1 System relaxes to steady state characterized by thermodynamics quantities depending only on the temperature S=1, T=1.56, fit: -0.51± 0.02 System displays spontaneous symmetry breaking accompanied by a singular behaviour of thermodynamic functions Tc(S=½)>Tc(S=1) TcD>Tc Measure of critical exponents α, β, γ, ν ISING UNIVERSALITY CLASS CONSERVED Tc(S=½) ≈2.60 (→ 2.27) Tc(S=1) ≈ 1.96 (→1.70) [2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, 46, 109 (2005)

  6. GEOMETRICAL PROPERTIES Bias → Sites corresponding to borders between clusters more frequently updated → Geometry of magnetic patterns affected Measure of spatial distribution of spin states as a function of T BOX-COUNTING FRACTAL DIMENSION dfD>dfHB T → Tc- κD<κHB S=1 - D Difference related to the way each thermal dynamics deals with fluctuations at small scales THE VERY EFFECTS OF BIASED DIFFUSIVE DYNAMICS CAN BE TRACKED DOWN IN GEOMETRY OF MAGNETIC CLUSTERS HB D S=½ D: diffusive dynamics HB: heat-bath dynamics, random updating [3] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, 49, 119 (2006)

  7. DIFFUSIVE PROPERTIES COUPLING RW-MAGNETIC SYSTEM RW ON ENERGY LANDSCAPE There exist energy barriers between n.n. sites whose height is lower when it is possible to obtain, via spin-flip a greater energy gain External parameter T is “dispersion parameter” tuning the roughness of energetic environment IN GENERAL, COUPLING MORE IMPORTANT AS CRITICAL POINT APPROACHED Two stochastic processes interacting: BRW diffusion and evolution of magnetic configuration Magnetic Lattice Visit Lattice [4] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, 48, 529 (2006)

  8. CORRELATION ENERGY • COVERING TIME TN(T) • DISTINCT SITES VISITED SN(T,n) • # RETURNS TO ORIGIN RN(T,n) Correlation energy >0 RW more likely to be found on boundaries between clusters CONVENTIONAL DIFFUSIVE REGIME RECOVERED, THOUGH TEMPERATURE DEPENDENT CORRECTIONS INTRODUCED Tc EXTREMAL POINT Large correlation length for magnetic lattice → highly inhomogeneous energy-landscape Effect larger for S=1 SN(T,n), L=240 • DIFFUSION SENSITIVE TO PHASE TRANSITION • SLOW THERMAL DYNAMICS

  9. DIFFUSIVE THERMAL DYNAMICS ON THE ERDÖS-RÉNYI RANDOM GRAPH Many physical, biological and social systems evidence complex topological properties Ising model prototype for phase transitions and cooperative behaviour: mimic wide range of phenomena N sites, (undirectly) connected pair-wise with probability p → average degree <z>=(N-1)p Connectivity of each node follows binomial distribution J/Tc = ½ ln(<z2>/(<z2>-2<z>)) ~ <z>/<z2> → Tc= 1 – p + Np ~ <z> Finite magnetization whenever <z2>≥ 2<z> [5] A. Bovier, V. Gayrard, J. Stat. Phys., 72, 643 (1993) [6] S.N. Dorogovstev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E, 66, 016104 (2002) [7] M. Leone, A. Vazquez, A. Vespignani, R. Zecchina, Eur. Phys. J. B, 28, 191 (2002) [8] L. De Sanctis, F. Guerra, arXiv:0801.4940v1 (2008)

  10. Glauber algorithm with random updating MAGNETIZATION AND SUSCEPTIBILITY Tc~ <z> independent of size Peak → Divergence thermodynamic limit Fluctuations scale with size N of the graph Best fit: Y = -1.12 X – 1.75 <z>= 10, 20 Compatible with Complete Graph Universality Class

  11. DIFFUSIVE THERMAL DYNAMICS N=800 <Z>=10, P=0.0125 <Z>=20, P=0.025 TcD ≈ 11.0 > 10 TcD ≈ 21.3 > 20 N=1600 <Z>=10, P=0.0063 <Z>=20, P=0.0125 TcD ≈ 11.1 > 10 TcD ≈ 21.4 > 20 INCREASE OF Tc ROBUST WITH RESPECT TO SPIN MAGNITUDE AND UNDERLYING TOPOLOGY Preliminary results suggest TcD only depends on <z> Less accurate data for the RG fail to show any deviations from conservation of universality

  12. APPLICATIONS TO SOCIAL SYSTEMS Population whose elements characterized by cultural trait, opinion, attitude… dichotomic variable (si=±1) Interaction between individuals i and j described by a potential, or cost function, reflecting the will to “agree” or “disagree” among the two J may also mirrors the strength of imitation within each subgroup If most acquaintances vote X, I am more likely to vote X as well, especially if degree of interaction J high RW may represent information exchange among the connected individuals, the reached individual is “activated” BRW → strategy: people in minority are more likely to be contacted CONDITIONS FOR A MAGNETIZED SYSTEM? DIFFUSIVE THERMAL DYNAMICS MORE EFFICIENT: IT REQUIRES LOWER INTERACTION CONSTANT FOR ONE OPINION TO PREVAIL, AS #BRWs GROWS LESS EFFICIENT Other possible strategies: greedy and reluctant algorithm [9] P. Contucci, I. Gallo, G. Menconi, to appear in Int. Jour. Mod. Phys. B [10] P. Contucci, C. Giardinà, C. Giberti, C. Vernia, Math. Mod. Appl. Sc., 15, 1349 (2005)

  13. CONCLUSIONS • INTRODUCTION OF CONSISTENT DIFFUSIVE THERMAL DYNAMICS • NON-CANONICAL EQUILIBRIUM STATES WITH LARGER TC • UNIVERSALITY CLASS CONSERVED • GEOMETRIC CHARACTERIZATION OF PHASE TRANSITION ABLE TO EVIDENCE BIASED-DIFFUSIVE CHARACTER • DIFFUSION ON ENERGY-LANDSCAPE WITH COUPLING: TEMPERATURE DEPENDENT CORRECTIONS • PRELIMINARY EXTENSION OF RESULTS ON RANDOM TOPOLOGY • APPLICATIONS IN SOCIAL SYSTEMS: POSSIBLE STRATEGIES [1] P. Buonsante, R. Burioni, D. Cassi, A. Vezzani, Diffusive Thermal Dynamics for the Ising ferromagnet, Phys. Rev. E, 66, 36121 (2002) [2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Diffusive Thermal Dynamics for the spin-S Ising ferromagnet, Eur. Phys. J. B, 46, 109 (2005) [3] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Random walks interacting with evolving energy landscapes, Eur. Phys. J. B, 48, 529 (2005) [4] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Fractal geometry of Ising magnetic patterns: signatures of criticality and diffusive dynamics, Eur. Phys. J. B, 49, 119 (2006)

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