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

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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

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

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

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)

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)

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)

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)

- 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

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)

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

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

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)

- 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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