Elena AgliariElena Agliari University of FreiburgUniversity of Freiburg

YEP 2008YEP 2008

Eurandom, Eindhoven, The Netherlands, March Eurandom, Eindhoven, The Netherlands, March 10-14 200810-14 2008

DIFFUSIVE THERMAL DYNAMICS DIFFUSIVE THERMAL DYNAMICS

FOR THE ISING MODEL FOR THE ISING MODEL

ON THE ERDÖS-RÉNYI RANDOM ON THE ERDÖS-RÉNYI RANDOM GRAPHGRAPH

SUMMARSUMMARYY

DIFFUSIVE THERMAL DYNAMICSDIFFUSIVE THERMAL DYNAMICS

- Motivations- Motivations

- How it works - How it works → BRW→ BRW

- Results on Regular Lattices- Results on Regular Lattices

Thermodynamics, Geometric, Diffusive Thermodynamics, Geometric, Diffusive PropertiesProperties

DIFFUSIVE THERMAL DYNAMICS ON THE ERDÖS-RÉNYI DIFFUSIVE THERMAL DYNAMICS ON THE ERDÖS-RÉNYI RGRG

- Extension of previous results- Extension of previous results

- Applications to Social systems- Applications to Social systems

Diffusive Dynamics Diffusive Dynamics → Strategy→ Strategy

DIFFUSIVE THERMAL DIFFUSIVE THERMAL DYNAMICSDYNAMICSMagnetic system evolves according to Magnetic system evolves according to relaxation dynamics relaxation dynamics → asymptotically → asymptotically drives it to equilibrium steady statedrives it to equilibrium steady state

Probability given configuration occurs Probability given configuration occurs proportional to Boltzmann factorproportional to Boltzmann factor

Relaxation dynamics (single spin-flip) Relaxation dynamics (single spin-flip)

Rule to select Rule to select sitesite

Rule to decide Rule to decide whether to flip whether to flip relevant spinrelevant spinPhysical interpretation: spin flips ascribed to coupling Physical interpretation: spin flips ascribed to coupling

magnetic system magnetic system && heat-bath heat-bathHEAT CAN BE INJECTED INTO A SYSTEM NON-UNIFORMLYHEAT CAN BE INJECTED INTO A SYSTEM NON-UNIFORMLY

INDEED, HEAT USUALLY PROPAGATES THROUGHOUT SAMPLE IN INDEED, HEAT USUALLY PROPAGATES THROUGHOUT SAMPLE IN DIFFUSIVE WAYDIFFUSIVE WAY

DIFFUSIVE CHARACTERDIFFUSIVE CHARACTER

DIFFUSION MORE LIKELY TOWARDS THOSE REGIONS WHERE ENERGY DIFFUSION MORE LIKELY TOWARDS THOSE REGIONS WHERE ENERGY VARIATIONS ARE MORE PROBABLE TO OCCURVARIATIONS ARE MORE PROBABLE TO OCCUR

BIASBIAS

Markov Markov chainchain

RANDOM- WALK (RW) THOUGHT OF AS A LOCALIZED RANDOM- WALK (RW) THOUGHT OF AS A LOCALIZED EXCITATION POSSIBLY INDUCING A SPIN-FLIP PROCESS AT EXCITATION POSSIBLY INDUCING A SPIN-FLIP PROCESS AT EVERY SITE IT VISITESEVERY SITE IT VISITES

Rw NON ISOTROPIC: BIAS TOWARDS SITES WHERE SPIN-FLIP Rw NON ISOTROPIC: BIAS TOWARDS SITES WHERE SPIN-FLIP MORE LIKELY TO OCCURMORE LIKELY TO OCCUR- Diffusive dynamics with isotropic hopping probabilities - Diffusive dynamics with isotropic hopping probabilities equivalent to single-spin-dynamics with random updateequivalent to single-spin-dynamics with random update

- Diffusive dynamics strictly local character, different from - Diffusive dynamics strictly local character, different from delocalized heat-bath energy exchangesdelocalized heat-bath energy exchanges

i

j

z

jj

Qs

G

jG

j

Tsp

TsspTjisP

0

)|,(

)|,()|,|,(

s

s

Each jump (zEach jump (zii+1)(2S+1) +1)(2S+1) optionsoptions

)]()([1

1)|,(

jHHjGe

Tsp sss

-No restrictions on the geometry of the underlying structure, No restrictions on the geometry of the underlying structure, neither on S Degrees of freedom neither on S Degrees of freedom

Spin-flips are the result of a Spin-flips are the result of a stochastic process featuring a stochastic process featuring a competition between energetic competition between energetic and entropic termand entropic term

PBCPBC

h=0, J costh=0, J cost[1] P. Buonsante, R. Burioni, D. Cassi, A. Vezzani, Phys. Rev. E, [1] P. Buonsante, R. Burioni, D. Cassi, A. Vezzani, Phys. Rev. E, 6666, 36121 (2002) , 36121 (2002)

[2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, [2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, 4646, 109 (2005) , 109 (2005)

jji

iij ssAJ

sH ,2

)(

THERMODYNAMIC PROPERTIESTHERMODYNAMIC PROPERTIES

System relaxes to steady state System relaxes to steady state characterized by characterized by thermodynamics quantities thermodynamics quantities depending only on the depending only on the temperaturetemperatureSystem displays spontaneous System displays spontaneous symmetry breaking symmetry breaking accompanied by a singular accompanied by a singular behaviour of thermodynamic behaviour of thermodynamic functionsfunctions

2/122 NEE

S=1, T=1.56, fit: -S=1, T=1.56, fit: -0.510.51±± 0.02 0.02

TTccDD>T>Tcc

TTcc(S=(S=½½)>T)>Tcc(S=1)(S=1)

S=S=11

S=S=½½

Tc(S=Tc(S=½½) ≈) ≈ 2.60 2.60 ((→ → 2.27)2.27)

Tc(S=1) ≈ 1.96 Tc(S=1) ≈ 1.96 ((→→ 1.70)1.70)

Measure of critical Measure of critical exponents exponents α, β, γ, α, β, γ, νν

ISING UNIVERSALITY ISING UNIVERSALITY CLASS CONSERVEDCLASS CONSERVED

[2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, [2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, 4646, 109 (2005), 109 (2005)

GEOMETRICAL PROPERTIESGEOMETRICAL PROPERTIES

Measure of spatial Measure of spatial distribution of spin distribution of spin states as a function of Tstates as a function of T

BOX-COUNTING FRACTAL BOX-COUNTING FRACTAL DIMENSIONDIMENSION

D: diffusive D: diffusive dynamicsdynamics

HB: heat-bath HB: heat-bath dynamics, random dynamics, random

updatingupdating

[3] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, [3] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, 4949, 119 (2006), 119 (2006)

Bias Bias → → Sites corresponding to borders between clusters Sites corresponding to borders between clusters more frequently updated more frequently updated → → Geometry of magnetic patterns Geometry of magnetic patterns affectedaffected

S=

1 -

S

=1

- DD

S=

S=

½½

HHBB

DD

c

cf T

TTAdsd )1(

T T → T→ Tcc--

ddffDD >>

ddffHBHB

κκDD << κκHBHB

Difference related to the way each thermal Difference related to the way each thermal dynamics deals with fluctuations at small dynamics deals with fluctuations at small scalesscales

THE VERY EFFECTS OF BIASED DIFFUSIVE THE VERY EFFECTS OF BIASED DIFFUSIVE DYNAMICS CAN BE TRACKED DOWN IN DYNAMICS CAN BE TRACKED DOWN IN GEOMETRY OF MAGNETIC CLUSTERSGEOMETRY OF MAGNETIC CLUSTERS

)/1log(

)(loglim

0 r

rNd

rf

DIFFUSIVE PROPERTIESDIFFUSIVE PROPERTIES

COUPLING COUPLING RW-MAGNETIC RW-MAGNETIC SYSTEMSYSTEM

RW ON ENERGY RW ON ENERGY LANDSCAPELANDSCAPE

SURWBRWSsN

N

ii

01

1

Magnetic Magnetic LatticeLattice

Visit Visit LatticeLattice

Two stochastic processes Two stochastic processes interacting: BRW diffusion interacting: BRW diffusion and evolution of magnetic and evolution of magnetic configurationconfiguration

There exist energy barriers between n.n. sites whose height There exist energy barriers between n.n. sites whose height is lower when it is possible to obtain, via spin-flip a greater is lower when it is possible to obtain, via spin-flip a greater energy gainenergy gain

External parameter T is “dispersion parameter” tuning the External parameter T is “dispersion parameter” tuning the roughness of energetic environmentroughness of energetic environment

[4] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, [4] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Eur. Phys. J. B, 4848, 529 (2006), 529 (2006)

IN GENERAL, COUPLING IN GENERAL, COUPLING MORE IMPORTANT AS MORE IMPORTANT AS CRITICAL POINT CRITICAL POINT APPROACHEDAPPROACHED

CONVENTIONAL DIFFUSIVE REGIME CONVENTIONAL DIFFUSIVE REGIME RECOVERED, THOUGH RECOVERED, THOUGH

TEMPERATURE DEPENDENT TEMPERATURE DEPENDENT CORRECTIONS INTRODUCEDCORRECTIONS INTRODUCED

TTcc EXTREMAL POINT EXTREMAL POINT Large correlation length for Large correlation length for

magnetic lattice magnetic lattice →→ highly highly inhomogeneous energy-landscapeinhomogeneous energy-landscape

Effect larger for S=1Effect larger for S=1

SSNN(T,n), L=240(T,n), L=240

- DIFFUSION SENSITIVE TO PHASE DIFFUSION SENSITIVE TO PHASE TRANSITIONTRANSITION

- SLOW THERMAL DYNAMICSSLOW THERMAL DYNAMICS

)()(),( nSTanTS NBRWN

• CORRELATION ENERGY CORRELATION ENERGY

• COVERING TIME TCOVERING TIME TNN(T)(T)

• DISTINCT SITES VISITED DISTINCT SITES VISITED SSNN(T,n)(T,n)

• # RETURNS TO ORIGIN # RETURNS TO ORIGIN RRNN(T,n)(T,n)

TTlocT )(~

RW more likely to be RW more likely to be found on boundaries found on boundaries between clustersbetween clusters

CorrelatCorrelation ion energy energy >0>0

DIFFUSIVE THERMAL DYNAMICS DIFFUSIVE THERMAL DYNAMICS

ON ON THE ERDÖS-RÉNYI RANDOM GRAPHTHE ERDÖS-RÉNYI RANDOM GRAPH

[5] A. Bovier, V. Gayrard, J. Stat. Phys., [5] A. Bovier, V. Gayrard, J. Stat. Phys., 7272, 643 (1993), 643 (1993)

[6] S.N. Dorogovstev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E, [6] S.N. Dorogovstev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E, 6666, 016104 (2002), 016104 (2002)

[7] M. Leone, A. Vazquez, A. Vespignani, R. Zecchina, Eur. Phys. J. B, [7] M. Leone, A. Vazquez, A. Vespignani, R. Zecchina, Eur. Phys. J. B, 2828, 191 (2002) , 191 (2002)

[8] L. De Sanctis, F. Guerra, arXiv:0801.4940v1 (2008)[8] L. De Sanctis, F. Guerra, arXiv:0801.4940v1 (2008)

Many physical, biological and social systems Many physical, biological and social systems evidence complex topological propertiesevidence complex topological properties

Ising model prototype for phase transitions Ising model prototype for phase transitions and cooperative behaviour: mimic wide and cooperative behaviour: mimic wide range of phenomena range of phenomena

N sites, (undirectly) connected pair-wise with probability p N sites, (undirectly) connected pair-wise with probability p → → average degree <z>=(N-1)paverage degree <z>=(N-1)p

Connectivity of each node follows binomial distributionConnectivity of each node follows binomial distribution

J/TJ/Tc c = ½ ln(<z= ½ ln(<z22>/(<z>/(<z22>-2<z>)) ~ <z>/<z>-2<z>)) ~ <z>/<z22> → T> → Tcc= 1 – p + Np ~ = 1 – p + Np ~ <z><z>

Finite magnetization whenever Finite magnetization whenever <z<z22>≥ 2<z>>≥ 2<z>

MAGNETIZATION AND MAGNETIZATION AND SUSCEPTIBILITYSUSCEPTIBILITY

TTcc ~ <z> independent ~ <z> independent of sizeof size

Peak Peak → → Divergence Divergence thermodynamic limitthermodynamic limit

Best fit: Best fit: Y = Y = -1.12-1.12 X – X – 1.751.75

<z>= 10, 20<z>= 10, 20

Compatible with Compatible with Complete Graph Complete Graph Universality ClassUniversality Class

Fluctuations scale Fluctuations scale with size N of the with size N of the

graphgraph

Glauber algorithm with random Glauber algorithm with random updatingupdating

221)(

TTMM

NTT

Less accurate data for the RG fail to show any deviations from Less accurate data for the RG fail to show any deviations from conservation of universalityconservation of universality

DIFFUSIVE THERMAL DYNAMICSDIFFUSIVE THERMAL DYNAMICS

N=800N=800

<Z>=10, P=0.0125<Z>=10, P=0.0125

<Z>=20, P=0.025<Z>=20, P=0.025

N=1600N=1600

<Z>=10, P=0.0063<Z>=10, P=0.0063

<Z>=20, P=0.0125<Z>=20, P=0.0125

Preliminary results suggest TPreliminary results suggest TccDD only depends on <z> only depends on <z>

TTccD D ≈ 11.1 > ≈ 11.1 >

1010

INCREASE OF TINCREASE OF Tcc ROBUST WITH RESPECT TO ROBUST WITH RESPECT TO SPIN MAGNITUDE AND UNDERLYING SPIN MAGNITUDE AND UNDERLYING TOPOLOGYTOPOLOGY

TTccD D ≈ 21.4 > ≈ 21.4 >

2020

TTccD D ≈ 11.0 > ≈ 11.0 >

1010TTccD D ≈ 21.3 > ≈ 21.3 >

2020

APPLICATIONS TO SOCIAL SYSTEMS APPLICATIONS TO SOCIAL SYSTEMS

[9] P. Contucci, I. Gallo, G. Menconi, to appear in Int. Jour. Mod. Phys. B [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., [10] P. Contucci, C. Giardinà, C. Giberti, C. Vernia, Math. Mod. Appl. Sc., 1515, ,

1349 (2005) 1349 (2005)

Population whose elements characterized by cultural trait, Population whose elements characterized by cultural trait, opinion, attitude… dichotomic variable (sopinion, attitude… dichotomic variable (sii=±1)=±1)

Interaction between individuals i and j described by a Interaction between individuals i and j described by a potential, or cost function, reflecting the will to “agree” or potential, or cost function, reflecting the will to “agree” or “disagree” among the two“disagree” among the two

J may also mirrors the strength of imitation within each subgroupJ may also mirrors the strength of imitation within each subgroup

RW may represent information exchange among the RW may represent information exchange among the connected individuals, the reached connected individuals, the reached individual is “activated”individual is “activated”

BRW BRW → → strategy: people in minority are more likely strategy: people in minority are more likely to be contactedto be contacted

If most acquaintances vote X, I am more likely If most acquaintances vote X, I am more likely to vote X as well, especially if degree of to vote X as well, especially if degree of interaction J highinteraction J high

CONDITIONS FOR A MAGNETIZED SYSTEM?CONDITIONS FOR A MAGNETIZED SYSTEM?

DIFFUSIVE THERMAL DYNAMICS MORE DIFFUSIVE THERMAL DYNAMICS MORE EFFICIENT: IT REQUIRES LOWER EFFICIENT: IT REQUIRES LOWER INTERACTION CONSTANT FOR ONE OPINION TO INTERACTION CONSTANT FOR ONE OPINION TO PREVAIL, AS #BRWs GROWS LESS EFFICIENTPREVAIL, AS #BRWs GROWS LESS EFFICIENT

Other possible strategies: greedy and Other possible strategies: greedy and reluctant algorithmreluctant algorithm

CONCLUSIONSCONCLUSIONS

• INTRODUCTION OF CONSISTENT DIFFUSIVE THERMAL INTRODUCTION OF CONSISTENT DIFFUSIVE THERMAL DYNAMICSDYNAMICS

• NON-CANONICAL EQUILIBRIUM STATES WITH LARGER TNON-CANONICAL EQUILIBRIUM STATES WITH LARGER TCC

• UNIVERSALITY CLASS CONSERVEDUNIVERSALITY CLASS CONSERVED

• GEOMETRIC CHARACTERIZATION OF PHASE TRANSITION GEOMETRIC CHARACTERIZATION OF PHASE TRANSITION ABLE TO EVIDENCE BIASED-DIFFUSIVE CHARACTERABLE TO EVIDENCE BIASED-DIFFUSIVE CHARACTER

• DIFFUSION ON ENERGY-LANDSCAPE WITH COUPLING: DIFFUSION ON ENERGY-LANDSCAPE WITH COUPLING: TEMPERATURE DEPENDENT CORRECTIONSTEMPERATURE DEPENDENT CORRECTIONS

• PRELIMINARY EXTENSION OF RESULTS ON RANDOM PRELIMINARY EXTENSION OF RESULTS ON RANDOM TOPOLOGYTOPOLOGY

• APPLICATIONS IN SOCIAL SYSTEMS: POSSIBLE APPLICATIONS IN SOCIAL SYSTEMS: POSSIBLE STRATEGIES STRATEGIES

[1] P. Buonsante, R. Burioni, D. Cassi, A. Vezzani, [1] P. Buonsante, R. Burioni, D. Cassi, A. Vezzani, Diffusive Thermal Dynamics for the IsingDiffusive Thermal Dynamics for the Ising ferromagnetferromagnet, Phys. Rev. E, 66, 36121 (2002) , Phys. Rev. E, 66, 36121 (2002)

[2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, [2] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Diffusive Thermal Dynamics for the spin-S Ising Diffusive Thermal Dynamics for the spin-S Ising ferromagnetferromagnet, Eur. Phys. J. B, 46, 109 (2005), Eur. Phys. J. B, 46, 109 (2005)

[3] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, [3] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Random walks interacting with evolving energy Random walks interacting with evolving energy landscapeslandscapes, Eur. , Eur. Phys. J. B, 48, 529 (2005)Phys. J. B, 48, 529 (2005)

[4] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, [4] E. Agliari, R. Burioni, D. Cassi, A. Vezzani, Fractal geometry of Ising magnetic patterns: Fractal geometry of Ising magnetic patterns: signatures of criticality and diffusive dynamicssignatures of criticality and diffusive dynamics, Eur. , Eur. Phys. J. B, 49, 119 (2006)Phys. J. B, 49, 119 (2006)