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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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
- 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
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
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
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)
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)
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)
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)
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
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)
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
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
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)
CONCLUSIONS
[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)