Exploring universality classes of nonequilibrium statistical physics

Universal scaling behavior is an attractive feature in statistical physics because a wide range of models can be classified purely in terms of their collective behavior due to a diverging correlation length. Scaling phenomenon has been observed in many branches of physics, chemistry, biology, economy ... etc., most frequently by critical phase transitions and surface growth. This is a basic science topic, the results can be used in applied sciences:Nonequilibrium critical phase transitions appear in models of: - Spatiotemporal intermittency: Z. Jabeen and N. Gupte, Phys. Rev. E 72 (2005) 016202

- Population dynamics : E. V. Albano, J. Phys. A 27 (1994) L881 (forest fire model )?

Nonequilibrium critical phase transitions appear in models of

- Epidemics spreading : T. Ligget, Interacting particle systems 1985

- Catalysis : Da-yin Hua, Phys. Rev E 70 (2004) 066101, - Itinerant electron systems : D. E. Feldman, Phys. Rev. Lett 95 (2005) 177201, - Cooperative transport : S. Havlin and D. ben-Avraham, Adv. Phys. 36 (1987) 695,

Nonequilibrium critical phase transitions appear in models of

- Biological control systems : K. Kiyono, et al., PRL95 (2005) 058101.

- Stock-prize fluctuations and markets: K. Kiyono, et al., PRL96 (2006) 068701,

- Meteorology and Climatology: O.Petres and D. Neelin, Nature Phys. 2 (2006)393

Nonequilibrium critical phase transitions appear in models of

- Enzyme biology:- Origin of life : G. Cardozo and J. F. Fontanari, Eur. Phys. J.55 (2006) 555.

- Brain : G. Werner : Biosystems, 90 (2007) 496,

- Plasma physics : C.A. Knapek et al. Phys. Rev. Lett 98 (2007) 015001.

Scaling laws may occur in “self-organized critical systems” (SOC)?

The term SOC usually refers to a mechanism of slow energy accumulation and fast energy redistribution, driving a system toward a critical state. Prototype: sandpile model

SOC mechanism has been proposed to model earthquakes, the evolution of biological systems, solar flare occurrence, fluctuations in confined plasma, snow avalanches and rain fall for example...

Self-tuning to critical point.

SOC models can be mapped onto ordinary nonequilibrium criticality of phase transition to absorbing phase

40000 grains dropped on center of 120 x 120 lattice with Zc=4

Add a grain of sand: z(x,y) → z(x,y) + 1 And avalanche if: z(x,y) > Zc: z(x,y) → z(x,y) - 4z( x 1, y) → z( x 1, y) + 1z( x, y 1) →z(x, y 1 ) + 1

Bak-Tang-Wiesenfeld sandpile model

Scaling in biological system

Life-span tends to lengthen--and metabolism slows down--in proportion to the quarter power of an animal's body weight.

Explanations based on scaling arguments:

In animals and plants, circulatory systems resemble branching fractal networks, and capillary size does not depend on organism size. G.West et al. Science (4 April 1997, p.122)

Optimal nuitrient distribution networkNature '99 Banavar et al. Similar to river network models

Multiscaling

Scaling in computer network system

Self similarity in internet networks

Heavy-tailed file size distribution in web trafic

Heavy-tailed distributed on/off processes on TCP

Veres et al ACM SIGCOMM Computer Communication ReviewVolume 30 , Issue 4 October 2000

Scaling at surfaces and interfaces

Family-Vicsek scaling:

Mapping between surfaces and reaction-diffusion models

Tumor growth following molecular-beam epitaxy class scaling behavior

For more details see the new book

Universality in nonequilibrium lattice systems, Theoretical foundations, World Scientific, Singapore, 1-295 oldal (2008). http://www.worldscibooks.com/physics/6813.html

Phase transition to absorbing state in branching and annihilating random walk (BARWn) models

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A (1+n) A, 2A 0

For n = 1 (n odd) below dc : A

reaction is generated due to fluctuationsdirected percolation (DP) universalitycontact process, epidemic spreading

For n = 2 (n even) the: A reaction is not generated, parity conserving (PC) universality class

Space-time evolution of universal, nonequilibrium reaction-diffusion models with zero density absorbing states

Unary particle production+ spreading without and with parity conservation: A (1+m)A, 2A 0

Binary particle production+ spreading coupled to slave modes without and with diffusion: 2A (2+m)A, 2A 0Drifting scaling exponents by varying the diffusion rate: G.Ódor, Phys. Rev. E 62 (2000) R3027. or two classes ?

The challenging binary production, diffusive pair contact process (PCPD)

Two zero particle density absorbing (odered) states without symmetry, (Carlon et al. PRE 2001). Numerical methods show new exponents, but results are controversial. No extra symmetries or conservation laws found to explain them

Bosonic field theories (allowing multiple site occupancy) have failed to describe the critical behavior. In the bosonic model variant in the active phase the particle density diverges.

Fermionic model show different critical behavior, but perturbative field theoretical renormalization did not find stable fixed point corresponding to the novel scaling behavior (2004). For a review see:M. Henkel and H. Hinrichsen, J. Phys. A37, R117-R159 (2004)

Challening problems Understanding the role symmetries and conservation laws in such system

In 1d two classes, non-universal or DP scaling ? Numerical confirmation following density decay is needed.

Proper correction-to scaling analysis to density decay: = At + Bt--x+ .. = ?, x= ?

More complex: nA (n+m) A type reaction -diffusion models with n > 2 show similar new universal behavior in low dimensions ...

Parallel algorithm realizations

Master-worker setup, Single Program Multiple Data (SPMD) algorithm.

The slave processes are completely identical and sequential. Minimal communication losses, easy program development