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STOCHASTIC RESONANCE IN EXTENDED STOCHASTIC RESONANCE IN EXTENDED STOCHASTIC RESONANCE IN EXTENDED STOCHASTIC RESONANCE IN EXTENDED
SYSTEMS: THE ROLE OF THE COUPLING SYSTEMS: THE ROLE OF THE COUPLING SYSTEMS: THE ROLE OF THE COUPLING SYSTEMS: THE ROLE OF THE COUPLING MECHANISMMECHANISMMECHANISMMECHANISM
Horacio S. Wio(a)Horacio S. Wio
Instituto de Fisica de Cantabria
Universidad de Cantabria-CSIC
(a) Electronic address: wio@ifca.unican.es
URL: http://www.ifca.unican.es/~wio/
XI LAWNP, Buzios, Brazil, Oct. 05-09, 2009
COLLABORATORS:
S. Bouzat 1 *
F. Castelpoggi 3 *
R. Deza 2
B. von Haeften 4 *
G. Izús 2 *
M. Kuperman 1 **
S. Mangioni 2 **S. Mangioni 2 **
J.A. Revelli 6
A. Sánchez 2 **
C. Tessone 5 *
1) Centro Atómico Bariloche & Instituto Balseiro , Argentina.
2) Universidad Nacional de Mar del Plata, Argentina.
3) CitiBank, Buenos Aires, Argentina.
4) Universidad de Vigo, Spain
5) ETH, Zurich, Swiss
6) Inst. Fisica de Cantabria, Spain
Organization of the Talk:
• Introduction: Stochastic Resonance in 0-d and spatially extended systems;
• Far from Equilibrium Potentials: Brief review;• Reaction-Diffusion Systems: Example of a scalar
system; Stochastic Resonance in Extended Systems: system;
• Stochastic Resonance in Extended Systems: (a) Non-local Interactions; (b) Bounded KPZ system; (c) KPZ plus non-local interaction;…
• Final Comments …
STOCHASTIC RESONANCE IN 0-D SYSTEMS:
L.Gammaitoni, P.Hänggi, P.Jung, F.Marchesoni; Rev. Mod. Phys.70, 223 (1998)
Nonlinear
System
TWO STATE THEORY:C.Nicolis, Tellus, 34, 1 (1982); B.McNamara, K.Wiesenfeld, Phys. Rev. A 39, 4854 (1989)
ASSUMPTIONS:
1. : probability of finding the system in
2. Non-stationary Master Equation for
(“adiabatic approx.”)
3. Perturb. up to 1st order in B,
Kramers-like approximation :
The knowledge of allows to obtain the correl. function
Its Fourier transform give us the Power Spectral Density
TWO STATE THEORY:
S(w) - The PSD results:
TWO STATE THEORY:
S(w) - The PSD results:
The Signal-to-Noise Ratio The Signal-to-Noise Ratio
(SNR) results:
FAR FROM EQUILIBRIUM POTENTIAL:R.Graham, in Instabilities and Nonequilibrium Structures, Eds. E. Tirapegui and D.Villaroel (D.Reidel,
Dordrecht,1987); H.S. Wio, in 4th.Granada Lectures in Computational Physics, Eds. J.Marro y P.Garrido (Springer-
Verlag, 1997), pg. 135
Dynamical Systems:
Gradient (or variational):
Relaxational or non-gradient:Relaxational or non-gradient:
Non-relaxational & non-gradient:
FAR FROM EQUILIBRIUM POTENTIAL:
Including stochastic terms:
Associated Fokker-Planck equation:
If: and
solution of Hamilton-Jacobi –like equation, independent of
(that is a solution of a 1st order pdf)
NEP Example: Scalar System
Balast resistor – Schlögl model:
NEP Example: Scalar System
FAR FROM FAR FROM FAR FROM FAR FROM
EQUILIBRIUM EQUILIBRIUM EQUILIBRIUM EQUILIBRIUM
POTENTIALPOTENTIALPOTENTIALPOTENTIAL
SR in Extended Systems:
SR in Extended Systems:
SR in Extended Systems:
Transitions rates (~ Kramer theory)
1st order in the “perturbation”
Signal-to-Noise Ratio:
Stochastic-Resonance in a Scalar System:
Non-Local Kernel
Stochastic-Resonance in a Scalar System:
Non-Local Kernel
Parameters:
D=1, L= 2 π
β = 0β = 0
β = 0.01 l = 0.05
β = 0.01 l = 0.2
β = 0.01 l = 0.3
Bounded KPZ:Variational formulation for the KPZ and related kinetic equations,
H.S.Wio, Int. J. Bif. Chaos 19, 2813 (2009)
Case of KPZ-like equation
Bounded KPZ:Variational formulation for the KPZ and related kinetic equations,
H.S.Wio, Int. J. Bif. Chaos 19, 2813 (2009)
Case of KPZ-like equation
with the nonequilibrium potential
fulfilling
Bounded KPZ:Variational formulation for the KPZ and related kinetic equations,
H.S.Wio, Int. J. Bif. Chaos 19, 2813 (2009)
Case of KPZ-like equation
with the nonequilibrium potential
fulfilling
The approximate form (expansion around a reference state)
allows to exploit all previous results in order to obtain the SNR
Bounded KPZ:Stochastic Resonance in Extended Systems: An Overview of Recent Results
for Systems with and without Nonequilibrium Potential, H.S.Wio, J.
Revelli, M.Rodriguez, R.Deza & G.Izús, Europ.Phys.J.B 69, 71 (2009)
Parameters:
λ = 0; 0.25; 0.5 (υ = 2; a= b= 30)
Bounded KPZ: Non-Local Kernel
Parameters:
λ = 0.1 (υ = 2; a= b= 30)
β = 0.01; l = 0.05; 0.25; 0.5, 1.
Final Comments :
• The knowledge of the NEP is extremely useful to analyze andunderstand the system’s dynamics (even when the NEP is notknown in full detail);
• The knowledge of the NEP, when accesible, allows us to:* clear understand the role played by each of the differentsystems parameters;
* analyze different forms to enhance and/or control thephenomenon of SR;phenomenon of SR;
* understand the physical origin of the different trends;
• Some future research lines:1. to analyze very general situations with selective or state
dependent coupling;2. to analyze the effect of other boundary conditions;3. to analyze the effect of other kind of noises: colored,
non-Gaussian, f -n ;4. to analyze the effect of other forms of coupling;
5. ……
RELEVANT PAPERS
1. Stochastic resonance in spatially extended systems, H.S.Wio, Phys. Rev. E 55, R3075 (1996).
2. Stochastic resonance in extended systems: enhancement due to coupling in a reaction-diffusion
model, F. Castelpoggi & H.S. Wio, Europhysics Letters 38, 91 (1997).
3. Stochastic resonance in extended bistable systems: the role of potential symmetry,
S.Bouzat & H.S.Wio, Phys.Rev.E 59, 5142 (1999)
4. Enhancement of stochastic resonance in distribute systems due to selective coupling,
B. von Haeften, R. Deza & H.S. Wio, Physical Review Letters 84, 404 (2000).
5. Stochastic Resonance in Spatially Extended Systems: the Role of Far From Equilibrium
Potentials; H.S. Wio, B. Von Haeften & S.Bouzat, Proc. 21st IUPAP INTERNATIONAL
CONFERENCE ON STATISTICAL PHYSICS, STATPHYS21, Physica A 306, 140-156 (2002).
6. Aspects of stochastic resonance in reaction--diffusion systems: The nonequilibrium-potential
approach, H.S.Wio & R.R.Deza, Europ.Phys.J-Special Topics (Invited) 146, 111 (2007).approach, H.S.Wio & R.R.Deza, Europ.Phys.J-Special Topics (Invited) 146, 111 (2007).
7. Stochastic Resonance in Extended Systems: An Overview of Recent Results for Systems with and
without Nonequilibrium Potential, H.S.Wio, J.A.Revelli, M.A.Rodriguez, R.R.Deza & G.G.Izús,
Proc. Stochastic Resonance 2008, L.Gammaitoni, P.Hänngi, F.Marchesoni, Eds.,
Europ.Phys.J.B 69, 71 (2009)
8. Variational formulation for the KPZ and related kinetic equations, H. S. Wio,
Int. J. Bif. Chaos 19 (8), 2813 (2009)
9. KPZ equation: Galilean-invariance violation, Consistency, and fluctuation--dissipation issues in
real-space discretization, H.S.Wio, J.A.Revelli, R.R.Deza & C.Escudero,
submitted to Europhys.Lett.
THANKS!THANKS!
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