Stationary states in 2D systems driven by L evy noises Bart lomiej - PowerPoint PPT Presentation

Stationary states in 2D systems driven by L´ evy noises Bart� lomiej Dybiec and Krzysztof Szczepaniec Department of Statistical Physics, Institute of Physics, Jagiellonian University Krak´ ow, Poland Unsolved Problems of Noise Barcelona, 16th July 2015 Supported by:

Motivation: Boltzmann-Gibbs distribution In the equilibrium � − E � P (state) ∝ exp , k B T T – system temperature, E – energy of the state. For an overdamped particle the Langevin equation is dx dt = − V ′ ( x ) + � 2 k B T ξ ( t ) . Particle’s energy is E = V ( x ) and the stationary distribution � − V ( x ) � P ( x ) ∝ exp k B T is fully determined by the potential V ( x ). Bart� lomiej Dybiec Stationary states in 2D

Motivation & Outlook Motivation Examination of stationary states for more general noises. Road map of presentation basic definitions: 1D α -stable noises, 2D α -stable noises. stationary states for 1D and 2D systems. Try to understand role of increasing spatial dimensionality, universalities of noise driven systems. Take home message 2D α -stable noises differs from their 1D analogs but systems driven by 2D α stable noises display universal properties. Bart� lomiej Dybiec Stationary states in 2D

Noise in 1D A random variable is X is stable if AX (1) + BX (2) d = CX + D , where X (1) and X (2) are independent copies of X , d = denotes equality in distributions. The random variable X is called strictly stable if D = 0. The random variable X is symmetric stable if it is stable and Prob { X } = Prob {− X } . The random variable is α -stable if C = ( A α + B α ) 1 /α where 0 < α � 2. The characteristic function of α -stable densities is − σ α | k | α � 1 − i β sign k tan πα  � � � exp + i µ k 2   if α � = 1 , � e ikX �  φ ( k ) = E = 1 + i β 2 � � � � exp − σ | k | π sign k ln | k | + i µ k   if α = 1 ,  where α ∈ (0 , 2], β ∈ [ − 1 , 1], σ > 0 and µ ∈ R . Bart� lomiej Dybiec Stationary states in 2D G. Samorodnitsky, and M. S. Taqqu, Stable NonGaussian Random Processes , (Chapman & Hall 1994).

1D Characteristic function  � − σ α | k | α � � � exp 1 − i β sign k tan πα + i µ k , for α � = 1 ,  2 φ ( k ) = � � 1 + i β 2 � � exp − σ | k | π sign k ln | k | + i µ k , for α = 1 ,  asymptotic behavior P ( x ) ∝ | x | − ( α +1) ( α < 2), 0.6 Gauss Cauchy Normal distribution ( α = 2 , β = 0) 0.5 α =0.5 0.4 ( x − µ ) 2 � � 1 P(x) 0.3 √ exp − , 2 σ 2 2 πσ 0.2 0.1 Cauchy distribution ( α = 1 , β = 0) 0 -4 -3 -2 -1 0 1 2 3 4 x σ 1 α =0.9 α =1.5 ( x − µ ) 2 + σ 2 , 0.45 0.45 π β =-1.0 β =-1.0 β =-0.5 β =-0.5 0.4 0.4 β =0.0 β =0.0 β =0.5 β =0.5 0.35 β =1.0 0.35 β =1.0 0.3 0.3 L´ evy-Smirnoff distribution (fully asymmetric, α = 1 0.25 0.25 2 , β = 1) P(x) P(x) 0.2 0.2 0.15 0.15 � σ � 1 0.1 0.1 � � 2 ( x − µ ) − 3 σ 2 exp 0.05 0.05 − . 2 π 2( x − µ ) 0 0 -20-15-10 -5 0 5 10 15 20 -6 -4 -2 0 2 4 6 x x Bart� lomiej Dybiec Stationary states in 2D

Noise 2D Analogously like in 1D: Random vector X = ( X 1 , . . . , X d ) is said to be a stable random vector in R d if for any positive numbers A and B , there is a positive number C and a vector D such that A X (1) + B X (2) d = C X + D , where X (1) and X (2) are independent copies of X , d = denotes equality in distributions. The vector X is called strictly stable if D = 0 . The vector X is symmetric stable if it is stable and Prob { X ∈ A } = Prob {− X ∈ A } for any Borel set A of R d . A random vector is α -stable if C = ( A α + B α ) 1 /α where 0 < α � 2. Bart� lomiej Dybiec Stationary states in 2D

2D e i � k , X � � � The characteristic function φ ( k ) = E of the α -stable vector X = ( X 1 , . . . , X d ) in R d is  � � Γ( d s ) + i � k , µ 0 � � S d |� k , s �| α � � exp − 1 − i sign( � k , s � ) tan πα  2    for α � = 1 ,  φ ( k ) = � � 1 + i 2 Γ( d s ) + i � k , µ 0 � � S d |� k , s �| α � � exp − π sign( � k , s � ) ln( � k , s � )     for α = 1 ,  where S d is a unit sphere in R d and Γ( · ) is a spectral measure. G. Samorodnitsky, and M. S. Taqqu, Stable NonGaussian Random Processes , (Chapman & Hall 1994). Bart� lomiej Dybiec Stationary states in 2D

Cauchy distribution α =1 For symmetric spectral measure concentrated on intersections of the axes with the unit sphere S 2 the bi-variate Cauchy ( α = 1) distribution is p ( x , y ) = 1 ( x 2 + σ 2 ) × 1 σ σ ( y 2 + σ 2 ) . π π For continuous and uniform 0.9 0.7 spectral measure 0.5 0.3 0.1 p ( x , y ) = 1 σ � 6 � 4 � 2 0 2 4 6 ( x 2 + y 2 + σ 2 ) 3 / 2 . 2 π Bart� lomiej Dybiec Stationary states in 2D

Equations in 1D The Langevin equation dx dt = − V ′ ( x ) + σζ α, 0 ( t ) , dx = − V ′ ( x ) dt + σ dL α, 0 ( t ) is associated with the fractional Smoluchowski-Fokker-Planck equation + σ α ∂ α p ( x , t ) ∂ p ( x , t ) ∂ V ′ ( x ) p ( x , t ) � � = ∂ | x | α ∂ t ∂ x ∂ V ′ ( x ) p ( x , t ) − σ α ( − ∆) α/ 2 p ( x , t ) . � � = ∂ x The fractional Riesz-Weil derivative is defined via its Fourier transform � ∂ α p ( x , t ) � � � − ( − ∆) α/ 2 p ( x , t ) = −| k | α F [ p ( x , t )] . F = F ∂ | x | α P. D. Ditlevsen, Phys. Rev. E 60 172 (1999). D. Schertzer and M. Larchevˆ eque, J. Duan, V. V. Yanowsky, S. Lovejoy, J. Math. Phys. 42 200 (2001). Bart� lomiej Dybiec Stationary states in 2D

Equations in 1D For α < 2, and V ( x ) = | x | c stationary states exist for c > 2 − α . Stationary states (if exist) have power-law asymptotics p st ( x ) ∝ | x | − ( c + α − 1) . For c = 2 the stationary density is the same as the stable distribution associated with the underlying noise. 4 x 4 and α = 1 For V ( x ) = 1 σ p st ( x ) = π ( σ 4 / 3 − σ 2 / 3 x 2 + x 4 ) . A. V. Chechkin, J. Klafter, V. Yu. Gonchar, R. Metzler and L. V. Tanatarov, Chem. Phys. 284 233 (2002); Phys. Rev. E 67 , 010102 (2003). B. Dybiec, I. M. Sokolov, A. V. Chechkin, J. Stat. Mech. P07008 (2010). Bart� lomiej Dybiec Stationary states in 2D

Stationary states (quartic – V ( x ) = x 4 / 4 – potential) For α = 2, the stationary states are of the Boltzmann-Gibbs type, i.e. P ( x ) ∝ exp[ − V ( x )]. 0.45 Cauchy √ 0.4 Gauss − x 4 2 � � 0.35 P 2 ( x ) = 4 ) exp . Γ( 1 4 0.3 0.25 P st (x) 0.2 For α < 2, stationary solutions are 0.15 no longer of the Boltzmann-Gibbs 0.1 type. For α = 1 0.05 0 -3 -2 -1 0 1 2 3 x 1 P 1 ( x ) = π ( x 4 − x 2 + 1) . A. V. Chechkin, J. Klafter, V. Yu. Gonchar, R. Metzler and L. V. Tanatarov, Chem. Phys. 284 233 (2002); Phys. Rev. E 67 , 010102 (2003). Bart� lomiej Dybiec Stationary states in 2D

Equations in 2D 2D Langevin equation d r dt = −∇ V ( r ) + σ ζ α ( t ) , d r = −∇ V ( r ) dt + σ d L α ( t ) . Especially interesting potentials are 2 r 2 = 1 2 ( x 2 + y 2 ), harmonic: V ( x , y ) = 1 4 r 4 = 1 4 ( x 2 + y 2 ) 2 . quartic: V ( x , y ) = 1 Bart� lomiej Dybiec Stationary states in 2D

Bivariate Gaussian 0.4 0.3 0.3 0.2 p(x,y) p(x,y) p(x,y) p(x,y) 0.2 0.1 0.1 0 0 0 0 1.5 1.5 0.5 0.5 1 1 x x 1 0.5 1 0.5 y y 1.5 0 1.5 0 2 0.4 2 0.3 1 0.3 1 0.2 0 0.2 0 0.1 -1 0.1 -1 -2 0 -2 0 -2 -1 0 1 2 -2 -1 0 1 2 2 ( x 2 + y 2 ) (left panel) and V ( x , y ) = 1 4 ( x 2 + y 2 ) 2 V ( x , y ) = 1 (right panel) subject to the bi-variate, uniform Gaussian white noise ( α = 2). Bart� lomiej Dybiec Stationary states in 2D

Equations in 2D The associated Smoluchowski-Fokker-Planck equation ∂ p ( r , t ) = ∇ · [ ∇ V ( r ) p ( r , t )] + σ α Ξ p ( r , t ) , ∂ t where Ξ is the fractional operator. ∇ · [ ∇ V ( r ) p ( r , t )] originates due to the deterministic force F ( r ) = −∇ V ( r ) acting on a test particle. For the bi-variate α -stable noise with the uniform spectral measure the fractional operator Ξ = − ( − ∆) α/ 2 . For the bi-variate α -stable noise with the discrete symmetric spectral measure (located on intersections of S 2 with axis) ∂ α ∂ α Ξ = ∂ | x | α + ∂ | y | α . S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications (Gordon and Breach, Yverdon, 1993). A. V. Chechkin, V. Y. Gonchar, and M. Szydlowski, Phys. Plasmas 9 , 78 (2002). Bart� lomiej Dybiec Stationary states in 2D

Bivariate Cauchy – parabolic potential 0.4 0.05 0.04 0.3 p(x,y) p(x,y) p(x,y) p(x,y) 0.03 0.2 0.02 0.1 0.01 0 0 0 0 0.5 1.5 2.5 2 0.5 1 1 1.5 1.5 x x 1 1 0.5 y y 2 2.5 0 0.5 1.5 0 2 0.4 3 0.05 2 0.04 1 0.3 1 0.03 0 0.2 0 0.02 -1 -1 0.1 0.01 -2 -2 0 -3 0 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 2 ( x 2 + y 2 ) with α = 1 (Cauchy noise). V ( x , y ) = 1 Bart� lomiej Dybiec Stationary states in 2D