Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
140 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Colored Stochastic Multiplicative Processes with Additive Noise Unveil a Third-Order PDE, Defying Conventional FPE and Fick-Law Paradigms (2404.14229v1)

Published 22 Apr 2024 in math.ST, cond-mat.stat-mech, math.PR, and stat.TH

Abstract: Research on stochastic differential equations (SDE) involving both additive and multiplicative noise has been extensive. In situations where the primary process is driven by a multiplicative stochastic process, additive white noise typically represents an intrinsic and unavoidable fast factor, including phenomena like thermal fluctuations, inherent uncertainties in measurement processes, or rapid wind forcing in ocean dynamics. This work focuses on a significant class of such systems, particularly those characterized by linear drift and multiplicative noise, extensively explored in the literature. Conventionally, multiplicative stochastic processes are also treated as white noise in existing studies. However, when considering colored multiplicative noise, the emphasis has been on characterizing the far tails of the probability density function (PDF), regardless of the spectral properties of the noise. In the absence of additive noise and with a general colored multiplicative SDE, standard perturbation approaches lead to a second-order PDE known as the Fokker-Planck Equation (FPE), consistent with Fick's law. This investigation unveils a notable departure from this standard behavior when introducing additive white noise. At the leading order of the stochastic process strength, perturbation approaches yield a \textit{third-order PDE}, irrespective of the white noise intensity. The breakdown of the FPE further signifies the breakdown of Fick's law. Additionally, we derive the explicit solution for the equilibrium PDF corresponding to this third-order PDE Master Equation. Through numerical simulations, we demonstrate significant deviations from outcomes derived using the FPE obtained through the application of Fick's law.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (32)
  1. J. Deutsch, “Probability distributions for multicomponent systems with multiplicative noise,” Physica A: Statistical Mechanics and its Applications 208, 445–461 (1994a).
  2. H. Nakao, “Asymptotic power law of moments in a random multiplicative process with weak additive noise,” Physical Review E 58 (1998), 10.1103/PhysRevE.58.1591.
  3. H. Fujisaka and T. Yamada, “A New Intermittency in Coupled Dynamical Systems,” Progress of Theoretical Physics 74, 918–921 (1985), https://academic.oup.com/ptp/article-pdf/74/4/918/5362597/74-4-918.pdf .
  4. T. Yamada and H. Fujisaka, “Effect of inhomogeneity on intermittent chaos in a coupled system,” Physics Letters A 124, 421–425 (1987).
  5. A. S. Pikovsky, “Statistics of trajectory separation in noisy dynamical systems,” Physics Letters A 165 (1992), 10.1016/0375-9601(92)91049-W.
  6. N. Platt, S. M. Hammel,  and J. F. Heagy, “Effects of additive noise on on-off intermittency,” Phys. Rev. Lett. 72, 3498–3501 (1994).
  7. D. Sornette, “Multiplicative processes and power laws,” Physical Review E 57, 4811–4813 (1998), arXiv:9708231 .
  8. D. Sornette and R. Cont, ‘‘Convergent multiplicative processes repelled from zero: Power laws and truncated power laws,” Journal de Physique II 7 (1997), 10.1051/jp1:1997169.
  9. A. Schenzle and H. Brand, “Multiplicative stochastic processes in statistical physics,” Phys. Rev. A 20, 1628–1647 (1979).
  10. R. Graham, M. Höhnerbach,  and A. Schenzle, “Statistical properties of light from a dye laser,” Phys. Rev. Lett. 48, 1396–1399 (1982).
  11. M. Levy and S. Solomon, “Power laws are logarithmic "boltzmann" laws,” International Journal of Modern Physics C 07, 595–601 (1996), https://doi.org/10.1142/S0129183196000491 .
  12. H. Takayasu, A.-H. Sato,  and M. Takayasu, “Stable infinite variance fluctuations in randomly amplified langevin systems,” Phys. Rev. Lett. 79, 966–969 (1997).
  13. M. Turelli, “Random environments and stochastic calculus,” Theoretical Population Biology 12, 140–178 (1977).
  14. J. Deutsch, “Probability distributions for one component equations with multiplicative noise,” Physica A: Statistical Mechanics and its Applications 208, 433–444 (1994b).
  15. S. Lepri, “Chaotic fluctuations in graphs with amplification,” Chaos, Solitons & Fractals 139, 110003 (2020).
  16. The general prescription is that there is a time \mathaccentV⁢b⁢a⁢r⁢016⁢τ\mathaccentV𝑏𝑎𝑟016𝜏\mathaccentV{bar}016\tauitalic_b italic_a italic_r 016 italic_τ such that, for any time t𝑡titalic_t, the instances of ξ𝜉\xiitalic_ξ at times t′>t+\mathaccentV⁢b⁢a⁢r⁢016⁢τsuperscript𝑡′𝑡\mathaccentV𝑏𝑎𝑟016𝜏t^{\prime}>t+\mathaccentV{bar}016\tauitalic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_t + italic_b italic_a italic_r 016 italic_τ are “almost statistically uncorrelated” with the instances of ξ𝜉\xiitalic_ξ at times t′<tsuperscript𝑡′𝑡t^{\prime}<titalic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_t. For “almost statistically uncorrelated” we mean that the joint probability density functions factorize up to terms O⁢(\mathaccentV⁢b⁢a⁢r⁢016⁢τ)𝑂\mathaccentV𝑏𝑎𝑟016𝜏O(\mathaccentV{bar}016\tau)italic_O ( italic_b italic_a italic_r 016 italic_τ ): pn⁢(ξ,t1′;ξ2,t2′;…;ξk,tk′;ξk+1,t1;…;ξn,th)=pk⁢(ξ,t1′;ξ2,t2′;…;ξk,tk′)⁢\tmspace+.1667⁢e⁢m⁢ph⁢(ξk+1,t1;…;ξn,th)+O⁢(\mathaccentV⁢b⁢a⁢r⁢016⁢τ)subscript𝑝𝑛𝜉superscriptsubscript𝑡1′subscript𝜉2superscriptsubscript𝑡2′…subscript𝜉𝑘superscriptsubscript𝑡𝑘′subscript𝜉𝑘1subscript𝑡1…subscript𝜉𝑛subscript𝑡ℎsubscript𝑝𝑘𝜉superscriptsubscript𝑡1′subscript𝜉2superscriptsubscript𝑡2′…subscript𝜉𝑘superscriptsubscript𝑡𝑘′\tmspace.1667𝑒𝑚subscript𝑝ℎsubscript𝜉𝑘1subscript𝑡1…subscript𝜉𝑛subscript𝑡ℎ𝑂\mathaccentV𝑏𝑎𝑟016𝜏p_{n}(\xi,t_{1}^{\prime};\xi_{2},t_{2}^{\prime};...;\xi_{k},t_{k}^{\prime};\xi% _{k+1},t_{1};...;\xi_{n},t_{h})=p_{k}(\xi,t_{1}^{\prime};\xi_{2},t_{2}^{\prime% };...;\xi_{k},t_{k}^{\prime})\tmspace+{.1667em}p_{h}(\xi_{k+1},t_{1};...;\xi_{% n},t_{h})+O(\mathaccentV{bar}016{\tau})italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_ξ , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; italic_ξ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; … ; italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; italic_ξ start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; … ; italic_ξ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ) = italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_ξ , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; italic_ξ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; … ; italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) + .1667 italic_e italic_m italic_p start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; … ; italic_ξ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ) + italic_O ( italic_b italic_a italic_r 016 italic_τ ) with k,h,n∈ℕ𝑘ℎ𝑛ℕk,h,n\in\mathbb{N}italic_k , italic_h , italic_n ∈ blackboard_N, k+h=n𝑘ℎ𝑛k+h=nitalic_k + italic_h = italic_n and ti′>tj+\mathaccentV⁢b⁢a⁢r⁢016⁢τsuperscriptsubscript𝑡𝑖′subscript𝑡𝑗\mathaccentV𝑏𝑎𝑟016𝜏t_{i}^{\prime}>t_{j}+\mathaccentV{bar}016{\tau}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_b italic_a italic_r 016 italic_τ. For example, p2⁢(ξ,t′;ξ2,t)=p1⁢(ξ,t′)⁢\tmspace+.1667⁢e⁢m⁢p1⁢(ξ2,t)+O⁢(\mathaccentV⁢b⁢a⁢r⁢016⁢τ)subscript𝑝2𝜉superscript𝑡′subscript𝜉2𝑡subscript𝑝1𝜉superscript𝑡′\tmspace.1667𝑒𝑚subscript𝑝1subscript𝜉2𝑡𝑂\mathaccentV𝑏𝑎𝑟016𝜏p_{2}(\xi,t^{\prime};\xi_{2},t)=p_{1}(\xi,t^{\prime})\tmspace+{.1667em}p_{1}(% \xi_{2},t)+O(\mathaccentV{bar}016{\tau})italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ξ , italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; italic_ξ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t ) = italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ξ , italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) + .1667 italic_e italic_m italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t ) + italic_O ( italic_b italic_a italic_r 016 italic_τ )..
  17. I. A. Lubashevsky, A. Heuer, R. Friedrich,  and R. Usmanov, “Continuous Markovian model for Lévy random walks with superdiffusive and superballistic regimes,” The European Physical Journal B 78, 207–216 (2010).
  18. The cause of this issue lies in the fact that x=0𝑥0x=0italic_x = 0 is both a stable attraction point for the unpertubed motion (ϵ⁢τ2=0italic-ϵsuperscript𝜏20\epsilon\tau^{2}=0italic_ϵ italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0) and a point where the perturbation vanishes. Hence, given that any perturbed trajectory will eventually arrive at x=0𝑥0x=0italic_x = 0, this is an accumulation point of any initial “ensemble”.
  19. M. Bianucci and P. Grigolini, “Nonlinear and non Markovian fluctuation-dissipation processes: A Fokker-Planck treatment,” The Journal of Chemical Physics 96, 6138–6148 (1992).
  20. M. Bianucci and R. Mannella, “Optimal FPE for non-linear 1D-SDE. i: Additive Gaussian colored noise,” Journal of Physics Communications 4, 105019 (2020).
  21. M. Bianucci, L. Bonci, G. Trefan, B. J. West,  and P. Grigolini, “Brownian motion generated by a two-dimensional mapping,” Physics Letters A 174, 377 – 383 (1993a).
  22. M. Bianucci, R. Mannella, X. Fan, P. Grigolini,  and B. J. West, “Standard fluctuation-dissipation process from a deterministic mapping,” Phys. Rev. E 47, 1510–1519 (1993b).
  23. M. Bianucci, B. J. West,  and P. Grigolini, “Probing microscopic chaotic dynamics by observing macroscopic transport processes,” Physics Letters A 190, 447 – 454 (1994).
  24. M. Bianucci, “Ordinary chemical reaction process induced by a unidimensional map,” Phys. Rev. E 70, 026107–1–12617–6 (2004).
  25. R. Zwanzig, ed., Nonequilibrium Statistical Mechanics (Oxford University Press, Oxford, 2001).
  26. P. Grigolini, “The projection approach to the Fokker-Planck equation: applications to phenomenological stochastic equations with colored noises,” in Noise in Nonlinear Dynamical Systems, Vol. 1, edited by F. Moss and P. V. E. McClintock (Cambridge University Press, Cambridge, England, 1989) Chap. 5, p. 161.
  27. M. Bianucci, R. Mannella, B. J. West,  and P. Grigolini, “From dynamics to thermodynamics: Linear response and statistical mechanics,” Phys. Rev. E 51, 3002–3022 (1995).
  28. M. Bianucci, “On the correspondence between a large class of dynamical systems and stochastic processes described by the generalized Fokker-Planck equation with state-dependent diffusion and drift coefficients,” Journal of Statistical Mechanics: Theory and Experiment 2015, P05016 (2015).
  29. M. Bianucci, M. Bologna,  and R. Mannella, “About the optimal FPE for non-linear 1d-SDE with Gaussian noise: the pitfall of the perturbative approach,” J. Stat. Phys.  (2023).
  30. M. Bianucci, “Using some results about the Lie evolution of differential operators to obtain the Fokker-Planck equation for non-Hamiltonian dynamical systems of interest,” Journal of Mathematical Physics 59, 053303 (2018), https://doi.org/10.1063/1.5037656 .
  31. \tmspace+.1667⁢e⁢m1⁢F1⁢(a;b;z):=\sum@⁢\slimits@k=0∞⁢akbk⁢k!assign\tmspace.1667𝑒subscript𝑚1subscript𝐹1𝑎𝑏𝑧\sum@superscriptsubscript\slimits@𝑘0subscript𝑎𝑘subscript𝑏𝑘𝑘\tmspace+{.1667em}_{1}F_{1}\left(a;b;z\right):=\sum@\slimits@_{k=0}^{\infty}% \frac{a_{k}}{b_{k}k!}+ .1667 italic_e italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a ; italic_b ; italic_z ) := start_POSTSUBSCRIPT italic_k = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_k ! end_ARG, where (x)n=x⁢(x+1)⁢…⁢(x+n−1)=Γ⁢(x+n)/Γ⁢(x)subscript𝑥𝑛𝑥𝑥1…𝑥𝑛1Γ𝑥𝑛Γ𝑥(x)_{n}=x(x+1)\ldots(x+n-1)=\Gamma(x+n)/\Gamma(x)( italic_x ) start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_x ( italic_x + 1 ) … ( italic_x + italic_n - 1 ) = roman_Γ ( italic_x + italic_n ) / roman_Γ ( italic_x ) is the Pochhammer symbol.
  32. M. Bianucci and M. Bologna, “About the foundation of the Kubo generalized cumulants theory: a revisited and corrected approach,” Journal of Statistical Mechanics: Theory and Experiment 2020, 043405 (2020).

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now