Spectral Decomposition of a Fokker–Planck Equation at Criticality

Bologna, Mauro; Beig, Mirza Tanweer Ahmad; Grigolini, Paolo; West, Bruce J.

doi:10.1007/s10955-015-1262-5

Cited by 4 publications

(3 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[16] and, even more remarkable, close to the eigenvalue of the first state of a Schrödinger-like equation that the authors of Ref. [20] propose to study the nonlinear stochastic process discussed herein. It is important to stress that using the assumption that is this eigenvalue leads us to 1/α = 1.4, namely α = 0.71 rather than α = 0.69.…”

Section: Stochastic Linearizationsupporting

confidence: 55%

“…It is important to stress that using the assumption that is this eigenvalue leads us to 1/α = 1.4, namely α = 0.71 rather than α = 0.69. Adopting intuitive arguments, we can explain stochastic linearization as a consequence of the fact that xP eq (x), due to symmetry, mainly overlaps only one of the eigenstates of the operator L, the first excited state [20].…”

Section: Stochastic Linearizationmentioning

confidence: 99%

“…The time-dependent solution to the FPE is taken up in [20] using a spectral decomposition method. It is sufficient for our purposes here to note that the contribution from the nonzero eigenvalues to the asymptotic PDF all vanish.…”

Section: B Infinitely Aged Correlation Functionmentioning

confidence: 99%

See 2 more Smart Citations

Critical slowing down in networks generating temporal complexity

Bologna

et al. 2015

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

We study a nonlinear Langevin equation describing the dynamic variable X(t), the mean field (order parameter) of a finite size complex network at criticality. The conditions under which the autocorrelation function of X shows any direct connection with criticality are discussed. We find that if the network is prepared in a state far from equilibrium, X(0)=1, the autocorrelation function is characterized by evident signs of critical slowing down as well as by significant aging effects, while the preparation X(0)=0 does not generate evident signs of criticality on X(t), in spite of the fact that the same initial state makes the fluctuating variable η(t)≡sgn(X(t)) yield significant aging effects. These latter effects arise because the dynamics of η(t) are directly dependent on crucial events, namely the re-crossings of the origin, which undergo a significant aging process with the preparation X(0)=0. The time scale dominated by temporal complexity, aging, and ergodicity breakdown of η(t) is properly evaluated by adopting the method of stochastic linearization which is used to explain the exponential-like behavior of the equilibrium autocorrelation function of X(t).

show abstract

Section: Stochastic Linearizationsupporting

confidence: 55%

Section: Stochastic Linearizationmentioning

confidence: 99%