TAILIEUCHUNG - Nonlinear Dynamics Part 11

Tham khảo tài liệu 'nonlinear dynamics part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Entropic Geometry of Crowd Dynamics 243 x 0 x 0 i j 1 . N 33 or corresponding Ito stochastic integral equation x t x 0 J0tdsẠ xi s s fdWj s Bij x s s 34 in which xi t is the variable of interest the vector Ai x t t denotes deterministic drift the matrix Bij x t t represents continuous stochastic diffusion fluctuations and W j t is an N-variable Wiener process . generalized Brownian motion 23 and dWj t Wj t dt - Wj t . The two Ito equations 33 - 34 are equivalent to the general Chapman-Kolmogorov probability equation see equation 35 below . There are three well known special cases of the Chapman- Kolmogorov equation see 23 1. When both Bij x t t and W t are zero . in the case of pure deterministic motion it reduces to the Liouville equation d dtP x t x t - - A x t t P x t x t . ox 2. When only W t is zero it reduces to the Fokker-Planck equation d dtP x 11 x t - - A x t t P x 11 x t ox 1 _ d2 í X 2ZeXMKMWx x t . 2 ij dx dx 3. When both Ai x t t and Bij x t t are zero . the state-space consists of integers only it reduces to the Master equation of discontinuous jumps dtP x t x t ỊdxW x x t P x t x t - ỊdxW x x t P x t x t . The Markov assumption can now be formulated in terms of the conditional probabilities P xi ti if the times ti increase from right to left the conditional probability is determined entirely by the knowledge of the most recent condition. Markov process is generated by a set of conditional probabilities whose probability-density P P x t x t evolution obeys the general Chapman-Kolmogorov integro-differential equation d 1 d2 dtP -XFl A x t t P 2 YFF ĩ B x t t P i dx 2 dxldxỈK j f Ịdx W x x xr t P - W xr xx t P including deterministic drift diffusion fluctuations and discontinuous jumps given respectively in the first second and third terms on the . . This general Chapman-Kolmogorov integro-differential equation 35 with its conditional probability density evolution P P x t x t is represented by our SFT-partition function 31 . 244 Nonlinear

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