ISSN:
1572-9613
Keywords:
Equivalent linearization
;
nonlinear stochastic partial differential equation
;
Kullback-Leibler entropy
;
Ginzburg-Landau model
;
relaxation from unstable state
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract We present a linearization procedure of a stochastic partial differential equation for a vector field (X i (t,x)) (t∈[0, ∞),x∈R d ,i=l,...,n): ∂ t X i (t,x)=b i (X(t, x)) +D, ΔX i (t, x) + σ i f i (t, x). HereΔ is the Laplace-Beltrami operator inR d , and (f i (t,x)) is a Gaussian random field with 〈f i (t,x)f j (t′,x′)〉 = δ ij δ(t − t′)δ(x − x′). The procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations. The linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the Kullback-Leibler entropy. The procedure is applied to the scalar-valued Ginzburg-Landau model in R1 withb 1(z) =μz - vz 3. Stationary values of mean, variance, and correlation length are calculated. They almost agree with exact ones ifμ ≲ 1.24 (ν 2θ 1 4 /D 1 1/3:=μ c . Whenμ⩾μ c , there appear quasistationary states fluctuating around one of the bottoms of the potentialU(z) = ∫b 1(z)dz. The second moment at the quasistationary states almost agrees with the exact one. Transient phenomena are also discussed. Half-width at half-maximum of a structure function decays liket −1/2 for small t. The diffusion term∂ x 2 X accelerates the relaxation from the neighborhood of an unstable initial stateX(0,x) ∼ 0.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01010500
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