Solve the fde for the characteristic functional \(\theta[\mathbf{y}() ; t\).

\[ \frac{\partial}{\partial t} \theta[\mathbf{y} ; t]=i \int_{\mathcal{D}} d \mathbf{x} y_{\alpha}(\mathbf{x})\left(\frac{1}{F r} g_{\alpha}(\mathbf{x})-\frac{\partial P_{0}}{\partial x_{\alpha}}\right) \theta[\mathbf{y} ; t] \]

governed by the truncated Hopf fde (9.40) driven solely by a constant external force

\[ F_{\alpha} \equiv \frac{1}{F r} g_{\alpha}-\frac{\partial P_{0}}{\partial x_{\alpha}} \]

The initial functional is the Gaussian

\[ \theta_{G}[\mathbf{y} ; t]=\exp \left\{-\frac{1}{2}(\mathbf{K} \cdot \mathbf{y}, \mathbf{y})+i(\mathbf{a}, \mathbf{y})\right\} \]

with correlation operator \(\mathbf{K}\) and zero mean \(\mathbf{a}=0\).

10.1.1: Solve the IVP for the fde governing \(\theta[\mathbf{y}]\) with Gaussian initial condition.

10.1.2: Show that the characteristic functional remains Gaussian and evaluate the effect of the external force on it.

Transcribed Image Text:

1 82 80 [y; t]+illaly, g; t, x]+ [y; 1, x] Re Ox30x8 Sya t] dxya(x) lim 820 () 0[y; 1] = [ dxya (x) {i x x x 5y3(x)ya (x) \ +i(- Fr Ja(x). -- axa [) 0 [y: 1}]} (9.40)