Effective stochastic behavior in dynamical systems with incomplete information

Complex systems are generally analytically intractable and difficult to simulate. We introduce a method for deriving an effective stochastic equation for a high-dimensional deterministic dynamical system for which some portion of the configuration is not precisely specified. We use a response function path integral to construct an equivalent distribution for the stochastic dynamics from the distribution of the incomplete information. We apply this method to the Kuramoto model of coupled oscillators to derive an effective stochastic equation for a single oscillator interacting with a bath of oscillators and also outline the procedure for other systems.

Figure 1

The vertices and propagator derived from the action $S_{\mathrm{bulk}}$. Compare with the diagrams from Ref. [7]. The open circle vertex represents initial condition terms which are the sampling corrections to the initial state. We label the coordinates of each vertex leg because these connections are nonlocal. $x=(\theta ,\omega ,t)$ and similarly for $x^{\prime }$.

Figure 2

The vertices and propagator derived from the action $S_{\phi }$. These vertices are nominally $O\left(1\right)$, but will contract with vertices from $S_{\mathrm{bulk}}$ and $S_{\mathrm{int}}$ such that moments will have $1/N$ scaling.

Figure 3

The vertices derived from $S_{\mathrm{int}}$. These are $O\left(1\right)$ but source the $\varphi$ operator, which means they can be regarded as counterterms. They will produce factors of $1/N$ in the overall moment for each such vertex.

Figure 4

Feynman diagrams for $\langle \delta \phi \left(t\right)\rangle$ derived directly from the action $S$ from Eq. (5). Dashed lines represnt $\varphi$, $\tilde{\varphi }$ variables; solid lines represent $\delta \phi \left(t\right)$, $\tilde{\phi }\left(t\right)$ variables.

Figure 5

$\delta \Omega \left(t\right)N$ vs $t$. Comparison of analytic and simulation results for different values of $K$ and $N=1000$. Bottom right figure shows the comparison with various values of $N$; $K=0.05$. Other parameters are $\gamma =0.05$, $\Omega =0.05$. Time step for simulation was $\delta t=0.1$ and the ensemble average was taken over 1 million samples. Dashed lines are the values indicated in the legend; adjoining solid lines are the analytic prediction.

Figure 6

Covariance of $d\phi \left(t\right)/dt$ at $t$ and $t_0=0$ s. Abscissa is $t$. Bottom right compares simulation and analytic prediction for different values of $N$ at $K=0.05$. All others compare analytic and simulation results for different values of $K$ at $N=1000$. Other parameters are $\gamma =0.05$, $\Omega =0.05$. Time step for simulation was $\delta t=0.1$ and the ensemble average was taken over 10 000 samples.

Figure 7

Covariance of $d\phi \left(t\right)/dt$ at $t$ and $t^{\prime }=100$ s, with $t_0=0$ s. Abscissa is $t$. Bottom right compares simulation and analytic prediction for different values of $N$ at $K=0.05$. All others compare analytic and simulation results for different values of $K$ at $N=1000$. Other parameters are $\gamma =0.05$, $\Omega =0.05$. Time step for simulation was $\delta t=0.1$ and the ensemble average was taken over 10 000 samples.

Figure 8

Variance of $\delta \phi (t+\delta t)-\delta \phi \left(t\right)$ for different values of $N$ with a comparison of simulation and analytic prediction (solid curve). Each curve has been normalized by $N$. The curves from bottom to top correspond to $N=10$, $N=100$, and $N=1000$, respectively. Other parameters are $\gamma =0.05$, $\Omega =0.05$, $K=0.05$. Deviations correspond to $O(1/N^2)$ corrections.Time step for simulation was $\delta t=0.1$ and the ensemble average was taken over 1 million samples.

Figure 9

Variance of $\delta \phi (t+\delta t)-\delta \phi \left(t\right)$ for different values of $K$ with a comparison of simulation (dashed lines) and analytic prediction (solid curves). Curves from bottom to top correspond to $K=0.01$, $K=0.02$, $K=0.05$, $K=0.06$, $K=0.08$, and $K=0.09$, respectively. Each curve has been normalized by $N$. Other parameters are $\gamma =0.05$, $\Omega =0.05$, $N=1000$. Time step for simulation was $\delta t=0.1$ and the ensemble average was taken over 1 million samples.