Stochastic resonance

Starting equation

Consider a time-dependent physical variable ${\displaystyle x(t)}$ (which, in the original climate papers, was the mean temperature of the Earth), described by a stochastic differential equation of the Langevin equation type:

$${\displaystyle {\frac {dx}{dt}}=-{\frac {\partial U}{\partial x}}+f(t)+\epsilon \cos(\omega t),}$$

where ${\displaystyle U(x)}$ is a potential function determining the system dynamics, ${\displaystyle f(t)}$ is a stochastic forcing corresponding to background noise, and ${\displaystyle \epsilon \cos(\omega t)}$ is the external periodic perturbation, of small amplitude (${\displaystyle \epsilon \ll 1}$), which in the climate studies corresponded to the Milankovitch cycles. It is assumed that ${\displaystyle U(x)}$ has a double-well shape (describable by a fourth-degree polynomial), i.e., it possesses two stable minima ${\displaystyle x_{-}}$ and ${\displaystyle x_{+}}$ (glacial and interglacial periods) separated by an unstable maximum ${\displaystyle x_{0}}$, corresponding to a potential barrier ${\displaystyle \Delta U}$ (for simplicity it is assumed that the two minima lie at the same level ${\displaystyle U(x_{\pm })}$). For ${\displaystyle f(t)}$ the simplest possible form is considered, namely that of Gaussian noise, with zero mean and Dirac delta correlation: ${\displaystyle \langle f(t)\rangle =0}$ and ${\displaystyle \langle f(t_{1})f(t_{2})\rangle =2D\delta (t_{2}-t_{1})}$.

In the absence of periodic perturbations (${\displaystyle \epsilon =0}$), it is well known from the theory of Markov processes that the variable ${\displaystyle x(t)}$ will fluctuate around the minimum points, with a variance proportional to the Gaussian noise intensity ${\displaystyle D}$. Occasionally the fluctuations will be strong enough to allow crossing of the potential barrier, and therefore the transition from one minimum state to the other. In the limit ${\displaystyle D\ll \Delta U}$, the mean transition frequency is given by the so-called Kramers formula:

$${\displaystyle r_{\pm }={\frac {1}{2\pi }}{\sqrt {-U^{''}(x_{0})U^{''}(x_{\pm })}}e^{-{\frac {\Delta U}{D}}},}$$ where ${\displaystyle U^{''}(x)}$ is the second derivative of ${\displaystyle U(x)}$.

Stochastic resonance

In the presence of periodic perturbations, the potential ${\displaystyle U}$ is replaced by the generalized potential ${\displaystyle W(x,t)=U(x)-\epsilon \cos \omega t}$. This means that the two potential wells vary in height, rising and falling, and therefore the barrier that the system must overcome to jump from one minimum to the other will at some moments be higher and at others lower. However, the oscillations are of small amplitude, so they are not able to completely remove the barrier and thus allow the system to transition between states on their own: the idea of stochastic resonance is that, if the perturbation frequency ${\displaystyle \omega }$ is in some way comparable with the mean transition frequency ${\displaystyle r_{\pm }}$, then the random fluctuations of the system will tend to synchronize with the external oscillations, making the transition more likely.

Using the explicit form of the potential ${\displaystyle U(x)=-{\frac {1}{2}}x^{2}+{\frac {1}{4}}x^{4}}$, it can be shown that, at least to first order approximation, the mean value of ${\displaystyle x(t)}$ becomes a periodic function of time with the same frequency as the external forcing:

$${\displaystyle \langle x(t)\rangle =A\cos(\omega t+\phi ),}$$

where the amplitude ${\displaystyle A}$ and the phase ${\displaystyle \phi }$ are given by the following relations:

$${\displaystyle A={\frac {\epsilon \langle x^{2}\rangle _{0}}{D}}{\frac {2r_{\pm }}{\sqrt {r_{\pm }^{2}+\omega ^{2}/4}}},\qquad \qquad \phi =-\arctan \left({\frac {\omega }{2r_{\pm }}}\right);}$$

with ${\displaystyle \langle x^{2}\rangle _{0}}$ the variance of ${\displaystyle x}$ in the absence of external perturbations (dependent on the noise intensity ${\displaystyle D}$).

The fundamental aspects are therefore:

• the system oscillations tend, on average, to synchronize with the external perturbations;
• the amplification effect is negligible unless the perturbation frequency ${\displaystyle \omega }$ is comparable with the transition frequency ${\displaystyle r_{\pm }}$ (dependent on the noise intensity), and it is maximal for ${\displaystyle \omega =2r_{\pm }}$;
• for fixed perturbation amplitude ${\displaystyle \epsilon }$ and frequency ${\displaystyle \omega }$, the amplitude ${\displaystyle A}$ is, for low noise intensities, a sharply increasing function of ${\displaystyle D}$, reaching a maximum peak at an intermediate noise level, and then becoming a decreasing function of ${\displaystyle D}$ for large noise values.