Frequency shift of Duffing oscillator with Boltzmann-distributed amplitude

Question

I have two coupled oscillators, namely a particle in an ion trap and an RLC circuit. The particle oscillates in this ion trap and induces charges in the RLC circuit, in turn the RLC circuit's voltage oscillates and drives the particle. Energy is dissipated via the resistor in the RLC circuit and at the same time added due to thermal noise.

The trap potential has a small nonlinear perturbation. The corresponding differential equations for the particle and circuit respectively are:

\begin{align} \ddot x + C_2 x + C_4 x^3 &= \kappa L I \\ \ddot I + \frac{1}{RC} \dot I + \frac{1}{LC} I &= \frac{1}{LC} I_{noise} + \frac{\kappa}{LC} \dot x \end{align}

Where $x$ is the displacement of the ion in the potential and $I$ is the current through the coil of the RLC circuit (one could also choose the current through the resistor or capacitor, it doesn't matter). $R$ is small enough that the eigenfrequency of the RLC circuit is the one of the undamped LC circuit. Additionally I am interested in the on-resonance case, namely that the undisturbed particle and circuit frequencies are the same: $ \omega_{0, circuit}^2 = \frac{1}{LC} = C_2 = \omega_{0,ion}^2$. The noise is white random noise, i.e. its autocorrelation time is a Dirac-distribution but it has a certain standard deviation $\sigma(I_{noise(t)}) > 0$.

The small nonlinearity in the trapping potential causes the eigenfrequency of the particle to shift slighlty. I see numerically that this is a function of $\kappa$. My question is: How does one calculate the frequency shift as a function of $\kappa$?

Answer 1

The principled way of introducing the temperature is by adding the damping and noise terms, related via the Einstein relation, so that the stationary distribution is the Boltzmann distribution. I.e., we start with $$ \ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \xi(t), $$ where $$ \langle \xi(t)\rangle = 0,\\ \langle \xi(t)\xi(t')\rangle = D\delta(t-t'), $$ formulate the equivalent Fokker-Planck equation, and its equilibrium solution will be a Boltzmann one, with the temperature expressed in terms of $D$ and $\delta$. This damping and driving thus may be not exactly the same as you already have implicitly in your system - they are a way to drive the system to the Boltzmann distribution. One could then add the driving terms to the Fokker-Planck equation, $\sim\cos(\omega t)$ to study the resulting response characteristic.

As a shortcut around it one could simply solve the equation for frequency and average the results over the Boltzmann distribution with the potential in the equation: $$ -\partial_x V(x) = -\alpha x -\beta x^3 $$

Update
In response to the expanded question: the straighforward approach would be to write a joint Fokker-Plank equation for the particle and the circuit, $$ \partial_t P(x, \dot{x}, I, \dot{I},t) = ... $$ and study its behavior. In particular, if $C_2$ and $C_4$ are both positive, the solution is globally stable and one can write the equilibrium Boltzmann-like solution of the FPE. Note that this is not a true Boltzmann distribution, since one cannot define a potential energy, as the coupling term contains time derivative $\dot{x}$.

I could suggest as a rough estimate to calculate the equilibrium state of the non-linear system in absence of noise, by first writing it as a system of the four first order equations in new variables $y=\dot{x},J=\dot{I}$: $$ \dot{x} = y,\\ \dot{y} = -C_2 x - C_4 x^3 + \kappa L I,\\ \dot{I} = J,\\ \dot{J} = -\frac{J}{RC} - \frac{I}{LC} + \frac{\kappa y}{LC}. $$ One can then perform the standard non-linear analysis by finding the equilibrium solutions of this system, linearizing it around this solution, finding the eigenfrequencies near this equilibrium, and, finally, writing the Boltzmann-like distribution near the equilibrium.