# A phase transition in a system driven by coloured noise

Abstract: For a system driven by coloured noise, we investigate the activation energy of escape, and the dynamics during the escape. We have performed analogue experiments to measure the change in activation energy as the power spectrum of the noise varies. An adiabatic approach based on path integral theory allows us to calculate analytically the critical value at which a phase transition in the activation energy occurs.

## Summary (2 min read)

### INTRODUCTION

- Gaussian white noise is a mathematical idealization which applies to a limited number of physical situations where noise has a broad power spectrum.
- More realistic models of random forces have also been proposed, and one of them, which is in some sense opposite to the white noise model, is quasi-monochromatic noise (QMN) [1, 2] .
- It has been attracting increasing attention in diverse scientific contexts, from crystal growth and current-induced desorption from crystal surfaces, to current switching in microstructures, and biological applications.
- For an overdamped system in a double well potential driven by QMN, a bifurcation in the optimal path (the optimal trajectory for departure from an initial attractor) was found theoretically [3] , leading to the prediction of a marked reduction in the mean escape time [1] .
- The authors show that this is a "continuous" transition which corresponds to spontaneous breaking of time symmetry of the most probable paths for escape from a metastable state.

### THE MODEL

- A simple picture of QMN is the noise which results from filtering white noise through a harmonic oscillator filter with natural frequency u; 0 and damping F -s).
- Here, £(t) represents Gaussian white noise of zero mean and intensity D, and the authors assume that F <C ^o- For weak noise /(t), the system mostly performs small fluctuations about one of the potential minima but, occasionally, a large fluctuation occurs in which the system switches from one potential well to another.
- The idea is to find the path x(i) which minimises the action 5'.
- This path is the solution of a variational problem which relates the optimal realization of the force £(£) (with a probability density functional which can be immediately found from ( 1)) and the trajectory of the coordinate (2), subject to the constraint that this trajectory starts at the potential minimum and ends at the saddle [1] .
- The action is the minimal value of the corresponding variational functional.

### H(x,p) = {woW

- The expressions for the generalised momenta are more complicated and the authors refer the interested reader to [3] .
- (8) and the phase space is reduced from six to four dimensions.the authors.
- Clearly such trajectories emerge in pairs.
- From (6) , they correspond to fastoscillating solutions x(i) of the original variational problem.
- When they provide the minimum to the action functional, this signals breaking of the time symmetry.

### THE DOUBLE WELL POTENTIAL

- For this potential, the authors were able to confirm the numerical results reported by Einchcomb and McKane.
- There is a critical value of F below which their adiabatic equations yield two solutions that provide extrema for the action.
- One of them gives a value for the action identical with that for white noise; the other corresponds to oscillatory motion.
- The authors note that the QMN-driven system provides a natural physical realization of the system discussed by Maier and Stein.

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