Switching-path distribution in multidimensional systems.
Summary (3 min read)
Introduction
- The theory of the distribution is developed and its direct measurement is performed in a micromechanical oscillator driven into parametric resonance.
- When the fluctuation intensity is small, for most of the time the system fluctuates about one of the stable states.
- In Sec. II the authors provide the qualitative picture of switching and give a preview of the central theoretical and experimental results.
- In the transition the system most likely moves first from the vicinity of qA1 to the vicinity of qS.
A. Shape of the switching-path distribution
- It follows from Eq. 3 that the overall probability flux along the MPSP is equal to the switching rate, d p12 q,t v = W12.
- The authors demonstrate that the cross-section of the ridge has a Gaussian shape.
- In such systems, attractors are periodic functions of time.
- For switching in systems with more than one dynamical variable, the formulation 26,27,31 no longer applies because it cannot be known in advance through what points the system will pass.
A. Model of a fluctuating system
- Here, vector K determines the dynamics in the absence of noise; K=0 at the stable state positions qA1, qA2 and at the saddle point qS.
- This is also the case for noise-driven extended systems, cf. Refs. 35–37 and papers cited therein.
- There exists extensive literature on numerical calculations of the switching rate and switching paths, cf. Refs. 38–42 and papers cited therein.
- The analysis is done separately for the cases where the observation point q lies within the attraction basins of the initially empty attractor A2 and the initially occupied attractor A1.
B. switching-path distribution in the initially unoccupied basin of attraction
- The authors start with the case where the observation point q lies in the basin of attraction of the initially empty state A2 far 051109-3 from the stationary states, q−qS , q−qA1,2 lD.
- For weak noise intensity, the system found at such q will most likely approach qA2 over time tr moving close to the noise-free trajectory q̇=K and will then fluctuate about qA2.
- In its central part it has the form of a normalized Gaussian peak centered at qA2, with typical size lD.
- Having passed through the region near the saddle point the system moves close to the deterministic downhill trajectory from S to A2, cf. Fig.
- The authors parametrize the deterministic section of the MPSP by its length counted off from qS and introduce a unit vector ̂ along the vector K on the MPSP and N−1 vectors perpendicular to it.
C. switching-path distribution in the initially occupied basin of attraction
- The case where the observation point q lies in the basin of attraction of the initially occupied state A1 is somewhat more complicated.
- Using condition H=0, one can associate SF 0 q with the mechanical action for reaching q from qA1; the motion formally starts at t→− from qA1, with momentum p=0 15 .
- And so should the auxiliary particle, too.
- The MPSP is thus given by the heteroclinic Hamiltonian trajectory that goes from the state qA1 ,p=0 to qS ,p=0 .
- From conservation of the stationary probability current it follows that the distribution p12 q , t should sharply increase near the saddle point.
D. switching-path distribution for systems with detailed balance
- The quasistationary solution of the forward FokkerPlanck equation 8 near a saddle point has been known since the work of Kramers 4 and Landauer and Swanson 47 .
- This demonstrates that the distribution p12 q , t does not diverge at the saddle point, but it contains a large factor D−1/2.
A. Device characteristics
- The authors measure the switching-path distribution using a high-Q microelectromechanical torsional oscillator Q 051109-5 =9966 driven into parametric resonance.
- The 2 m gap underneath the movable plate is created by etching away a sacrificial silicon oxide layer.
- The dc voltage Vdc 1 V is much larger than the amplitude Vac 141 mV of sinusoidal modulation and the random noise voltage Vnoise.
- The renormalization of this term by the nonlinear terms in Eq. 22 for example, 0 −2F cos t is small for their device and is disregarded.
- Torsional oscillations of the top plate are detected capacitively by the other electrode.
B. Transformation to slow variables and parametric resonance
- Equation q̇=K gives the downhill section of the MPSP of the oscillator.
- The remaining two parameters are extracted from the parametric resonance of the oscillator for close to 2 1.
- The theoretical optimal escape path in Fig. 2 is calculated with the above parameter values.
A. Measured switching-path distribution
- When white noise is added to the excitation voltage, the system can occasionally overcome the activation barrier and switch from one stable state to the other.
- There are no adjustable parameters since all device parameters are accurately determined from the harmonic and parametric resonances of the oscillator without noise in the excitation as described in the previous section.
- Motion near the saddle point is dominated by diffusion.
- Figure 6 b plots the area under the Gaussian distribution versus the reciprocal measured velocity on the MPSP, for different cross sections.
- In the basin of attraction to A2 but not too close to A2, much of the probability distribution carries the switching current.
C. Lack of time reversal symmetry in a driven oscillator
- Another generic feature of the observed distribution is characteristic of systems far from thermal equilibrium.
- The authors data show that the uphill section of the MPSP, which is formed by fluctuations, the dissipationreversed path, and the downhill noise-free path from the saddle to the stable state are all distinct.
- The switching-path distribution was shown theoretically to have a shape of a narrow ridge in phase space.
- All parameters of the oscillator, including the nonlinearity constant, were directly measured.
- Measuring the switching trajectories can help to determine the model globally, far from the stable states.
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Citations
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References
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Frequently Asked Questions (10)
Q2. What is the probable fluctuational path from the saddle point to the attractor?
For equilibrium systems, the most probable fluctuational path uphill, i.e., from an attractor to the saddle point, is the time reversal of the fluctuation-free downhill path from the saddle point back to the attractor.
Q3. What is the effect of adding white noise to the excitation voltage?
When white noise is added to the excitation voltage, the system can occasionally overcome the activation barrier and switch from one stable state to the other.
Q4. What are the parameters used for the calculation of the MPSP?
Calculation of the MPSP requires a number of device parameters including , 1, the parametric modulation amplitude ke, and the nonlinear constant nonlinear=3 /8 1 48 .
Q5. How many s2 is the parametric modulation amplitude ke?
the parametric modulation amplitude ke is determined from the bifurcation frequencies 2 b1,2=2 1 p, where p= ke 2 − 4 1 2 1/2 /2 1. This gives ke=1.94 107 s−2.
Q6. How many dimensionless constants are in Eqs. 26 and 25?
27Using these measured device parameters, the dimensionless constants contained in K in Eqs. 26 and 25 can be directly calculated to be E=176.349, =4.968, and =0.9367.
Q7. What is the condition that p12 q, t be maximal on the MP?
From conservation of the stationary probability current it follows that the distribution p12 q , t should sharply increase near the saddle point.
Q8. What is the renormalization of the term cos t?
The renormalization of this term by the nonlinear terms in Eq. 22 for example, 0−2F cos t is small for their device and is disregarded.
Q9. What is the typical relaxation time of the MPSP?
If the typical relaxation time is smaller than or of the order of the modulation period, the MPSPs are well-synchronized and periodically repeat in time.
Q10. How many paths were followed in switching between the two dynamical variables?
The paths followed in switching between these states were accumulated and their distribution in the space of the two dynamical variables the oscillation quadratures was obtained.