Stochastic Resonance
Stochastic Resonance Alden Astwood
Stochastic Resonance Alden Astwood
June 11, 2009
Ingredients Stochastic Resonance
1
Bistable (or multistable) System, like a double well. Double Well Potential
V(x)
Alden Astwood
x
Ingredients Stochastic Resonance
1
Bistable (or multistable) System, like a double well.
Alden Astwood
2
Subthreshold Periodic Tilting Force, F = a cos(ωt).
3
Thermal noise, hξ(t)ξ(t + τ )i = 2D0 δ(τ ), D0 = kT /mγ.
V(x)
Double Well Potential
x
Matching Time Scales Stochastic Resonance Alden Astwood
With no tilting, the characteristic transition rate is the Kramers rate: RK =
1 p 00 V (xmin )V 00 (xmax )exp(−∆V /D0 ) 2π
We have two time scales: TK = 1/RK and Tω = 2π/ω. SR occurs when these two processes work together: Tω = 2TK .
Power Spectral Density Stochastic Resonance Alden Astwood
Related to the Fourier transform of the autocorrelation function: Z ∞ S(Ω) = hx(t)x(t + τ )ie −iΩτ dτ −∞
If x(t) is not stationary, an average over t is required. For white noise, the power spectrum is a constant.
Example: Damped, Driven Oscillator Stochastic Resonance
mγ x˙ = −ω 2 x + R(t) + a cos(ωt) S(Ω) is a Lorentzian plus a delta function at Ω = ω. Power Spectral Density
S(Ω)
Alden Astwood
ω Ω
Signal to Noise Ratio Stochastic Resonance Alden Astwood
Power Spectral Density looks like: S(Ω) = N(Ω) + Sδ(Ω − ω) SNR = amplitude of delta function / amplitude of noise at the driving frequency: SNR = S/N(ω). Or, possibly in a decibel scale: 10 log10 (S/N). For the oscillator above, the SNR is proportional to 1/D0 . Stochastic Resonance is characterized by a maximum in the SNR for a particular value of D0 .
Two State Model Stochastic Resonance Alden Astwood
dP+ = R− (t)P− − R+ (t)P+ dt dP− = R+ (t)P+ − R− (t)P− dt P+ = probability to be in the right well. P− = probability to be in the left well. R− = transition rate from left well to right well. R+ = transition rate from right well to left well. Exact solution in general can be written down in terms of integrals over the rates.
Kindergarten model Stochastic Resonance Alden Astwood
If there is no driving force, then R+ (t) = R− (t) = R0 . You learned how to solve this in kindergarten: 1 1 −2R0 t P± = + e P± (0) − 2 2 The power spectrum S(Ω) is a Lorentzian with half-width 2RK .
But We’re Not in Kindergarten... Stochastic Resonance Alden Astwood
Need time dependent rates to model the tilting signal. Start with the Kramers rate for V0 = ∆V0 (x/c)4 − 2(x/c)2 : 1 p 00 V (xmin )V 00 (xmax )exp(−∆V /D0 ) 2π 4∆V0 R0 = √ exp(−∆V /D0 ) c 2 2π
RK =
Plug in the new potential V = V0 (x) − ax cos(ωt): R± (t) ≈ R0 exp[∓ac cos(ωt)/D0 ] Expand: R± (t) ≈ R0 [1 ∓ ac cos(ωt)/D0 ]
SNR for the two state model and Stochastic Resonance (finally!) Stochastic Resonance Alden Astwood
Delta function appears in S(Ω) at Ω = ω. HeightRof the Lorentzian is diminished such that total power S(Ω)dΩ is the same. SNR ≈
∆V0 a2 exp(−∆V0 /D0 ). πD02
SNR goes to zero for very small or very large D0 , but what about in between?
Stochastic Resonance
Alden Astwood
The SNR exhibits a peak at a critical value of the noise. This peak is the characteristic property of Stochastic Resonance. Signal to Noise Ratio
SNR
Stochastic Resonance
D0
How is this related to last week’s questions? Stochastic Resonance Alden Astwood
Same overdamped particle. Multistable rather than bistable potential. Tilt is static, but can be threshold or superthreshold. Thermal AND spatial noise.
Stochastic Resonance Alden Astwood
June 11, 2009
Ingredients Stochastic Resonance
1
Bistable (or multistable) System, like a double well. Double Well Potential
V(x)
Alden Astwood
x
Ingredients Stochastic Resonance
1
Bistable (or multistable) System, like a double well.
Alden Astwood
2
Subthreshold Periodic Tilting Force, F = a cos(ωt).
3
Thermal noise, hξ(t)ξ(t + τ )i = 2D0 δ(τ ), D0 = kT /mγ.
V(x)
Double Well Potential
x
Matching Time Scales Stochastic Resonance Alden Astwood
With no tilting, the characteristic transition rate is the Kramers rate: RK =
1 p 00 V (xmin )V 00 (xmax )exp(−∆V /D0 ) 2π
We have two time scales: TK = 1/RK and Tω = 2π/ω. SR occurs when these two processes work together: Tω = 2TK .
Power Spectral Density Stochastic Resonance Alden Astwood
Related to the Fourier transform of the autocorrelation function: Z ∞ S(Ω) = hx(t)x(t + τ )ie −iΩτ dτ −∞
If x(t) is not stationary, an average over t is required. For white noise, the power spectrum is a constant.
Example: Damped, Driven Oscillator Stochastic Resonance
mγ x˙ = −ω 2 x + R(t) + a cos(ωt) S(Ω) is a Lorentzian plus a delta function at Ω = ω. Power Spectral Density
S(Ω)
Alden Astwood
ω Ω
Signal to Noise Ratio Stochastic Resonance Alden Astwood
Power Spectral Density looks like: S(Ω) = N(Ω) + Sδ(Ω − ω) SNR = amplitude of delta function / amplitude of noise at the driving frequency: SNR = S/N(ω). Or, possibly in a decibel scale: 10 log10 (S/N). For the oscillator above, the SNR is proportional to 1/D0 . Stochastic Resonance is characterized by a maximum in the SNR for a particular value of D0 .
Two State Model Stochastic Resonance Alden Astwood
dP+ = R− (t)P− − R+ (t)P+ dt dP− = R+ (t)P+ − R− (t)P− dt P+ = probability to be in the right well. P− = probability to be in the left well. R− = transition rate from left well to right well. R+ = transition rate from right well to left well. Exact solution in general can be written down in terms of integrals over the rates.
Kindergarten model Stochastic Resonance Alden Astwood
If there is no driving force, then R+ (t) = R− (t) = R0 . You learned how to solve this in kindergarten: 1 1 −2R0 t P± = + e P± (0) − 2 2 The power spectrum S(Ω) is a Lorentzian with half-width 2RK .
But We’re Not in Kindergarten... Stochastic Resonance Alden Astwood
Need time dependent rates to model the tilting signal. Start with the Kramers rate for V0 = ∆V0 (x/c)4 − 2(x/c)2 : 1 p 00 V (xmin )V 00 (xmax )exp(−∆V /D0 ) 2π 4∆V0 R0 = √ exp(−∆V /D0 ) c 2 2π
RK =
Plug in the new potential V = V0 (x) − ax cos(ωt): R± (t) ≈ R0 exp[∓ac cos(ωt)/D0 ] Expand: R± (t) ≈ R0 [1 ∓ ac cos(ωt)/D0 ]
SNR for the two state model and Stochastic Resonance (finally!) Stochastic Resonance Alden Astwood
Delta function appears in S(Ω) at Ω = ω. HeightRof the Lorentzian is diminished such that total power S(Ω)dΩ is the same. SNR ≈
∆V0 a2 exp(−∆V0 /D0 ). πD02
SNR goes to zero for very small or very large D0 , but what about in between?
Stochastic Resonance
Alden Astwood
The SNR exhibits a peak at a critical value of the noise. This peak is the characteristic property of Stochastic Resonance. Signal to Noise Ratio
SNR
Stochastic Resonance
D0
How is this related to last week’s questions? Stochastic Resonance Alden Astwood
Same overdamped particle. Multistable rather than bistable potential. Tilt is static, but can be threshold or superthreshold. Thermal AND spatial noise.
