Chapter Three Theoretical Description Of Stochastic Resonance 24

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1 Table of Contents List of Abbreviations and Symbols 5 Chapter One Introduction The Phenomenon of the Stochastic Resonance The Purpose of the Study 10 Chapter Two The Experimental Set-up The Electric Circuit Experimental Realisation Methods of Signal Characterisation Stochastic Resonance Measures Spectral Amplification Signal-to-Noise Ratio Experimental Set-up 19 Chapter Three Theoretical Description Of Stochastic Resonance Effect Basics System with Double-well Potential System Response Stochastic Resonance Characteristics Spectral Amplification Signal-to-Noise Ratio Stochastic Resonance in Continuous Bistable System Fokker-Planck Description Floquet Approach Expressions for Stochastic Resonance Characteristics Expression for Spectral Amplification 37

2 Expression for Signal-to-Noise Ratio Results of Simulations Intrawell Motion Contribution Linear Response Approximation Concluding Remarks Ferroelectric TGS Crystal as a System Displaying Stochastic Resonance Frequency Scaling 47 Chapter Four Experimental Results Signatures of Stochastic Resonance Synchronisation and Signal Enhancement Behaviour of Spectral Amplification Behaviour of Signal-to-Noise Ratio Discussion Characterisation of Stochastic Resonance Frequency Dependences Discussion Amplitude Dependences Discussion Temperature Dependence of Stochastic Resonance Behaviour Behaviour of Stochastic Resonance Measures at Different Temperatures of Ferroelectric TGS Frequency Scaling Discussion 77 Chapter Five Conclusions and Outlook Outlook 81 References 83

3 5 List of Abbreviations and Symbols: A surface area A 0 amplitude of periodic modulation A A ~ 0 0 = ax m scaled amplitude a,b parameters of double-well potential AS C 0 C F c k d D D m asymptotic linear capacitance ferroelectric capacitance Fourier coefficient sample thickness noise intensity noise intensity that maximises system response D D ~ = scaled noise intensity 2 ax m E electric field strength eq. equation f frequency f ( x ) = V ( x ) / m scaled first derivative of the potential V(x) g n H(t) i expansion coefficient Heaviside step function imaginary unit k, m, n indexes 0 K xx ( t) correlation function

4 6 L 0 (t) unperturbed Fokker-Planck operator L ext (t) Fokker-Planck operator of periodic perturbation L * (t) adjoint Fokker-Planck operator m mass M n p {p µ } Floquet modes complex valued amplitudes of the system response probability density P Ρ(X,t Y, s) P P 1 P A P n Q polarisation transition probability density power integrated power of the delta-like peak at the frequency f=ω total power of the modulation signal in the absence of noise integrated power of δ-peaks of the n-th frequency component electric charge Q F r K R S(ω) S N (ω) SNR t, s, τ time t 0 electric charge of nonlinear capacitance C F Kramer s rate ohmic resistance output spectral density spectral density of noise signal-to-noise ratio initial time ~ at t = scaled time γ T T K T Ω TGS period period of Kramer s hopping period of periodic modulation triglycine sulfate C 0 voltage drop over C 0 C F voltage drop over C F G R driving voltage (periodic modulation) voltage drop over R

5 7 v V(x) V(x) x(t) x 0 =x(t 0 ) x m velocity double-well potential height of the potential barrier one-dimensional time-dependent coordinate initial condition coordinate of the potential minima x (D) periodic component of the system response x x~ = scaled variable x δx(t) X(ω) x m X(t), Y(s) Z χ(t) χ(ω) δ φ γ η ϕ λ n system response within linear response approximation Fourier transform of x(t) state vectors impedance response function Fourier transform of χ(t) delta-function phase shift viscous friction spectral amplification phase eigenvalue of Fokker-Planck operator µ Floquet eigenvalue Θ Θ τ smpl ω 0 ω k, ω n Ω temperature accuracy of temperature measurement sample rate angular frequency discrete angular frequency angular frequency of external periodic modulation ~ Ω Ω = γ scaled angular frequency a ξ(t) Gaussian white noise