Generalized Nyquist theorem
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1 Non-Equilibrium Statistical Physics Prof. Dr. Sergey Denisov WS 215/16 Generalized Nyquist theorem by Stefan Gorol & Dominikus Zielke
2 Agenda Introduction to the Generalized Nyquist theorem Derivation of the theorem - Dissipation - Fluctuations - Relation between dissipation and fluctuations Further Applications - General strategy to obtain the fluctuation spectrum of a dissipative system - 1 st example: Brownian motion - 2 nd example: Electric field fluctuations and radiation (Planck s law) Summary and outlook Stefan Gorol & Dominikus Zielke March 21, 216 2
3 Introduction to the Generalized Nyquist theorem Based on the Nyquist theorem for the white noise of a resistor: V 2 = 2 π k BT R dω with the mean square of the voltage V 2, Boltzmann s constant k B, temperature T, angular frequency ω and resistance R Relates a fluctuating force of a system at thermal equilibrium with energy dissipation First stated (in its generalized form) by [Callen&Welton] (1951) Applicable to linear dissipative systems with small perturbations System is dissipative if it is capable of absorbing energy when subjected to a time periodic perturbation System is linear if the power dissipation is quadratic in the magnitude of the perturbation (e.g. electrical resistor: P = I 2 R) Stefan Gorol & Dominikus Zielke March 21, 216 3
4 Derivation of the theorem Dissipation 1/7 Suppose: Ĥ(t) = Ĥ + V (t) ˆQ Ĥ Hamiltonian of non-perturbated system (time independent) V (t) ˆQ time depentent perturbation Furthermore suppose: E 1, E 2,... are eigenfunctions of Ĥ with corresponding eigenvalues E 1, E 2,... Now solve the Schrödinger Equation i t φ(t) S = Ĥ φ(t) S Use linear perturbation theory in the interaction picture ( ) iĥt φ(t) S = exp φ(t) I Stefan Gorol & Dominikus Zielke March 21, 216 4
5 Derivation of the theorem Dissipation 2/7 The Schrödinger Equation turns into φ(t) I = 1 i V (t) exp ( ) ( iĥt ˆQ exp Expand φ(t) I in terms of E n φ(t) I = a n(t) E n n ) iĥt φ(t) I The projection on E m gives ȧ m(t) = 1 ( ) it(em E n) V (t)a n(t) exp E m ˆQ E n i n Stefan Gorol & Dominikus Zielke March 21, 216 5
6 Derivation of the theorem Dissipation 3/7 Let the system be in an initial state E n and V (t) = V cos (ωt) ȧ m(t) = 1 ( i V (Em En) cos (ωt) exp it } {{ } =:ω mn Integration yields a m(t) = 1 [ 2 V E m ˆQ E exp (i(ωmn ω)t) 1 n (ω mn ω) The probability p for a transition n m is p mn = V 2 Em ˆQ E n 2 exp (i(ωmn ω)t) 1 (ω mn ω) + ) E m ˆQ E n ] exp (i(ωmn + ω)t) 1 (ω mn + ω) + exp (i(ωmn + ω)t) 1 (ω mn + ω) 2 Stefan Gorol & Dominikus Zielke March 21, 216 6
7 Derivation of the theorem Dissipation 4/7 Blue Curve: Small time t Red Curve: Large time t For large times t, the transition probability turns into δ-functions. For ω mn ω, which is the case when E m E n ω, one obtains ) exp (i(ωmn ω)t) 1 (ω mn ω) + exp (i(ωmn + ω)t) 1 (ω mn + ω) ( 2 4 sin2 (ω mn ω) t 2 (ω mn ω) 2 Use lim t dω mn sin 2 ((ω mn ω)t/2) (ω mn ω) 2 t = π 2 Stefan Gorol & Dominikus Zielke March 21, 216 7
8 Derivation of the theorem Dissipation 5/7 We can identify from the above integral in the limit t : sin 2 ((ω mn ω)t/2) lim t (ω mn ω) 2 = π δ(ωmn ω)t 2 Therefore the transition probability p mn can be expressed as: p mn = π 2 V 2 Em ˆQ E n 2 δ(e m E n ω)t The transition rate Γ = dp mn/dt then is: Γ mn,absorb = π 2 V 2 Em ˆQ E n 2 δ(e m E n ω) For E m E n ω it follows similarly Γ mn,emission = π 2 V 2 Em ˆQ E n 2 δ(e m E n + ω) Stefan Gorol & Dominikus Zielke March 21, 216 8
9 Derivation of the theorem Dissipation 6/7 Multiplying the emission and absorbance rate with ω and ω respectively, yields the total dissipated power. P S,n = πω [ En 2 V 2 + ω ˆQ E n 2 ρ(e n + ω) En ω ˆQ E ] n 2 ρ(e n ω) Suppose: f(e, T ) is the distribution functions of densely distributed initial energy states: P total = πω [ E 2 V 2 de ρ(e)f(e,t ) + ω ˆQ E 2 ρ(e + ω) ] E ω ˆQ E 2 ρ(e ω) Stefan Gorol & Dominikus Zielke March 21, 216 9
10 Derivation of the theorem Dissipation 7/7 The total average dissipated power of an impedanze Z(ω) can be written as P total = 1 2 V 2 R(ω) Z(ω) 2 Thus in quantum mechanical systems, one can identify R(ω) E Z(ω) 2 = πω de ρ(e)f(e)[ + ω ˆQ E 2 ρ(e + ω) ] de ρ(e)f(e) E ω ˆQ E 2 ρ(e ω) Supposing Boltzmann distribution, one ends up with ω R Z 2 = πω( ) 1 exp ( ωβ) de f(e)ρ(e)ρ(e + ω) E + ω ˆQ E 2 Stefan Gorol & Dominikus Zielke March 21, 216 1
11 Derivation of the theorem Fluctuations 1/3 Suppose thermal equilibrium and thus This also follows from Ĥ being hermitian: E n ˆQ En = i E n ˆQ En = ( En Ĥ ˆQ E n E n ˆQĤ E n ) = ie There are still fluctuations given by E n ˆQ2 E n = E ˆQ Em E ˆQ En n m ( En ˆQ E n E n ˆQ E n ) = m = E i n [Ĥ, ˆQ ] E i m E m [Ĥ, ˆQ ] E n m = 1 2 (E n E m) 2 Em ˆQ E n 2 m Stefan Gorol & Dominikus Zielke March 21,
12 Derivation of the theorem Fluctuations 2/3 Suppose again densely distributed energy levels [ E n ˆQ2 E n = 1 En 2 de m ρ(e m)(e n E m) 2 Em ˆQ E n 2 de m ρ(e m)(e n E m) 2 Em ˆQ E ] n 2 E n + Substitution of E m E n = ω yields [ E n ˆQ2 E n = 1 dω ρ(e n ω)( ω) 2 En ω ˆQ E n 2 +ρ(e n + ω)( ω) 2 En + ω ˆQ E n 2] Stefan Gorol & Dominikus Zielke March 21,
13 Derivation of the theorem Fluctuations 3/3 As before, suppose distributed initial energy states. [ E ˆQ2 E S = dω ω 2 de ρ(e)f(e, T ) ρ(e ω) E ω ˆQ E 2 + ρ(e + ω) E + ω ˆQ E 2 ] =: dω I Q (ω) = Q 2 From complex alternating current calculation, we know the force-response equation The Wiener-Khinchin theorem yields I Q (ω) V (ω) = Z(ω) Q(ω) I V (ω) {}}{{}}{ V (ω) 2 = Z(ω) 2 Q(ω) 2 V 2 = dω I V (ω) = dω Z(ω) 2 I Q (ω) Stefan Gorol & Dominikus Zielke March 21,
14 Derivation of the theorem Relation between dissipation and fluctuations 1/1 Thus we can identify (see above) ˆV 2 = dω Z 2 [1 + exp ( ωβ)] ω 2 de f(e)ρ(e)ρ(e+ ω) E + ω ˆQ E 2 As a reminder, we the following solution before: R Z 2 = πω( ) 1 exp ( ωβ) de f(e)ρ(e)ρ(e + ω) E + ω ˆQ E 2 By seperating the integral term de ρ(e)ρ(e + ω) E + ω ˆQ E 2 R 1 = ( ) πω Z 2 1 exp( ωβ) Inserting this into V 2 yields the Generalized Nyquist theorem and the Nyquist relation V 2 = 2 ( ) 1 dω R(ω) ω π kb T ω V 2 = 2 dω R(ω)k B T exp ( ωβ) 1 π Stefan Gorol & Dominikus Zielke March 21,
15 Further Applications General strategy to obtain the fluctuation spectrum of a dissipative system 1/1 Generalize the Hamiltonian from the derivation: Modify force-response-equation: Ĥ(t) = Ĥ + V (t) ˆQ Ĥ(t) = Ĥ + F (t) ˆx V (ω) = Z(ω) Q(ω) F (ω) = Z(ω) v(ω) Establish a differential equation of motion for v(t) Fourier transform the equation (aka. harmonic analysis) Specify the complex impedance Z(ω) of the system Use Generalized Nyquist theorem to calculate the mean square of the fluctuating physical quantity: A 2 (t) Bonus: Spectral decomposition A 2 (t) = dωi A (ω) (follows with Wiener-Khinchin theorem see e. g. [Reif, p.69]) yields intensity spectrum I A (ω) Stefan Gorol & Dominikus Zielke March 21,
16 Further Applications 1 st example: Brownian motion 1/3 Brownian particle with mass m is immersed in fluid with frictional constant f Langevin equation of motion in one dimension has a random and a systematic part F ran(t) f v(t) =! m v(t) Define Fourier Transforms of F ran(t) and v(t) F ran(t) = F ran(ω)e iωt dω and v(t) = ṽ(ω)e iωt dω Apply Fourier Transforms to equation of motion F ran(ω)e iωt dω f ṽ(ω)e iωt dω = m iωṽ(ω)e iωt dω Stefan Gorol & Dominikus Zielke March 21,
17 Further Applications 1 st example: Brownian motion 2/3 Since integral kernels must be equal, the force-response-equation has the form F ran(ω) = (f + imω) ṽ(ω) The impedance can thus be defined as Z(ω) = f + imω Application of the Generalized Nyquist theorem yields for k B T ω Fran 2 2 dω Re{Z(ω)}k B T = dω π I Fran (ω) = const White spectrum I Fran (ω) {}}{ 2 π k BT f Since the integral is divergent, it follows, that f = const. cannot be valid for high frequencies cut-off-frequency (see e.g. [Weber]) Stefan Gorol & Dominikus Zielke March 21,
18 Further Applications 1 st example: Brownian motion 3/3 The velocity and the displacement are fluctuating as well Use spectral decomposition to determine e. g. x 2 Applying FT on v(t) = dx(t) yields ṽ(ω) = iω x(ω) dt The force response-equation becomes with Wiener-Khinchin theorem (WK) F ran(ω) = (f + imω) iω x(ω) F ran(ω) 2 = ( f 2 + (mω) 2) ω 2 x(ω) 2 WK IFran (ω) = ( f 2 + (mω) 2) ω 2 I x(ω) Hence the mean square of the dispacement is 2 x 2 I Fran (ω) = I x(ω)dω = (f 2 + (mω) 2 ) ω 2 dω = π k BT f (f 2 + (mω) 2 ) ω 2 dω This integral is again divergent Introduce lower cut-off frequency to experimentally check the theory Stefan Gorol & Dominikus Zielke March 21,
19 Further Applications 2 nd example: electric field fluctuations and radiation (Planck s law) 1/5 Consider an accelerated non-relativistic point charge with mass m and charge e Total radiated power (see [Jackson, ch.14) is P (t) = 2 e 2 3 c 3 v(t)2 (Larmor formula) Establish equation of motion for point charge in electric field in one dimension ee x(t) + F rad (t)! = m v(t) F rad (t) takes account for energy loss due to radiation Use energy conservation to specify F rad (t) t 2 t 2 F rad(t) v(t) dt = 2 e 2 P (t) dt = 3 c 3 t 2 v v dt t 1 t 1 t 1 Stefan Gorol & Dominikus Zielke March 21,
20 Further Applications 2 nd example: electric field fluctuations and radiation (Planck s law) 2/5 Integration by parts yields (for periodic motion) t 2 t 1 t 2 2 e 2 F rad v dt = 3 c 3 t 1 = {}}{ 2 e 2 v v dt = 3 c 3 [ v v] t 2 (Abraham-Lorentz) equation of motion then reads t1 ee x(t) + 2 e 2 v(t) = m v(t) 3 c3 { ( )} Solutions for E x : v(t) =, v exp 3mc 3 2e 2 t t 2 t 1 v v dt Frad = 2 e 2 3 c 3 v Unphysical nontrivial (runaway) solution in contradiction with periodic motion Equation of motion only useful when radiated energy is small compared to total energy (see extensive discussion in [Jackson, ch.16]) Stefan Gorol & Dominikus Zielke March 21, 216 2
21 Further Applications 2 nd example: electric field fluctuations and radiation (Planck s law) 3/5 Define Fourier Transforms of E x(t) and v(t): E x(t) = Ẽ x(ω)e iωt dω and v(t) = Apply Fourier Transformation to Abraham-Lorentz equation e Ẽ x(ω)e iωt dω + 2 e 2 3 c 3 ω 2 ṽ(ω)e iωt dω = m ṽ(ω)e iωt dω iωṽ(ω)e iωt dω Since integral kernels must be equal, the force-response-equation has the form ( ) eẽx(ω) = ω 2 2 e 2 3 c 3 + iωm ṽ(ω) Stefan Gorol & Dominikus Zielke March 21,
22 Further Applications 2 nd example: electric field fluctuations and radiation (Planck s law) 4/5 The impedance of the system can thus be defined as Z(ω) = ω 2 2 e 2 3 c 3 + iωm Application of the Generalized Nyquist theorem yields e 2 Ex 2 = 2 π = ( dω ω 2 2 e 2 3 c 3 ω ( 1 ω exp dω 4 e 2 ω 3 3 πc 3 k B T ( ) ( 1 ) ω exp k B T 1 ) 1 ) Stefan Gorol & Dominikus Zielke March 21,
23 Further Applications 2 nd example: electric field fluctuations and radiation (Planck s law) 5/5 The radiated power per unit area is given by the absolute value of the Poynting-Vector S S = c 4π E B E B = c 4π E B B= E c = c 4π E E c = E2 4π Taking the mean value yields S = 3 E2 x 4π isotropic radiation = 3E2 x 4π With this definition and with Ex 2 from the Generalized Nyquist theorem, one obtains Planck s law ( ) S = ω 3 π 2 c 3 dω 2 + ( ω 3 ) ω exp k B T 1 First term: divergent zero point contribution Second term: convergent contribution of black body radiation Stefan Gorol & Dominikus Zielke March 21,
24 Summary and outlook Generalized Nyquist theorem: Correlates a property of a system in thermal equilibrium (e. g. voltage fluctuations) with a parameter which characterizes an irreversible process (e. g. electrical resistance) Nice way to do non-equilibrium thermodynamics and rediscover underlying laws like the white spectrum of Brownian motion or Planck s law with less effort compared to traditional derivations Generalized Nyquist theorem is one way to formulate the fluctuation-dissipation theorem Other ways involve other formalisms like for example response functions, generalized susceptibility and retarded Green functions Since they all describe the relation between fluctuation and dissipation, they are all connected (mostly via Fourier Transformation) Stefan Gorol & Dominikus Zielke March 21,
25 Bibliography [Callen&Welton]: Herbert B. Callen and Theodore A. Welton: Irreversibility and Generalized Noise in Physical Review Volume 38, Number 1, July 1, 1951 [Jackson]: John David Jackson: Classical Electrodynamics (3 rd edition), John Wiley & Sons, Inc., 1999 [Reif] Frederick Reif, Statistische Physik und Theorie der Wärme, 3. Auflage, Walter de Gruyter, 1987 [Weber] J. Weber: Fluctuation Dissipation theorem in Phys. Review, Volume 11, Number 6, March 15, 1956 Stefan Gorol & Dominikus Zielke March 21,
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