GeneralizedNyquisttheorem Non-EquilibriumStatisticalPhysicsProf.Dr.SergeyDenisovWS2015/16

Prof. Dr. Sergey Denisov WS 2015/16

Generalized Nyquist theorem

by

Stefan Gorol & Dominikus Zielke

• 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^stexample: Brownian motion

-

2^ndexample: Electric field fluctuations and radiation (Planck’s law)

• Summary and outlook

• Based on the Nyquist theorem for the white noise of a resistor:

hV²i= 2 πkBT R

Z

∞ 0

dω

• with the mean square of the voltagehV²i, Boltzmann’s constantk_B, temperature T, angular frequencyωand resistanceR

• 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=I²R)

Suppose:

H(t) = ˆˆ H0+V(t) ˆQ

Hˆ0≡ Hamiltonian of non-perturbated system (time independent) V(t) ˆQ ≡ time depentent perturbation Furthermore suppose:

|E1i,|E2i,. . .are eigenfunctions ofHˆ0with corresponding eigenvaluesE1,E2,. . . Now solve the Schr ¨odinger Equation

i~∂

∂t|φ(t)i_S= ˆH|φ(t)i_S Use linear perturbation theory in the interaction picture

|φ(t)i_S= exp

−i ˆH₀t

~

|φ(t)i_I

Dissipation 2/7

The Schr ¨odinger Equation turns into

|φ(t)i˙ _I= 1 i~V(t) exp

i ˆH0t

~

Qˆexp

−i ˆH0t

~

|φ(t)i_I

Expand|φ(t)iIin terms of|Eni

|φ(t)i_I=

X

n

an(t)|E_ni

The projection on|Emigives a˙m(t) = 1

i~

X

n

V(t)an(t) exp

_it(E

m−En)

~

hEm|Q|Eˆ ni

Let the system be in an initial state|EniandV(t) =V0cos (ωt)

˙

am(t) = 1 i~

V0cos (ωt) exp

it(Em−En)

~

| {z }

=:ω_mn

hEm|Q|Eˆ ni

Integration yields am(t) =−1

2~

V0hEm|Q|Eˆ ni

h

_{exp (i(ωmn−ω)t)−1}

(ωmn−ω) +exp (i(ωmn+ω)t)−1 (ωmn+ω)

i

The probabilitypfor a transitionn→mis

pmn= 1 4~²|V0|²

hEm|Q|Eˆ ni

2 ·

exp (i(ωmn−ω)t)−1

(ωmn−ω) +exp (i(ωmn+ω)t)−1 (ωmn+ω)

2

Dissipation 4/7

Blue Curve: Small timet Red Curve: Large timet

For large timest→ ∞, the transition probability turns intoδ-functions. Forωmn≈ω, which is the case whenEm−En≈~ω, one obtains

exp (i(ωmn−ω)t)−1

(ωmn−ω) +exp (i(ωmn+ω)t)−1 (ωmn+ω)

2

≈4 sin² (ωmn−ω)^t₂

(ωmn−ω)²

Use

t→∞lim

Z

∞

−∞

dωmnsin²((ωmn−ω)t/2) (ωmn−ω)²t =π

2

We can identify from the above integral in the limitt→ ∞:

lim

t→∞

sin²((ωmn−ω)t/2) (ωmn−ω)² =π

2δ(ωmn−ω)t

Therefore the transition probabilitypmncan be expressed as:

pmn= π 2~

|V₀|²

^hEm|Q|Eˆ ni

2·δ(Em−En−~ω)t

The transition rateΓ = dpmn/dtthen is:

Γmn,absorb= π 2~

|V₀|²

^hEm|Q|Eˆ _ni

2 ·δ(E_m−En−~ω)

ForEm−En≈ −~ωit follows similarly Γmn,emission= π

2~

|V₀|²

^hEm|Q|Eˆ _ni

2·δ(Em−En+~ω)

Dissipation 6/7

Multiplying the emission and absorbance rate with−~ωand~ωrespectively, yields the total dissipated power.

PS,n= πω 2 |V₀|²

h

^hEn+~ω|Q|Eˆ _ni

2ρ(En+~ω)

−

^hEn−~ω|Q|Eˆ ni

2ρ(En−~ω)

i

Suppose:f(E, T)is the distribution functions of densely distributed initial energy states:

Ptotal=πω 2 |V₀|²

Z

^∞

−∞

dE ρ(E)f(E,T)

h

^hE+~ω|Q|Eiˆ

2ρ(E+~ω)

−

^hE−~ω|Q|Eiˆ

2ρ(E−~ω)

i

The total average dissipated power of an impedanzeZ(ω)can be written as Ptotal=1

2|V0|² R(ω)

|Z(ω)|²

Thus in quantum mechanical systems, one can identify R(ω)

|Z(ω)|² =πω

Z

∞ 0

dE ρ(E)f(E)

h

^hE+~ω|Q|Eiˆ

2ρ(E+~ω)

−

Z

∞

~ω

dE ρ(E)f(E)

^hE−~ω|Q|Eiˆ

2ρ(E−~ω)

i

Supposing Boltzmann distribution, one ends up with R

|Z|² =πω 1−exp (−~ωβ)

Z

∞

0

dE f(E)ρ(E)ρ(E+~ω)|hE+~ω|Q|Ei|ˆ ²

Fluctuations 1/3

Suppose thermal equilibrium and thus

hEn|Q|E˙ˆ ni= 0

This also follows fromHˆ0being hermitian:

hEn|Q|E˙ˆ ni= i

~

hEn|Hˆ0Q|Eˆ ni − hEn|QˆHˆ0|Eni

= iE0

~

hEn|Q|Eˆ ni − hEn|Q|Eˆ ni

= 0

There are still fluctuations given by hEn|Q˙ˆ²|Eni=

X

m

hEn|Q|E˙ˆ mihEm|Q|E˙ˆ ni

=

X

m

hE_n|i

~

_ˆ

H₀,Qˆ

|E_mihE_m|i

~

_ˆ

H₀,Qˆ

|E_ni

= 1

~²

X

m

(En−Em)²

hEm|Q|Eˆ ni

2

Suppose again densely distributed energy levels

hEn|Q˙ˆ²|Eni= 1

~²

Z

En 0

dEmρ(Em)(En−Em)²

hEm|Q|Eˆ ni

2

+

Z

∞ En

dEmρ(Em)(En−Em)²

hEm|Q|Eˆ ni

2

Substitution ofEm−En=~ωyields

hE_n|Q˙ˆ²|E_ni=1

~

Z

∞ 0

dω ρ(En−~ω)(~ω)²

^hEn−~ω|Q|Eˆ _ni

2

+ρ(En+~ω)(~ω)²

hEn+~ω|Q|Eˆ ni

2

i

Fluctuations 3/3

As before, suppose distributed initial energy states.

hE|Q˙ˆ²|Ei_S=

Z

∞ 0

dω~ω²

Z

∞ 0

dE ρ(E)f(E, T)

h

ρ(E−~ω)

^hE−~ω|Q|Eiˆ

2

+ρ(E+~ω)

hE+~ω|Q|Eiˆ

2

i

=:

∞

Z

0

dω IQ˙(ω) =hQ˙²i

From complex alternating current calculation, we know the force-response equation V(ω) =Z(ω) ˙Q(ω)

The Wiener-Khinchin theorem yields

⇒ h

I_V(ω)

z }| {

|V(ω)|²i=|Z(ω)|²

IQ˙(ω)

z }| {

h|Q(ω)|˙ ²i ⇒ hV²i=

∞

Z

0

dω IV(ω) =

∞

Z

0

dω|Z(ω)|²IQ˙(ω)

Thus we can identify (see above)

hVˆ²i=

Z

∞ 0

dω|Z|²[1 + exp (−~ωβ)]~ω²

Z

∞ 0

dE f(E)ρ(E)ρ(E+~ω)

hE+~ω|Q|Eiˆ

2

As a reminder, we the following solution before:

R

|Z|² =πω 1−exp (−~ωβ)

Z

^∞

0

dE f(E)ρ(E)ρ(E+~ω)|hE+~ω|Q|Ei|ˆ ²

By seperating the integral term

Z

^∞

0

dE ρ(E)ρ(E+~ω)|hE+~ω|Q|Ei|ˆ ²= R πω|Z|²

1 1−exp(−~ωβ)

Inserting this intohV²iyields the Generalized Nyquist theorem and the Nyquist relation

hV²i= 2

Z

∞

dω R(ω) ω

₁

+ 1

_k

BT~ω

⇒ hV²i= 2

Z

∞

dω R(ω)k T

General strategy to obtain the fluctuation spectrum of a dissipative system 1/1

• Generalize the Hamiltonian from the derivation:

H(t) = ˆˆ H₀+V(t) ˆQ →H(t) = ˆˆ H₀+F(t) ˆx

• Modify force-response-equation:

V(ω) =Z(ω) ˙Q(ω)→F(ω) =Z(ω)v(ω)

• Establish a differential equation of motion forv(t)

• Fourier transform the equation (aka. harmonic analysis)

• Specify the complex impedanceZ(ω)of the system

• Use Generalized Nyquist theorem to calculate the mean square of the fluctuating physical quantity:hA²(t)i

• Bonus: Spectral decompositionhA²(t)i=

R

^∞

0 dωIA(ω)(follows with

Wiener-Khinchin theorem see e. g.[Reif,p.690]) yields intensity spectrumIA(ω)

• Brownian particle with massmis immersed in fluid with frictional constantf

• Langevin equation of motion in one dimension has a random and a systematic part Fran(t)−f·v(t)=^! mv(t)˙

• Define Fourier Transforms ofFran(t)andv(t)

Fran(t) =

∞

Z

−∞

F˜ran(ω)e^iωtdωandv(t) =

∞

Z

−∞

˜

v(ω)e^iωtdω

• Apply Fourier Transforms to equation of motion

∞

Z

−∞

F˜ran(ω)e^iωtdω−f

∞

Z

−∞

˜

v(ω)e^iωtdω=m

∞

Z

−∞

iω˜v(ω)e^iωtdω

1^stexample: Brownian motion 2/3

• Since integral kernels must be equal, the force-response-equation has the form F˜ran(ω) = (f+ imω) ˜v(ω)

• The impedance can thus be defined as

Z(ω) =f+ imω

• Application of the Generalized Nyquist theorem yields forkBT~ω

hF_ran² i ≈ 2 π

Z

∞ 0

dω Re{Z(ω)}k_BT =

Z

∞ 0

dω

I_Fran(ω)

z }| {

2 πkBT f

• IFran(ω) = const→White spectrum

• Since the integral is divergent, it follows, thatf= const.cannot be valid for high frequencies→cut-off-frequency (see e.g.[Weber])

• The velocity and the displacement are fluctuating as well

• Use spectral decomposition to determine e. g.hx²i

• Applying FT onv(t) =^dx(t)_dt yields˜v(ω) =iωx(ω)˜

• The force response-equation becomes with Wiener-Khinchin theorem (WK) F˜ran(ω) = (f+ imω) iωx(ω)˜

⇒ h|F˜ran(ω)|²i = f²+ (mω)²

ω²h|˜x(ω)|²i

WK⇒ IFran(ω) = f²+ (mω)²

ω²Ix(ω)

• Hence the mean square of the dispacement is

hx²i=

∞

Z

0

Ix(ω)dω=

∞

Z

0

IFran(ω)

(f²+ (mω)²)ω²dω=

∞

Z

0 2 πkBT f (f²+ (mω)²)ω²dω

• This integral is again divergent→Introduce lower cut-off frequency to

2^ndexample: electric field fluctuations and radiation (Planck’s law) 1/5

• Consider an accelerated non-relativistic point charge with massmand chargee

• Total radiated power (see[Jackson,ch.14) is

P(t) = 2 3

e²

c³v(t)˙ ²(Larmor formula)

• Establish equation of motion for point charge in electric field in one dimension eEx(t) +Frad(t)=^! mv(t)˙

• Frad(t)takes account for energy loss due to radiation

• Use energy conservation to specifyFrad(t)

t₂

Z

t₁

Frad(t)v(t) dt=−

t₂

Z

t₁

P(t) dt=−2 3

e² c³

t₂

Z

t₁

˙ vv˙dt

• Integration by parts yields (for periodic motion)

t2

Z

t₁

F_radvdt=−2 3

e² c³

t2

Z

t₁

˙

vv˙dt=−2 3

e² c³





=0

z }| {

[ ˙v v]^t_t²

1−

t2

Z

t₁

¨ v vdt





^⇒Frad=2 3

e² c³ ¨v

• (Abraham-Lorentz) equation of motion then reads eEx(t) +2

3 e²

c³v(t) =¨ mv(t)˙

• Solutions forEx≡0 : ˙v(t) =

n

0,

XX

XX XX X

˙ v0exp

3mc³ 2e² t

o

• 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])

2^ndexample: electric field fluctuations and radiation (Planck’s law) 3/5

• Define Fourier Transforms ofEx(t)andv(t):

Ex(t) =

∞

Z

−∞

E˜x(ω)e^iωtdωandv(t) =

∞

Z

−∞

˜

v(ω)e^iωtdω

• Apply Fourier Transformation to Abraham-Lorentz equation

e

∞

Z

−∞

E˜x(ω)e^iωtdω+2 3

e² c³

∞

Z

−∞

−ω²v(ω)e˜ ^iωtdω=m

∞

Z

−∞

iω˜v(ω)e^iωtdω

• Since integral kernels must be equal, the force-response-equation has the form

e ˜Ex(ω) =

ω²2 3

e² c³+ iωm

˜ v(ω)

• The impedance of the system can thus be defined as Z(ω) =ω²2

3 e² c³ + iωm

• Application of the Generalized Nyquist theorem yields

he²E²_xi= 2 π

∞

Z

0

dω ω²2 3

e² c³~ω 1

2+ 1

exp _k^~ω

BT

−1

!

=

∞

Z

0

dω4 3

e²~ω³ πc³

1

2+ 1

exp _k^~ω

BT

−1

!

2^ndexample: 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-VectorS

S=

c 4πE×B

E⊥B= c

4πE·B^B=

E

=c c 4πE·E

c =E² 4π

isotropic radiation

= 3E_x² 4π

• Taking the mean value yields

⇒ hSi=3hE_x²i 4π

• With this definition and withhE_x²ifrom the Generalized Nyquist theorem, one obtainsPlanck’s law

hSi= ~ π²c³

Z

∞ 0

dω ω³

2 + ω³

exp _k^~ω

BT

−1

!

• First term: divergent ”zero point” contribution

• 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)

• [Callen&Welton]: Herbert B. Callen and Theodore A. Welton:Irreversibility and Generalized Noisein Physical Review Volume 38, Number 1, July 1, 1951

• [Jackson]: John David Jackson:Classical Electrodynamics(3^rdedition), John Wiley & Sons, Inc., 1999

• [Reif]Frederick Reif,Statistische Physik und Theorie der W ¨arme, 3. Auflage, Walter de Gruyter, 1987

• [Weber]J. Weber:Fluctuation Dissipation theoremin Phys. Review, Volume 110, Number 6, March 15, 1956