Fluctuation-Dissipation Theorem (FDT)

Transcription

1 Fluctuation-Dissipation Theorem (FDT) NGL July 6, 7 Abstract This is a short writeup with erivation of fluctuation-issipation theorem (FDT) in classical an quantum mechanics. 1 Classical mechanics We use Lecture 1 from Liouville equation for f(γ, t), Γ = (q, p): f t + ilf = (1) where il = {...,H} is the Poisson bracket. Consier a small perturbation to a stationary Hamiltonian H an a steay-state istribution f (Γ) = e H (Γ)/T /Z: H(Γ,t) = H (Γ) X(Γ)F(t) () il = il + i L (3) f(γ,t) = f (Γ) + f(γ,t) (4) Let us fin f using perturbation theory to the first orer. We have ( ) t + il f(γ,t) = i Lf (Γ) (5) since il f =. Consier the left-han sie: i Lf (Γ) = {f,xf(t)} = F(t){f,X} (6) 1

2 Since f (Γ) = e H (Γ)/T /Z, Γ f = f Γ H (7) H {f,x} = f {H,X} = 1 ( ) X H T f (8) t where Γ enotes a vector consisting of / q i an / p i an the subscript enotes time evolution accoring to unperturbe Hamiltonian H. Thus, enoting Ẋ(Γ) = (X/t), we have The solution of equation (5) is f(γ,t) = i Lf (Γ) = 1 T F(t)f Ẋ (9) = 1 T t e il (t t ) i L(t )f (Γ) t e il (t t ) F(t )f (Γ)Ẋ(Γ) (1) Consier operator A(Γ) an let us fin its perturbe average value using the foun perturbation f: A(t) = ΓA(Γ)f(Γ,t) = A + ΓA(Γ) f(γ, t) = A + 1 t F(t ) ΓA(Γ)e il (t t ) f (Γ)Ẋ(Γ) (11) T Now we use some trickery with operators: A(t) = e ilt A() therefore A (t) = A ()e ilt therefore A(t) = A()e ilt. We use the fact that A is real an L is hermitian (why?). We get A(t) = A + 1 T t F(t ) Γ A(Γ t t (Γ ))Ẋ(Γ ) (1) where Γ t t (Γ ) is the phase coorinates propagate through time t t using H. A(t) = A + 1 T t F(t ) A(t t )Ẋ() (13)

3 The response function α AX (τ) = 1 T A(τ)Ẋ() = 1 T vali only for τ >. In the last equation we use τ A(τ)X() (14) t A(τ + t)x(t) = = Ȧ(τ + t)x(t) + A(τ + t)ẋ(t) (15) From now on, we rop subscript. The linear response is escribe by α AB (t) = 1 T A(t)B() (16) t If A X, then α(τ) = 1 X(τ)X(), the response function is relate to the correlation of X. Taking Fourier transforms an using Wiener- T τ Khintchine relation: an (X ) ω α(ω) = For ω = we have X(τ)X() e iωτ τ = Re α(τ)e iωτ τ = 1 T = 1 ( X X e iω ) + iω T T = 1 T X + iω T (X ) ω ω T Im α(ω = ) = 1 T X()X(τ) e iωτ τ (17) τ X(τ)X() eiωτ τ ( X X ) = (δx) T The FDT is the imaginary part of the previous equation or X(τ)X() e iωτ τ X()X(τ) e iωτ τ (18) (19) Im α(ω) = ω T (X ) ω () (X ) ω = T ω 3 Im α(ω) (1)

4 We can consier the fluctuations in X to be cause by ranom force F. Using (X ) ω = α(ω) (F ) ω, we obtain (F ) ω = T Im α(ω) ω α(ω) = T ( ) 1 ω Im () α (ω) Note that this force is ficticious (however, it can be very real see Example in Section.1 below). Note that is usually applie to issipative systems. The conservative Hamiltonian H belongs to a super-system with conserve energy, an the issipative system is just a part of it (see the abovementione example). For space-epenent response, α(r, t), treat spatial coorinate as an inex, e.g. X(r,t) A(t), X(,t) B(t). α(r,t) = 1 T X(r,t)X(, ) (3) t Examples in classical mechanics.1 Dampe 1D oscillator Consier a riven oscillator of mass M at frequency ω, with amping force νp. The equation of motion is or, in frequency omain, where MẌ = Mω X MνẊ + F(t) (4) α(ω) = X(ω) = α(ω)f(ω) (5) 1 M(ω ω iων) The spectrum of the force is (F ) ω = T ( ) 1 ω Im α (ω) (6) = TMν (7) The force is elta-correlate, which coul be ue, e.g., to bombarment by point particles. Let us obtain this result from microscopic consieration by 4

5 eriving an expression for ν. Assume that the oscillator has transverse area A an is immerse in a gas of ensity N, temperature T an consisting of molecules of mass m M. The hamiltonian H escribes the system which inclues both the oscillator an the gas, so there is no total energy loss. Assume, however, that there is so much gas that the temperature of the gas oes not change as the energy is transfere from oscillator to the gas. Momentum transfere by a single particle is p = mv. The number of particles of velocity v falling on the oscillator surface in unit time is λ(v) = Avf(v)v, so that the average force is Amv f(v)v. The velocity has to be positive for particles falling from the left, an negative for particles falling from the right. Assume the oscillator has velocity V. Then the istribution that it sees is f(v) = N e (v V ) v t N ( 1 + vv ) πvt πvt v t e v v t (8) where v t = T/m an we expane the exponent in a series an neglecte terms O(V /vt ) m/m (we anticipate that V will be of the orer of thermal velocity T/M). We use v n e v ( ) v n + 1 t v = n/ vt n+1 Γ (9) The average force from particles falling from the left sie is The total force is F sie (V ) = Am = NAm πvt v f(v)v = NAm πvt ( πv 3 t v ( 1 + vv v t ) + V v t e v v t v ) (3) F(V ) = F sie (V ) F sie ( V ) = 8NAmV v t π (31) so that ν = 8NAmv t πm = 8NA mt M π (3) 5

6 Let us fin inepenently the spectrum of the elta-correlate ranom force ue to bombarment by particles. Spectrum ue to particles with velocity v is (F ) ω = ( p) λ(v) = (mv) Avf(v)v. Force ue to particles falling from the left an from the right has the spectrum (F ) ω = (F ) ω = (mv) Avf(v)v = 8NAm v 3 e v v t v = 16NAm vt 3 πvt π We see that (F ) ω = MTν, as expecte.. Johnson-Nyquist noise = 16NATmv t (33) π Consier an open circuit consisting of an impeance Z(ω) (we use physics notation i = j, e iωt in this example). The change in energy ue to fluctuations of voltage an charge at ens of the impeance is H(t) = QV (t), where voltage plays the role of the external force. Ohm s law is Q(ω) = α(ω)v (ω), where α(ω) = i/(ωz(ω)). Thus, the spectrum is (Q ) ω = T ω Im ( i ωz ) (34) (V ) ω = T ω Im [iωz (ω)] = TRe Z(ω) (35) The experimentally measure spectrum per unit frequency f is (V ) ω, which is 4TRe Z(ω). The current spectrum is (I ) ω = (V ) ω / Z(ω) = TRe (1/Z (ω)) (36) 3 Quantum mechanics Use Lecture 3 from the same lecture notes (too lazy to copy them, I use classical mechanics mostly anyway). 6