08. Brownian Motion Gerhard Müller University of Rhode Island, [email protected]

08. Brownian Motion Gerhard Müller University of Rhode Island, Gmuller@Uri.Edu

University of Rhode Island DigitalCommons@URI

Nonequilibrium Statistical Physics Physics Course Materials

10-19-2015 08. Brownian Motion Gerhard Müller University of Rhode Island, [email protected]

Follow this and additional works at: http://digitalcommons.uri.edu/ nonequilibrium_statistical_physics Part of the Physics Commons Abstract Part eight of course materials for Nonequilibrium Statistical Physics (Physics 626), taught by Gerhard Müller at the University of Rhode Island. Entries listed in the table of contents, but not shown in the document, exist only in handwritten form. Documents will be updated periodically as more entries become presentable.

Recommended Citation Müller, Gerhard, "08. Brownian Motion" (2015). Nonequilibrium Statistical Physics. Paper 8. http://digitalcommons.uri.edu/nonequilibrium_statistical_physics/8

This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Nonequilibrium Statistical Physics by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. Contents of this Document [ntc8]

8. Brownian Motion

• Relevant time scales (collisions, relaxation, observations) • Einstein’s theory • Smoluchovski equation with link to Fokker-Planck equation • Einstein relation (example of fluctuation-dissipation relation) • Fick’s law for particle current • Fourier’s law for heat current • Thermal diffusivity [nex117] • Shot noise (e.g. electric current in vacuum tube) • Campbell’s theorem [nex37] • Critically damped ballistic galvanometer [nex70] • Langevin’s theory (on most contracted level of description) • White noise • Brownian motion and Gaussian white noise [nln20] • Wiener process [nsl4] • Autocorrelation function of Wiener process [nex54] • Ballistic and diffusive regimes of Langevin solution • Langevin’s equation: attenuation without memory [nln21] • Formal solution of Langevin equation [nex53] • Velocity correlation function of Brownian particle I [nex55] • Mean-square displacement of Brownian particle [nex56], [nex57], [nex118] • Ergodicity [nln13] • Intensity spectrum and spectral density (Wiener-Khintchine theorem) [nln14] • Fourier analysis of Langevin equation • Velocity correlation function of Bownian particle II [nex119] • Generalized Langevin equation: attenuation with memory [nln22] • Fluctuation-dissipation theorem • Velocity correlation function of Brownian particle III [nex120] • Brownian harmonic oscillator I: Fourier analysis [nex121] • Brownian harmonic oscillator II: position correlation function [nex122] • Brownian harmonic oscillator III: contour integrals [nex123] • Brownian harmonic oscillator IV: velocity correlations [nex58] • Brownian harmonic oscillator V: formal solution for velocity [nex59] • Brownian harmonic oscillator VI: nonequilibrium correlations [nex60] • Generalized Langevin equation inferred from microscopic dynamics • Brownian motion: levels of contraction and modes of description [nln23] [nex117] Thermal diffusivity

A solid wall of very large thickness and lateral extension (assumed to occupy all space at z > 0) is brought into contact with a heat source at its surface (z = 0). The wall is initially in thermal equilibrium at temperature T0. The heat source is kept at the higher temperature T1. The contact is established at time t = 0. Show that the temperature profile inside the wall depends on time as follows: z T (z) = T0 + (T1 − T0)erfc √ , 2 DT t where DT = λ/cV is the thermal diffusivity, λ the thermal conductvity, and cV the specific heat.

Solution: [nex37] Campbell processes. P Consider a stationary stochastic process of the general form Y (t) = k F (t − tk), where the times tk are distributed randomly with an average rate λ of occurrences. Campbell’s theorem then yields the following expressions for the mean value and the autocorrelation function of Y :

Z ∞ Z ∞ hY i = λ dτ F (τ), hhY (t)Y (0)ii ≡ hY (t)Y (0)i − hY (t)ihY (0)i = λ dτ F (τ)F (τ + t). −∞ −∞ Apply Campbell’s theorem to calculate the average current hIi and the current autocorrelation P −αt function hhI(t)I(0)ii for the shot noise process I(t) = k F (t − tk) with F (t) = qe Θ(t).

Solution: [nex70] Critically damped ballistic galvanometer.

The response of a critically damped ballistic galvanometer to a current pulse at t = 0 is Ψ(t) = cte−γt. Consider the situation where the galvanometer experiences a steady stream of independent P random current pulses, X(t) = k Ψ(t−tk), where the tk are distributed randomly with an average rate n of occurrences. (a) Find the average displacement hXi of the galvanometer. (b) Find the autocorrelation function for the displacement of the galvanometer, hhX(t)X(0)ii, and the associated spectral density ΦXX (ω).

Solution: Brownian motion and Gaussian white noise [nln20]

Gaussian white noise: completely factorizing stationary process.

• Pw(y1, t1; y2, t2) = Pw(y1)Pw(y2) if t2 6= t1 (factorizability) 1 y2 • Pw(y) = √ exp − (Gaussian nature) 2πσ2 2σ2 • hy(t)i = 0 (no bias)

• hy(t1)y(t2)i = Iwδ(t1 − t2) (whiteness) 2 2 • Iw = hy i = σ (intensity)

Brownian motion: Markov process.

• Discrete time scale: tn = n dt.

• Position of Brownian particle at time tn: zn.

• P (zn, tn) = P (zn, tn|zn−1, tn−1) P (zn−1, tn−1) (white-noise transition rate). | {z } Pw(yn)δ(yn−[zn−zn−1])

• Speciﬁcation of white-noise intensity: Iw = 2Ddt. n X • Sample path of Brownian particle: z(tn) = y(ti). i=1 • Position of Brownian particle: n X – mean value: hz(tn)i = hy(ti)i = 0. i=1 n 2 X – variance: hz (tn)i = hy(ti)y(tj)i = 2Dndt = 2Dtn. i,j=1

Gaussian white noise with intensity Iw = 2Ddt is used here to generate the diﬀusion process discussed previously [nex26], [nex27], [nex97]: 2 1 (z − z0) P (z, t + dt|z0, t) = √ exp − . 4πDdt 4Ddt

Sample paths of the diffusion process become continuous in the limit dt → 0 (Lindeberg condition). However, in the present context, we must use dt τR, where τR is the relaxation time for the velocity of the Brownian particle. On this level of contraction, the velocity of the Brownian particle is nowhere defined in agreement with the result of [nex99] that the diffusion process is nowhere differentiable. Wiener process [nls4]

Speciﬁcations:

1. For t0 < t1 < ··· , the position increments ∆x(tn, tn−1), n = 1, 2,... are independent random variables.

2. The increments depend only on time diﬀerences: ∆x(tn, tn−1) = ∆x(dtn), where dtn = tn − tn−1. 3. The increments satisfy the Lindeberg condition: 1 lim P[∆x(dt) ≥ ] = 0 for all > 0 dt→0 dt

The diﬀusion process discussed previously [nex26], [nex27], [nex97], 2 1 (x − x0) P (x, t + dt|x0, t) = √ exp − , 4πDdt 4Ddt is, for dt → 0, a realization of the Wiener process. Sample paths of the Wiener process thus realized are everywhere continuous and nowhere diﬀer- entiable.

n n X X W (tn) = ∆x(dti), tn = dti. i=1 i=1

[adapted from Gardiner 1985] [nex54] Autocorrelation function of the Wiener process.

Use the regression theorem, Z Z hx(t)x(t + dt)|[0, 0]i = dx1 dx2 x1x2P (x2, t + dt|x1, t)P (x1, t|0, 0), for the conditional probability distribution P (x+∆x, t+dt|x, t) = (4πD dt)−1/2 exp(−(∆x)2/4D dt) to show that the autocorrelation function of the Wiener process depends only on one time even though it is non-stationary. On what time does it depend and how?

Solution: Attenuation without memory [nln21]

Langevin equation for Brownian motion: dv m + γv = f(t). dt

Random force (uncorrelated noise):

Sff (ω) = 2γkBT,Cff (t) = 2γkBT δ(t).

1.0 2.0 Γ=0.5 Γ=0.5 0.8 1.5 T T B

B 0.6 k k L L t Ω 1.0 H H ff ff 0.4 C S 0.5 0.2

0.0 0.0 -4 -2 0 2 4 0 1 2 3 4 Ω t

Stochastic variable (velocity):

2γk T k T S (ω) = B ,C (t) = B e−(γ/m)t. vv γ2 + m2ω2 vv m

1.0 2.0 m=1 Γ=1 m=1 0.8 1.5 T T B B

k 0.6 k

L Γ=

2 m L Ω 1.0 t H H vv

vv 0.4 S C Γ=1 0.5 Γ=3 0.2 Γ=2 Γ=3 0.0 0.0 -4 -2 0 2 4 0 1 2 3 4 Ω t [nex53] Formal solution of Langevin equation

Consider the Langevin equation for a Brownian particle of mass m constrained to move along a straight line and subject to a drag force −γv and a white-noise random force f(t):

dx dv γ 1 = v, = − v + f(t). dt dt m m Calculate via formal integration the functional dependence of velocity v(t) and position x(t) on the random force f(t) for initial conditions x(0) = 0 and v(0) = v0.

Solution: [nex55] Velocity correlation function of Brownian particle I

Consider a Brownian particle of mass m constrained to move along a straight line and subject to a drag force −γv and a white-noise random force f(t). Calculate the velocity autocorrelation function hv(t1)v(t2)i0 of a Brownian particle for t1 > t2 as a conditional average from the formal solution (see [nex53]) Z t −γt/m 1 0 −(γ/m)(t−t0) 0 v(t) = v0e + dt e f(t ) m 0 of the Langevin equation with a white noise random force of intensity Iw. Show that for t1 > t2 γ/m the result only depends on the time diﬀerence t1 − t2. Use equipartition to determine the temperature dependence of the random-force intensity If .

Solution: [nex56] Mean−square displacement of Brownian particle I

Consider a Brownian particle of mass m constrained to move along a straight line and subject to a drag force −γv and a white-noise random force f(t). Calculate the mean-square displacement hx2(t)i of a Brownian particle via integration with initial condition x(0) = 0 from the (stationary) velocity autocorrelation function calculated in [nex55]:

kBT hv(t )v(t )i = e−(γ/m)|t1−t2|. 1 2 m

Solution: [nex57] Mean−square displacement of Brownian particle II

Z t m −(γ/m)t 1 −(γ/m)(t−s) x(t) = v0 1 − e + ds 1 − e f(s) γ γ 0 of the Langevin equation with a white-noise random force of intensity If = 2kBT γ by taking a thermal average over initial velocities.

Solution: [nex118] Mean-square displacement of Brownian particle III

Consider the Langevin equation for a Brownian particle of mass m constrained to move along a straight line and subject to a drag force −γx˙ and a white-noise random force f(t):

mx¨ = −γx˙ + f(t).

Construct from it the second-order linear ODE,

d2 d m hx2i + γ hx2i = 2k T, dt2 dt B

2 2 by using equipartition, hx˙ i = kBT/m. Then solve this ODE for initial conditions dhx i/dt|0 = 0 2 2 and hx i|0 = 0. Identify the quadratic time-dependence of hx i in the ballistic regime, t m/γ, and the linear time dependence in the diﬀusive regime, t m/γ.

Solution: Ergodicity [nln13]

Consider a stationary process x(t). Quantities of interest are expectation values related to x(t).

• Theoretically, we determine ensemble averages: hx(t)i, hx2(t)i, hx(t)x(t + τ)i are independent of t. • Experimentally, we determine time averages: x(t), x2(t), x(t)x(t + τ) are independent of t.

Ergodicity: time averages are equal to ensemble averages. Implication: the ensemble average of a time average has zero variance. The consequences for the correlation function . C(t1 − t2) = hx(t1)x(t2)i − hx(t1)ihx(t2)i are as follows (set τ = t2 − t1 and t = t1): Z +T Z +T 2 2 1 hx i − hxi = lim 2 dt1 dt2 [hx(t1)x(t2)i − hx(t1)ihx(t2)i] T →∞ 4T −T −T 1 Z +2T = lim 2 dτ C(τ)(2T − |τ|) T →∞ 4T −2T 1 Z +2T |τ| = lim dτ C(τ) 1 − = 0. T →∞ 2T −2T 2T

Necessary condition: lim C(τ) = 0. τ→∞ Z ∞ Suﬃcient condition: dτ C(τ) < ∞. 0 t2

τ=2

τ=1

τ=0 t1

T τ=−1

τ=−2 T Intensity spectrum and spectral density [nln14]

Consider an ergodic process x(t) with hxi = 0.

. Z T Fourier amplitude:x ˜(ω, T ) = dt eiωtx(t) ⇒ x˜(−ω, T ) =x ˜∗(ω, T ). 0

. 1 2 Intensity spectrum (power spectrum): Ixx(ω) = lim |x˜(ω, T )| . T →∞ T . 1 Z T Correlation function: Cxx(τ) = hx(t)x(t + τ)i = lim dt x(t)x(t + τ). T →∞ T 0 Z +∞ . iωτ Spectral density: Sxx(ω) = dτ e Cxx(τ). −∞

Wiener-Khintchine theorem: Ixx(ω) = Sxx(ω). Proof: Z T Z T 1 0 −iωt0 0 iωt Ixx(ω) = lim dt e x(t ) dt e x(t) T →∞ T 0 0 1 Z T Z T −τ Z T −τ = lim dτ eiωτ dt0x(t0)x(t0 + τ) + e−iωτ dt x(t)x(t + τ) T →∞ T 0 0 0 Z T 1 Z T −τ = lim 2 dτ cos ωτ dt x(t)x(t + τ) T →∞ 0 T 0 Z ∞ Z +∞ iωτ = 2 dτ cos ωτCxx(τ) = dτ e Cxx(τ) = Sxx(ω). 0 −∞ t’ T−τ T τ= t’−t

T−τ τ=t−t’ t T [nex119] Velocity correlation function of Brownian particle II

Consider the Langevin equation for the velocity of a Brownian particle of mass m constrained to move along a straight line, dv m = −γv + f(t), dt 0 0 where γ is the damping constant and f(t) is a white-noise random force, hf(t)f(t )i = If δ(t − t ), with intensity If = 2kBT γ in thermal equilibrium as shown in [nex55]. (a) Convert the diﬀerential equation for v(t) into an algebraic equation for the Fourier amplitude . v˜(ω) = R dt eiωtv(t). (b)Use the Wiener-Khintchine theorem (see [nln14]) to calculate the spectral density Svv(ω) and, via inverse Fourier transform, the velocity correlation function hv(t)v(t0)i in thermal equlibrium.

Solution: Attenuation with memory [nln22]

Generalized Langevin equation for Brownian harmonic oscillator: Z t dx 0 0 0 1 2 −(γ/m)t m + dt α(t − t )x(t ) = f(t), α(t) = mω0e . dt −∞ ω0

Random force (correlated noise): 2k T γm2ω2 S (ω) = B 0 ,C (t) = k T mω2e−(γ/m)t. ff γ2 + m2ω2 ff B 0

1.0 2.0

0.8 Γ2m<Ω0 2

2 1.5 0 0 Ω Ω 0.6 T B Tm k B k

L 1.0 L Ω t

H 0.4 H ff ff

S Γ2m=Ω0 Γ2m<Ω C 0 0.5 Γ2m>Ω0 0.2 Γ2m=Ω0 Γ2m>Ω0 0.0 0.0 -4 -2 0 2 4 0 1 2 3 4

ΩΩ0 Ω0t

Stochastic variable (position):

2kBT γ Sxx(ω) = 2 2 2 2 2 2 , m (ω0 − ω ) + γ ω  k T γ h γ i p B − 2m t 2 2 2  mω2 e cos ω1t + 2mω sin ω1t , ω1 = ω0 − γ /4m > 0  0 1  γ kB T − t γ Cxx(t) = 2 e 2m 1 + t , ω0 = γ/2m mω0 2m  k T − γ t h γ i p  B 2m 2 2 2  2 e cosh Ω1t + sinh Ω1t , Ω1 = γ /4m − ω0 > 0 mω0 2mΩ1

6 1.0 0.8 5 Γ2m>Ω0

T 0.6 Γ2m>Ω0 B

4 k T 2 B

Γ =Ω 0 0.4 k 2m 0 L 3 Ω Ω m H 0.2 L

Γ2m<Ω0 t xx H S 2 xx 0.0 C Γ2m=Ω0 1 -0.2 Γ2m<Ω0 -0.4 0 -2 -1 0 1 2 0 2 4 6 8 10

ΩΩ0 Ω0t [nex120] Velocity correlation function of Brownian particle III

The generalized Langevin equation for a particle of mass m constrained to move along a sraight line, dv Z t m = − dt0α(t − t0)v(t0) + f(t), dt −∞ is known to produce the following expression for the spectral density of the velocity: Z ∞ Sff (ω) . iωt Svv(ω) = 2 , αˆ(ω) = dt e α(t),Sff (ω) = 2kBT <[ˆα(ω)], |αˆ(ω) − iωm| 0 where the relation between the random-force spectral density, Sff (ω), and the Laplace-transformed attenuation function,α ˆ(ω), is dictated by the ﬂuctuation-dissipation theorem. The special case of Brownian motion (see [nex55], [nex119]) uses attenuation without memory: α(t − t0) = 2γδ(t − t0)Θ(t − t0). Calculate the velocity correlation function, hv(t)v(t0)i, of the Brow- nian particle in thermal equilibrium from the above expression for Svv(ω) via contour integration in the plane of complex ω.

Solution: [nex121] Brownian harmonic oscillator I: Fourier analysis

The Brownian harmonic oscillator is speciﬁed by the Langevin-type equation,

mx¨ + γx˙ + kx = f(t),

2 where m is the mass of the particle, γ represents attenuation without memory, k = mω0 is the spring constant, and f(t) is a white-noise random force. Calculate the spectral density Sxx(ω) of the position coordinate via Fourier analysis and by using the result Sff (ω) = 2kBT γ for the random-force spectral density as dictated by the ﬂuctuation-dissipation theorem.

Solution: [nex122] Brownian harmonic oscillator II: position correlation function

The Brownian harmonic oscillator is speciﬁed by the Langevin-type equation, mx¨+γx˙ +kx = f(t), 2 where m is the mass of the particle, γ represents attenuation without memory, k = mω0 is the spring constant, and f(t) is a white-noise random force. 2 2 2 2 2 2 (a) Start from the result Sxx(ω) = 2γkBT/[m (ω0 − ω ) + γ ω ] for the spectral density of the position coordinate as calculated in [nex121] to derive the position correlation function  k T − γ t h γ i  B 2m  mω2 e cos ω1t + 2mω sin ω1t Z +∞  0 1 . dω  γ −iωt kB T − t γ hx(t)x(0)i = e Sxx(ω) = 2m mω2 e 1 + 2m t −∞ 2π  0  γ h γ i  kB T − 2m t  2 e cosh Ω1t + sinh Ω1t mω0 2mΩ1

p 2 2 2 2 2 2 for the cases ω1 = ω0 − γ /4m > 0 (underdamped), ω0 = γ /4m (critically damped), and p 2 2 2 Ω1 = γ /4m − ω0 > 0 (overdamped), respectively. 2 (b) Plot Sxx(w) versus ω/ω0 and hx(t)x(0)imω0/kBT versus ω0t with three curves in each frame, one for each case. Use Mathematica for both parts and supply a copy of the notebook.

Solution: [nex123] Brownian harmonic oscillator III: contour integrals

The generalized Langevin equation for the Brownian harmonic oscillator,

Z t dx 0 0 0 1 2 −(γ/m)t m + dt α(t − t )x(t ) = f(t), α(t) = mω0e , dt −∞ ω0

2 where α(t) is the attenuation function, mω0 the spring constant, and f(t) a correlated-noise random force, is known to produce the following expression for the spectral density of the position coordinate: 2 Z ∞ Sff (ω)/ω0 iωt Sxx(ω) = 2 , αˆ(ω) = dt e α(t),Sff (ω) = 2kBT <[ˆα(ω)], |αˆ(ω) − iωm| 0 where the relation between the random-force spectral density, Sff (ω), and the Laplace-transformed attenuation function,α ˆ(ω), is dictated by the ﬂuctuation-dissipation theorem. (a) Calculate Sff (ω) and determine its singularity structure. (b) Determine Sxx(ω) and its singularity structure for the cases γ/2m < ω0 (underdamped), γ/2m = ω0 (critically damped), and γ/2m > ω0 (overdamped). . R +∞ −iωt (c) Calculate hx(t)x(0)i = −∞ (dω/2π)e Sxx(ω) via contour integration for the two cases of underdamped and overdamped attenuation.

Solution: [nex58] Brownian harmonic oscillator IV: velocity correlations

The Brownian harmonic oscillator is speciﬁed by the Langevin-type equation, mx¨+γx˙ +kx = f(t), 2 where m is the mass of the particle, γ represents attenuation without memory, k = mω0 is the spring constant, and f(t) is a white-noise random force. 2 (a) Find the velocity spectral density by proving the relation Svv(ω) = ω Sxx(ω) and using the result from [nex121] for the position spectral density Sxx(ω). (b) Find the velocity correlation function by proving the relation hv(t)v(0)i = −(d2/dt2)hx(t)x(0)i and using the result from [nex122] for the position correlation function. Distinguish the cases p 2 2 2 p 2 2 2 2 2 2 ω1 = ω0 − γ /4m > 0, Ω1 = γ /4m − ω0 > 0, and ω0 = γ /4m for underdamped, overdamped, and critically damped motion, respectively. (c) Plot Svv(ω) versus ω/ω0 and hv(t)v(0)im/kBT versus ω0t with three curves in each frame, one for each case.

Solution: [nex59] Brownian harmonic oscillator V: formal solution for velocity

Convert the Langevin-type equation, mx¨+γx˙ +kx = f(t), for the overdamped Brownian harmonic 2 oscillator with mass m, damping constant γ, spring constant k = mω0, and white-noise random force f(t) into a second-order ODE for the stochastic variable v(t). Then show that

2 Z t −Γt ω0 −Γt 1 0 0 −Γ(t−t0) 0 v(t) = v0e c(t) − x0e sinh Ω1t + dt f(t )e c(t − t ) Ω1 m 0

p 2 2 with Γ = γ/2m,Ω1 = Γ − ω0, c(t) = cosh Ω1t − (Γ/Ω1) sinh Ω1t is a formal solution for initial conditions x(0) = x0 and v(0) = v0.

Solution: [nex60] Brownian harmonic oscillator VI: nonequilibrium correlations

Use the formal solution for the velocity,

2 Z t −Γt ω0 −Γt 1 0 0 −Γ(t−t0) 0 v(t) = v0e c(t) − x0e sinh Ω1t + dt f(t )e c(t − t ), Ω1 m 0

p 2 2 with Γ = γ/2m,Ω1 = Γ − ω0, c(t) = cosh Ω1t − (Γ/Ω1) sinh Ω1t of the Langevin-type equation, mx¨ + γx˙ + kx = f(t), for the overdamped Brownian harmonic oscillator with mass m, damping 2 constant γ, spring constant k = mω0, initial conditions x(0) = x0 and v(0) = v0, and white-noise random force f(t) with intensity If to calculate the velocity correlation function hv(t2)v(t1)i for the nonequilibrium state. Then take the limit t1, t2 → ∞ with 0 < t2 − t1 < ∞ to recover the result of [nex58] for the stationary state.

Solution: Brownian motion: panoramic view [nln23]

• Levels of contraction (horizontal) • Modes of description (vertical)

−→ contraction −→

relevant N-particle 1-particle conﬁguration space phase space phase space space

dynamical {x , p } x, p x variables i i

theoretical Hamiltonian Langevin Einstein framework mechanics theory theory

... for generalized Langevin Langevin dynamical Langevin equation equation variables equation (for dt τR) (for dt τR)

... for quant./class. Fokker-Planck Fokker-Planck probability Liouville equation (Ornstein- equation (diﬀusion distribution equation Uhlenbeck process) process)

• Here dt is the time step used in the theory and τR is the relaxation time associated with the drag force the Brownian particle experiences. • The generalized Langevin equation is equivalent to the Hamiltonian equation of motion for a generic classical many-body system and equivalent to the Heisenberg equation of motion for a generic quantum many- body system.