08. Brownian Motion. University of Rhode Island. Gerhard Müller University of Rhode Island,

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1 University of Rhode Island Nonequilibrium Statistical Physics Physics Course Materials Brownian Motion Gerhard Müller University of Rhode Island, Follow this and additional works at: 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, "8. Brownian Motion" (215). Nonequilibrium Statistical Physics. Paper 8. This Course Material is brought to you for free and open access by the Physics Course Materials at It has been accepted for inclusion in Nonequilibrium Statistical Physics by an authorized administrator of For more information, please contact

2 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 [nex7] Langevin s theory (on most contracted level of description) White noise Brownian motion and Gaussian white noise [nln2] 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]

3 Generalized Langevin equation: attenuation with memory [nln22] Fluctuation-dissipation theorem Velocity correlation function of Brownian particle III [nex12] 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 [nex6] Generalized Langevin equation inferred from microscopic dynamics Brownian motion: levels of contraction and modes of description [nln23]

4 [nex117] Thermal diffusivity A solid wall of very large thickness and lateral extension (assumed to occupy all space at z > ) is brought into contact with a heat source at its surface (z = ). The wall is initially in thermal equilibrium at temperature T. The heat source is kept at the higher temperature T 1. The contact is established at time t =. Show that the temperature profile inside the wall depends on time as follows: ( ) z T (z) = T + (T 1 T )erfc 2, D T t where D T = λ/c V is the thermal diffusivity, λ the thermal conductvity, and c V the specific heat.

5 [nex37] Campbell processes. Consider a stationary stochastic process of the general form Y (t) = k F (t t k), where the times t k 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 : Y = λ dτ F (τ), Y (t)y () Y (t)y () Y (t) Y () = λ dτ F (τ)f (τ + t). Apply Campbell s theorem to calculate the average current I and the current autocorrelation function I(t)I() for the shot noise process I(t) = k F (t t k) with F (t) = qe αt Θ(t).

6 [nex7] Critically damped ballistic galvanometer. The response of a critically damped ballistic galvanometer to a current pulse at t = is Ψ(t) = cte γt. Consider the situation where the galvanometer experiences a steady stream of independent random current pulses, X(t) = k Ψ(t t k), where the t k are distributed randomly with an average rate n of occurrences. (a) Find the average displacement X of the galvanometer. (b) Find the autocorrelation function for the displacement of the galvanometer, X(t)X(), and the associated spectral density Φ XX (ω).

7 Brownian motion and Gaussian white noise [nln2] Gaussian white noise: completely factorizing stationary process. P w (y 1, t 1 ; y 2, t 2 ) = P w (y 1 )P w (y 2 ) if t 2 t 1 (factorizability) ( ) 1 P w (y) = exp y2 (Gaussian nature) 2πσ 2 2σ 2 y(t) = (no bias) y(t 1 )y(t 2 ) = I w δ(t 1 t 2 ) I w = y 2 = σ 2 (intensity) (whiteness) Brownian motion: Markov process. Discrete time scale: t n = n dt. Position of Brownian particle at time t n : z n. P (z n, t n ) = P (z n, t n z n 1, t n 1 ) P (z }{{} n 1, t n 1 ) P w(y n)δ(y n [z n z n 1 ]) Specification of white-noise intensity: I w = 2Ddt. n Sample path of Brownian particle: z(t n ) = y(t i ). Position of Brownian particle: n mean value: z(t n ) = y(t i ) =. variance: z 2 (t n ) = i=1 i=1 n y(t i )y(t j ) = 2Dndt = 2Dt n. i,j=1 (white-noise transition rate). Gaussian white noise with intensity I w = 2Ddt is used here to generate the diffusion process discussed previously [nex26], [nex27], [nex97]: 1 P (z, t + dt z, t) = exp ( (z z ) ) 2. 4πDdt 4Ddt Sample paths of the diffusion process become continuous in the limit dt (Lindeberg condition). However, in the present context, we must use dt τ R, where τ R 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.

8 Wiener process [nls4] Specifications: 1. For t < t 1 <, the position increments x(t n, t n 1 ), n = 1, 2,... are independent random variables. 2. The increments depend only on time differences: x(t n, t n 1 ) = x(dt n ), where dt n = t n t n The increments satisfy the Lindeberg condition: 1 lim P[ x(dt) ɛ] = for all ɛ > dt dt The diffusion process discussed previously [nex26], [nex27], [nex97], 1 P (x, t + dt x, t) = exp ( (x x ) ) 2, 4πDdt 4Ddt is, for dt, a realization of the Wiener process. Sample paths of the Wiener process thus realized are everywhere continuous and nowhere differentiable. W (t n ) = n x(dt i ), t n = i=1 n dt i. i=1 [adapted from Gardiner 1985]

9 [nex54] Autocorrelation function of the Wiener process. Use the regression theorem, x(t)x(t + dt) [, ] = dx 1 dx 2 x 1 x 2 P (x 2, t + dt x 1, t)p (x 1, t, ), 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?

10 Attenuation without memory [nln21] Langevin equation for Brownian motion: m dv dt + γv = f(t). Random force (uncorrelated noise): S ff (ω) = 2γk B T, C ff (t) = 2γk B T δ(t). Sff Ω kbt Γ.5 Cff t kbt Γ Stochastic variable (velocity): S vv (ω) = 2γk BT γ 2 + m 2 ω 2, C vv(t) = k BT m e (γ/m)t. Svv Ω kbt m 1 Γ 2 Γ 3 Γ Cvv t m kbt m Γ 1.2 Γ 2 Γ

11 [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 dt = v, dv dt = γ m v + 1 m f(t). 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() = and v() = v.

12 [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 v(t 1 )v(t 2 ) of a Brownian particle for t 1 > t 2 as a conditional average from the formal solution (see [nex53]) v(t) = v e γt/m + 1 m t dt e (γ/m)(t t ) f(t ) of the Langevin equation with a white noise random force of intensity I w. Show that for t 1 > t 2 γ/m the result only depends on the time difference t 1 t 2. Use equipartition to determine the temperature dependence of the random-force intensity I f.

13 [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 x 2 (t) of a Brownian particle via integration with initial condition x() = from the (stationary) velocity autocorrelation function calculated in [nex55]: v(t 1 )v(t 2 ) = k BT m e (γ/m) t1 t2.

14 [nex57] Mean square displacement of Brownian particle II 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 x 2 (t) of a Brownian particle from the formal solution (see [nex53]) x(t) = v m γ ( 1 e (γ/m)t) + 1 γ t ( ds 1 e (γ/m)(t s)) f(s) of the Langevin equation with a white-noise random force of intensity I f = 2k B T γ by taking a thermal average over initial velocities.

15 [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 γẋ and a white-noise random force f(t): Construct from it the second-order linear ODE, mẍ = γẋ + f(t). m d2 dt 2 x2 + γ d dt x2 = 2k B T, by using equipartition, ẋ 2 = k B T/m. Then solve this ODE for initial conditions d x 2 /dt = and x 2 =. Identify the quadratic time-dependence of x 2 in the ballistic regime, t m/γ, and the linear time dependence in the diffusive regime, t m/γ.

16 Ergodicity [nln13] Consider a stationary process x(t). Quantities of interest are expectation values related to x(t). Theoretically, we determine ensemble averages: x(t), x 2 (t), x(t)x(t + τ) are independent of t. Experimentally, we determine time averages: x(t), x 2 (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(t 1 t 2 ). = x(t 1 )x(t 2 ) x(t 1 ) x(t 2 ) are as follows (set τ = t 2 t 1 and t = t 1 ): x 2 x 2 1 = lim T 4T 2 = lim T = lim T 1 4T 2 1 2T +T T +2T 2T +2T 2T Necessary condition: lim C(τ) =. τ Sufficient condition: +T dt 1 dt 2 [ x(t 1 )x(t 2 ) x(t 1 ) x(t 2 ) ] T dτ C(τ) <. dτ C(τ)(2T τ ) ( dτ C(τ) 1 τ ) =. 2T t 2 τ=2 τ=1 τ= t 1 T τ= 1 T τ= 2

17 Intensity spectrum and spectral density [nln14] Consider an ergodic process x(t) with x =. Fourier amplitude: x(ω, T ). = T dt e iωt x(t) x( ω, T ) = x (ω, T ). Intensity spectrum (power spectrum): I xx (ω) =. 1 lim T T x(ω, T ) 2. Correlation function: C xx (τ) =. 1 x(t)x(t + τ) = lim T T Spectral density: S xx (ω). = + dτ e iωτ C xx (τ). Wiener-Khintchine theorem: I xx (ω) = S xx (ω). Proof: I xx (ω) = 1 lim T T = 1 lim T T T T T dt e iωt x(t ) T τ [e iωτ dτ T = lim 2 dτ cos ωτ 1 T T = 2 dτ cos ωτc xx (τ) = dt e iωt x(t) T dt x(t )x(t + τ) + e iωτ T τ T τ + dt x(t)x(t + τ) dτ e iωτ C xx (τ) = S xx (ω). dt x(t)x(t + τ). ] dt x(t)x(t + τ) T t T τ τ= t t τ=t t T T τ t

18 [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, m dv = γv + f(t), dt where γ is the damping constant and f(t) is a white-noise random force, f(t)f(t ) = I f δ(t t ), with intensity I f = 2k B T γ in thermal equilibrium as shown in [nex55]. (a) Convert the differential equation for v(t) into an algebraic equation for the Fourier amplitude ṽ(ω). = dt e iωt v(t). (b)use the Wiener-Khintchine theorem (see [nln14]) to calculate the spectral density S vv (ω) and, via inverse Fourier transform, the velocity correlation function v(t)v(t ) in thermal equlibrium.

19 Attenuation with memory [nln22] Generalized Langevin equation for Brownian harmonic oscillator: m dx dt + t dt α(t t )x(t ) = 1 ω f(t), α(t) = mω 2 e (γ/m)t. Random force (correlated noise): S ff (ω) = 2k BT γm 2 ω 2 γ 2 + m 2 ω 2, C ff(t) = k B T mω 2 e (γ/m)t. 2. Sff Ω kbtω Γ 2m Ω Γ 2m Ω Γ 2m Ω Cff t kbtmω Γ 2m Ω Γ 2m Ω Γ 2m Ω Stochastic variable (position): 2k B T γ S xx (ω) = m 2 (ω 2 ω 2 ) 2 + γ 2 ω, 2 ] k B T e γ mω 2 2m [cos t ω 1 t + γ 2mω 1 sin ω 1 t, ω 1 = ω 2 γ 2 /4m 2 > k C xx (t) = B T e [ γ mω 2 2m t 1 + γ t], ω 2m = γ/2m ] k B T e γ mω 2 2m [cosh t Ω 1 t + γ 2mΩ 1 sinh Ω 1 t, Ω 1 = γ 2 /4m 2 ω 2 > Sxx Ω kbt Γ 2m Ω Γ 2m Ω Γ 2m Ω Cxx t mω 2 kbt.8.6 Γ 2m Ω.4.2. Γ 2m Ω.2 Γ 2m Ω

20 [nex12] Velocity correlation function of Brownian particle III The generalized Langevin equation for a particle of mass m constrained to move along a sraight line, m dv t dt = dt α(t t )v(t ) + f(t), is known to produce the following expression for the spectral density of the velocity: S vv (ω) = S ff (ω) ˆα(ω) iωm 2, ˆα(ω). = dt e iωt α(t), S ff (ω) = 2k B T R[ˆα(ω)], where the relation between the random-force spectral density, S ff (ω), and the Laplace-transformed attenuation function, ˆα(ω), is dictated by the fluctuation-dissipation theorem. The special case of Brownian motion (see [nex55], [nex119]) uses attenuation without memory: α(t t ) = 2γδ(t t )Θ(t t ). Calculate the velocity correlation function, v(t)v(t ), of the Brownian particle in thermal equilibrium from the above expression for S vv (ω) via contour integration in the plane of complex ω.

21 [nex121] Brownian harmonic oscillator I: Fourier analysis The Brownian harmonic oscillator is specified by the Langevin-type equation, mẍ + γẋ + kx = f(t), where m is the mass of the particle, γ represents attenuation without memory, k = mω 2 is the spring constant, and f(t) is a white-noise random force. Calculate the spectral density S xx (ω) of the position coordinate via Fourier analysis and by using the result S ff (ω) = 2k B T γ for the random-force spectral density as dictated by the fluctuation-dissipation theorem.

22 [nex122] Brownian harmonic oscillator II: position correlation function The Brownian harmonic oscillator is specified by the Langevin-type equation, mẍ+γẋ+kx = f(t), where m is the mass of the particle, γ represents attenuation without memory, k = mω 2 is the spring constant, and f(t) is a white-noise random force. (a) Start from the result S xx (ω) = 2γk B T/[m 2 (ω 2 ω 2 ) 2 + γ 2 ω 2 ] for the spectral density of the position coordinate as calculated in [nex121] to derive the position correlation function x(t)x(). = + dω 2π e iωt S xx (ω) = ] k B T e γ mω 2 2m [cos t ω 1 t + γ 2mω 1 sin ω 1 t k B T e [ γ mω 2 2m t 1 + γ 2m t] k B T e γ mω 2 2m [cosh t Ω 1 t + γ 2mΩ 1 sinh Ω 1 t for the cases ω 1 = ω 2 γ2 /4m 2 > (underdamped), ω 2 = γ 2 /4m 2 (critically damped), and Ω 1 = γ 2 /4m 2 ω 2 > (overdamped), respectively. (b) Plot S xx (w) versus ω/ω and x(t)x() mω/k 2 B T versus ω t with three curves in each frame, one for each case. Use Mathematica for both parts and supply a copy of the notebook. ]

23 [nex123] Brownian harmonic oscillator III: contour integrals The generalized Langevin equation for the Brownian harmonic oscillator, m dx t dt + dt α(t t )x(t ) = 1 f(t), α(t) = mω ω e 2 (γ/m)t, where α(t) is the attenuation function, mω 2 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: S xx (ω) = S ff (ω)/ω 2 ˆα(ω) iωm 2, ˆα(ω) = dt e iωt α(t), S ff (ω) = 2k B T R[ˆα(ω)], where the relation between the random-force spectral density, S ff (ω), and the Laplace-transformed attenuation function, ˆα(ω), is dictated by the fluctuation-dissipation theorem. (a) Calculate S ff (ω) and determine its singularity structure. (b) Determine S xx (ω) and its singularity structure for the cases γ/2m < ω (underdamped), γ/2m = ω (critically damped), and γ/2m > ω (overdamped). (c) Calculate x(t)x() =. + (dω/2π)e iωt S xx (ω) via contour integration for the two cases of underdamped and overdamped attenuation.

24 [nex58] Brownian harmonic oscillator IV: velocity correlations The Brownian harmonic oscillator is specified by the Langevin-type equation, mẍ+γẋ+kx = f(t), where m is the mass of the particle, γ represents attenuation without memory, k = mω 2 is the spring constant, and f(t) is a white-noise random force. (a) Find the velocity spectral density by proving the relation S vv (ω) = ω 2 S xx (ω) and using the result from [nex121] for the position spectral density S xx (ω). (b) Find the velocity correlation function by proving the relation v(t)v() = (d 2 /dt 2 ) x(t)x() and using the result from [nex122] for the position correlation function. Distinguish the cases ω 1 = ω 2 γ2 /4m 2 >, Ω 1 = γ 2 /4m 2 ω 2 >, and ω2 = γ 2 /4m 2 for underdamped, overdamped, and critically damped motion, respectively. (c) Plot S vv (ω) versus ω/ω and v(t)v() m/k B T versus ω t with three curves in each frame, one for each case.

25 [nex59] Brownian harmonic oscillator V: formal solution for velocity Convert the Langevin-type equation, mẍ+γẋ+kx = f(t), for the overdamped Brownian harmonic oscillator with mass m, damping constant γ, spring constant k = mω 2, and white-noise random force f(t) into a second-order ODE for the stochastic variable v(t). Then show that v(t) = v e Γt c(t) ω2 Ω 1 x e Γt sinh Ω 1 t + 1 m t dt f(t )e Γ(t t ) c(t t ) with Γ = γ/2m, Ω 1 = Γ 2 ω 2, c(t) = cosh Ω 1t (Γ/Ω 1 ) sinh Ω 1 t is a formal solution for initial conditions x() = x and v() = v.

26 [nex6] Brownian harmonic oscillator VI: nonequilibrium correlations Use the formal solution for the velocity, v(t) = v e Γt c(t) ω2 Ω 1 x e Γt sinh Ω 1 t + 1 m t dt f(t )e Γ(t t ) c(t t ), with Γ = γ/2m, Ω 1 = Γ 2 ω 2, c(t) = cosh Ω 1t (Γ/Ω 1 ) sinh Ω 1 t of the Langevin-type equation, mẍ + γẋ + kx = f(t), for the overdamped Brownian harmonic oscillator with mass m, damping constant γ, spring constant k = mω 2, initial conditions x() = x and v() = v, and white-noise random force f(t) with intensity I f to calculate the velocity correlation function v(t 2 )v(t 1 ) for the nonequilibrium state. Then take the limit t 1, t 2 with < t 2 t 1 < to recover the result of [nex58] for the stationary state.

27 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 variables {x i, p i } x, p x theoretical Hamiltonian Langevin Einstein framework mechanics theory theory... for generalized Langevin Langevin dynamical Langevin equation equation variables equation (for dt τ R ) (for dt τ R )... 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 τ R 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 manybody system.