Diffusion in a Logarithmic Potential Anomalies, Aging & Ergodicity Breaking

Transcription

1 Diffusion in a Logarithmic Potential Anomalies, Aging & Ergodicity Breaking David Kessler Bar-Ilan Univ. E. Barkai (Bar-Ilan) A. Dechant (Augsburg) E. Lutz (Augsburg) PRL, (2010) arxiv: arxiv:

2 A Simple Problem A particle diffusing in an external potential V (x) Probability Density Function W (x, t) satisfies the Fokker-Planck equation: [ 2 W (x, t) = D t x ] k B T x V (x) W (x, t) Equilibrium Density: W eq = 1 Z e V (x)/k BT ; Z = e V (x)/k BT What happens when V (x) V 0 ln x at large x? Then W eq x V 0/k B T. Power-Law Tail Divergent moments!

3 Applications This kind of logarithmically growing potential appears in many systems Charges in the vicinity of a long uniformly charged polymer Manning (1969) Tracers in a 1-d driven fluid Levine, Mukamel & Schütz (2005) Dynamics of bubbles in dsdna Fogedby & Metzler (2007) Vortex dynamics Chevanis & Lemou (2007) Diffusive spreading of momenta in optical lattices Castin, Dalibard & Cohen- Tannoudji (1991); Marksteiner, Ellinger & Zoller (1996); Douglas, Gergamini & Renzoni (2006)

4 What s wrong with divergent moments? Consider the spreading of momenta in an optical lattice: Castin, Dalibard & Cohen-Tannoudji (1991) V (p)/k B T = U 0 2 ln(1 + p2 ) Describes cooling effect (drift toward zero p) For 1 < U 0 < 3, the average kinetic energy p 2 /2m =, which is unphysical. For U 0 < 1, Z = There is no equilibrium distribution. What then?

5 What does the computer say? We solve numerically the time-dependent Fokker-Planck eqn. δ(x) initial condition with V (x)/k B T = ln(1 + x 2 ). for a 10 0 W(x,t) t = 4Dt = Dt = Dt = Dt = Dt = x As t, W (x, t) does approach W eq (x) (blue trace). At any finite t, the power-law tail of W (x, t) is cut off beyond some x c which grows with t

6 Turning the picture into an equation Want to describe the large t, large x behavior of W (x, t) V (x)/k B T U 0 ln x /a + o(1) for x Note that for large x, the spatial part of the Fokker-Planck equation becomes homogeneous [ ] 2 W (x, t) = D t x 2 + U 1 0 W (x, t) xx Assume scaling solution: W (x, t) (4Dt/a 2 ) γ F(z), where z x/ 4Dt For 1 x 4Dt, W (x, t) W eq (x) Z 1 (x/a) U 0 This gives F z U 0 for z 1 γ = U 0 /2 F satisfies F + ( ) U0 z + 2z F ( ) U0 z 2 2U 0 F = 0

7 Scaling Solution W (x, t) (4Dt/a 2 ) U 0/2 F(z); F + ( ) U0 z + 2z F z x/ 4Dt ( ) U0 z 2 2U 0 F = 0 Solution: ( Γ U0 +1 F = Z 1 z U 0 Γ 2, z2 ) ( U0 +1) 2 ) Γ(α, x) is the incomplete Gamma function, Γ(α, x) = x tα 1 e t dt x α 1 e x, x 1. W (x, t) has a Gaussian falloff for x 4Dt

8 Data Collapse W (x, t) (4Dt/a 2 ) U 0/2 F(z); z x/ 4Dt For V (x) = ln(1 + x 2 ), i.e. U 0 = 2, plot DtW (x, t) vs. z D t W(x,t) t = 4Dt = Dt = Dt = Dt = Dt = x / (4Dt) 1/2

9 Second moment W eq useless for calculating x 2. x 2 = 0 x2 W (x, t)dx is dominated by x 4Dt, grows with time! To leading order, for 1 < U 0 < 3, x 2 is determined by the scaling function F(z). x 2 0 x 2 (4Dt/a 2 ) U 0/2 F(x/ 4Dt)dx = a 3 (4Dt/a 2 ) 3/2 U 0/2 = ( 4Dt a ) 3/2 U0 /2 For U 0 1, x 2 t, as in normal diffusion 0 z 2 F(z)dz a 3 (3/2 U 0 /2)Γ((U 0 + 1)/2)Z For U 0 3, x 2 t 0, as with usual" potentials.

10 Infinite Covariant Density Similarly, all sufficiently high moments are determined by the scaling function, F. All sufficiently small moments are determined by W eq. We can think of F as defining a non-normalizable density: Ṡo, W ICD (z, t) a(4dt/a 2 ) 1/2 U 0/2 F(z) x q 2(4Dt) q/2 z q W ICD (z, t)dz; q > U 0 1 We call this the Infinite Covariant Density Infinite: It is non-normalizable, since W ICD z U 0 for small z. Covariant: It has a prefactor of t 1/2 U 0 /2. Density: It can be used to calculate moments. Plays a role dual to that of the equilibrium density x q = x q W eq (x)dx q < U 0 1

11 U 0 < 1 When U 0 < 1, this is NO normalizable equilibrium density, since the decay at large x is too slow. We expect x 2 t, independent of U 0, so that here we assume a scaling function of the form W (x, t) (4Dt/a 2 ) 1/2 G(z) Same scaling as with free diffusion, U 0 = 0. Fokker-Planck equation gives ( ) G U0 + z + 2z G ( ) U0 z 2 2 G = 0 Solution: 1 G(z) = aγ((1 U 0 )/2) z U 0 e z G(z) is a normalizable density 2

12 U 0 < 1 (cont d) W (x, t) (4Dt/a 2 ) 1/2 G(z); G(z) = 1 aγ((1 U 0 )/2) z U 0 e z 2 For small z, W (x, t) (4Dt/a 2 ) (U 0 1)/2 (x/a) U 0 aγ((1 U 0 )/2 (4Dt/a 2 ) (U 0 1)/2 e V (x)/k BT aγ((1 U 0 )/2 In central region, then e V (x)/k BT W (x, t) (4Dt/a 2 ) (U 0 1)/2 aγ((1 U 0 )/2, which approximately solves the Fokker-Planck equation to leading order in 1/t.

13 U 0 < 1, Numerical test We look at V (x)/k B T = 1 4 ln(1 + x2 ) (U 0 = 1/2) (4Dt) 1/4 W(x,t) t = 4Dt = Dt = Dt = Dt = (4Dt) 1/2 W(x,t) t = 4Dt = Dt = Dt = Dt = x x / (4Dt) 1/2

14 Breaking of ergodicity The system exhibits anomalous diffusion for 1 < U 0 < 3. What about ergodicity? Consider X 2 (t) x 2 (t), where x 1 t the random walk x(t). t 0 x(t )dt is the time-average of For paths which start at the origin, x = 0 For ergodic systems, lim t x(t) = x eq = 0 for every path, so lim t X 2 (t) is a measure of ergodicity breaking.

15 And the numerics say... Look at x(t) for a single trajectory, V (x)/k B T = ln(1 + x 2 ), (U 0 = 2). No sign of ergodicity!

16 General Formalism X 2 (t) is calculated from correlation function C(t 2, t 1 ) x(t 2 )x(t 1 ) : X 2 (t) = 2 t t2 t 2 dt 2 dt 1 C(t 2, t 1 ) 0 C(t 2, t 1 ) can be calculated in terms of x(x 1, τ) dx 2x 2 P (x 2, τ x 1, 0), the mean x at time τ after starting at x 1 : C(t 2, t 1 ) = 0 dx 1 x(x 1, t 2 t 1 ) W (x 1, t 1 ) W (x, t) is the solution of the Fokker-Planck equation with δ(x) initial condition x(x 1, τ) is given by the solution of a backward time-dependent Fokker-Planck eqn. á la Szabo, Schulten & Schulten, (1980) [ 2 τ x(x 1, τ) = D x 2 1 V (x 1 ) k B T ] x(x 1, τ) ; x(x 1, 0) = x 1 x 1 For normal" potentials, x(x 1, τ) c 0 e c 1τ, X 2 (t) t 1.

17 Scaling Solution [ 2 τ x(x 1, τ) = D x 2 1 V (x 1 ) k B T ] x(x 1, τ) ; x(x 1, 0) = x 1 x 1 For logarithmic potential, assume x(x 1, τ) (4Dτ/a 2 ) 1/2 g(x 1 / 4Dτ) Solution: g(z) = ( aγ(3/2) Γ((3 + U 0 )/2) z1+u 0 e z 2 3 M 2, 3 + U ) 0, z 2 2 M(a, b, x) is the Kummer M function From this, we can get the correlation function: f(0) = 1 C(t 2, t 1 ) x 2 (t 1 ) f ( ) t2 t 1 ; 1 < U 0 < 3 t 1 f(r) r U 0 /2 for r 1. (super)aging! ( x 2 (t) is growing in time)

18 x 2 (t) From C(t 2, t 1 ) we can finally get X 2 (t) = x 2 (t). X 2 (t) t 1 ; U 0 < 1 x 2 (t) free diffusion ergodicity breaking t 3 U 0 2 ; 1 < U 0 < 3 x 2 (t) ERGODICITY BREAKING t 3 U 0 2 ; 3 < U 0 < 5 NOT x 2 slow approach to ergodicity t 1 ; U 0 > 5 NOT x 2 normal ergodic behavior

19 And the numerics say... Look at X 2 (t) for V (x)/k B T = U 0 /2 ln(1 + x 2 ), for different U 0 s. FIG. 3. In the nonergodic phase k B T > U 0 /3, x 2 (t) in-

20 Summary Anomalous diffusion, aging, ergodicity breaking Appearance of Infinite Covariant Density, in addition to Boltzmann Calculations can also be done via eigenfunction decomposition of Fokker-Planck eqn. (see poster of Andreas Dechant!) All this without heavy-tailed waiting times, without disorder, without many-body physics Even simple systems can do complicated things! For optical lattice system, where p diffuses, what is distribution of x(t)? Levy? Marksteiner, Ellinger & Zoller (1996) Experimentally observable? Douglas, Bergamini & Renzoni (2006) PRL, (2010) arxiv: arxiv: