Diffusion far from equilibrium
Peter Reimann
Universität Bielefeld
Fakultät für Physik
1. The harmonic oscillator bath
2. General heat baths
3. Example: thermal diffusion
4. Overdamped dynamics
5. The Fokker-Planck equation
6. Equilibrium conditions
7. First passage time moments
8. Diffusion on the tilted washboard
1. The harmonic oscillator bath .
.Consider the classical Hamiltonian H = HS + HB
HS = HS (x, p, t) =
p2
2m
+ U (x, t)
system
2 N
2 2
P
Mi ω i
Pi
ci x
+
X
−
HB = HB (X, P, x) =
i
2Mi
2
M ω2
i i
i=1
N n 2
P
Pi
Mi ωi2 2
2 Xi
N
P
o
x2
2
N
P
c2i
Mi ωi2
+
−x
ci Xi +
i=1
{z
} | i=1{z } | i=1
{z }
|
...... oscillator bath ........ coupling .. “counter term”
=
2Mi
Mi, ωi, ci : model parameters
Bath dynamics: Ẋi =
∂H
MiẌi = Ṗi = − ∂X
i
∂H
∂Pi
=
Pi
Mi
(i = 1, ..., N )
= −Miωi2 Xi − Mci ωx2
⇒
i i
i (0)
Xi(t) = Xi(0) cos(ωit) + P
Mi ωi sin(ωi t)
Rt
ci
.................................. + Miωi 0 ds x(s) sin(ωi(t − s))
i x(t)
Partial integration & Yi(t) := Xi(t) − cM
ω2
i i
⇒
i(0)
Yi(t) = Yi(0) cos(ωit) + P
Miωi sin(ωi t)
ci R t
.................................. − M ω2 0 ds ẋ(s) cos(ωi(t − s))
i i
System dynamics: ẋ =
∂H
∂p
=
p
m
N
P
ci Xi − Mci ωx2
i=1
{z i i }
|
............................................................. Y i(t)
0
mẍ = ṗ = − ∂H
=
−U
(x, t) +
∂x
1
2
⇒ Generalized Langevin equation
Z t
ds ẋ(s) η̂(t−s) + ξ(t)
mẍ(t) + U 00(x(t), t) = −
mẍ(t) + U (x(t), t)
| 0
{z
}
|
{z
}
...... system dynamics .................. bath effects
........................................... dissipation ................ noise
η̂(t) :=
N
P
i=1
ξ(t) :=
c2i
Mi ωi2
N n
P
cos(ωit) = η̂(−t)
ciYi(0) cos(ωit) +
i=1
memory kernel
ci
Mi ωi Pi(0) sin(ωi t)
o
• same time inversion symmetry as original Hamiltonian
dynamics!
• bath initial conditions: origin of randomness
• For any finite N : η̂(t) and ξ(t) quasi-periodic (Poincaré
recurrences)
⇒ N → ∞ required, such that η̂(t) → 0 for t → ∞
(N → ∞ before t → ∞ as usual for heat baths).
• Special case: Mi, ωi, ci such that for N → ∞
η̂(t) → 2 η δ(t)
(Ohmic bath)
⇒
mẍ(t) + U 0(x(t), t) = −η ẋ(t) sign(t) + ξ(t)
Langevin equation with single model parameter η
(“friction coefficient”); usually only t ≥ 0 of interest.
ξ(t) =
N
P
i=1
n
3
o
i Pi(0)
ciYi(0) cos(ωit) + cM
sin(ωit) = ξ(t, φ0)
i ωi
i x(0)
Yi(0) := Xi(0) − cM
ω2
i i
φ0 := {Y1(0), ..., YN (0), P1(0), ..., PN (0)}
t ≤ 0: assume U (x, t) ≡ U0(x) & system+bath at
thermal equilibrium (temperature T ) ⇒
ρ(x(0), p(0), φ0) = Z −1e−H(x(0),p(0),φ0)/kT
o
N n 2
P
Mi ωi2 2
Pi (0)
p20
H(x0, p0, φ0) = 2m + U0(x0) +
2Mi + 2 Yi (0)
i=1
t > 0: dynamics of actual interest with x(0) = x0,
p(0) = p0 ⇒ relevant conditional probability density
ρ̃(φ0) := ρ(φ0|x0, p0) = Z̃ −1e−H(x0,p0,φ0)/kT
• hξ(t)i :=
R
⇒
dφ0 ξ(t, φ0) ρ̃(φ0) = 0
• hξ(t) ξ(s)i = kT η̂(t − s)
(FDT 2)
• Higher order correlations ⇒ Gaussian random process
For Ohmic bath: η̂(t) = 2 η δ(t) ⇒
hξ(t) ξ(s)i = 2 η kT δ(t − s)
Gaussian white noise: uncorrelated, no “memory”; bath
much “faster” than system (separation of time scales);
Single model parameter η =
ˆ coupling-strength.
4
Conceptual remarks
• Same model but without “counter 2term”
2 P c
i
“renormalized potential” U (x, t) − x2
M ω2
⇒
i i
• For t ≤ 0 system+bath (weakly) coupled to
“superbath” (⇒ canonical distribution)
• For t > 0 superbath removed (⇒ Hamiltonian
dynamics); U (x, t) of actual interest “switched on”
⇒ in general system+bath out of equilibrium; time
inversion symmetry broken.
• Non-equilibrium theory based on Hamiltonian and
equilibrium statistical mechanics without any further
assumptions or approximations.
• Harmonic oscillator bath alone is integrable: not
generic. System+bath is generic due to anharmonic
U (x, t) and/or superbath.
5
Historical remarks
• Model which can be mapped to a harmonic oscillator
bath but without any fluctuation effects (T = 0):
H. Lamb, On a peculiarity of the wave-system due to
the free vibrations of a nucleus in an extended medium,
Proc. London. Math. Soc. 32 (1900) 208.
• Harmonic oscillator model for an oscillating electrical
dipole under blackbody irradiation:
A. Einstein and L. Hopf, Statistische Untersuchung
der Bewegung eines Resonators in einem Strahlungsfeld,
Ann. Phys. (Leipzig) 33 (1910) 1105.
• Exactly our above model with a harmonic potential
U (x) but without determining the statistical properties
of the fluctuations ξ(t):
N. N. Bogolyubov, Elementary example for
establishing statistical equilibrium in a system coupled
to a thermostat, in: On some statistical methods in
mathematical physics, Publ. Acad. Sci. Ukr. SSR,
1945, pp. 115–137, [In Russian].
• The latter issue, together with a quantum mechanical
transcription of the harmonic oscillator bath model:
V. B. Magalinskii, Dynamical model in the theory
of the Brownian motion, Sov. Phys. JETP 9 (1959)
1381, [JETP 36 (1959) 1942].
6
• Subsequent re-inventions, refinements, and generalizations of the model:
R. J. Rubin, Statistical dynamics of simple cubic
lattices. Model for the study of Brownian motion, J.
Math. Phys. 1 (1960) 309.
R. P. Feynman and F. L. Vernon, The theory of
a general quantum systems interacting with a linear
dissipative system, Ann. Phys. (New York) 24 (1963)
118.
P. Ullersma, An exactly solvable model for Brownian
motion, Physica (Utrecht) 32 (1966) 27, 56, 74, and
90.
R. Zwanzig, Nonlinear generalized Langevin equations,
J. Stat. Phys. 9 (1973) 215.
A. O. Caldeira and A. J. Leggett, Quantum
tunneling in dissipative systems, Ann. Phys. (New
York) 149 (1983) 374, erratum: Ann. Phys. (New
York) 153 (1984) 445.
G. W. Ford, J. T. Lewis, and R. F. O’Connell,
Quantum Langevin equation, Phys. Rev. A 37 (1988)
4419.
H. Grabert, P. Schramm, and G.-L. Ingold,
Quantum Brownian motion: The functional integral
approach, Phys. Rep. 168 (1988) 115.
2. General heat baths .
H
. = H S + HB ,
HB =
N
P
i=1
Pi2
2Mi
HS =
p2
2m
7
+ U (x, t)
+ UB (x, X1, ..., XN )
1 ,...,XN )
Bath dyn.: MiẌi = − ∂UB (x,X
∂Xi
φ0 := {Xi(0), Pi(0)}N
i=1
(i = 1, ..., N )
arbitrary but fixed
Consider x(t) as an arbitrary but fixed function of t ≥ 0
⇒ Xi(t) = Fi(t, [x(t0)]t0, φ0) ,
i = 1, ..., N , t ≥ 0
exists and unique: in general complicated chaotic
behavior. Closed analytical solution only for harmonic
oscillator bath, otherwise only “formal solution”.
1 (t),...,XN (t))
System dyn.: mẍ(t)+U 0(x(t), t) = − ∂UB (x(t),X∂x
∂
= − ∂x
UB (x(t), F1(t, [x(t0)]t0, φ0), ..., FN (t, [x(t0)]t0, φ0))
=: f (t, [x(t0)]t0, φ0)
mẍ(t) + U 0(x(t), t) = f (t, [x(t0)]t0, φ0)
|
{z
}
|
{z
}
...... system dynamics ........... bath effects
• Bath dynamics eliminated/integrated out/projected
• Self-consistency condition for x(t), t ≥ 0
• Non-local in time (memory)
• Bath initial conditions φ0: origin of randomness
8
t ≤ 0: assume U (x, t) ≡ U0(x) & system+bath at
thermal equilibrium (temperature T ).
t > 0: dynamics of actual interest with x(0) = x0,
p(0) = p0 ⇒ relevant conditional probability density
ρ̃(φ0) := ρ(φ0|x0, p0) = Z̃ −1e−H(x0,p0,φ0)/kT
Consider
hf (t, [x(t0)]t0, φ0)i
• Independent of φ0
:=
R
dφ0 ρ̃(φ0) f (t, [x(t0)]t0, φ0)
• [x(t0)]t0 uniquely fixed by [ẋ(t0)]t0 and x(0).
⇒ hf (t, [x(t0)]t0, φ0)i = f¯(t, x(0), [ẋ(t0)]t0)
Assumption: bath spatially homogeneous, i.e. for all ∆
UB (x + ∆, X1 + ∆, ..., XN + ∆) = UB (x, X1, ..., XN )
• Often the case
• Not necessary but simplifies the calculations
⇒ For any given t and [ẋ(t0)]t0 the averaged bath effect
satisfies
f¯(t, x(0), [ẋ(t0)]t ) = f¯(t, x(0) + ∆, [ẋ(t0)]t ) for all ∆
0
0
⇒ hf (t, [x(t0)]t0, φ0)i = f¯(t, 0, [ẋ(t0)]t0)
9
hf (t, [x(t0)]t0, φ0)i = f¯(t, 0, [ẋ(t0)]t0) =
R t δf¯(t,0,[ẋ(t0)≡0]t0)
0
t
¯
f (t, 0, [ẋ(t ) ≡ 0]0) + 0 ds
ẋ(s) + O(ẋ2)
δ ẋ(s)
• f¯(t, 0, [ẋ(t0) ≡ 0]t0) = hf (t, [x(t0) ≡ 0]t0, φ0)i = 0
•
δ f¯(t,0,[ẋ(t0)≡0]t0 )
δ ẋ(s)
= lim 1 hf (t, [x(t0) = Θ(t0 − s)]t0, φ0)i
→0
is a function of t − s only, which we denote as −η̂(t − s).
• Assume separation of time scales: System “slow”, bath
“fast” ⇒
O(ẋ2) negligible
⇒ Generalized Langevin equation
mẍ(t) + U 0(x(t), t) = f (t, [x(t0)]t0, φ0)
Z t
ds ẋ(s) η̂(t−s) + ξ(t)
mẍ(t) + U 00(x(t), t) = −
mẍ(t) + U (x(t), t)
| 0
{z
}
|
{z
}
...... system dynamics .................. bath effects
........................................... dissipation ................ noise
ξ(t) := f (t, [x(t0)]t0, φ0) − hf (t, [x(t0)]t0, φ0)i
⇒ hξ(t)i = 0
ξ(t) = ξ(t, [x(t0)]t0, φ0) = ξ(t, x(0), [ẋ(t0)]t0, φ0) =
= ξ(t, 0, [ẋ(t0)]t0, φ0) = ξ(t, 0, [ẋ(t0) ≡ 0]t0, φ0) + O(ẋ)
Separation of time scales ⇒ O(ẋ) negligible:
– Bad regarding single realizations ξ(t)
– Good regarding statistical properties of ξ(t)
—(=
ˆ van Kampen’s criticism of Kubo’s response theory)
10
Theorem:
If a thermal bath gives rise to an additive dissipation term
which is a linear functional of the system velocity and an
additive noise term which is independent of the system
dynamics
then all statistical properties of the noise are uniquely
fixed by the temperature and the dissipation term.
Proof:
Assume two such baths with identical temperatures and
dissipation terms but with different noises.
Then it is possible to construct a periodically operating
device which pumps energy from one bath into the other,
and this even after infinitesimal temperature changes ⇒
contradiction to the second law of thermodynamics.
Details: P. Reimann, Chem. Phys. 268 (2001) 337.
Apply theorem to Generalized Langevin equation
Rt
0
mẍ(t) + U (x(t), t) = − 0 ds ẋ(s) η̂(t−s) + ξ(t)
⇒ ξ(t) same as for a harmonic oscillator bath with
the same η̂(t) and T though the true microscopic bath
dynamics may be completely different!
Invoking separation of time scales once more ⇒ Ohmic
bath η̂(t) = 2 η δ(t) with single model parameter η ⇒
x(t) Markov process (no memory effects, local in time).
3.
Example: thermal diffusion .
.
11
Langevin equation for U (x, t) = −x F and Ohmic bath:
..........mẍ(t) = −η ẋ(t) + F + ξ(t)
Gaussian noise, hξ(t)i = 0, hξ(t) ξ(s)i = 2 η kT δ(t − s)
Example: Brownian motion of sphere (radius r) in a
fluid (viscosity ν): η = 6πν r (Stokes friction)
Auxiliary variable y(t) := x(t) − x(0) − t F/η ⇒
mÿ(t) = −η ẏ(t) + ξ(t) , y(0) = 0 , ẏ(0) = ẋ(0)−F/η
R t 0 − η (t−t0) 0
η
1
−m
t
⇒ ẏ(t) = ẏ(0) e
+ m 0 dt e m
ξ(t )
η
R t 00 R t00 0 − η (t00−t0) 0
1−e− m t
1
⇒ y(t) = ẏ(0) η/m + m 0 dt 0 dt e m
ξ(t )
|
{z
}
η
R t 0 R t 00 − η (t00−t0) 0
1−e− m t
1
ξ(t )
⇒ y(t) = ẏ(0) η/m + m 0 dt t0 dt e m
|
{z
}
η
Rt 0 0
η
0
1−e− m t
1
⇒ y(t) = ẏ(0) η/m + η 0 dt ξ(t ) [1 − e− m (t−t )]
⇒ hx(t)i = x(0) + t F/η + hy(t)i
η
...... hy(t)i =
−mt
ẏ(0) 1−e
η/m
hx(t)i
t
t→∞
................ lim
=
m
η [ẋ(0)
η
− F/η] [1 − e− m t]
= F/η
⇒ h[x(t) − hx(t)i]2i = h[y(t) − hy(t)i]2i =
η
η
m −2 m t
3m
2m − m t
−
...... = 2 kT
t
−
+
e
η
2η
η
2η e
.....
h[x(t)−hx(t)i]2 i
lim
2t
t→∞
.......... D0 = kT /η
=
hx2 (t)i−hx(t)i2
lim
2t
t→∞
=: D0
Einstein relation (in 1d)
m only relevant for transients !
Summary
of 1 - 3 : .
.
H = H S + HB ,
HB =
N
P
i=1
Pi2
2Mi
HS =
p2
2m
12
+ U (x, t)
+ UB (x, X1, ..., XN )
Separation of time scales & UB translation invariant
⇒ Generalized Langevin equation
Z t
ds ẋ(s) η̂(t−s) + ξ(t)
mẍ(t) + U 00(x(t), t) = −
mẍ(t) + U (x(t), t)
| 0
{z
}
|
{z
}
...... system dynamics .................. bath effects
........................................... dissipation ................ noise
ξ(t) same as for a harmonic oscillator bath with the same
η̂(t) and T (microscopic dynamics may be completely
different!):
• ξ(t) Gaussian random process
• hξ(t)i = 0
• hξ(t) ξ(s)i = kT η̂(t − s)
(FDT 2)
Invoking separation of time scales once more ⇒
mẍ(t) + U 0(x(t), t) = −η ẋ(t) + ξ(t) Markov process
ξ(t) Gaussian white noise
hξ(t) ξ(s)i = 2 η kT δ(t − s)
single model parameter η =
ˆ coupling-strength
U (x, t) = −xF ⇒
hx2 (t)i−hx(t)i2
lim
2t
t→∞
= D0 =
kT
η
13
4.
Overdamped dynamics
.
Consider Langevin equation with white Gaussian noise:
mẍ(t) + U 0(x(t), t) = −η ẋ(t) + ξ(t)
hξ(t) ξ(s)i = 2 η kT δ(t − s)
For the typically very small systems one has in mind,
and for which thermal fluctuations play any notable role,
inertia effects are negligible (η ∝ l , m ∝ l3),
mẍ(t) ' 0 ⇒
η ẋ(t) = −U 0(x(t), t) + ξ(t)
• mathematically subtle, singular limit
• extension to generalized Langevin equation unknown
• note that m appears neither in D0 = kT /η
• nor in hξ(t) ξ(s)i = 2 η kT δ(t − s)
5.
The Fokker-Planck equation
.
14
Consider average P (x, t) := hδ(x − x(t))i
over many realizations of η ẋ(t) = −U 0(x(t), t) + ξ(t)
If U 0(x, t) ≡ 0: free thermal diffusion
⇒
∂
∂t P (x, t)
If ξ(t) ≡ 0:
∂
∂t P (x, t)
=
=
∂2
D0 ∂x2 P (x, t)
,
U 0 (x(t),t)
ẋ(t) = −
,
η
∂
−ẋ(t) ∂x
δ(x−x(t))
D0 = kT /η
P (x, t) = δ(x−x(t)) ⇒
=
U 0 (x(t),t) ∂
η
∂x δ(x−x(t))
∗ variable x and function x(t) have nothing to do with each other ∗
0
∂ U (x(t),t)
∂x {
η
0
∂ U (x,t)
∂x {
η
δ(x−x(t))} =
n 0
o
U (x,t)
∂
∂
⇒ ∂t P (x, t) = ∂x
P (x, t)
η
=
δ(x−x(t))}
=
ˆ Liouville eq.
General case: by linear superposition
n 0
o
U (x,t)
∂
kT ∂ 2
∂
P (x, t) = ∂x
P (x, t) + η ∂x2 P (x, t)
∂t
η
{z
}
|
{z
} |
........................... drift term ............ diffusion term
..............Fokker-Planck equation (in 1d)
• More rigogous derivation: H. Risken, The Fokker-Planck
Equation, Springer, Berlin, 1984
• Historical account: N. G. van Kampen, Die FokkerPlanck-Gleichung, Phys. Bl. 53 (1997) 1012
15
6.
Equilibrium conditions
.
Recall:
t ≤ 0: U (x, t) ≡ U0(x) & system+bath at equilibrium.
t > 0: U (x, t) of actual interest “switched on”.
Thermal equilibrium
⇔ U (x, t) = U (x) (t-independent)
⇔ P (x, t) = P (x) = Z −1 e−U (x)/kT
& hẋ(t)i = 0
Non-equilibrium:
(i) Relaxation towards equilibrium:
(i) hẋ(t)i → 0 for t → ∞
(i) Example: Sect. 7
(ii)
(i)
(i)
(i)
(i)
External driving imposed via
U (x, t) and/or boundary conditions.
Sufficient: hẋ(t)i 6→ 0 for t → ∞
Example 1: U (x, t) = −x F , F 6= 0 (Sect. 5)
Example 2: Sect. 8
7.
First passage time moments
.
16
Consider a static potential, U (x, t) = U (x) for t > 0,
with U (x) = ∞ for x ≤ q ⇔ reflecting boundary at q
When passes x(t) the point b > q for the first time,
given x(0) = a ∈ (q, b) ? (Non-equilibrium problem)
∂
∂t P (x, t)
= Γx P (x, t) , P (x, 0) = δ(x − a)
0
1 ∂
∂
Γx := η ∂x U (x) + kT ∂x
Fokker-Planck operator
FPE:
Formal solution: P (x, t) = eΓxtδ(x − a)
n-th moment of first passage times:
R
R∞
d b
n
n
ht (a → b)i = 0 dt t − dt q dx P (x, t) =
R
R∞
n d b
= − 0 dt t dt q dx eΓxtδ(x − a) =
† R∞
R
n d b
= − 0 dt t dt q dx δ(x − a) eΓxt 1 =
† † R∞
R
∞
d
= − 0 dt tn dt
eΓat 1 = −Γ†a 0 dt tn eΓat 1
† If htn(a → b)i < ∞ then tn eΓat 1 → 0 for t → ∞
⇒ by partial integration for n ≥ 1:
† R
∞
htn(a → b)i = n 0 dt tn−1 eΓat 1
⇒
Γ†ahtn(a → b)i = −n htn−1(a → b)i
...........n = 1, 2, 3, ... ,
ht0(a → b)i ≡ 1
17
n
With
τ
(x)
:=
ht
(x → b)i:
n
.
Γ†x τn(x) = −n τn−1(x)
=
U 0(x) 0
− η τn(x)
00
+ kT
τ
η n (x)
d −U (x)/kT 0
e
τn(x)
= D0 eU (x)/kT dx
Without proof (plausible): τn(x) is a decreasing function
for x ∈ (q, b), approaching zero for x → b ⇒
τn(x) =
n
D0
Zb
dy eU (y)/kT
x
n = 1, 2, 3, ... ,
Zy
dz τn−1(z) e−U (z)/kT
q
τ0(x) ≡ 1
• general & exact
• long history: Schrödinger 1915; Pontryagin, Andronov,
Vitt 1933, etc. See Chapter VII in P. Hänggi, P. Talkner,
and M. Borkovec, Reaction Rate theory: Fifty years after
Kramers, Rev. Mod. Phys. 62 (1990) 252
18
8.
Diffusion on the tilted washboard
.
Model: .
η ẋ(t) = −U 0(x(t)) + ξ(t)
U (x) = V (x) − x F
V (x + L) = V (x)
hξ(t) ξ(s)i = 2 η kT δ(t − s)
.
Observables: .
hẋi := lim
hx(t)i
t
hx2(t)i − hx(t)i2
t→∞
D := lim
t→∞
2t
.
Applications: .
• real Brownian particle
• Brownian motion in a “traveling periodic potential”
(pump) of the form U (x − vt)
• systems in d ≥ 2 with periodic entropy barriers
• Josephson junctions
• biophysical processes (intracellular transport, neural
activity)
• matter-antimatter asymmetry (sphaleron)
• etc.
18
Further Applications:
• diffusion of atoms and molecules on crystal surfaces
• particle separation by electrophoresis
• rotating dipoles in external fields
• rotation of molecules in solids
• motion of fluxons in superconductors
• superionic conductors
• charge density waves
• plasma accelerators
• synchronization phenomena in electrical circuits (Adler
equation), phase locked loops, and laser gyroscopes
19
Known Results
For V (x) = 0
⇔
U (x) = −x F [Einstein 1905]
hẋi = F/η
D = D0 := kT /η
For F = 0 ⇔ U (x) = V (x) [Lifson and Jackson 1962]
hẋi = 0
D = RL
dx
0
L
D0
R L dx
eV (x)/kT
0
L
e−V (x)/kT
.............. Cauchy-Schwarz inequality ⇒ D ≤ D0
In general [Stratonovich 1958]
1 − e−LF/kT
hẋi = R
L dx V (x) R L dy − V (x−y)+yF
kT
e kT 0 D0 e
0 L
⇒
non-equilibrium system ⇔ F 6= 0
D = ???
20
...............................Step 1
Choose F > 0 =
ˆ reflecting boundary at q = −∞
Consider first passage times t(a → b) and their
“dispersion” ∆t(a → b) := t(a → b) − ht(a → b)i
21
By definition t(a → c) = t(a → b) + t(b → c) , a < b < c
x(t) Markov: t(a → b), t(b → c) statistically independent
⇒
ht(a → c)i = ht(a → b)i + ht(b → c)i
⇒ h∆t2(a → c)i = h∆t2(a → b)i + h∆t2(b → c)i
Periodicity: t(a → b), t(a+L → b+L) identically distrib.
CLT: t(x0 → x0 + nL) for large n Gaussian distributed
with mean n ht(x0 → x0 + L)i and
dispersion n h∆t2(x0 → x0 + L)i
Discrete “states” xm := x0 + m · nL, m = 0, ±1, ±2, ...
Continuous and discretized dynamics yield the same
•
hẋi := limt→∞ hx(t)i
t
hx2 (t)i−hx(t)i2
limt→∞
2t
•
D :=
•
statistics of t(xm → xm+1)
The statistics of t(xm → xm+1) fixes all properties of the
discretized process (provided e−nLF/kT negligible) and is
in turn fixed by
ht(x0 → x0 + L)i and h∆t2(x0 → x0 + L)i
⇒ If two processes x(t) have the same ht(x0 → x0 + L)i
and h∆t2(x0 → x0 + L)i then hẋi and D will also be
same in the two cases.
...............................Step 2
22
First passage time moments for a reflecting boundary at
q = −∞ (F > 0) : ht0(a → b)i = 1 ,
Rb Rx
V (x)−V (y)−(x−y)F
n
n−1
n
kT
(y → b)i e
ht (a → b)i = D0 dx dy ht
a
−∞
Note that ∆t(a → b) := t(a → b) − ht(a → b)i implies
h∆t2(a → b)i = ht2(a → b)i − ht(a → b)i2
Choose V (x) = 0 ⇔ U (x) = −xF
hẋi =
F
η
=
D = D0 =
⇒
L
ht(x0 →x0 +L)i
kT
η
=
2
L2 h∆t (x0 →x0 +L)i
2 ht(x0→x0 +L)i3
for any x0
Consider an arbitrary periodic V (x).
Whatever ht(x0 → x0 + L)i and h∆t2(x0 → x0 + L)i are,
we can reproduce them by some auxiliary process with
V (x) = 0 (η, T , F may be different)
Step 1: then also hẋi and D will be same in the two
cases.
⇒ for any periodic V (x) and F > 0:
hẋi =
D =
L
ht(x0 → x0 + L)i
L2 h∆t2(x0 → x0 + L)i
2 ht(x0 → x0 + L)i3
23
...............................Step 3
hẋi =
L
ht(x0→x0 +L)i
ht(a → b)i =
Rx
−∞
=
Rx
dy e
a
dx e
−V (y)+yF
kT
x−L dy e
⇒
Rb
1−e−LF/kT
Rx
dy
−∞ D0
P∞ R x
=
RL
0
L
Z
0
L
I±(x)
D = D0
[Stratonovich 1958]
dy ±V (x)∓V (x∓y)−yF
kT
e
D0
Similarly one obtains from D =
R x0+L dx
x0
−V (y−nL)+(y−nL)F
kT
1−e−LF/kT
hẋi = R x0+L dx
x0
−V (y)+yF
kT
−V (x−y)+(x−y)F
kT
dy e
1 − e−LF /kT
I±(x) :=
e
n=0 x−L dy e
=
−V (y)+yF
kT
V (x)−xF
kT
2
L2 h∆t (x0 →x0 +L)i
2 ht(x0 →x0 +L)i3
I±(x) I+(x) I−(x)
hR
i3
x0 +L dx
I (x)
x0
L ±
L
for arbitrary x0 , ± , F
η ẋ(t) = −V 0(x(t)) + F + ξ(t)
.
24
hx2(t)i − hx(t)i2
D = lim
t→∞
2t
R x0+L dx
I (x) I+(x) I−(x)
x0
L ±
= D0
hR
i3
x0 +L dx
I (x)
x0
L ±
I±(x) :=
RL dy
0
D0
e
±V (x)∓V (x∓y)−yF
kT
V 0(x) ≡ 0 ⇒ D = D0 = kT /η
D0
F =0 ⇒D=
RL dx RL dy V (x)−V (y)
kT
L
L e
0
[Einstein 1905]
[Lifson und Jackson 1962]
0
Dimensionless units: η = 1 , V (x) = sin(x)
20
D =kT=0.01
D/D
0
0
10
D =kT=0.1
0
0
0,6
1
1,4
F
.... enhanced diffusion
25
Universality and Scaling
1st Assumption:
For small x:
V (x) − xF = −µ x3 − x
= F − Fc small
µ = −V 000(0)/6 > 0
2nd Assumption: kT LFc
D=
R L/2 dx 2
I (x) I− (x)
−L/2 L +
D0 hR L/2
i3
dx I (x)
−L/2 L +
D = D0
3
L µ
kT
, I±(x) :=
RL dy
D0
0
e
±V (x)∓V (x∓y)−yF
kT
2/ 3 R ∞
˜2(x, ˜) I(−x,
˜
dx
I
˜)
−∞
R∞
˜ ˜)]3
[ −∞ dx I(x,
{z
}
|
G(˜
)
.
˜ :=
.
µ1/3 [kT ]2/3
∞
R
˜ ˜) := dy e−x
I(x,
0
0,08
G(˜)
0,06
3
3
0,04
+(x−y ) −y ˜
0,02
0
−5
0
˜
5
10
26
Generalization
For x ∈ [−L/2, L/2]:
V (x) − xF = −µ x |x|γ−1 − x
η ẋ(t) = µ γ |x(t)|γ−1 + + ξ(t)
x(0) > 0, = ξ = 0:
⇒
2 > γ > 4/3
D = D0
˜ :=
η x2−γ (t)−x2−γ (0)
t = µγ
2−γ
γ
L µ
kT
3− γ4
R∞
˜2(x, ˜) I(−x,
˜
dx
I
˜)
−∞
3
γ2 −1
2γ−1
kT
S(˜
)
+
Lγ µ
γ (2−γ )
µ1/γ [kT ]1−1/γ
∞
R
−x|x|γ−1 +(x−y )|x−y|γ−1 −y ˜
˜
I(x, ˜) := dy e
0
S(˜
≥ 0) := 0
S(˜
≤ 0) :=
2π
γ (γ−1)
For ˜ −1 :
R∞
|˜
/γ|
2−γ
γ−1
n
γ o
exp 2(γ − 1)|˜
/γ| γ−1
2
˜2(x, ˜) I(−x,
˜
dx
I
˜
)
=
S
(˜
)/2
−∞
27
.
22−γ γ(2 − γ) Lγ µ
D(˜
max) =
˜max ' −γ
27
2−γ
2γ(γ − 1)
η
ln
γ
L µ
kT
1−1/γ
Dimensionless units: γ = 3/2, η = µ = Fc = 1, L = 2
⇒ D(˜
max) = 0.111...
0,1
-2
D =kT=10
0
10
-3
10
-4
10
-5
D
0
0,8
F
0,9
1
28
Giant acceleration of thermal diffusion
z
h
|
h(y)
0
L
|
0
y
g
Spherical particle in a liquid:
3
r = 1 mm
, ρparticle − ρliquid = 10 g/cm
η = 6πνwaterr , T = 293o K
h(y + L0) = h(y) − h0 , h0 = 1.5 cm ,
h(y) ∼ −µ0 y |y|γ−1 − 0
L0 = 10 cm
D0 = kT /η ' 2 · 10−10 mm2/s
D ' 0.5 mm2/s
for γ = 3,
D ' 2 · 104 mm2/s
for γ = 3/2, 0 = max
.........D ' 1014 D0
0 = 0
29
Summary
.
η ẋ(t) = −V 0(x(t)) + F + ξ(t)
V (x + L) = V (x)
hξ(t) ξ(s)i = 2 η kT δ(t − s)
.
F = 0: equilibrium system, D ≤ D0.
F 6= 0: paradigmatic non-equilibrium system;
exact analytical expression for D ⇒
giant acceleration, universality and scaling of diffusion.
Reference:
P. Reimann, C. Van den Broeck, H. Linke, P. Hänggi,
J.M. Rubi, and A. Perez-Madrid
Phys. Rev. E 65 (2002) 031104