Ohm’s law and the pinball machine

Diffusion, réponse linéaire et transport ohmique dans des systèmes hamiltoniens ouverts

S. De Bièvre (Université de Lille)

Grenoble

Décembre 2006

Ohm’s law and the pinball machine

•

^{Ohm’s law:}

V = RI or E ~ = ρ~j or ~v =

^q_m^τ

E. ~ m d~v

dt = q E ~ − m

τ ~v, ~v(t) ∼ qτ

m E ~ (t → ∞ ).

•

The pinball machine (or the inelastic Lorentz gas)

Towards a Hamiltonian model for Ohm’s law?

A continuum model: friction

L. Bruneau and S.D.B.

A Hamiltonian model for linear friction in a homogeneous medium, Commun. Math. Phys. 229, 511-542, 2002

THE MODEL: a particle interacting with local harmonic degrees of freedom

H(X) = HS+HR +R

Rⁿ ρ(x−qS, y)qR(x, y) dxdy.

HS = ^p

2 S

2 + V (qS)^. HR = ¹₂ R

R

`p²_R(x, y) + |∇yqR(x, y)|²´

dxdy.

THE RESULTS

V = 0 ⇒ limt→+∞ q˙S(t) = 0.

V = −F q ⇒ limt→+∞ q˙S(t) = v_F ∼ ^F_γ .

¨

qS = −∇V (qS) − γq˙S.

The particle experiences a friction force linear in its velocity, due to the presence of the field. Its behaviour is therefore Ohmic.

Diffusion and transport in a 1 -d inelastic Lorentz gas

S.D.B., P. Parris, A. Silvius, Physica D, 208, 96-114 (2005) and Phys. Rev. B 73, 014304 (2006) THE MODEL : a classical Holstein polaron

00 0 11 1 00

11 00 11

00 0 11

1 01

00000000 0000 11111111 1111

00000000 11111111

00000000 11111111

00000000 11111111

00000000 11111111 00000000

0000 11111111 1111

00000000 11111111

00000000 11111111 00000000

11111111 00000000

0000 11111111 1111

A one-dimensional periodic array (with period

a

) of identical oscillators of frequency

ω

. The particle interacts with the oscillator at

ma

if it is within a distance

σ <

^a₂^.

H = 1

2 p

²_S

+ X

m∈Z

1

2 p

²_m

+ ω

²

q

_m²

+ α X

m

q

_m

n

_m

(q

S

) − F q

S

.

⁽¹⁾

where

n

_m

(q

S

)

vanishes outside the interaction region associated with the oscillator at

ma

and is equal to unity inside it.

THE DYNAMICS (no external field)

The particle moves at constant speed, except when entering or leaving the

interaction region, when the oscillator displacement serves as a potential barrier:

energy conservation then decides whether the particle reverses direction or not and how its speed changes. Two examples of what may happen:

00 11

0000 1111

−L−1 L+1

000000 111111

00 11 00 0 11 1

00 11

−L−1 L+1

00 0 11 1

00 11

−L−1 L+1

00 0 11 1

−L−1 L+1

00 11

0000 1111

0000 00 1111 11

NUMERICAL STUDY We injected a thermal distribution of particles (inverse temperature

β

) into the array of oscillators, also in equilibrium at the same temperature. We computed

h q

_S²

(t) i

and observed

h q

_S²

(t) i ∼ 2Dt,

where the diffusion constant

D

depends on the temperature and on the two dimensionless parameters of the model:

• E

_B

/E

0: here

E

_B

=

₂^α_ω²² is the binding energy and

E

0

= σ

²

ω

^2.

• 2σ/L

^{: here}

L = a − 2σ

is the size of the non-interacting region in a cell.

Diffusion constants as a function of β and system parameters

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5 10

-7 10

-5 10

-3 10

-1 10

1 10

3 10

5 10

7 10

9

10 -7 10

-5 10

-3 10

-1 10

1 10

3 10

5 10

7 10

9

x 10 4

x 1 5.0

E B

/ E 0

2.0

0.5

x 10 2 2 / L

0.5

1.0

2.0

10.0

D/D

0 L

D/D

0 H

E B

x 10 -2 0.2

High temperature:

D ∼ D

_H⁰

(βE

_B

)

^−5/2

D

_H⁰

= 2

q

9EBa² 32π

EB

E⁰

Low temperature:

D ∼ D

_L⁰

(βE

_B

)

^−3/4

D

_L⁰

=

_σ^a

Γ(3/4)

q

EBa² 2π²

TWO REGIMES – TWO DYNAMICS :

•

High temperatures Traversal time

<<<

oscillator period.

Typical potential energy barrier

∆ ∼ √

2E

_B

kT <<<

particle energy. A

thermalized particle will pass through many interaction regions in succession before slowing down and undergoing a velocity reversing (or randomizing) kick back up to thermal velocities. More precisely, a particle of momentum

p

will take a time

τ (p) =

_12E^p3_B^a_E0 to travel an average distance

ℓ(p) =

_16E^p4_B^a_E0. Adiabatic regime:

the random potential seen by the particle typically changes adiabatically with respect to the particle’s net motion (cfr. polaron).

•

Low temperatures: Traversal time

>>>

oscillator period.

Random walk, hopping motion.

Turning on the field: F > 0

Work in progress with P. Lafitte and P. Parris

QUESTION: Does the particle reach a limiting drift velocity, defined as

v

_F

:= lim

t→+∞

v

_F

(t) := lim

t→+∞

h q

S

(t, F ) i t

and is the latter linear in

F

, at least for small

F

? In other words,

v

_F

∼ µF

^{? If so,}

does the Einstein relation

µ = βD

hold? In fancier terms, is linear response valid in this model, and if so, does the Kubo formula hold?

Turning on the field: F > 0

Work in progress with P. Lafitte and P. Parris

QUESTION: Does the particle reach a limiting speed

v

_F and is the latter linear in

F

, at least for small

F

? In other words,

v

_F

∼ µF

? If so, does the Einstein relation

µ = βD

hold? In fancier terms, is linear response valid in this model, and if so, does the Kubo formula hold?

ANSWER 1: (GOOD NEWS) There is good reason to believe that linear response is valid at finite times and, according to Kubo’s formula, it yields

v

_F

(t)

F = h q

S

(t, F ) i

F t = β h q

_S²

(t, F = 0) i

2t + O

_t

(F ).

Taking

F → 0

first, the right hand side has a limit as

t

tends to infinity, since in absence of the field the motion is diffusive. This yields the Einstein relation:

µ := lim

t→∞

lim

F→0

h q

S

(t, F ) i

F t = βD.

This we have tested numerically at high temperatures with the following results:

0 2 4 6 8x10 6

0 2 4 6 8 10x10

5

0 1 2 3 4 x10

6 0 1 2 3x10

3

0 1 2 3 4 x10

6 0

1 2 3 x10

5

0 2 4 6 8x10

6

0 1 2 3 4x10

5

ω^t

(d) (c)

(b)

〈q(t)〉

ω^t

(a)

〈q(t)〉

(a)

βE

_B

= 0.015

^(c)

βE

_B

= 0.50

EB

E⁰

= 0.5,

²_L^σ

= 0.5

(triangles)

EB

E⁰

= 5,

²_L^σ

= 0.5

^(cercles)

(b)

βE

_B

= 0.020

^(d)

βE

_B

= 0.70

EB

E⁰

= 0.5,

²_L^σ

= 2

^(diamants)

EB

E⁰

= 5,

²_L^σ

= 2

^(carrés).

CONCLUSIONS The mean displacement is clearly linear in time for very long times.

In addition, the mean drift speed

v

_F

(t)

computed from the above data turns out to be linear in the applied field and independent of

t

^.

0

1

2

3

4

5

6

7x10 -5 0.00

0.03 0.06 0

1

2

3

4

5

6

7

8x10 -4 0

1 2 3x10

-3

vF /v0

Force F/F 0 vF

/v0

So the systems do seem to have a well-defined low field mobility.

BUT HERE IS SOME BAD NEWS

ANSWER 2: (BAD NEWS) This can’t possibly be true. Indeed, for any

F > 0

^{, you}

should expect

h q

S

(t) i ∼ 1

2 F t

²

,

for large enough

t

^.

ANSWER 2: (BAD NEWS) This can’t possibly be true. Indeed, for any

F > 0

^{, you}

should expect

h q

S

(t) i ∼ 1

2 F t

²

,

for large enough

t

^.

WHY? Because the energy loss experienced by high speed particles when

scattering off the oscillators cannot possibly be sufficiently efficient to compensate the energy

F a

gained from the field.

PROOF 1: “Seeing is believing”

10 3

10 4 10

4 10

5 10

6 10

7

〈q(t)〉 /F

ω^t

F / F 0

0.00267

0.00226

0.00191

0.00161

0.00115

βE

_B

= 0.5, E

_B

/E

0

= 5, 2σ/L = 0.5

. Log-log plot of

h q

S

(t, F ) i

^against

t

^.

The parts of the graphs parallel to the dashed line correspond to behaviour linear in time. All graphs have the same mobility, as promised! But for all forces, at large times, the displacements is no longer linear in time.

PROOF 2: “Cogito, ergo sum” In absence of the oscillators, any particle

accelerates indefinitely, since it picks up an energy

F a

from the field each time it traverses a cell. On the other hand, when a fast particle of momentum

p

scatters off a thermalized oscillator, the average energy loss of the particle is

h ∆ε i = −

^4E_p^B2E0. So the oscillators are inefficient in slowing down fast particles.

In particular, at high temperatures, where the typical particle energy ¹₂

p

^{2 is of the}

order of

kT

one certainly needs a small enough field to keep the particles from accelerating indefinitely (runaway):

F a << E

0

(E

B

β).

But even so, in a thermal distribution of particles, there are always some that are very fast, and those won’t be slowed down by the oscillators. So eventually, their contribution to

h q

S

(t) i

will dominate and yield the ¹₂

F t

^{2 behaviour.}

Estimating the breakdown time The fraction of particles that runs away is at high temperatures

P

_r

∼

exp

−

^2βE_{F a}^BE0

p 8πβE

_B

E

0

/F a << 1

They contribute to

h q

S

(t, F ) i

a term of order

(F a)

^3/2

t

²

e

^−β

2EbE0

√

F a

8βE

_b

E

0

π

whose quadratic behaviour in time will dominate the displacement for times beyond a critical time

t

_c

∼ p

8βE

_B

E

0

πe

2βEB E0

F a

(F a)

^−1/2

.

0.0

1.0x10 -3

2.0x10 -3

3.0x10 -3 10

2 10

3 10

4 10

5 10

6 10

7 10

8

runawaytime

F

βE

_B

= 0.5, E

_B

/E

0

= 5, 2σ/L = 0.5

. Runaway times as a function of

F

^.

t

_c

= 1

F

^1/2

exp(b/F ).

Turning on the field: F > 0

QUESTION: Does the particle reach a limiting speed

v

_F and is the latter linear in

F

, at least for small

F

? In other words,

v

_F

∼ µF

? If so, does the Einstein relation

µ = βD

hold? In fancier terms, is linear response valid in this model, and if so, does the Kubo formula hold?

ANSWER: Yes, for times

µ << t << t

_c

(F )

, there exists a non-zero mobility

µ

satisfying

v

_F

(t) = µF + o(F ),

where the error term is uniform for times satisfying

µ << t << t

_c

(F )

. And as far as the Einstein relation is concerned, we find:

The Einstein relation

0 5 10 15

0.6 0.8 1.0 1.2

/ D

Parameter Set Number

µ/βD

^{values for}

9

different systems, each at two high temperatures

βE

_B

= 0.015

^and

βE

_B

= 0.020

^.