pablo rodriguez-lopez parodrilo@gmail - centro de … casimir entropy between spheres pablo...

Negative Casimir entropy between spheres

Pablo [email protected]

UCM

21 - September - 2011

Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 1 / 28

[P. Rodriguez–Lopez, PRB 84, 075431 (2011).]

Collaborators

Ricardo Brito (UCM, Spain)Rodrigo Soto (U. Chile, Chile)Thorsten Emig (U Paris-Sud, France)Sahand Jamal Rahi (MIT, USA)Adolfo Grushin (CSIC, Spain)Alberto Cortijo (CSIC, Spain)


Outline

1 Introduction

2 Multiscattering Formalism

3 Two spheresPerfect Metal ModelPlasma ModelDrude Model

4 Discussion

5 Conclusions


Introduction

Negative entropy found ina system of Drude parallelplates [V. B. Bezerra et. al. PRA 65,

052113 (2002).] [Gert-Ludwig Ingold et. al.

PRE 80, 041113 (2009).]

Also found in perfect metalsphere–plate system. [A.

Canaguier-Durand, et. al. PRA 82, 012511

(2010).]


2 spheres

Geometry under consideration:

dR1 R2

[P. Rodriguez–Lopez, PRB 84, 075431 (2011).]

R1 = R2 = R

Asymptotic limit: Rd � 1


Multiscattering Formalism

F = kBT∞∑

n=0

′

log |1− N(κn)|

S = −∂TF F = −∂dF

T1 T2U12

U21[T. Emig, et. al. PRL 99(17), 170403 (2007).] [S. J. Rahi, et. al. PRD 80, 085021 (2009).]

[O. Kenneth and I. Klich. PRB 78(1), 014103 (2008).] [A. Lambrecht, et. al. NJP 8(10), 243 (2006).]

1 N = T1U12T2U21

2 Ti(κ)= Scattering matrix of each ith object, for spheres (P = E,M):

TPP′`m,`′m′(κ) = − i`(κR)∂R(Ri`(nκR))−αP∂R(Ri`(κR))i`(nκR)

k`(κR)∂R(Ri`(nκR))−αP∂R(Rk`(κR))i`(nκR)δ``′δmm′δPP′

αE = ε αM = µ n =√εµ

3 Uij(κ) = Translation matrix from object i to object j.4 κn = n/λT = Matsubara frequency.


Perfect metal Spheres

The susceptibilities are:

εi(icκ)→∞ µi(icκ) = µ0

Dominant contributions of T matrix in the Far Distance Limit:

TMM1m,1m′ = −q3

3

(Rd

)3

TEE1m,1m′ =

2q3

3

(Rd

)3

q = κ d is the adimensional frequency.Same power exponent.Different signs.


Perfect metal spheres - Helmholtz Free Energy

2 4 6 8

0.5

1.0

1.5

2.0FF0

dλT

∀T

T = 0

T → ∞

F0 = −143~cR6

16πd7 λT =~c

2πkBT


Perfect metal spheres - Entropy

2 4 6 8

-0.2

0.2

0.4

0.6

0.8

1.0SScl

dλT

∀T

T → ∞

Scl =15kBR6

4d6 λT =~c

2πkBT



1 10

0.001

0.1

SScl

dλT

r → 0

r =Rd

λT =~c

2πkBT



1 10

0.001

0.1

SScl

dλT

r → 0

r = 0.1

r =Rd

λT =~c

2πkBT



1 10

0.001

0.1

SScl

dλT

r → 0

r = 0.1

r = 0.2

r =Rd

λT =~c

2πkBT



1 10

0.001

0.1

SScl

dλT

r → 0

r = 0.1

r = 0.2

r = 0.3

r =Rd

λT =~c

2πkBT



1 10

0.001

0.1

SScl

dλT

r → 0

r = 0.1

r = 0.2

r = 0.3

r = 0.4

r =Rd

λT =~c

2πkBT



1 10

0.001

0.1

SScl

dλT

r → 0

r = 0.1

r = 0.2

r = 0.3

r = 0.4

r = 0.41

r =Rd

λT =~c

2πkBT



1 10

0.001

0.1

SScl

dλT

r → 0

r = 0.1

r = 0.2

r = 0.3

r = 0.4

r = 0.41

r = 0.45

r =Rd

λT =~c

2πkBT


Perfect metal spheres - Force

2 4 6 8 10

0.5

1.0

1.5

2.0FF0

dλT

∀T

T = 0

T → ∞

F0 = −1001~cR6

16πd8 λT =~c

2πkBT


Plasma Model Spheres


εi(icκ) = ε0 +4π2

λPκ2 µi(icκ) = µ0


TMM1m,1m′ = −q3

3

(Rd

)3 (1− 3y−1 coth (y) + 3y−2) TEE

1m,1m′ =2q3

3

(Rd

)3

y = 2π RλP

Same power exponent.Different signs.


10 20 30 40 50

0.2

0.4

0.6

0.8

1.0TMM1m,1m′(y)

y


2 4 6 8

-0.2

0.2

0.4

0.6

0.8

1.0SScl

dλT

λP → 0

λP → ∞

T → ∞

limλP→0

TMM1m,1m′ = −q3

3

(Rd

)3lim

λP→∞TMM

1m,1m′ = 0



0 1 2 3 40.0

0.5

1.0

1.5

2.0

2.5

3.0λP

R

dλT

S = 0

∂dS = 0

∂TS = 0


Drude Model Spheres


εi(icκ) = ε0 +4π2

λ2Pκ

2 + πcκσ

µi(icκ) = µ0


TMM1m,1m′ = −4πq4

45Rσc

(Rd

)4

TEE1m,1m′ =

2q3

3

(Rd

)3

TMM1m,1m′ subdominant compared with TEE

1m,1m′ .Different signs.When σ →∞⇒ Plasma Model, but it is a singular limit.


Drude Model Spheres

2 4 6 8 10

0.2

0.4

0.6

0.8

1.0SScl

dλT

∀T

T → ∞


Discussion

1 Why intervals of negative entropy appear?2 Is 3rd Law of Thermodynamics verified?

min S(T) = limT→0

S(T)

3 Is 2nd Law of Thermodynamics verified? There exist processeswhose dynamics tends to decrease the entropy of the system?

2 4 6 8

-0.2

0.2

0.4

0.6

0.8

1.0SScl

dλT

∀T

T → ∞

0 1 2 3 40.0

0.5

1.0

1.5

2.0

2.5

3.0λP

R

dλT

S = 0

∂dS = 0

∂TS = 0


Why intervals of negative entropy appear?

U =

(UEE UEM

UME UMM

)T =

(TEE OO TMM

)sgn

[TEE] = −sgn

[TMM]

Then, in the Far Distance Limit

log |1− N| ≈ −Tr [N] = EEE + EEM + EME + EMM

with Eij = −Tr[Tii

1Uij12T

jj2U

ji21

]{

EEE = −Tr[TEE

1 UEE12 TEE

2 UEE21

]< 0 ⇒ {EEE,EMM} < 0

EEM = −Tr[TEE

1 UEM12 TMM

2 UME21

]> 0 ⇒ {EEM,EME} > 0

When ∂T |EEE + EMM| < ∂T |EEM + EME|, intervals of S < 0 appear.


Is 3rd Law verified?

min S(T) = limT→0

S(T)

Krein Formula

βF = −∞∑

n=0

′ [log∣∣∆ + κ2

n

∣∣+

n∑α=1

log |Tα|+ log |1− N|

]

FV = −kBT∑∞

n=0′ log

∣∣∆ + κ2n

∣∣ ∝ V

Fi = −kBT∑∞

n=0′ log |Ti| ∝ Vi

FC = −kBT∑∞

n=0′ log |1− N| ∝ R6

d7

SC

SV= 0⇒ min S(T) = min SV(T) = lim

T→0S(T) = 0


Is 2nd Law verified?

In an isolated system, for any process:

dS ≥ 0

Microcanonical ensemble.However, T = cte⇒ Canonical ensemble2nd Law in Canonical ensemble: dF ≤ 0

S can verify either dS > 0 or dS < 0, but dF ≤ 0.The unique consequence of S < 0 interval is a nonmonotonicity ofthe force with T

∂F∂T

= − ∂2F∂T∂d

=∂S∂d


Conclusions

1 An interval of negative entropy appearsPerfect Metal spheresLow penetration lenght Plasma model spheres.

in an interval of distances and temperatures in the far distancelimit.

2 This interval appears because of the cross polarization terms(EEM,EME) of EM Casimir energy. It does not have a scalaranalogous.

3 Third Law is verified (Krein formula).4 Second Law is verified, we study the Canonical ensemble, not the

Microcanonical.5 Negative entropy interval produces nonmonotonicies of the

Casimir force with the temperature.


THANK YOU FOR YOUR ATTENTION

2 4 6 8

0.5

1.0

1.5

2.0FF0

dλT

∀T

T = 0

T → ∞

2 4 6 8 10

0.5

1.0

1.5

2.0FF0

dλT

∀T

T = 0

T → ∞

2 4 6 8

-0.2

0.2

0.4

0.6

0.8

1.0SScl

dλT

∀T

T → ∞

∂F∂T

= − ∂2F∂T∂d

=∂S∂d


pablo rodriguez-lopez parodrilo@gmail - centro de … casimir entropy between spheres pablo...

Documents