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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)
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 2 / 28
Outline
1 Introduction
2 Multiscattering Formalism
3 Two spheresPerfect Metal ModelPlasma ModelDrude Model
4 Discussion
5 Conclusions
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 3 / 28
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).]
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 4 / 28
2 spheres
Geometry under consideration:
dR1 R2
[P. Rodriguez–Lopez, PRB 84, 075431 (2011).]
R1 = R2 = R
Asymptotic limit: Rd � 1
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 5 / 28
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.
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 6 / 28
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.
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 7 / 28
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 8 / 28
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 9 / 28
Perfect metal spheres - Entropy
1 10
0.001
0.1
SScl
dλT
r → 0
r =Rd
λT =~c
2πkBT
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 10 / 28
Perfect metal spheres - Entropy
1 10
0.001
0.1
SScl
dλT
r → 0
r = 0.1
r =Rd
λT =~c
2πkBT
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 11 / 28
Perfect metal spheres - Entropy
1 10
0.001
0.1
SScl
dλT
r → 0
r = 0.1
r = 0.2
r =Rd
λT =~c
2πkBT
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 12 / 28
Perfect metal spheres - Entropy
1 10
0.001
0.1
SScl
dλT
r → 0
r = 0.1
r = 0.2
r = 0.3
r =Rd
λT =~c
2πkBT
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 13 / 28
Perfect metal spheres - Entropy
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 14 / 28
Perfect metal spheres - Entropy
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 15 / 28
Perfect metal spheres - Entropy
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 16 / 28
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 17 / 28
Plasma Model Spheres
The susceptibilities are:
εi(icκ) = ε0 +4π2
λPκ2 µi(icκ) = µ0
Dominant contributions of T matrix in the Far Distance Limit:
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.
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 18 / 28
10 20 30 40 50
0.2
0.4
0.6
0.8
1.0TMM1m,1m′(y)
y
Plasma Model Spheres
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 19 / 28
Plasma Model Spheres
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 20 / 28
Drude Model Spheres
The susceptibilities are:
εi(icκ) = ε0 +4π2
λ2Pκ
2 + πcκσ
µi(icκ) = µ0
Dominant contributions of T matrix in the Far Distance Limit:
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.
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 21 / 28
Drude Model Spheres
2 4 6 8 10
0.2
0.4
0.6
0.8
1.0SScl
dλT
∀T
T → ∞
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 22 / 28
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 23 / 28
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.
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 24 / 28
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 25 / 28
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
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 26 / 28
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.
Pablo Rodriguez-Lopez (UCM) Negative Casimir entropy between spheres 21 - September - 2011 27 / 28