x - CBPF

Anomalous Carnot cycle
Evaldo M. F. Curado
CBPF & INCT-SC
•
Collaborators: Fernando D. Nobre, Andre M. C. Souza,
Roberto F. S. Andrade, José Andrade, Veit Schwammle,
Andre Moreira, G. F. T. da Silva
• Fokker-Planck equation
Linear Fokker-Planck equation:
⇥P (x, t)
=
⇥t
dφ
F (x) = −
dx
⇥{F (x)P (x, t)}
⇥ 2 P (x, t)
+D
⇥x
⇥x2
Linear Fokker-Planck equation:
⇥P (x, t)
=
⇥t
dφ
F (x) = −
dx
⇥{F (x)P (x, t)}
⇥ 2 P (x, t)
+D
⇥x
⇥x2
Linear Fokker-Planck equation:
⇥P (x, t)
=
⇥t
⇥{F (x)P (x, t)}
⇥ 2 P (x, t)
+D
⇥x
⇥x2
dφ
F (x) = −
dx
H-theorem -> FPE and Boltzmann-Gibbs entropy
F =U
TS ; U =
dx (x)P (x, t) ; S =
kB
BG
dx P (x, t) ln P (x, t)
Linear Fokker-Planck equation:
⇥P (x, t)
=
⇥t
⇥{F (x)P (x, t)}
⇥ 2 P (x, t)
+D
⇥x
⇥x2
dφ
F (x) = −
dx
H-theorem -> FPE and Boltzmann-Gibbs entropy
F =U
TS ; U =
dF
dt
0
dx (x)P (x, t) ; S =
Pstat (x) / e
kB
(x)/D
BG
dx P (x, t) ln P (x, t)
stationary and
stable solution
general solution - Fokker-Planck equation
F (x) =
• time dependence
P (x, t) =
1
2 D(1
e
2t )/k
e
kx
kx2
2D(1 e 2t )
• normal diffusion (t << 1)
P (x, t) =
•
t >> 1
1
4 Dt/k
e
kx2
4Dt
⇥(x
⇥(x
D
2
⇥x⇤) ⇤ =
k
2Dt
⇥x⇤) ⇤ =
k
2
P (x) = ⇤
1
2 D/k
e
kx2
2D
⇥
general solution - Fokker-Planck equation
F (x) =
• time dependence
P (x, t) =
1
2 D(1
e
2t )/k
e
kx
kx2
2D(1 e 2t )
BG entropy
• normal diffusion (t << 1)
P (x, t) =
•
t >> 1
1
4 Dt/k
e
kx2
4Dt
⇥(x
⇥(x
D
2
⇥x⇤) ⇤ =
k
2Dt
⇥x⇤) ⇤ =
k
2
P (x) = ⇤
1
2 D/k
e
kx2
2D
⇥
nonlinear Fokker-Planck
Equations
nonlinear Fokker-Planck equation phenomenological equations
Porous media equation
(M. Muskat - 1937)
⇤(x
2
⇤x⌅) ⌅ ⇥ t
2
+1
⇥P (x, t)
⇥ 2 P (x, t)
=D
⇥t
⇥x2
nonlinear Fokker-Planck equation phenomenological equations
Porous media equation
(M. Muskat - 1937)
⇤(x
•
2
⇤x⌅) ⌅ ⇥ t
⇥P (x, t)
⇥ 2 P (x, t)
=D
⇥t
⇥x2
2
+1
A. R. Plastino and A. Plastino, Physica A 222 (1995) 347;
C. Tsallis and Bukman D. J., PRE 54 (1996) R2197
nonlinear Fokker-Planck equation phenomenological equations
Porous media equation
(M. Muskat - 1937)
⇤(x
•
2
⇤x⌅) ⌅ ⇥ t
⇥P (x, t)
⇥ 2 P (x, t)
=D
⇥t
⇥x2
2
+1
A. R. Plastino and A. Plastino, Physica A 222 (1995) 347;
C. Tsallis and Bukman D. J., PRE 54 (1996) R2197
⇥P (x, t)
=
⇥t
⇥{F (x)P (x, t)}
⇥ 2 P (x, t)
+D
⇥x
⇥x2
dφ
F (x) = −
dx
⇥P (x, t)
=
⇥t
⇥{F (x)P (x, t)}
⇥ 2 P (x, t)
+D
⇥x
⇥x2
dφ
F (x) = −
dx
stationary solution ( = 2
P (x) = C[1
= (1/D)[C q
(1
1
/(2
q)
q)⇥(x)]1/(1
q)]
q)
(C is a positive constant)
⇥P (x, t)
=
⇥t
⇥{F (x)P (x, t)}
⇥ 2 P (x, t)
+D
⇥x
⇥x2
dφ
F (x) = −
dx
stationary solution ( = 2
P (x) = C[1
= (1/D)[C q
(1
1
/(2
q)
q)⇥(x)]1/(1
q)]
q)
(C is a positive constant)
same distribution that maximizes Tsallis entropy
with the external constraint (x) !
NLFPE <-> Tsallis entropy!
FPE
entropy
LFP
F =U
Gaussian
T SBG
1 2
(x) = kx
2
FPE
entropy
LFP
SBG
F =U
T SBG
MEP
Gaussian
1 2
(x) = kx
2
FPE
entropy
LFP
SBG
F =U
NLFP
T SBG
F =U
MEP
Gaussian
1 2
(x) = kx
2
q-Gaussian
T Sq
FPE
entropy
LFP
SBG
F =U
NLFP
T SBG
MEP
F =U
Sq
MEP
Gaussian
1 2
(x) = kx
2
q-Gaussian
T Sq
how could it be derived?
•
•
Langevin equation
master equation
Langevin equation
master equation
master equation -> NL Fokker-Planck equation
∞
!
∂P (n, t)
[P (m, t)wm,n (t) − P (n, t)wn,m (t)]
=
∂t
m=−∞
•
nonlinear transition rates -> nonlinear FokkerPlanck equations
m,n (t)
m,n (P, t)
!
"
∂{F (x)Ψ[P (x, t)]}
∂
∂P (x, t)
∂P (x, t)
=−
+
Ω[P (x, t)]
∂t
∂x
∂x
∂x
master equation -> NL Fokker-Planck equation
∞
!
∂P (n, t)
[P (m, t)wm,n (t) − P (n, t)wn,m (t)]
=
∂t
m=−∞
•
nonlinear transition rates -> nonlinear FokkerPlanck equations
m,n (t)
m,n (P, t)
!
"
∂{F (x)Ψ[P (x, t)]}
∂
∂P (x, t)
∂P (x, t)
=−
+
Ω[P (x, t)]
∂t
∂x
∂x
∂x
master equation -> NL Fokker-Planck equation
∞
!
∂P (n, t)
[P (m, t)wm,n (t) − P (n, t)wn,m (t)]
=
∂t
m=−∞
•
nonlinear transition rates -> nonlinear FokkerPlanck equations
m,n (t)
m,n (P, t)
!
"
∂{F (x)Ψ[P (x, t)]}
∂
∂P (x, t)
∂P (x, t)
=−
+
Ω[P (x, t)]
∂t
∂x
∂x
∂x
nonlinear Fokker-Planck equation
!
"
∂{F (x)Ψ[P (x, t)]}
∂
∂P (x, t)
∂P (x, t)
=−
+
Ω[P (x, t)]
∂t
∂x
∂x
∂x
Ω[P ] > 0
!
Ψ[P ] > 0
∞
dx P (x, t) =
−∞
!
dφ
F (x) = −
dx
∞
dx P (x, t0 ) = 1
(∀t)
−∞
[P ] = P
[P ] = const.
FPE
master equation -> NLFPE
[P ] = P
Ω[P ] = DP µ−1
NLFPE - Plastino and
Plastino, Physica A 1995
EMFC & FD Nobre, PRE 2003,
FD Nobre, EMFC & G Rowlands, Physica A 2004
H theorem - general case
S[P ] =
⇥
⇥
d2 g
≤0
2
dP
g[P (x, t)] dx
H theorem - general case
S[P ] =
⇥
g[P (x, t)] dx
⇥
d2 g
≤0
2
dP
1 d2 g[P ]
⇥[P ]
=
2
dP
[P ]
dF
dt
0
H theorem - general case
S[P ] =
⇥
g[P (x, t)] dx
⇥
d2 g
≤0
2
dP
1 d2 g[P ]
⇥[P ]
=
2
dP
[P ]
dF
dt
0
H theorem - general case
S[P ] =
⇥
g[P (x, t)] dx
⇥
d2 g
≤0
2
dP
1 d2 g[P ]
⇥[P ]
=
2
dP
[P ]
dF
dt
0
H theorem - general case
S[P ] =
⇥
g[P (x, t)] dx
NLFP
⇥
d2 g
≤0
2
dP
1 d2 g[P ]
⇥[P ]
=
2
dP
[P ]
S[P ]
F =U
T S[P ]
MEP
dF
dt
0
stationary solution
- Frank, Chavannis, Nobre, EMFC
•
type II disordered supercondutors overdamped motion of interacting
vortices
•
overdamped motion
µ⇥vi =
X
⇥ ri
J(⇥
⇥rj ) + F⇥ e (⇥ri ) + (⇥ri , t)
j6=i
⇥ r)
J(⇥
f0 K1 (|⇥r|/ )r̂ (repulsive interaction)
h⌘i = 0
kTh
h i=
µ
2
•
overdamped motion
µ⇥vi =
X
⇥ ri
J(⇥
⇥rj ) + F⇥ e (⇥ri ) + (⇥ri , t)
j6=i
⇥ r)
J(⇥
•
f0 K1 (|⇥r|/ )r̂ (repulsive interaction)
h⌘i = 0
kTh
h i=
µ
2
continuous description
P(r~1 , r~2 , · · · , r~N , t)
X
P
⇥ i ( f⇥i P + kB Th ⇥
⇥ i P)
µ
=
⇥
t
i
•
overdamped motion
µ⇥vi =
X
⇥ ri
J(⇥
⇥rj ) + F⇥ e (⇥ri ) + (⇥ri , t)
j6=i
⇥ r)
J(⇥
•
f0 K1 (|⇥r|/ )r̂ (repulsive interaction)
h⌘i = 0
kTh
h i=
µ
2
continuous description
X
P
⇥ i ( f⇥i P + kB Th ⇥
⇥ i P)
µ
=
⇥
t
i
P(r~1 , r~2 , · · · , r~N , t)
P(r~1 , r~2 , · · · , r~N , t) ! ⇢(~r, t)
⇥
µ
=
⇥t
⇤
r
Z
⇤r
d2 r0 J(⇤
r⇤0 )
(2)
(single particle density)
(⇤r, r⇤0 , t) + F⇤ e (⇤r) (⇤r, t) + kTh r2 (⇥r, t)
•
overdamped motion
µ⇥vi =
X
⇥ ri
J(⇥
⇥rj ) + F⇥ e (⇥ri ) + (⇥ri , t)
h⌘i = 0
j6=i
⇥ r)
J(⇥
•
kTh
h i=
µ
f0 K1 (|⇥r|/ )r̂ (repulsive interaction)
2
continuous description
X
P
⇥ i ( f⇥i P + kB Th ⇥
⇥ i P)
µ
=
⇥
t
i
P(r~1 , r~2 , · · · , r~N , t)
P(r~1 , r~2 , · · · , r~N , t) ! ⇢(~r, t)
⇥
µ
=
⇥t
⇤
r
Z
⇤r
d2 r0 J(⇤
r⇤0 )
(2)
(⇤r, r⇤0 , t) + F⇤ e (⇤r) (⇤r, t) + kTh r2 (⇥r, t)
⇢(2) (~r, ~r 0 , t) ' ⇢(~r, t)⇢(~r 0 , t)
Z
⇥r
d2 r0 J(⇥
r⇥0 ) (r⇥0 , t) ⇡
(single particle density)
⇥ (⇥r, t)
ar
a⌘
Z
~ r)/2
d2 r ~r · J(~
= 2⇡f0
3
⇥
⇥
µ
=
⇥t
⇥x
F~ e =
•
⇢ 
⇥
a
⇥x
A(x)
⇥2
+ kTh 2
⇥x
A(x)x̂
for Th = 0 and A(x) =
⇥stat (x) =
2a
(x2e
x ( > 0)
x2 ), |x| < xe
⇥
⇥
µ
=
⇥t
⇥x
F~ e =
•
⇢ 
⇥
a
⇥x
A(x)
⇥2
+ kTh 2
⇥x
A(x)x̂
for Th = 0 and A(x) =
⇥stat (x) =
[⇢] = ⇢
2a
(x2e
x ( > 0)
x2 ), |x| < xe
a
⌦[⇢] = ⇢
µ
1 d2 g[⇢]
⌦[⇢]
=
2
d⇢
[⇢]
⇥
⇥
µ
=
⇥t
⇥x
F~ e =
•
⇢ 
⇥
a
⇥x
A(x)
⇥2
+ kTh 2
⇥x
A(x)x̂
for Th = 0 and A(x) =
⇥stat (x) =
[⇢] = ⇢
✓
S[⇢] = k 1
2a
(x2e
x ( > 0)
x2 ), |x| < xe
1 d2 g[⇢]
⌦[⇢]
=
2
d⇢
[⇢]
a
⌦[⇢] = ⇢
µ
Z
1
1
dx ⇢(x)2
◆
⇥
⇥
µ
=
⇥t
⇥x
F~ e =
•
⇢ 
⇥
a
⇥x
A(x)
⇥2
+ kTh 2
⇥x
A(x)x̂
for Th = 0 and A(x) =
⇥stat (x) =
[⇢] = ⇢
✓
S[⇢] = k 1
2a
(x2e
x ( > 0)
x2 ), |x| < xe
a
⌦[⇢] = ⇢
µ
Z
1
1
dx ⇢(x)2
◆
1 d2 g[⇢]
⌦[⇢]
=
2
d⇢
[⇢]
Z
U=
(x)⇢(x)dx
molecular dynamics simulations (2d)
Lx = 100
N = 800
Ly = 20 (pbc)
↵ = 10
3
f0
molecular dynamics simulations (2d)
Lx = 100
N = 800
Ly = 20 (pbc)
↵ = 10
3
f0
0.5
MD
Fokker-Planck
ρ(x)
0.4
T=0.0
0.3
0.2
0.1
0
-200
-100
⇣
hx2 i / t2/3 t
2
⌫+1
⌘
0
x
100
Pstat / x2e
200
x2
effective temperature
N ⇡f0
k✓ ⌘ D =
Ly
2
= n⇡f0
2
effective temperature
N ⇡f0
k✓ ⌘ D =
Ly
2
= n⇡f0
2
portanto...
@P (x, t)
µ
=
@t
@[A(x)P (x, t)]
@
+ 2D
@x
@x
⇣ ⌘
⇤
xe 2
2
2
Pst (x) =
(xe x ) =
4k⇥⇤
4k⇥
⇤
(xe = (3k✓ /↵)1/3 )
⇢
@P (x, t)
[ P (x, t)]
@x
⇣ x ⌘2
⇤
portanto...
@P (x, t)
µ
=
@t
@[A(x)P (x, t)]
@
+ 2D
@x
@x
⇣ ⌘
⇤
xe 2
2
2
Pst (x) =
(xe x ) =
4k⇥⇤
4k⇥
⇤
(xe = (3k✓ /↵)1/3 )
Z
s
=1
k
u=
Z
xe
dx
xe
xe
2
dx [Pst (x)] = 1
xe
⇢
@P (x, t)
[ P (x, t)]
@x
⇣ x ⌘2
⇤
3
2/3
5
✓
2
↵
k✓
x2
32/3
Pst (x) =
( ⇤2 )1/3 (k⇥)2/3
2
10
◆1/3
,
portanto...
@P (x, t)
µ
=
@t
@[A(x)P (x, t)]
@
+ 2D
@x
@x
⇣ ⌘
⇤
xe 2
2
2
Pst (x) =
(xe x ) =
4k⇥⇤
4k⇥
⇤
(xe = (3k✓ /↵)1/3 )
Z
s
=1
k
Z
xe
xe
2
dx [Pst (x)] = 1
xe
⇢
@P (x, t)
[ P (x, t)]
@x
⇣ x ⌘2
⇤
3
2/3
5
✓
2
↵
k✓
x2
32/3
u=
dx
Pst (x) =
( ⇤2 )1/3 (k⇥)2/3
2
10
xe
"
✓ 2 ◆1/2 #
3
⇥
s(u, ) = k 1
5 10u
◆1/3
,
portanto...
@P (x, t)
µ
=
@t
@[A(x)P (x, t)]
@
+ 2D
@x
@x
⇣ ⌘
⇤
xe 2
2
2
Pst (x) =
(xe x ) =
4k⇥⇤
4k⇥
⇤
(xe = (3k✓ /↵)1/3 )
Z
s
=1
k
Z
xe
xe
2
dx [Pst (x)] = 1
xe
⇢
@P (x, t)
[ P (x, t)]
@x
⇣ x ⌘2
⇤
3
2/3
5
✓
2
↵
k✓
◆1/3
x2
32/3
u=
dx
Pst (x) =
( ⇤2 )1/3 (k⇥)2/3
2
10
xe
"
✓ ◆
✓ 2 ◆1/2 #
⇥s
1
3
⇥
s(u, ) = k 1
=
5 10u
⇥u ↵
,
Legendre
• free energy
u(s, ↵) ! f (✓, ↵)
Legendre
• free energy
u(s, ↵) ! f (✓, ↵)
f (✓, ↵) = u
✓s ;
)
df =
sd✓ + d↵
Legendre
• free energy
u(s, ↵) ! f (✓, ↵)
f (✓, ↵) = u
✓s ;
)
df =
sd✓ + d↵
35/3
f (✓, ↵) =
(↵ 2 )1/3 (k✓)2/3 k✓
10
✓ ◆
✓ ◆
@f
@f
= s;
=
@✓ ↵
@↵ ✓
and so on...
• calor e trabalho
du = ⇥Q + ⇥W = ⇤ds + ⌅d
Q = ⇥ds
2/3
3
=
10
⇥W = ⇤d
2
✓
k✓
↵ 2
◆2/3
- isothermal process
Z
⇣
⌘
32/3
2/3
Q=
✓ds =
(k✓ )
↵i 1/3 ↵f 1/3
5
si
Z f
⇣
⌘
35/3
2/3
1/3
1/3
W =
⌅d =
(k⇥⇤)
f
i
10
i
uf
sf
⇣
32/3
2/3
ui = Q + W =
(k⇥⇤)
10
f
1/3
i
1/3
⌘
- adiabatic process
s
=1
k
uf
3
2/3
5
✓
ui = W =
2
◆1/3
⇤
k⇥
Z f
i
!
⇥
⇥d = ⇥(
= cte (adiabático)
f
i)
0.5
c
→
➙
Q2
b
σ/λ
Q1
W
0.3
→
d
0.2
0
➙
2
0.4
1
2
a
3
2
αλ
4
efficiency
W
Q1 Q2
=
=
=1
Q1
Q1
⇥2
⇥1
(0 ⇥
⇥ 1)
- ciclo de Carnot é válido com a temperatura
efetiva (T = 0)
Obrigado
•
EMFC, F. D. Nobre, PRE 67 (2003) 021107
•
F. D. Nobre, EMFC, G. Rowlands, Physica A (2004) 109
•
V. Schwammle, EMFC, F. D. Nobre, Eur. Phys. J. B 58 (2007) 159
•
V. Schwammle, F. D. Nobre, EMFC, PRE 76 (2007) 041123
•
V. Schwammle, EMFC, F. D. Nobre, Eur. Phys. J. B 70 (2009) 107
•
J. S. Andrade, G. F. T. Silva, A. A. Moreira, F. D. Nobre, EMFC, PRL 105 (2010)
260601
•
M. S. Ribeiro, F. D. Nobre, EMFC, Entropy 13 (2011) 1928
•
M. S. Ribeiro, F. D. Nobre, EMFC, EPJB 85 (2012) 399
•
M. S. Ribeiro, F. D. Nobre, EMFC, PRE 85 (2012) 021146
•
F. D. Nobre, A. M. C. Souza, EMFC, PRE 86 (2012) 061113