Time asymmetric driving and entropy production Juan MR Parrondo - - PowerPoint PPT Presentation
Time asymmetric driving and entropy production Juan MR Parrondo (GISC and Universidad Complutense de Madrid) Micromachines Reversible transport (1998) Hidden pumps (M. Esposito, 2014) Carnot cycles (Leo Granger and Johannes Hoppenau, 2015)
Juan MR Parrondo
(GISC and Universidad Complutense de Madrid)
Micromachines Reversible transport (1998) Hidden pumps (M. Esposito, 2014) Carnot cycles (Leo Granger and Johannes Hoppenau, 2015)
Kyoto, July 28th 2015
Non-feedback Feedback
Non-symmetric Symmetric
JMRP, PRE (1998). Reversible ratchets as Brownian particles in an adiabatically changing periodic potential.
An overdamped Brownian particle in a driven periodic potential:
V (0; λ) = V (L; λ) ˙ x(t) = −V (x; λ(t)) + ξ(t)
Quasistatic limit: ˙ λ(t) → 0 ⇒ ρ(x, t) = 1 Z(λ(t))e−βV (x;λ(t)) Zero current, BUT... λ(t), 0 ≤ t ≤ τ
ρ±(x; λ) = e±βV (x;λ) Z±
J = Z τ δJ(t)
Not an exact differential
Integrated current:
W = Z τ δW(t)
Exact differential
Total work:
W(t) = kT
(t)dt
J(t) = L dx x dx+(x; (t))
(t)dt
J = Z τ δJ(t)
crossing x 0 to the right,
2 4 6 8
α
0.0 0.1 0.2 0.3
η
2 4 6 8
F
0.00 0.02 0.04 0.06
η
Efficiency:
1 2 3 4
Irreversible ratchet Reversible ratchet Irreversible ratchet
J = Z τ δJ(t)
dλ(t)
Line integral (it only depends on the path in the parameter space)
Time asymmetry is not enough to induce reversible transport
time
Vmax(t)
In a flashing ratchet with an asymmetric potential J=0
J(t) = L dx x dx+(x; (t))
(t)dt
1
2
a
1
2
a
Ea = ∞ E1 E2
p2 p1
1
2
a
Ea
1
2
a
Ea = ∞
1
2
a
1
2
a
Ea
barrier state
An auxiliary state a
modulated barriers
. PRE 2015)
λt A pump biases a transition, i.e., creates an effective force. The idea: design a protocol such that, at some coarse-grain level (hidden pump), the dynamics of the network is identical to that of an autonomous Markovian system.
chemical motors and Maxwell demons (Horowitz, Sagawa, JMRP . PRL 2013).
R L R L R R L L
∆E
L R R L
spatial diffusion chemistry /demon
λt
time
pi (t + ∆t) = pi(t) + X
j6=i
(wj!i − wi!j) pj(t)∆t
makes a cycle in every ∆t
λt
time
Markovianity at the coarse grained level: A cyclic protocol with period In each cycle a small amount of probability is transferred. Low entropy production: Protocol must be slow compared to the kinetics of the hidden states:
∆t (hidden rates)−1
∆t ⌧ 1/wi→j
1
2
a
1
2
a
Ea = ∞
p1e−β(Ea−E1)
E1 E2
p2 p1
1
2
a
Ea
1
2
a
Ea = ∞
1
2
a
1
2
a
We transfer an amount of probability from 1 to 2:
Irreversible leak!
Ea
barrier state
p1e−β(Ea−E1) ' p1w1→2∆t
An auxiliary state a
modulated barriers
makes a cycle in every ∆t
λt
time
Poissonian rate
barrier states
Eb E(0)
b
E(1)
b
E(2)
b
Ea
barrier states
E(0)
a
E(1)
a
E(2)
a
f) e) d) c) b) a) g)
1 2
a
b
Ea Eb
E1 E2
(a) (b)
e−β(E(1)
a
−E1) = w1→2∆t
w1→2 w2→1 = e−β(E2−E1−F eff
21 )
F eff
21 ≡ E(2) b
− E(1)
a
Effective force induced by the pump
λt
T ˙ S(cg)
tot = J12F (eff) 12
−
δ ˙ Fij ≥ 0 T ˙ Stot = −
δ ˙ Fij ≥ 0
Real entropy production: Entropy production of the coarse-grained description:
T ˙ S(cg)
tot ≥ T ˙
Stot
It is even possible to have zero entropy production with finite current!
~ = time reversal
No hidden driving.
(overdamped systems + no magnetic fields)
Entropy production: Coarse grained entropy production:
p(x, x0; {λt}) = X
y,y0
p(x, y; x0, y0; {λt})
Marginal probability distribution
One can prove:
Two assumptions!
Stot = k
(x,y)
p(x, y; x, y; {λt}) ln p(x, y; x, y; {λt}) p(˜ x, ˜ y; ˜ x, ˜ y; {˜ λt})
S(cg)
tot = k
p(x, x; {λt}) ln p(x, x; {λt}) p(˜ x, ˜ x; {˜ λt})
S(cg)
tot ≤ Stot
λt
T ˙ S(cg)
tot = J12F (eff) 12
−
δ ˙ Fij ≥ 0 T ˙ Stot = −
δ ˙ Fij ≥ 0
Real entropy production: Entropy production of the coarse-grained description:
T ˙ S(cg)
tot ≥ T ˙
Stot
It is even possible to have zero entropy production with finite current!
2 F eff
21
1 B
(b)
A
= µA − µB > 0);
The pump induces the production of A.
2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.000 0.002 0.004 0.006 0.008 0.010 2 4 6 8 10 12
1 a, 2 b 400 1 b, 2 a 400
Ea Eb Stot Stot
num
Stot
cg
a) b) c) d) e) f) g)
wreac
21
6 2 5 1 3 4 Fext
(c)
Feff
Finite current with zero entropy production
Entropy Temperature
∆S Th
Tc
∆Sh
B < 0
∆Sc
B > 0
700 600 500 400 300 200
25 20 15 10 5
1.25 1.00 0.75 0.50 0.25
25 20 15 10 5
700 600 500 400 300 200
0.0 0.2
Isothermal compression Adiabatic compression Isothermal expansion Adiabatic expansion
(1) (2) (3) (4)
Stiffness Temperature Time Th Tc
A B C D
∆S/k Fκ (×10−3µm2) κ (pN/µm) Tpart (K) τ κ (pN/µm) Tpart (K)
, Rica. Brownian Carnot engine. arXiv:1412.1282v3 (2015).
Entropy Temperature
∆S Th
Tc
∆Sh
B < 0
∆Sc
B > 0
Hoppenau, Granger, Dinis, JMRP
v u bath
1.Hoppenau, J., Niemann, M. & Engel, A. Carnot process with a single particle. Phys Rev E 87, 062127 (2013).
v(2) v(1) v(3) piston x L0 Lf Tf t x0
Entropy Temperature
∆S Th
Tc
∆Sh
B < 0
∆Sc
B > 0
∆Sc
B
∆Sh
B h∆Sh
Bi
h∆Sc
Bi
Carnot line
ρ(∆Sc
B, ∆Sh B) = const
−S∗ S∗
Hoppenau, Granger, Dinis, JMRP
Driven systems apparently perform much better than autonomous systems. Time asymmetry is not enough to have reversible transport. Most likely reversible trajectories have finite power.
. Reversible ratchets as Brownian particles in an adiabatically changing periodic
, Blanco, Cao, Brito. Efficiency of Brownian motors. Europhys Lett 43, 248 (1998).
. Imitating Chemical Motors with Optimal Information Motors.
. Stochastic thermodynamics of hidden pumps. Phys Rev E 91, 052114 (2015).
, Rica. Brownian Carnot engine. arXiv:1412.1282v3 (2015).
. Efficiency of finite-time small Carnot engines: optimal designs and fluctuations. In preparation.