Carnot Cycle at Finite Power and Attainability of Maximal Efficiency

Armen Allahverdyan(Yerevan Physics Institute)

-- Introduction: heat engines, Carnot cycle

-- Non-equilibrium Carnot cycle

-- Analysis: PRL 2013.

-- Coauthors: Karen Hovhannisyan,

S. Gevorkian, A. Melkikh

Cyclic engine

Work-source

1T 2T

1 2T THot bath Cold bath

1 2 0W Q Q Work: Power:

1/ 1W Q Efficiency:

/ cycleW

output / input

1Q2Q

W

Challenge: to make engines both powerful and efficient

U. Seifert, Rep. Prog. Phys. '12 Benenti, Casati, Prosen, Saito, arxiv:1311.4430

B. Andresen, Angew Chem '11

Carnot cycle: useless in practice: 4 times slow

2T

2S

1T

1S

Thermally isolated and slow

Isothermal and slow

1 1 1 2( )Q T S S

2 2 1 2( )Q T S S

1 2 1 1 2 1( ) / ( ) /Q Q Q T T T

Carnot = maximal efficiency

1 1/1( , )H Te H

2 2/2( , )H Te H

1 2( , )H

2 1( , )H

1T 2T

Non-equilibrium Carnot cycle

Engine: density matrix and Hamiltonian

2cycle relax

tH

tH

H,

1 1/1( , )H Te H

2 2/2( , )H Te H

1 2( , )H

2 1( , )H

Work-source and baths act separately easy to derive work and heat

)(tr 211/

111 HHeW TH

)(tr 121/

111 HHeQ TH

Maximize W over dynamics ,t tH H

n+1 energy levels and the temperatures are fixed

1T 2T

n degenerate states: energy concentration

nlnoptimized energy gaps

Sudden changes are optimal1 2H H 2cycle relax

Work and efficiency

1 2( ) ln (1)W T T n O

n>>1 number of levels

Relaxation time ?

)ln

1(

nOCarnot

Ad hoc: system-bath (interaction) Hamiltonian is fine-tuned to system Hamiltonian

Realistic

ln n >>1 number of particles

2cycle relax

BBSS HHH

n

kkkS wwEwwH

1

||||0

bath: 2-level systems

0],[ BSBS HHH )ln( nOrel

)ln

1(

nOCarnot

there is

)ln(2

nOW

rel

An example of fine-tuned system-bath Hamiltonian

realistic works for anyBSH SH

0,| wEw ,| 1 Ewn ,|

Unstructured data-base search (computationally complex)

)(nOrel )

ln(

2 n

nO

W

rel

Power zero

Vogl, Schaller, Brandes, PRA'10

Farhi, Gutman, PRA '98

Grover, PRL '97

Levinthal’s problem for protein folding Zwanzig, PNAS '95

Reduce W, resolve the degeneracy

0

0

E

E

( 0)0.3

( 0)CarnotW

W

92.0)0(

Carnot

s1relax

45.0)0(

)0(

W

W

The reason of not reaching Carnot efficiency for realistic system-bath interaction is computational complexity

Conclusions

Protein models as sub-optimal Carnot engine

Non-reachability of Carnot efficiency at a large power is not a law of nature: there is a fine-tuned interaction that achieves this.