1.3 Ratchet Efficiency
5
If L is chosen to be smaller but close to L 0 , then the wheel will move forward
very slowly, lifting the weight. Let us calculate the efficiency of the engine. If the
ratchet performs N + clockwise jumps and N − counterclockwise jumps, the total
work done is (N + − N − )Lθ and the amount of heat taken from bath 1 is (N + −
N − )(Lθ + ). Therefore, the efficiency is
η =
Lθ
Lθ +
(1.4)
and, in the limit L → L 0 (or zero power), the efficiency converges to that of a
Carnot cycle, (1.3):
η → η C = 1 −
T 2
T 1
.
(1.5)
1.2.1 Parrondo Criticism
Parrondo in his paper together with Español, [14] showed that Feynman’s analysis
contains some misguided aspects: Since the engine is simultaneously in contact with
reservoirs at different temperatures, it can never work in a reversible way. As a
consequence, the engine can never achieve the efficiency of a Carnot cycle, not
even in the limit of zero power (infinitely slow motion), in contradiction with the
conclusion reached in the Lectures. In spite of this criticism the Feynman analysis,
besides its pedagogical interest, has been the inspiration of a currently very active
research field on transport induced by Brownian motion in asymmetric potentials
[15, 16].
1.3 Ratchet Efficiency
The existence of reversible ratchets is of extreme importance for designing Brownian motors with high efficiency. We saw in the previous section that Feynman
calculated under very simple assumptions the efficiency of a ratchet, finding that
it is equal to Carnot efficiency in the quasistatic limit, and Parrondo revealed
the inconsistency of this arguments by proving the intrinsic irreversibility of the
system under consideration. Most of the ratchets proposed in the literature are
also intrinsically irreversible, [17] and their efficiency turns out to be very low,
whereas reversible ratchets posses a comparatively high efficiency. Parrondo et al.
[18] calculated analytically and numerically the efficiency of different types of
Brownian motors. They found that motors based on flashing ratchets present a low
efficiency and an unavoidable entropy production. On the other hand, a certain class
5
If L is chosen to be smaller but close to L 0 , then the wheel will move forward
very slowly, lifting the weight. Let us calculate the efficiency of the engine. If the
ratchet performs N + clockwise jumps and N − counterclockwise jumps, the total
work done is (N + − N − )Lθ and the amount of heat taken from bath 1 is (N + −
N − )(Lθ + ). Therefore, the efficiency is
η =
Lθ
Lθ +
(1.4)
and, in the limit L → L 0 (or zero power), the efficiency converges to that of a
Carnot cycle, (1.3):
η → η C = 1 −
T 2
T 1
.
(1.5)
1.2.1 Parrondo Criticism
Parrondo in his paper together with Español, [14] showed that Feynman’s analysis
contains some misguided aspects: Since the engine is simultaneously in contact with
reservoirs at different temperatures, it can never work in a reversible way. As a
consequence, the engine can never achieve the efficiency of a Carnot cycle, not
even in the limit of zero power (infinitely slow motion), in contradiction with the
conclusion reached in the Lectures. In spite of this criticism the Feynman analysis,
besides its pedagogical interest, has been the inspiration of a currently very active
research field on transport induced by Brownian motion in asymmetric potentials
[15, 16].
1.3 Ratchet Efficiency
The existence of reversible ratchets is of extreme importance for designing Brownian motors with high efficiency. We saw in the previous section that Feynman
calculated under very simple assumptions the efficiency of a ratchet, finding that
it is equal to Carnot efficiency in the quasistatic limit, and Parrondo revealed
the inconsistency of this arguments by proving the intrinsic irreversibility of the
system under consideration. Most of the ratchets proposed in the literature are
also intrinsically irreversible, [17] and their efficiency turns out to be very low,
whereas reversible ratchets posses a comparatively high efficiency. Parrondo et al.
[18] calculated analytically and numerically the efficiency of different types of
Brownian motors. They found that motors based on flashing ratchets present a low
efficiency and an unavoidable entropy production. On the other hand, a certain class