Bibliography
179
e
−W/k B T
= e
− /k B T
(F.21)
A rigorous formal derivation can be found in the original Jarzynski’s papers [5, 6].
An excelent article on Jarzynski’s equality illustrated by simple examples was given
by, Híjar and Zárate, [7].
F.1.3.2 Crooks Theorem
Many theoretical developments based on the Jarzynski equality have appeared.
Many of them represent equivalent formulations adopting different perspectives.
Most noted is the so-called Crooks theorem [8], usually formulated as
P F (W )
P B (−W )
= exp [(W − )/k B T ]
(F.22)
where the left-hand side is the ratio of the probability of work W when going
from the initial to the final state (forward probability) to the probability of work
−W when going from the final to the initial state (backward probability). The
equivalence of the Jarzynski and Crooks formulations can be readily demonstrated
[8]. Among other theoretical developments we mention extensions to cover the
Langevin dynamics, when the system is in equilibrium with a thermal bath Imparato
and Peliti [9].
Bibliography
1. Mogilner, A., Elston, T., Wang, H., Oster, G.: Molecular motors: examples. In: Fall, C.,
Marland, E., Tyson, J., Wagner, J. (eds.) Joel Keizer’s Computational Cell Biology, Chapter 12,
12..6.4 p. 348, Eq. (12.71): Modeling Chemical Reactions (2002). https://doi.org/10.1016/
S0370-1573(01)00081-3
2. Park, J.-M., Chun, H.-M., Noh, J.D.: Efficiency at maximum power and efficiency fluctuations
in a linear Brownian heat-engine model. Phys. Rev. E 94, 012127 (2016)
3. Kesimoto, K.: Langevin equation and thermodynamics. Prog. Theor. Phys. Suppl. 130, 17 (1998)
4. Cubero, D., Renzoni, F.: Brownian Ratchets, p. 99. Cambridge University Press, Cambridge
(2016)
5. Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690
(1997)
6. Jarzynski, C.: Equilibrium free-energy differences from nonequilibrium measurements: a
master-equation approach. Phys. Rev. E 56, 5018–5035 (1997)
7. Híjar, H., Ortiz de Zárate, J.M.: ıJarzynski’s equality illustrated by simple examples. Eur. J.
Phys. 31, 1097 (2010)
8. Crooks, G.E.: Entropy production fluctuation theorem and the nonequilibrium work relation for
free energy differences. Phys. Rev. E 60, 2721 (1999)
9. Imparato, A., Peliti, L.: Work-probability distribution in systems driven out of equilibrium. Phys.
Rev. E 72, 046114 (2005)
179
e
−W/k B T
= e
− /k B T
(F.21)
A rigorous formal derivation can be found in the original Jarzynski’s papers [5, 6].
An excelent article on Jarzynski’s equality illustrated by simple examples was given
by, Híjar and Zárate, [7].
F.1.3.2 Crooks Theorem
Many theoretical developments based on the Jarzynski equality have appeared.
Many of them represent equivalent formulations adopting different perspectives.
Most noted is the so-called Crooks theorem [8], usually formulated as
P F (W )
P B (−W )
= exp [(W − )/k B T ]
(F.22)
where the left-hand side is the ratio of the probability of work W when going
from the initial to the final state (forward probability) to the probability of work
−W when going from the final to the initial state (backward probability). The
equivalence of the Jarzynski and Crooks formulations can be readily demonstrated
[8]. Among other theoretical developments we mention extensions to cover the
Langevin dynamics, when the system is in equilibrium with a thermal bath Imparato
and Peliti [9].
Bibliography
1. Mogilner, A., Elston, T., Wang, H., Oster, G.: Molecular motors: examples. In: Fall, C.,
Marland, E., Tyson, J., Wagner, J. (eds.) Joel Keizer’s Computational Cell Biology, Chapter 12,
12..6.4 p. 348, Eq. (12.71): Modeling Chemical Reactions (2002). https://doi.org/10.1016/
S0370-1573(01)00081-3
2. Park, J.-M., Chun, H.-M., Noh, J.D.: Efficiency at maximum power and efficiency fluctuations
in a linear Brownian heat-engine model. Phys. Rev. E 94, 012127 (2016)
3. Kesimoto, K.: Langevin equation and thermodynamics. Prog. Theor. Phys. Suppl. 130, 17 (1998)
4. Cubero, D., Renzoni, F.: Brownian Ratchets, p. 99. Cambridge University Press, Cambridge
(2016)
5. Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690
(1997)
6. Jarzynski, C.: Equilibrium free-energy differences from nonequilibrium measurements: a
master-equation approach. Phys. Rev. E 56, 5018–5035 (1997)
7. Híjar, H., Ortiz de Zárate, J.M.: ıJarzynski’s equality illustrated by simple examples. Eur. J.
Phys. 31, 1097 (2010)
8. Crooks, G.E.: Entropy production fluctuation theorem and the nonequilibrium work relation for
free energy differences. Phys. Rev. E 60, 2721 (1999)
9. Imparato, A., Peliti, L.: Work-probability distribution in systems driven out of equilibrium. Phys.
Rev. E 72, 046114 (2005)