Erratum: “Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws” [J. Chem. Phys. 141, 024117 (2014)] Matteo Polettini and Massimiliano Esposito Citation: The Journal of Chemical Physics 142, 229901 (2015); doi: 10.1063/1.4922305 View online: http://dx.doi.org/10.1063/1.4922305 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics J. Chem. Phys. 142, 204106 (2015); 10.1063/1.4921558 Erratum: “Thermodynamic properties of bulk and confined water” [J. Chem. Phys. 141, 18C504 (2014)] J. Chem. Phys. 141, 249903 (2014); 10.1063/1.4904482 Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws J. Chem. Phys. 141, 024117 (2014); 10.1063/1.4886396 Erratum: “On the fluctuation theorem for the dissipation function and its connection with response theory” [J. Chem. Phys.128, 014504 (2008)] J. Chem. Phys. 128, 249901 (2008); 10.1063/1.2943320 Erratum: “Incomplete descriptions and relevant entropies” [Am. J. Phys. 67 (12), 1078–1090 (1999)] Am. J. Phys. 68, 1060 (2000); 10.1119/1.1286861
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THE JOURNAL OF CHEMICAL PHYSICS 142, 229901 (2015)
Erratum: “Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws” [J. Chem. Phys. 141, 024117 (2014)] Matteo Polettinia) and Massimiliano Esposito Complex Systems and Statistical Mechanics, University of Luxembourg, Campus Limpertsberg, 162a avenue de la Faïencerie, L-1511 Luxembourg, Luxembourg
(Received 19 May 2015; accepted 26 May 2015; published online 11 June 2015) [http://dx.doi.org/10.1063/1.4922305] In Sec. III D of Ref. 1, we defined the scaled rate constants
K±ρ
(1)
j± = k± z ·∇± = [Z]−1 J±.
(2)
k±ρ = [Z]
σ ∇σ,±ρ −1
and the scaled currents We failed to notice that if the total concentration of the reaction [Z] varies in time, then so do the scaled rate constants that, in fact, are not constant. This has some minor implications for the results derived in Secs. IV C and IV D as regards the transient behavior of closed chemical networks. In the rest of the paper, dealing with open systems, it is assumed that one substance, typically water, is so abundant that all concentrations are relative to its concentration and [Z] is approximately constant in time, and thus are not affected by this problem. eq Let [Z]eq be the total concentration at equilibrium and k±ρ be the scaled rate constants at equilibrium. Section V C discussing detailed balance is concentrated on steady states; hence, the discussion remains unchanged, with the specification that in Eq. (59), one should define the force in terms of the scaled rates at equilibrium, F = ln
k+eq . k−eq
(59′)
Interestingly, circulation calculated Eq. (60) is not affected because it is independent of [Z], 0 = c · F = ln
eq eq k+ρ . . . k +ρ n 1
(3)
eq eq k−ρ . . . k −ρ n 1
= ln
K+ρ1 . . . K+ρ n − cρ ∇σ, ρ ln[Z]eq K−ρ1 . . . K−ρ n σ, ρ
(4)
= ln
k+ρ1 . . . k+ρ n , k−ρ1 . . . k −ρ n
(5)
where we used the fact that a cycle c is a null vector of the stoichiometric matrix. Hence the conditions under which detailed balance holds are verified at all times. In Sec. V D, our mistake leads to additional terms to the entropy production rate (EPR), which are total time derivatives not affecting the steady state behavior. The second passage in Eq. (69) has to be replaced by ∆G = RT ∇ ln
k+ k−eq z j+ = −RT ln + RT ln , z eq j− k− k+eq
(69′)
where we used modified Eqs. (59′) and (61). The latter term can be expressed as RT ln
eq k+ρ k −ρ eq k−ρ k +ρ
= −RT ln
[Z] ∇σ, ρ . [Z]eq σ
(6)
Then, the EPR reads j+ [Z] T Σ = − j · ∆G = RT ( j+ − j−) · ln + N˙ ln , j− [Z]eq
(70′)
a)Electronic mail: [email protected]
0021-9606/2015/142(22)/229901/2/$30.00
142, 229901-1
© 2015 AIP Publishing LLC
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229901-2
M. Polettini and M. Esposito
J. Chem. Phys. 142, 229901 (2015)
where the total rate is given by N˙ = d ln[Z]/dt. Leaving the definition in Eq. (72) untouched, then also Eqs. (71) and (73) remain unaltered, which shows that T[Z]Σ is a total time derivative. Interestingly, as it is hereby defined, the EPR is not necessarily positive, because of an additional term that accounts for the total systems’ size increase. An alternative definition of entropy production per volume would then be Σ ′ = R(J+ − J−) · ln
J+ . J−
(7)
This quantity is positive and it can be easily proven to be a total time derivative. Its role and behavior will be discussed in a future publication. We thank Riccardo Rao for noticing the error and for useful comments. 1M. Polettini and M. Esposito, “Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws,” J. Chem. Phys. 141,
024117 (2014).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 144.173.6.37 On: Tue, 11 Aug 2015 02:28:20
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 144.173.6.37 On: Tue, 11 Aug 2015 02:28:20
THE JOURNAL OF CHEMICAL PHYSICS 142, 229901 (2015)
Erratum: “Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws” [J. Chem. Phys. 141, 024117 (2014)] Matteo Polettinia) and Massimiliano Esposito Complex Systems and Statistical Mechanics, University of Luxembourg, Campus Limpertsberg, 162a avenue de la Faïencerie, L-1511 Luxembourg, Luxembourg
(Received 19 May 2015; accepted 26 May 2015; published online 11 June 2015) [http://dx.doi.org/10.1063/1.4922305] In Sec. III D of Ref. 1, we defined the scaled rate constants
K±ρ
(1)
j± = k± z ·∇± = [Z]−1 J±.
(2)
k±ρ = [Z]
σ ∇σ,±ρ −1
and the scaled currents We failed to notice that if the total concentration of the reaction [Z] varies in time, then so do the scaled rate constants that, in fact, are not constant. This has some minor implications for the results derived in Secs. IV C and IV D as regards the transient behavior of closed chemical networks. In the rest of the paper, dealing with open systems, it is assumed that one substance, typically water, is so abundant that all concentrations are relative to its concentration and [Z] is approximately constant in time, and thus are not affected by this problem. eq Let [Z]eq be the total concentration at equilibrium and k±ρ be the scaled rate constants at equilibrium. Section V C discussing detailed balance is concentrated on steady states; hence, the discussion remains unchanged, with the specification that in Eq. (59), one should define the force in terms of the scaled rates at equilibrium, F = ln
k+eq . k−eq
(59′)
Interestingly, circulation calculated Eq. (60) is not affected because it is independent of [Z], 0 = c · F = ln
eq eq k+ρ . . . k +ρ n 1
(3)
eq eq k−ρ . . . k −ρ n 1
= ln
K+ρ1 . . . K+ρ n − cρ ∇σ, ρ ln[Z]eq K−ρ1 . . . K−ρ n σ, ρ
(4)
= ln
k+ρ1 . . . k+ρ n , k−ρ1 . . . k −ρ n
(5)
where we used the fact that a cycle c is a null vector of the stoichiometric matrix. Hence the conditions under which detailed balance holds are verified at all times. In Sec. V D, our mistake leads to additional terms to the entropy production rate (EPR), which are total time derivatives not affecting the steady state behavior. The second passage in Eq. (69) has to be replaced by ∆G = RT ∇ ln
k+ k−eq z j+ = −RT ln + RT ln , z eq j− k− k+eq
(69′)
where we used modified Eqs. (59′) and (61). The latter term can be expressed as RT ln
eq k+ρ k −ρ eq k−ρ k +ρ
= −RT ln
[Z] ∇σ, ρ . [Z]eq σ
(6)
Then, the EPR reads j+ [Z] T Σ = − j · ∆G = RT ( j+ − j−) · ln + N˙ ln , j− [Z]eq
(70′)
a)Electronic mail: [email protected]
0021-9606/2015/142(22)/229901/2/$30.00
142, 229901-1
© 2015 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 144.173.6.37 On: Tue, 11 Aug 2015 02:28:20
229901-2
M. Polettini and M. Esposito
J. Chem. Phys. 142, 229901 (2015)
where the total rate is given by N˙ = d ln[Z]/dt. Leaving the definition in Eq. (72) untouched, then also Eqs. (71) and (73) remain unaltered, which shows that T[Z]Σ is a total time derivative. Interestingly, as it is hereby defined, the EPR is not necessarily positive, because of an additional term that accounts for the total systems’ size increase. An alternative definition of entropy production per volume would then be Σ ′ = R(J+ − J−) · ln
J+ . J−
(7)
This quantity is positive and it can be easily proven to be a total time derivative. Its role and behavior will be discussed in a future publication. We thank Riccardo Rao for noticing the error and for useful comments. 1M. Polettini and M. Esposito, “Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws,” J. Chem. Phys. 141,
024117 (2014).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 144.173.6.37 On: Tue, 11 Aug 2015 02:28:20
