Equilibrium states are used as limit states to define thermodynamically reversible processes. When these processes are understood in terms of statistical physics, these limit states can change with time due to thermal fluctuations. For macroscopic systems, the changes are insignificant on ordinary time scales and what little change there is can be suppressed by macroscopically negligible, entropy-creating dissipation. For systems of molecular sizes, the changes are large on short time scales. They can only sometimes be suppressed with significant entropy-creating dissipation, and this entropy creation is unavoidable if any process is to proceed to completion. As a result, at molecular scales, thermodynamically reversible processes are impossible in principle. Unlike the macroscopic case, they cannot be realized even approximately when we account for all sources of dissipation, and argumentation invoking them on molecular scales can lead to spurious conclusions.

1.
Such as in
Hendrick C.
Van Ness
,
Understanding Thermodynamics
(
McGraw-Hill
,
New York
,
1969
; reprinted by Dover, New York, 1983), pp.
19
22
.
2.
Christopher
Jarzynski
, “
Nonequilibrium equality for free energy differences
,”
Phys. Rev. Lett.
78
(
14
),
2690
2693
(
1997
).
For a precise characterization of thermodynamically reversible processes and a critical historical survey, see
John D.
Norton
, “
The Impossible Process: Thermodynamic Reversibility
,”
Studies in History and Philosophy of Modern Physics
55
(2016), pp.
43
61
.
3.
R.
Kawai
,
J. M. R.
Parrondo
, and
C.
Van den Broeck
, “
Dissipation: The phase-space perspective
,”
Phys, Rev. Lett.
98
,
080602
(
2007
).
4.

To see this, note that for small changes we have dFsys=d(UsysTSsys)=dUsysTdSsys=dUsys+dUenv=dUtot=0, where the heat passed to the environment in the reversible process is dQsys= TdSsys, which equals the energy change in the environment dUenv.

5.

For a small, reversible change we have dFsys= dUsys − ΤdUsys= dUsys − dQ = −dW, so that −dFsys/dx = dW/dx = X.

6.
Albert
Einstein
, “
On a heuristic viewpoint concerning the production and transformation of light
,”
Ann. Phys.
17
,
132
148
(
1905
).
7.

To connect with the usual statement of the canonical distribution, if Vph,tot is the volume of the full phase space accessible to the system, then the canonical distribution is p = exp(−E/kT)/ Z(Vph,tot) and the probability that the system is in subvolume Vph is equal to Z(Vph)/ Z(Vph,tot).

8.

The intermediate states can never be completely inaccessible or the process could not proceed. Rather the process design must be such as to make them accessible only with arbitrarily small probability.

9.
Equations (15) and (16) with a term ln[1 + Pfin)/Pinit)] give slightly higher dissipation than the corresponding formulas (22) and (23) of an earlier paper [
John D.
Norton
, “
All shook up: Fluctuations, Maxwell's Demon and the thermodynamics of computation
,”
Entropy
15
,
4432
4483
(
2013
)], which instead have a term ln[P(λfin)/P(λinit)]. The latter formulae presumed that the process ends in a way that prevents return to the initial state. In the absence of a non-dissipative way of preventing this return, the newer formulae provide a better limit.
10.
See supplementary material at http://dx.doi.org/10.1119/1.4966907 for “Appendix: Moving a Brownian particle”; also available at <http://philsci-archive.pitt.edu/12202/>.
11.
John D.
Norton
, “
Waiting for Landauer
,”
Stud. Hist. Philos. Mod. Phys.
42
,
184
198
(
2011
), Sec. 7.5.
12.

These two probabilities are to be read as follows: over the longer term in which the gas-piston system fully explores the phase space accessible to it, it comes to an equilibrium with probability P(h0) of the initial compressed h-state and probability P(h1) of the final, expanded h-state.

13.
Leo
Szilard
, “
On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings
,” (1929), in
The Collected Works of Leo Szilard: Scientific Papers
(
MIT Press
,
Cambridge, MA
,
1972
), pp.
120
129
.
14.
For a survey and collection of works, see
Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing
, edited by
Harvey S.
Leff
and
Andrew
Rex
(
Institute of Physics Publishing
,
Bristol and Philadelphia
,
2003
).
15.
L.
Szilard
, Ref. 13, pp. 122–123. Szilard proposes further “It is best to imagine the mass of the piston as large and its speed sufficiently great, so that the thermal agitation of the piston at the temperature in question can be neglected.” This attempt to suppress fluctuations requires the piston to be in a state far from equilibrium, incompatible with a reversible process. See J. D. Norton, Ref. 9, p. 4459.

Supplementary Material

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