On thermodynamic and microscopic reversibility Page: 2 of 5
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On Thermodynamic and Microscopic Reversibility
Thermodynamic reversibility
The concept of a thermodynamically reversible process was crucial to the 19th century
development of thermodynamics [1, 2]. A reversible thermodynamic process is one
that can proceed equally well either forward or backward, which can occur only if the
dissipation, the total entropy change, is zero.
We need to consider a system driven from one thermodynamic equilibrium
to another by a changing external perturbation. We can denote the controllable
parameters of the system as A(t) for t E [a, b], and the overall experimental protocol,
describing the time course of the control, as A. We assume that the control parameters
are unchanging outside the time interval [a, b], so that the system starts at equilibrium
at time a and relaxes back to equilibrium at some time T b. Every protocol also has
a conjugate, time-reversed protocol A where the system begins in thermal equilibrium
at A(b) and the control parameters retrace the same series of changes, in reverse, back
to A(a).
The second law, the central tenet of thermodynamics, states that the mean total
entropy change for any such protocol is non-negative,
(Osta)A ;> 0 . (1)
The total entropy change includes the entropy change of the system and any
concomitant entropy change of the environment, for instance due to exchange of energy
in the form of heat. The average is taken over many realizations of the experiment.
The second law defines the directionality of time [3, 4], and appears to be the only
physical law that breaks time symmetry in a non-trivial manner. The direction of
time we call the future is that in which the entropy increases (on average).
A thermodynamically reversibly process is one with zero net entropy change.
Because there is no time directionality in this case, both the forward and reversed
protocol can be performed without any dissipation. In practice, this can only be
achieved with a quasi-static process, where the active perturbation of the system is
spread over a long time span, so that the system is always very close to equilibrium [5].
Microscopic reversibility
In the early decades of the 20th century, it became apparent, from considerations
of chemical kinetics and quantum mechanics, that a type of dynamic, stochastic
reversibility must hold on the microscopic level [6 11]. For instance, in 1924 G. N.
Lewis stated [8]:
Corresponding to every individual process there is a reverse process, and in a
state of equilibrium the average rate of every process is equal to the average
rate of its reverse process.
Expressed another way, this Principle of Microscopic Reversibility at Equilib-
rium [10, 11] states that the probability of observing any trajectory at equilibrium is
equal to that of observing the time reversed trajectory, under the same conditions,
PFX A]=P[X A]. (2)
Here, X denotes a trajectory of the system through phase space, X denotes the
conjugate, time reversed trajectory, A denotes the fixed external constraints of the
system and P [ X I A ] is the probability of observing the given trajectory.2
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Crooks, Gavin E. On thermodynamic and microscopic reversibility, article, July 12, 2011; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc832244/m1/2/: accessed March 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.