Because of the possibility of the phenomenon of quenched disorder I
wish to amend my definition of thermodynamic entropy to properly
account for this possibility when the experimental equilibration time
scale is much shorter than the time scale over which the quenched
disorder anneals.
Thermodynamic entropy: The average minimal amount of further information
required to exactly determine the exact microscopic state of a
macroscopic physical system given *both* a precise specification of that
system's macroscopic state *and* given the understanding that the
universe of possible microscopic states is to include only those
microscopic states that are mutually dynamically accessible under the
system's dynamical law of evolution (subject to an infinitesimal strength
random perturbation in that dynamical law).
The conceptual (not practical) algorithm for calculating the
thermodynamic entropy is as follows:
1A.) Exactly specify the system's macroscopic state. This specification
is assumed given.
1B.) Determine each disjoint universe {i} of dynamically mutually
accessible microscopic states i. This is done as follows:
a) Pick a microscopic state consistent with the macrostate.
b) Use the system's dynamical law of evolution (the Liouville
equation, the Schrodinger equation, etc.) found from its
Hamiltonian and evolve the picked microstate in time over a time
interval equal to the 'resolving time' of the experimental
apparatus over which the observations are to be made. Include in
the evolution law a random infinitesimal noise perturbation that
respects any macroscopically conserved quantities (maybe energy,
particle number, volume etc., but the exact set of such fixed
variables will depend in the details of just which quantities are
being constrained in the macrostate description) that may exist
by virtue of the specified macroscopic description.
c) Collect the sequence of microscopic states visited by the
dynamical evolution above and consider them all as belonging to
the universe of dynamically accessible microscopic states.
d) Repeat steps b) and c) *many* times but each with a different
realization of the random infinitesimal noise process but with
the *same* initially picked microstate chosen in step a) and take
the union of the sets of microstates visited over all repetitions
of the evolution.
e) Continue with the repetitions in step d) until no more new
microstates are being included in the union set of all visited
microstates. The resulting union set is the universe of all
microscopic states dynamically accessible to the microstate
initially picked microstate in step a).
2.) Find the distribution (density matrix for a quantum system) of
microscopic states {p_i} for the system for the universe of
assessible states found in step 1B.) by maximizing the functional:
S({p_i}) = - SUM(i,p_i*log(p_i)) over all possible distributions
that are consistent with the specified macrostate and involve only
those mutually dynamically assessible microscopic states. The
resulting distribution {p_i} is the proper microstate distribution
for that universe of microstates, and the resulting maximum S({p_i})
value is the actual entropy for that distribution over that universe
of microstates.
3.) Average the resulting maximal S value from 2.) from each of the
disjoint universes of mutually dynamical accessible microstates over
all of them. All the other disjoint universes of microstates which
are consistent with the macrostate are found by repeating steps 1B)
and 2) where the initial microstate picked in 1Ba) is taken to not
belong to any of the previously found universes. When there are no
more microstates that have not not been previously found, then all
the microstates that are consistent with the macrostate will have
been found and these microstates will all have been partitioned into
disjoint subsets that define the various disjoint universes of
separately dynamically accessible microstates.
The average value of S averaged over all these universes of mutually
dynamically accessible microstates consistent with the macroscopic
description *is* the thermodynamic entropy of the system.
This whole algorithm *greatly* simplifies if there is no quenched
disorder and there is only one universe of mutually dynamically
accessible microstates consistent with the macrostate. In this case we
just do step 1A) to get the macrostate description and then consider all
microstates consistent with it as the relevant universe of accessible
microstates. We then jump to step 2) for that single universe. Step 3)
is skipped since there is only one resulting value of S from step 2)
which does not have to be averaged any further.
Notice that this definition and this algorithm make the thermodynamic
entropy an objective function of the macrostate description. It also
has the property that for a macrostate description that makes the system
isolated from the rest of the universe, the system's entropy will
monotonically grow with time (in accordance with the 2nd law) until the
condition of equilibrium is achieved. If the entropy for this latter
case is to be calculated as a function of time where some parts of the
macroscopic description are considered to be time dependent and whose
values are allowed to be determined by the evolution of the underlying
microscopic dynamics (rather than as an externally specified macroscopic
value frozen by the macrostate description), then an additional
modification of the algorithm for finding S is in order. In this case
the initial value of S at time t = t_i is calculated according to the
above algorithm. Then for each of the universes of dynamically
accessible microstates their initially determined distribution {p_i}
(this is a density matrix if we have a quantum system) for that universe
is evolved forward to a later time, say, t = t_f by using the dynamical
evolution law (the Liouville equation/Newton's laws in classical
mechanics or the Schrodinger equation in quantum mechanics). In both
the quantum (because of unitarity) and classical (because of Liouville's
theorem) cases the forward-evolved distributions (density matrices) will
have exactly the same entropy as their initial values do. But these
forward-evolved distributions do not represent the actual subsequent
entropy at the later time t_f. Rather, the only purpose of these
forward-evolved distributions is to calculate the new averages for those
macroscopic parameters that are allowed to change with time by the
system's external constraints. Once these new mean values in the
time-variable macroscopic variables are found and they are combined with
the externally fixed macroscopic variables, then the new future
macrostate is then specified. Using this new macrostate the entropy is
recalcuated by re-maximizing S({p_i}) as in step 2.), above, over all
possible microstate distributions (density matrices) consistent with the
new updated macrostate. This new re-maximized S value cannot be less
than the S value of forward-evolved distribution because that
distribution is one of the set of distributions over which the S value
is remaximized. Since the forward-evolved S value is the same as the S
value at t = t_0 we see that the re-maximized S value has not decreased.
Then the whole collection of re-maximized S values for each of the
disjoint universes of microstates are reaveraged (only if necessary,
because of possible quenched disorder making a plurality of such
universes) and this resulting S value is the value of S at t = t_f.
If we wish to find S(t_f) for the simpler case where there is no
quenched disorder then we only have one universe of microstates to
consider. In this case we just take the {p_i} distribution which
maximizes S({p_i}) at t = t_i and forward-evolve it to t = t_f and use
the resulting distribution to calculate the new average values for those
macroscopic parameters that happen to be 'floating' w.r.t. the system's
constraints, and then these parameters together with the initial fixed
macroscopic constraints define the new macrostate. Then S is
re-maximized using the new macrostate parameters. The new re-maximized
S value is the thermodynamic entropy of the system at t = t_f.
Again S(t_i) <= S(t_f) and equality occurs in equilibrium since then the
macroscopic parameters are no longer changing and re-maximizing S has
no effect on the S value and doesn't change {p_i} distribution either.