of 17
PHYSICAL REVIEW RESEARCH
3
, 033115 (2021)
Harnessing fluctuations in thermodynamic computing via time-reversal symmetries
Gregory Wimsatt,
1
,
*
Olli-Pentti Saira,
2
,
Alexander B. Boyd,
1
,
Matthew H. Matheny,
2
,
§
Siyuan Han,
3
,

Michael L. Roukes,
2
,
and James P. Crutchfield
1
,
2
,
#
1
Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, California 95616, USA
2
Condensed Matter Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA
3
Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA
(Received 29 January 2021; accepted 21 June 2021; published 5 August 2021)
We experimentally demonstrate that highly structured distributions of work emerge during even the simple
task of erasing a single bit. These are signatures of a refined suite of time-reversal symmetries in distinct
functional classes of microscopic trajectories. As a consequence, we introduce a broad family of conditional
fluctuation theorems that the component work distributions must satisfy. Since they identify entropy production,
the component work distributions encode the frequency of various mechanisms of both success and failure during
computing, as well giving improved estimates of the total irreversibly dissipated heat. This new diagnostic tool
provides strong evidence that thermodynamic computing at the nanoscale can be constructively harnessed. We
experimentally verify this functional decomposition and the new class of fluctuation theorems by measuring
transitions between flux states in a superconducting circuit.
DOI:
10.1103/PhysRevResearch.3.033115
I. INTRODUCTION
Physics dictates that all computing is subject to sponta-
neous error. These days, this truism repeatedly reveals itself:
despite the once-predictable miniaturization of nanoscale
electronics, computing performance increases have dramati-
cally slowed in the last decade or so. In large measure, this
is due to the concomitant rapid decrease in the number of
information-bearing physical degrees of freedom, rendering
information storage and processing increasingly susceptible
to corruption by thermal fluctuations. Said simply, all physical
computing is thermodynamic.
Controlling the production of fluctuations and removing
heat pose key technological challenges to further progress.
Practically, the challenge remains of how to probe and di-
agnose information processing in overtly noisy systems. The
following introduces trajectory class fluctuation theorems
to do this by identifying the thermodynamic signature of
successful and failed information processing. It then experi-
mentally demonstrates how this is practically implemented in
a new microscale platform for thermodynamic computing.
*
gwwimsatt@ucdavis.edu
osaira@caltech.edu
abboyd@ucdavis.edu
§
matheny@caltech.edu

han@ku.edu
roukes@caltech.edu
#
Corresponding author: chaos@ucdavis.edu
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International
license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
Only recently have tools appeared that precisely describe
what tradeoffs exist between thermodynamic resources and
useful information processing—these are highly reminiscent
of the centuries-old puzzle of how Maxwell’s “very observant
and neat-fingered” demon uses its “intelligence” to convert
disorganized heat energy to useful work [
1
]. In our modern
era, his demon has led to the realization that information
itself is physical [
2
4
]—or, most constructively, that infor-
mation is a thermodynamic resource (see [
5
] and references
therein). This opened up the new paradigm of
thermodynamic
computing
[
6
] in which fluctuations play a positive role in
efficient information processing on the nanoscale. We now
conceptualize this via
information engines
: physical systems
that are driven by, manipulate, store, and dissipate energy,
but simultaneously generate, store, lose, communicate, and
transform information. In short, information engines combine
traditional engines comprised of heat, work, and other familiar
reservoirs with what we now call
information reservoirs
[
7
,
8
].
Reliable thermodynamic computing requires detecting and
controlling fluctuations in informational and energetic re-
sources and in engine functioning. To do so requires a new
generation of diagnostic tools. For these, we appeal to fluctua-
tion theorems that capture exact time-reversal symmetries and
predict entropy production leading to irreversible dissipation
[
9
15
]. As the following demonstrates, these place us on the
doorstep of the very far-from-equilibrium thermodynamics
needed to understand the physics of computing. And, in turn,
the physical principles of how nature processes information in
the service of biological functioning and survival have begun
to emerge.
Proof-of-concept experimental tests have been carried
out in several substrates: probing biomolecule free energies
[
16
18
], work expended during elementary computing (bit
erasure) [
19
24
], and Maxwellian demons [
25
]. That said, the
2643-1564/2021/3(3)/033115(17)
033115-1
Published by the American Physical Society
GREGORY WIMSATT
et al.
PHYSICAL REVIEW RESEARCH
3
, 033115 (2021)
FIG. 1. Inner plot sequence: Erasure protocol (Table
I
) evolution of position distribution Pr(
x
) from simulation. Potential
V
(
x
,
t
s
)at
substage boundary times
t
s
,
s
=
0
,
1
,
2
,
3
,
4. Starting at
t
=
t
0
, the potential evolves clockwise, ending at
t
=
t
4
in the same configuration
as it starts:
V
(
x
,
t
0
)
=
V
(
x
,
t
4
). However, the final position distribution Pr(
x
) predominantly indicates the
R
state. The original one bit of
information in the distribution at time
t
=
t
0
has been erased. Outer plot sequence: Substage work distributions from simulation Pr(
W
s
,
C
s
)
during substages
s
: (1) Barrier Drop, (2) Tilt, (3) Barrier Raise, (4) Untilt. During each substage
s
, distributions are given for up to three
substage trajectory classes
C
s
: red consists of trajectories always in the
R
state, orange trajectories always in the
L
state, and blue the rest,
spending some time in each state.
suite of theoretical predictions and contemporary principles
(Appendix
A
) far outstrips experimental validation to date.
To close the gap, we show how to diagnose thermody-
namic computing on the nanoscale by explaining the signature
structures in work distributions generated during informa-
tion processing. Previous efforts explored features in work
and heat distributions that track the mesoscale evolution of
asystem’s
informational states
;seeRefs.[
26
,
27
] and Ap-
pendix
A
. Here, we show that functional and nonfunctional
informational-state evolutions can be identified by appropriate
conditioning, and that their thermodynamics obey a suite of
trajectory-class fluctuation theorems. As such, the latter give
accurate bounds on work, entropy production, and dissipation
for computing subprocesses. The result is a practical tool
that employs mesoscopic (work) measurements to diagnose
microscopic thermodynamic computing. For simplicity, and
to make direct contact with previous efforts, we demonstrate
the tools on Landauer erasure [
2
] of a bit of information in a
superconducting flux qubit.
II. MODEL SYSTEM
As a reference, we first explore the thermodynamics of
bit erasure in a simple model: a particle with position and
momentum in a double-well potential
V
(
x
,
t
) and in contact
with a heat reservoir at temperature
T
.(RefertoFig.
1
.)
An external controller adds or removes energy from a work
reservoir to change the form of the potential
V
(
·
,
t
) via a pre-
determined
erasure protocol
{
(
β
(
t
)
(
t
)
)
:0

t

τ
}
.
β
(
t
)
and
δ
(
t
) change one at a time piecewise linearly through four
protocol substages: (i)
drop barrier
, (ii)
tilt
, (iii)
raise barrier
,
and (iv)
untilt
. (See Appendix
B
.) The system starts at time
t
=
0 in the equilibrium distribution for a double-well
V
(
x
,
0)
at temperature
T
.
We u s e
underdamped
Langevin dynamics to simulate this
model:
dx
=
v
dt
,
md
v
=
2
k
B
T
γ
r
(
t
)
dt
(
x
V
(
x
,
t
)
+
γ
v
)
dt
,
(1)
where
k
B
is Boltzmann’s constant,
γ
is the coupling between
the heat reservoir and the system,
m
is the particle’s mass, and
TABLE I. Erasure protocol.
Stage
Drop Barrier
Tilt
Raise Barrier
Untilt
t
s
t
0
t
1
t
2
t
3
t
4
β
(
t
)
|
t
1
t
t
1
t
0
|
0
|
t
t
2
t
3
t
2
|
1
|
δ
(
t
)
|
0
|
t
t
1
t
2
t
1
|
1
|
t
4
t
t
4
t
3
|
033115-2
HARNESSING FLUCTUATIONS IN THERMODYNAMIC ...
PHYSICAL REVIEW RESEARCH
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, 033115 (2021)
r
(
t
) is a memoryless Gaussian random variable with

r
(
t
)
=
0 and

r
(
t
)
r
(
t

)
=
δ
(
t
t

).
The default potential,
V
(
·
,
0)
=
V
(
·
), has two symmetric
wells separated by a barrier. Following common practice we
call the two wells, from negative to positive position, the
left
(
L
) and
right
(
R
)
informational states
, respectively. Initially
being equiprobable, the informational states associated with
each of the two wells thus contain 1 bit of information [
28
].
The erasure protocol is designed so that the particle ends
in the
R
state with high probability, regardless of its initial
state. Simulating the protocol 3
.
5
×
10
6
times, 96
.
2% of the
particles were successfully erased into the
R
state. Thus, as
measured by the Shannon entropy, the initial 1 bit of infor-
mation was reduced to 0.231 bits. We intentionally designed
the protocol to fail frequently at erasure to better illustrate our
main results on diagnosing success
and
failure. But, crucially,
the results we present hold for arbitrarily successful erasure
protocols.
At all other times
t
,
V
(
·
,
t
) has either one or two local
minima, naturally defining metastable regions for a particle to
be constrained and gradually evolve toward local equilibrium.
We therefore define the informational states at time 0

t

τ
to be the metastable regions, labeling them
R
and, if two exist,
L
—from most positive to negative in position.
Since the protocol is composed of four simple substages,
we coarse-grain the system’s response by its activity during
each substage at the level of its informational state. Specif-
ically, for each substage, we assign one of three
substage
trajectory classes
: the system (i) was always in the
R
state,
(ii) was always in the
L
state, or (iii) spent time in each.
Sometimes there is only one informational state, and so the
latter two classes are not achievable for all substages.
III. WORK CHARACTERIZATION
We then focus on a single mesoscopic observable—the
thermodynamic work expended during erasure. An individual
realization generates a trajectory of system microstates, with
W
(
t
,
t

) being the work done on the system between times
0

t
<
t


τ
; see Appendix
H
.Let
W
s
=
W
(
t
s
1
,
t
s
) denote
the work generated during substage
s
, and
C
s
denotes the
substage trajectory class. Figure
1
(outer plot sequence) shows
the corresponding substage work distributions Pr(
W
s
,
C
s
) ob-
tained from our simulations. (See Appendix
I
.)
The drop-barrier and tilt substage work distributions are
rather simple, being narrow and unimodal. The raise-barrier
distributions have some asymmetry, but they are also simi-
larly simple. However, the untilt work distributions (farthest
right in Fig.
1
) exhibit unusual features that are significant
for understanding the intricacies of erasure. Trajectories that
spend all of the untilt substage in either the
R
state or the
L
state form peaks at the most positive (red) and negative
(orange) work values, respectively. This is because the
R
-state
well is always increasing in potential energy while the
L
-state
well is always decreasing during untilt. In contrast, the other
trajectories contribute a log-linear ramp of work values (blue)
dependent on the time spent in each. The ramp’s positive slope
signifies that more time is typically spent in the
R
stateinthis
last set of trajectories.
FIG. 2. Rear (purple): total work distribution of all trajectories
Pr(
W
total
) during erasure simulation: a histogram generated from
3
.
5
×
10
6
trials for
W
total
[
6
,
4] over 201 bins. Inset (gray): typ-
ical unimodal work distribution illustrated for a spatially translated
thermally driven simple harmonic oscillator. Three front plots: work
distributions Pr(
W
total
,
C
4
) for the trajectory classes
C
4
determined by
the untilt trajectory partition in simulation: The red work distribution
(middle) is that of Success trajectories, the orange (rear) is that of Fail
trajectories, and the blue (front) is that of the remaining, Transitional
trajectories.
Looking at the total work
W
total
=
W
(0
) generated for
each trajectory over the course of the entire erasure protocol,
we observe the strikingly complex and structured distribution
Pr(
W
total
) shown in Fig.
2
(rear). There are two clear peaks
at the most positive and negative work values separated by a
ramp. This highly structured work distribution, generated by
bit erasure, contrasts sharply with the unimodal work distribu-
tions common in previous studies of fluctuation theorems; see,
for example, Fig.
2
(inset) for the work distribution generated
by a thermodynamically driven simple harmonic oscillator
translated in space or Fig.
2
in Ref. [
13
]. The total average
work was 0
.
634
k
B
T
, satisfying Landauer’s bound by being
greater than the informational-state Shannon entropy decrease
of 0
.
769
×
ln 2
k
B
T
=
0
.
533
k
B
T
.
We can understand the mechanisms behind this structure
when decomposing Fig.
2
(rear)’s total work distribution
under the untilt substage trajectory classes
C
4
. We label tra-
jectories that spend all of the untilting substage in the
R
state
as
Success
since, via the previous substages, they reach the
intended
R
state by the untilting substage and remain there
until the protocol’s end. Similarly, trajectories that spend all of
the untilt substage in the
L
state are labeled
Fai l
.Theremain-
ing trajectories are labeled
Transitional
, since they transition
between the two informational states during untilt, potentially
succeeding or failing to end in the
R
state.
Figure
2
’s three front plots show the work distribution for
each of these three trajectory classes. Together they recover
033115-3
GREGORY WIMSATT
et al.
PHYSICAL REVIEW RESEARCH
3
, 033115 (2021)
the total work distribution over all trajectories shown in Fig.
2
(rear). However, now the thermodynamic contributions to the
total from the functionally distinct component trajectories are
made apparent.
IV. TRAJECTORY-CLASS FLUCTUATION THEOREM
Exploring the mesoscale dynamics of erasure revealed sig-
natures of a “thermodynamics” for each trajectory that is
uniquely associated with successful or failed information pro-
cessing. We now introduce the underlying fluctuation theory
from which the trajectory thermodynamics follow. Key to this
is comparing system behaviors in both forward and reverse
time [
9
15
]. (See Appendixes
C
and
F
.)
This suite of trajectory-class fluctuation theorems (TCFTs)
applies to arbitrary classes of system microstate trajectories
obtainable during a thermodynamic transformation. Impor-
tantly, they interpolate between Jarzynki’s equality [
11
] and
Crooks’ detailed fluctuation theorem [
13
] as the trajectory
class varies from the entire ensemble of trajectories to a single
particular trajectory, respectively. Accordingly, they unify a
wide range of other previously established fluctuation theo-
rems. (See Appendix
C
.)
One TCFT presents a lower bound on the average work

W

C
over any measurable subset
C
of the ensemble of system
microstate trajectories
Z
, where
W
is the total work for a
trajectory:

W

C


F
+
k
B
T
ln
P
(
C
)
R
(
C
R
)
=
W

min
C
,
(2)
with

F
the change in equilibrium free energy over the pro-
tocol,
P
(
C
) the probability of realizing the class
C
during
the protocol, and
R
(
C
R
) the probability of obtaining the time
reverse of class
C
under the time-reverse protocol. As detailed
in Appendix
G
, this allows accurate estimation of the work
generated for trajectory classes with narrow work distribu-
tions, such as the Success and Fail classes of erasure, even
with limited knowledge (low sampling) of system response
under the protocol and its time reverse.
The TCFTs lead to additional consequences. First, they
more strongly bound the average work over all trajectories
compared to the equilibrium free-energy change

F
. Second,
they provide a new expression for obtaining equilibrium free-
energy changes:

F
=−
k
B
T
ln
(
P
(
C
)
R
(
C
R
)

e
W
/
k
B
T

C
)
.
(3)
Remarkably, this only requires statistics for any particular
class
C
and its reverse
C
R
to produce the system’s free-energy
change. Since rare microstate trajectories may generate suffi-
ciently negative works that dominate the average exponential
work, this leads to a substantial statistical advantage over
direct use of Jaryznski’s equality

F
=−
k
B
T
ln

e
W
/
k
B
T

Z
for estimating free energies [
29
]. (See Appendix
C
.)
The erasure protocol’s 96
.
2% success rate is reflected in
the Success class’s dominance in the work distributions of
Fig.
2
. We can adjust the protocol to exhibit an arbitrarily
higher success rate while still maintaining high efficiency, i.e.,
keeping the total average work close to Landauer’s bound.
When done, we still observe the same qualitative features
of the work distribution: two peaks separated by a log-linear
ramp, each associated with the Success, Fail, and Transitional
classes. Though, of course, the probabilities of the Fail and
Transitional classes become arbitrarily small.
That said, the contributions of the Fail and Transitional
classes to various fluctuation theorems—such as, Jarzynski’s
equality to mention one—remain significant since the work
generated by those classes compensates by becoming increas-
ingly negative. In fact, the contribution to the exponential
average work of the Success class only approaches the value
1
/
2 out of the required value of 1 when averaging over all
trajectories. Thus, while the probabilities of the Fail and Tran-
sitional classes can become arbitrarily small by considering
Erasure protocols with higher success rates, we cannot ignore
the existence of the rare events due to Transitional and Fail tra-
jectories unless we employ particular fluctuation theorems, in
particular, a TCFT. (Again, see Appendix
C
.) In this way, one
sees that the TCFT provides a detailed diagnosis of successful
and failed information processing and of the associated ener-
getics.
V. REALIZING THERMODYNAMIC COMPUTING
To explore these predictions, we selected a supercon-
ducting flux qubit composed of paired Josephson junctions
[Fig.
3(a)
], resulting in a double-well nonlinear potential that
supports information storage and processing [Fig.
3(b)
]. Ap-
pendix
J1
explains the physics underlying their nonlinear
equations of motion, comparing the similarities and differ-
ences with our model’s idealized Langevin dynamics.
Despite control protocols for double-well potentials that
perform accurate and efficient bit erasure [
30
], we run the
flux qubit in a mode that yields imperfect erasure [Fig.
3(c)
].
As with the simulations, our intention is to illustrate how
trajectory classes and the TCFT can be used to diagnose and
interpret success and failure in microscopic information pro-
cessing using only mesoscopic measurements of work, which
is done more clearly by increasing the probabilities of rare
events.
Interplay between the linear geometric magnetic and
nonlinear Josephson inductances gives rise to a potential land-
scape that can be controlled with external bias fluxes. It is
natural to call the
φ
x
and
φ
xdc
fluxes, threading the differential
mode and the small SQUID loop, respectively, the
tilt
and
barrier controls
. [See the caption to Fig.
3(a)
.] Appendix
J
presents a derivation of the flux qubit potential and details
its calibration. All experiments presented here were carried
out at a temperature of 500 mK so that the effect of quantum
fluctuations is negligible.
To execute an erasure protocol, we first choose an
information-storage state with a tall barrier and two equal-
depth wells. The two-dimensional potential for this at the
calibrated device parameters is depicted in Fig.
3(b)
.Weim-
plement the bit erasure protocol as a time-domain deformation
imposed by the two control fluxes that starts and ends at
the storage configuration. In contrast to the simulation, the
flux qubit maintains two metastable regions and, hence, two
informational states
L
and
R
at all times, though they are
033115-4
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FIG. 3. Superconducting implementation of metastable memory and bit erasure driven by thermal fluctuations: (a) Optical micrograph of
a gradiometric flux qubit with control lines and local magnetometers for state readout. The flux
φ
x
, threading the large U-shaped differential-
mode loop, controls the potential’s tilt, and the flux
φ
xdc
, threading the small SQUID loop, controls the potential barrier height. Currents in the
barrier control
and
tilt control
lines modulate those fluxes. (b) Calculated potential energy landscape at the beginning of the erasure protocol;
see Eqs. (
J1
)and(
J2
). (c, top) Sequence of tilt and barrier control waveforms implementing bit erasure, and (c, bottom) sample of resulting
magnetometer traces tracking the system’s internal state. Note that for this experiment, in contrast to the simulation, two possible informational
states were present at all times during the protocol. So, though the barrier was reduced sufficiently to allow transitions to the target
R
state,
the trace is attracted to either a positive or negative value at all times. (d) Work distributions Pr(
W
total
|
C
4
) over trajectories conditioning on the
Success, Fail, and Transitional classes. Experimental distributions obtained from 10
5
protocol repetitions.
shallow enough to allow transitions as the barrier drops. The
amplitudes of the control waveforms in reduced units are
small; see Fig.
3(c)
. Due to this, the microscopic energetics
change linearly as a function of the control fluxes.
We use a local dc-SQUID magnetometer to continuously
monitor the trapped flux state in the device—
Readout
1in
Fig.
3(a)
. The digitized signal has a rise time of 100
μ
s,
after which the two logical states are discriminated virtually
without error. A typical magnetometer trace
V
(
t
) acquired
during the execution of the erasure protocol is shown in
Fig.
3(c)
. We operate the magnetometer with a low-amplitude
AC current bias at 10 MHz to avoid an increase in the effective
temperature during continuous readout of the flux state due to
wide-band electromagnetic interference.
To collect work statistics, we repeat the erasure protocol
10
5
times. We identify the logical-state transitions from the
magnetometer traces as zero-crossings, recording the direc-
tion
δ
i
—sign convention:
+
1(
1) for an
L
-to-
R
(
R
-to-
L
)
transition—and the time
t
i
relative to the start of the protocol.
We evaluate a single-shot work estimate
W
=
i
δ
i
U
LR
(
t
i
),
where
U
LR
(
t
)
=
U
R
(
t
)
U
L
(
t
) is the biasing of the poten-
tial minima at time
t
i
. Making use of the linearity of the
system energetics and the choice of offsets and compensa-
tion coefficients, we find
U
LR
(
t
)
=
A
[
φ
x
(
t
)
φ
x
(0)], with the
coefficient
A
=
210 K
×
k
B
evaluated from the calibrated po-
tential. The above work estimate based on the logical-state
transitions is an accurate estimate of the true microscopic
work assuming that the timescales for the state transitions and
for changes in the control parameters are much slower than
the intrawell equilibration. (See Appendix
H
.)
The total work distribution estimated from the flux qubit
experiments is shown as the rearmost distribution in Fig.
3(d)
.
Using the previous microstate trajectory partitioning into
the
Success, Fail,
and
Transitional
trajectory classes reveals
a decomposition of the total work distribution given by
Fig.
3(d)
(three front panels). The close similarity with our
simulations (Fig.
2
) is notable, especially given the rather sub-
stantial differences between the simulated system (idealized
double-well potential, thermal noise, exactly one-dimensional
system, etc.) and the experimental system (complex potential
in two dimensions, nonideal fluctuations, etc.).
Apriori
it
is not clear that the TCFT predictions should apply so di-
rectly and immediately to the real-world qubit, that is until
one recalls that trajectory-class membership is a topological
033115-5
GREGORY WIMSATT
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PHYSICAL REVIEW RESEARCH
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property and that trajectories carry their probabilities and so
their thermodynamics.
In point of fact, these differences serve to emphasize the
descriptive power and robustness of the mesoscopic-work
TCFT: Despite substantial differences in system detail, they
successfully diagnose the information-processing classes of
microscopic trajectories.
Indeed, looking to thermodynamic transformations beyond
bit erasure, the essential requirement of our analysis is for the
protocol to be slow enough compared to the timescales of os-
cillations due to the potential and of the thermal fluctuations.
With this, the system is always near metastable equilibrium—
what we call
metastable quasistationarity
. This ensures that
in an amount of time, small on the timescale of the protocol,
the system visits every point in the potential in proportion
to the metastable distribution for the metastable region it
occupies. Since the work rate for a particle is determined by
the rate of change in potential at its location, the work rate
must be that of the metastable distribution’s. We then need
only describe which metastable region a particle is in as a
function of time to characterize its total work. The specific
shape and dimensionality of these metastable regions are then
insignificant for determining the shape and qualitative features
of the total work distribution. Under these conditions, there is
substantial robustness. This will be especially helpful when
using the TCFT to monitor thermodynamic computing in bi-
ological systems where, in many cases, information-bearing
degrees-of-freedom cannot be precisely modeled.
VI. CONCLUSION
We experimentally demonstrated that work fluctuations
generated by information engines are highly structured.
Nonetheless, they strictly obeyed a suite of time-reversal
symmetries—the trajectory-class fluctuation theorems intro-
duced here. The latter are direct signatures of how a system’s
informational states evolve, and they identify functional and
nonfunctional microscopic trajectory bundles. We showed
that the trajectory-class fluctuation theorems naturally in-
terpolate between Jarzynski’s integral and Crooks’ detailed
fluctuation theorems, providing a unified diagnostic probe of
nonequilibrium thermodynamic transformations that support
information processing.
The trajectory-class fluctuation theorems gave a detailed
thermodynamic analysis of the now-common example of eras-
ing a bit of information as an external protocol manipulated
a stochastic particle in a double-well potential (simulation)
and the stochastic state of a flux qubit (experiment). To give
insight into the new level of mechanistic analysis possible,
we briefly discussed the untilt trajectory-class partitioning.
Though ignoring other protocol stages, this was sufficient to
capture the basic trajectory classes that generate the overall
work distribution’s features. Partitioning on informational-
state occupation times during barrier raising and untilting—an
alternative used in follow-on studies—yields an even more
incisive decomposition of the work distributions and diagnosis
of informational functioning. Practically, the corresponding
bounds on thermodynamic resources obtained via the TCFT
also improve on current estimation methods. The net result is
that trajectory-class fluctuation analysis can be readily applied
to debug thermodynamic computing by engineered or biolog-
ical systems.
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
ACKNOWLEDGMENTS
We thank C. Jarzynski, D. Mandal, and P. Riechers for
helpful discussions. As an External Faculty member, J.P.C.
thanks the Santa Fe Institute and the Telluride Science Re-
search Center for their hospitality during visits. This material
is based upon work supported by, or in part by, the US Army
Research Laboratory and the US Army Research Office under
Contracts No. W911NF-13-1-0390 No. W911NF-18-1-0028
and No. W911NF-21-1-0048.
J.P.C., O.-P.S., M.L.R., and G.W. conceived of the project.
A.B.B., G.W., and J.P.C. developed the theory. S.H. provided
the flux qubit and the experimental and analytical methods
for its calibration. M.H.M., M.L.R., and O.-P.S. designed,
implemented, and carried out the experiments. G.W. and
J.P.C. performed the calculations. J.P.C., O.-P.S., M.L.R., and
G.W. drafted the manuscript. J.P.C. and M.L.R. supervised the
project.
The authors declare that they have no competing financial
interests.
APPENDIX A: PRINCIPLES OF THERMODYNAMIC
COMPUTING: A RECENT SYNOPSIS
A number of closely related thermodynamic costs of
computing have been identified, above and beyond the
house-
keeping heat
that maintains a system’s overall nonequilibrium
dynamical state. First, there is the
information-processing
Second Law
[
31
] that extends Landauer’s original bound on
erasure [
2
] to dissipation in general computing and properly
highlights the central role of information generation measured
via the physical substrate’s dynamical Kolmogorov-Sinai en-
tropy. It specifies the minimum amount of energy that must be
supplied to drive a given amount of computation forward. Sec-
ond, when coupling thermodynamic systems together, even a
single system and a complex environment, there are transient
costs as the system synchronizes to, predicts, and then adapts
to errors in its environment [
32
34
]. Third, the very mod-
ularity of a system’s organization imposes thermodynamic
costs [
35
]. Fourth, since computing is necessarily far out of
equilibrium and nonsteady state, there are costs due to driv-
ing transitions between information-storage states [
36
]. Fifth,
there are costs to generating randomness [
37
], which is itself
a widely useful resource. Finally, by way of harnessing these
principles, new strategies for optimally controlling nonequi-
librium transformations have been introduced [
30
,
38
40
].
APPENDIX B: MICROSCOPIC STOCHASTIC
THERMODYNAMICAL SYSTEM
For concreteness, we concentrate on a one-dimensional
system: a particle with position and momentum in an external
potential
V
(
x
,
t
) and in contact with a heat reservoir at temper-
ature
T
. An external controller adds or removes energy from
a work reservoir to change the form of the potential
V
(
·
,
t
)via
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a predetermined
erasure protocol
{
(
β
(
t
)
(
t
)
)
:0

t

τ
}
.
(See Appendix
H
for details on the alternative definitions of
work.) The potential takes the form
V
(
x
,
t
)
=
ax
4
b
0
β
(
t
)
x
2
c
0
δ
(
t
)
x
,
with constants
a
,
b
0
,
c
0
>
0. During the erasure protocol,
β
(
t
)
and
δ
(
t
) change one at a time piecewise-linearly through four
protocol substages: (i)
drop barrier
, (ii)
tilt
, (iii)
raise barrier
,
and (iv)
untilt
, as shown in Table
I
. The system starts at time
t
=
0 in the equilibrium distribution for a double-well
V
(
x
,
0)
at temperature
T
. Being equiprobable, the informational states
associated with each of the two wells thus contain 1 bit of
information [
28
]. The effect of the control protocol on the
system potential and system response is graphically displayed
in Fig.
1
.
We model the erasure physical information processing with
underdamped
Langevin dynamics:
dx
=
v
dt
,
md
v
=
2
k
B
T
γ
r
(
t
)
dt
(
x
V
(
x
,
t
)
+
γ
v
)
dt
,
where
k
B
is Boltzmann’s constant,
γ
is the coupling between
the heat reservoir and the system,
m
is the particle’s mass, and
r
(
t
) is a memoryless Gaussian random variable with

r
(
t
)
=
0 and

r
(
t
)
r
(
t

)
=
δ
(
t
t

).
For comparison to experiment, we simulated erasure
with the following parameters, sufficient to fully specify
the dynamics:
γτ/
m
=
500, 2
k
B
T
τ
2
a
/
(
mb
0
)
=
2
.
5
×
10
5
,
b
2
0
/
(4
ak
B
T
)
=
7, and
8
a
/
b
3
0
c
0
=
0
.
4. The resulting poten-
tial, snapshot at times during the erasure substages, is shown
in Fig.
1
(inner plot sequence).
Reliable information processing dictates that we set
timescales so that the system temporarily, but stably, stores
information. To support metastable-quasistatic behavior at all
times, the relaxation rates of the informational states are much
faster than the rate of change of the potential, keeping the
system near metastable equilibrium throughout. The entropy
production for such protocols tends to be minimized.
APPENDIX C: TRAJECTORY-CLASS FLUCTUATION
THEOREM AND INTERPRETATION
Here, we describe the trajectory-class fluctuation theorems,
explaining several of their possible implications and exploring
their application to both the simulations and flux qubit exper-
iment. Their derivations are given in Appendix
F
.
First, we treat each system trajectory
z
as a function from
time between 0 and
τ
(the time interval of a control protocol)
to the set of possible system microstates. Then consider a for-
ward process distribution
P
, defined by the probabilities of the
system microstate trajectories
Z
due to an initial equilibrium
microstate distribution evolving forward in time under the
control protocol. Then, the reverse process distribution
R
is
determined by preparing the system in equilibrium in the final
protocol configuration and running the reverse protocol. The
reverse protocol is the original protocol conducted in reverse
order but also with objects that are odd under time reversal,
like magnetic fields, negated. The time-reversal of a trajectory
is
z
R
(
t
)
=
[
z
(
τ
t
)]
R
, where for a microstate
z
the time-
reverse
z
R
is simply
z
but with time-odd components (e.g.,
momentum or spin) negated. In other words, time-reversing a
trajectory runs the trajectory backwards while also negating
all time-odd components of the microstates. For a measurable
subset of trajectories
C
Z
, called a
trajectory class
,letthe
class average
·
C
denote an average over the ensemble of for-
ward process trajectories conditioned on the trajectories being
in the class
C
.Let
P
(
C
) and
R
(
C
R
) denote the probabilities
of observing the class
C
in the forward process and the reverse
class
C
R
={
z
R
|
z
C
}
in the reverse process, respectively.
We first introduce a
trajectory-class fluctuation theorem
(TCFT) for the
class-averaged exponential work

e
W
/
k
B
T

C
:

e
W
/
k
B
T

C
=
R
(
C
R
)
P
(
C
)
e

F
/
k
B
T
,
(C1)
with

F
the system equilibrium free-energy change. We also
introduce a
class-averaged work
TCFT:

W

C
=

F
+
k
B
T
(
D
KL
[
P
(
Z
|
C
)
||
R
(
Z
R
|
C
R
)]
+
ln
P
(
C
)
R
(
C
R
)
)
.
(C2)
This employs the Kullback-Liebler divergence
D
KL
[
·
] taken
between forward and reverse process distributions over all
class trajectories
z
C
, conditioned on the forward class
C
and reverse class
C
R
, respectively. If we disregard this diver-
gence, which is non-negative and can generally be difficult to
obtain experimentally, we then find the lower bound

W

min
C
on the class-averaged work of Eq. (
2
).
If we choose the class
C
to consist of only a single trajec-
tory, we recover detailed fluctuation theorems. For example,
Eq. (
C1
) then simplifies to Crooks’ detailed fluctuation theo-
rem [
13
]:
e
W
(
z
)
/
k
B
T
=
R
(
z
R
)
P
(
z
)
e

F
/
k
B
T
.
(C3)
If, however, we take
C
to be the entire set of trajectories
Z
,
we recover integral fluctuation theorems. In this case, Eq. (
C1
)
simplifies to Jarzynski’s equality [
11
]:

e
W
/
k
B
T

Z
=
e

F
/
k
B
T
,
(C4)
exploiting the fact that
Z
R
=
Z
and
P
(
Z
)
=
R
(
Z
)
=
1.
Furthermore, many other fluctuation theorems can be seen
as special cases of the TCFT. In particular, Eq. (9) of Ref. [
41
]
is closely related to Eq. (
C1
). Having a nearly identical form,
the former is a special case of the latter in that the correspond-
ing classes consist of trajectories defined by restrictions on
the visited microstates up to only a finite number of times.
Similarly, Eqs. (6) and (7) of Ref. [
42
] derive from Eqs. (
C1
)
and (
2
) by considering a system with negligible contact with
the thermal bath during the protocol and coarse-graining on
features of the visited microstate at a single time. Along the
same lines, Eqs. (7) and (8) of Ref. [
20
] are obtained from
Eq. (
C1
) by considering a bit erasure process and trajectory
classes corresponding to ending in the target well and in the
opposite well, respectively. And, letting the trajectory class be
all trajectories that yield a particular value of obtained work
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during the forward process, Eq. (
C1
) reduces to Crooks’ work
fluctuation theorem [
13
]:
P
(
W
)
R
(
W
)
=
e
(
W

F
)
/
k
B
T
,
(C5)
where
P
(
W
) and
R
(
W
) are the probabilities of obtaining
values
W
and
W
for the work when running the forward
and reverse process, respectively. Finally, yet other fluctuation
theorems arise directly from the TCFT by particular class
choices [
16
,
27
,
43
,
44
].
In this way, one sees that the TCFT is a suite that spans the
space of fluctuation theorems between the extreme of the de-
tailed theorems, which require very precise information about
an individual trajectory, and the integral theorems, which de-
scribe the system’s entire trajectory ensemble. It thus unifies
a wide range of existing (and future) fluctuation theorems.
Appendix
F
below provides proofs.
APPENDIX D: EMPIRICAL USE IN STATISTICAL
ESTIMATION
Beyond the synthesis of distinct fluctuation theorems, the
TCFT is empirically useful in greatly improving sampling and
errors in statistical estimation. And, this is its primary role
here—a diagnostic tool for thermodynamic computing. We
can rearrange Eq. (
C1
) to obtain Eq. (
3
)—an expression for
estimating equilibrium free-energy changes:

F
=−
k
B
T
ln
(
P
(
C
)
R
(
C
R
)

e
W
/
k
B
T

C
)
.
(D1)
Thus, to estimate free energy one sees that statistics are
needed for only one particular class and its reverse. Generally,
this gives a substantial statistical advantage over direct use of
Jaryznski’s equality:

F
=−
k
B
T
ln

e
W
/
k
B
T

Z
,
since rare microstate trajectories may generate negative work
values that dominate the average exponential work [
29
]. The
problem is clear in the case of erasure. Recall from Fig.
2
(three front panels) that Fail trajectories generate the most
negative work values. In the limit of higher success-rate pro-
tocols that maintain low entropy production, failures generate
more and more negative works, leading them to dominate
when estimating average exponential works despite becoming
negligible in probability.
In contrast, to efficiently determine the change in equi-
librium free energy from Eq. (
3
), its form indicates that one
should choose a class that (i) is common in the forward pro-
cess, (ii) has a reverse class that is common in the reverse
process, and (iii) generates a narrow work distribution. This
maximizes the accuracy of statistical estimates for the three
factors on the right-hand side. For example, while the equilib-
rium free-energy change in the case of our erasure protocol is
theoretically simple (zero), the Success class fits the criteria.
We can then monitor the class-averaged work in excess of
its bound:
E
C
=
W

C
−
W

min
C
=
k
B
TD
KL
[
P
(
Z
|
C
)
||
R
(
Z
R
|
C
R
)
]

0
.
FIG. 4. Flux qubit experiment work fluctuations: Crooks’ work
fluctuation theorem prediction (green dashed line), measured values
(blue), and 1-
σ
statistical errors (red).
The inequality in Eq. (
2
) is a refinement of the equilib-
rium Second Law and therefore the bound

W

min
C
generally
provides a more accurate estimate of the average work of
trajectories in a class compared to the equilibrium free-energy
change

F
. More precisely, as we will see below, an average
of the excess
E
C
over all classes
C
in a partition of trajectories
must be smaller than the dissipated work

W
−

F
.For
trajectory classes with narrow work distributions, this can be a
significant improvement. We can see this by Taylor expanding
the left-hand side of Eq. (
C1
) about the mean dimensionless
work

W
/
k
B
T

C
. This shows that Eq. (
2
) becomes an equality
when the variance and higher moments vanish. Appendix
G
below delves more into moment approximations. In any case,
trajectory classes with narrow work distributions have small
excess works
E
C
.
APPENDIX E: FLUCTUATIONS IN THERMODYNAMIC
COMPUTING: THE CASE OF ERASURE
Before applying the TCFT to analyze thermodynamic fluc-
tuations during erasure, we first explore both Jarzynski’s
equality Eq. (
C4
) and Crooks’ work fluctuation theorem
Eq. (
C5
).
Since the erasure protocol is cyclic, the change in equi-
librium free energy

F
vanishes. Jarzynski’s equality then
predicts that the average exponential work

e
W
/
k
B
T

Z
must
be 1. From simulation, we obtain a value of 1
.
0025
±
5
×
10
5
, which is very close to the predicted value. From ex-
periment, we obtain a value of 0
.
89
±
5
×
10
5
—within 10%
of the prediction, but it falls somewhat outside the expected
error.
Crooks’ work fluctuation theorem predicts that the quantity
ln
(
P
(
W
)
/
R
(
W
)
)
must equal
β
W
at each
W
. We verify this
experimentally by building probability histograms for
P
(
W
)
and
R
(
W
), taking their log ratios, and plotting against
their binned work values expressed in units of
k
B
T
. Figure
4
shows that the experiment follows the theoretical prediction
quite closely. However, as for the case of Jarzynski’s equality,
the experimental results are not all within expected errors.
The discrepancies appear to arise in the statistical errors in
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