The Thermodynamic Uncertainty Theorem Kyle J. Ray1Alexander B. Boyd2 3 4yGiacomo Guarnieri5zand James P. Crutcheld6x 1Complexity Sciences Center and Physics Department

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The Thermodynamic Uncertainty Theorem

Kyle J. Ray,1, ∗Alexander B. Boyd,2, 3, 4, †Giacomo Guarnieri,5, ‡and James P. Crutchﬁeld6, §

1Complexity Sciences Center and Physics Department,

University of California at Davis, One Shields Avenue, Davis, CA 95616, USA

2Corresponding author

3The Division of Physics, Mathematics and Astronomy,

California Institute of Technology, Pasadena, CA 91125, USA

4School of Physics, Trinity College Dublin, College Green, Dublin 2, D02 PN40, Ireland

5Dahlem Center for Complex Quantum Systems,

Freie Universit¨at Berlin, 14195 Berlin, Germany

6Complexity Sciences Center and Physics Department,

University of California at Davis, One Shields Avenue, Davis, CA 95616

Thermodynamic uncertainty relations (TURs) express a fundamental tradeoﬀ between the preci-

sion (inverse scaled variance) of any thermodynamic current by functionals of the average entropy

production. Relying on purely variational arguments, we signiﬁcantly extend these inequalities by

incorporating and analyzing the impact of higher statistical cumulants of entropy production within

a general framework of time-symmetrically controlled computation. This allows us to derive an ex-

act expression for the current that achieves the minimum scaled variance, for which the TUR bound

tightens to an equality that we name Thermodynamic Uncertainty Theorem (TUT). Importantly,

both the minimum scaled variance current and the TUT are functionals of the stochastic entropy

production, thus retaining the impact of its higher moments. In particular, our results show that,

beyond the average, the entropy production distribution’s higher moments have a signiﬁcant eﬀect

on any current’s precision. This is made explicit via a thorough numerical analysis of swap and

reset computations that quantitatively compares the TUT against previous generalized TURs. Our

results demonstrate how to interpolate between previously-established bounds and how to identify

the most relevant TUR bounds in diﬀerent nonequilibrium regimes.

Keywords: thermodynamic uncertainty relation, entropy production, nonequilibrium steady state, current

Introduction. The past few decades witnessed sub-

stantial technological advances in miniaturization that,

today, have culminated in experimental realizations of

nanoscale thermal machines [1–8]. These devices ex-

hibit three fundamental features. First, they operate un-

der nonequilibrium conditions, i.e., either by keeping the

system in a nonequilibrium steady-state by means of volt-

age or temperature biases or through the application of

an external time-dependent control protocol. This im-

plies that a certain amount of entropy production Σ—

which quantiﬁes the amount of irreversible dissipation

associated with nonequilibrium processes—is always gen-

erated [9]. Crucially, the entropy production limits heat

engine and refrigerator performance, constrains the phys-

ical mechanisms underlying complex biological function-

ing [10], and is the central quantity in Landauer’s infor-

mation erasure—the keystone of the bridge between ther-

modynamics and communication theory. Second, due to

their microscopic nature, the ﬂuctuations of all thermo-

dynamic quantities (such as heat, work, and so on) be-

come as signiﬁcant as their average values. Last, but

not least, the laws of quantum mechanics have impor-

tant repercussions for ﬂuctuations, which generally have

both thermal and quantal origins [11–14].

Since Onsager’s and Kubo’s pioneering discovery of the

ﬂuctuation-dissipation theorem (FDT) [15–17], deter-

mining the universal properties of ﬂuctuations in out-

of-equilibrium processes, as well as their role in dissipa-

tion, has been a cornerstone of stochastic thermodynam-

ics. In the ‘90s, Jarzynski and Crooks generalized the

FDT through the ﬂuctuation relations (FRs) [18–29]. At

the microscopic scale, the FRs reﬁne the famous Second

Law of Thermodynamics hΣi ≥ 0 by determining the

full distribution of stochastic thermodynamic quantities

and thus their ﬂuctuations. That is, the FRs replaced

the familiar Second law inequality with an equality from

which the Second Law is easily derived through Jensen’s

inequality.

More recently, a third milestone was crossed by con-

necting thermodynamic ﬂuctuations out of equilibrium

to dissipation. These broad results, called thermo-

dynamic uncertainty relations (TURs), were originally

discovered in nonequilibrium steady-states of classical

time-homogeoneous Markov jump-processes satisfying lo-

cal detailed balance [30, 31]. Today, though, TURs

have been generalized to ﬁnite-time processes [32–34],

periodically-driven systems [35–41], Markovian quantum

systems undergoing Lindblad dynamics [42–44], and au-

tonomous classical [33, 45] and quantum [45–50] systems

in steady-states close to linear response.

arXiv:2210.00914v3 [cond-mat.stat-mech] 25 Nov 2022

In all these, TURs bound the ﬂuctuations of

any (time-reversal anti-symmetric stochastic) thermody-

namic quantity Jas a function of the average entropy

production hΣi:

2

J≡var(J)

hJi2≥f(hΣi).(1)

with hJiand var(J) = hJ2i − hJi2being the average

and variance of J, respectively. In this way, the scaled

variance 2

Jcan be seen as the inverse of current J’s

signal-to-noise ratio or precision. Since fis generally

a monotonically-decreasing function, TURs express the

trade-oﬀ that increased precision in Jinevitably comes

at the cost of more dissipation. Such a no-free-lunch

statement echoes that from the above-mentioned Second

Law. It diﬀers critically, however, as it includes ﬂuctua-

tions of many thermodynamic quantities of interest.

Our work signiﬁcantly advances this line of inquiry

by analyzing the impact of higher statistical moments

of the entropy production on the signal-to-noise ratio of

thermodynamic currents Jin two state-of-the-art sce-

narios; namely, bit swap and reset protocols. We quan-

tify the impact of such higher moments onto the r.h.s.

of Eq. (1)—i.e. f(hΣi)—leading to replacing the latter

with:

2

J≥g(hfmin(Σ)i),(2)

where g(x)≡x−1−1. With this, the bound becomes

a functional of the stochastic entropy production Σ dis-

tribution. And so, critically, it accounts for all higher

moments.

Notably, these higher moments have become a focus of

attention [51] since they are particularly germane to dis-

sipation management in nanoscale devices—devices that

must be designed to tolerate large and potentially de-

structive ﬂuctuations. For example, entropy production

variance and skewness determine the probability of ex-

periencing a trajectory (i.e., an experimental run) that

generates extreme dissipated heat ﬂowing through the

system. This, naturally, can damage or disrupt the oper-

ation of these new classes of microscopic quantum hard-

ware [52].

In both examples, we compute Eq. (2), and quantita-

tively compare it against several diﬀerent previously de-

rived TUR bounds. In particular, it is important to stress

that g(hfmin(Σ)i) appearing in our Eq. (2) agrees and

coincides with a recent result obtained in Ref. [53], al-

beit via a completely diﬀerent derivation. Our approach,

moreover, focuses on realizing the bound by also ﬁnding

an explicit expression for the minimum-variance current

Jmin(~s) that saturates Eq. (2).

This minimum depends sensitively on the entropy pro-

duction’s higher-order ﬂuctuations. In much the same

way that ﬂuctuation theorems [19, 54, 55] reframe the

Second Law from an inequality to an equality, the TUT

replaces the bounds set by TUR with a saturable equal-

ity. Applying the TUT to thermodynamic simulations of

fundamental bit swap and reset computations, we demon-

strate that current ﬂuctuations can depart substantially

from previous bounds set by TURs.

Minimum Scaled Variance Current. Currents Jare

observations of a system Sthat ﬂip sign under time rever-

sal: When a movie of a ﬂowing river is reversed, the pos-

itive currents become negative and visa versa. Formally,

a system trajectory is the sequence ~s ≡s0sdt ···sτ−dtsτ,

where each st∈ S is the system’s state at time t. The

current associated with a reversed trajectory R(~s)≡

s†

τs†

τ−dt ···s†

dts†

0is minus the current of the forward tra-

jectory:

J(R(~s)) = −J(~s).(3)

The system Smay be inﬂuenced by an external con-

trol parameter λtat every time t, thereby performing

a computation over the time interval t∈(0, τ). Under

time-symmetric control λ†

τ−t=λtand conjugation of

the distribution under the operation Pr(s, τ) = Pr(s†,0)

the probability of a reverse trajectory is exponentially

damped by the entropy production [56]:

Pr(R(~s),−Σ) = e−ΣPr(~s, Σ).(4)

This is the Detailed Fluctuation Theorem (DFT) for a

Time-Symmetrically Controlled Computation (TSCC).

(Reference [57] details its derivation and scope.) This

DFT [54, 55, 58] includes NESS dynamics for which

the control parameter is constant λt=λt0. It can also

describe, as explored here, computations that begin in

equilibrium and are then allowed to relax after the appli-

cation of a time-symmetric control signal. These latter

symmetries are ubiquitous in computing [59].

The symmetry imposed by the TSCC imbues J’s

statistics with special properties in stochastic nonequi-

librium systems when compared with the entropy pro-

duction Σ. To address this, we derive the exact form of

the current Jmin that achieves minimum scaled variance

for any TSCC.

Given a TSCC operating over the time interval [0, τ],

described by probability distribution Pr(~s, Σ) over state

trajectories ~s and entropy productions Σ, our task is to

ﬁnd a current function J(~s) = −J(R(~s)) of the state

trajectories ~s that minimizes this scaled variance. The

proof that this can be done is found in Ref. [57]. We

provide a short overview here.

The demonstration develops over two steps. First, we

identify a special class of entropy-conditioned currents,

that can be expressed as functions of the entropy pro-

duction, containing a current with the minimum scaled

variance. Then, we use the TSCC detailed ﬂuctuation

theorem Eq. (4) and the calculus of variations to de-

rive the primary result—the exact form of this minimum

current:

Jmin(~s) = hJ2

mini

hJminitanh (Σ(~s)/2) .(5)

While Eq. (5) may appear self-referential, it is worth

noting that any real constant may be chosen for the ra-

tio hJ2

mini

hJmini. The resulting current will achieve the min-

imum variance as dictated by the following Thermody-

namic Uncertainty Theorem.

Theorem 1 (Thermodynamic Uncertainty Theorem).

In a TSCC, the scaled variance of any current Jis

bounded below by the the scaled variance of Jmin , given

by:

2

Jmin =1

htanh(Σ/2)i−1.(6)

Since this minimum is achievable by a well-deﬁned cur-

rent Jmin, it sets the tightest possible bound that can

be determined using all moments of the entropy produc-

tion distribution. We note that this expression contrasts

with other past TURs [56, 60–62] in that it is an equality

rather than an inequality. It speciﬁes the achievable min-

imum scaled variance since it depends on the underlying

physical system’s entropy production distribution Pr(Σ).

Equation (6) agrees and coincides with Eq. (16) of

Ref. [53], which was derived through a completely diﬀer-

ent method; namely, by means of a “Hilbert uncertainty

relation” that leverages the reproducing element implied

by the Riesz representation theorem for any Hilbert space

equipped with an appropriate inner product. This uncer-

tainty relation was shown to reduce to the an equation

of the form of Eq. (2). Our approach is complementary

as it investigates how the relation stems from thermody-

namics through a detailed ﬂuctuation theorem (DFT).

While reaching the same conclusion, our method has two

additional merits. First, it clearly highlights its relation

to other previously generalized TUR bounds, also ob-

tained from the requirement of a DFT. Thus, it gives

a quantitative comparisons, as shown shortly. Second,

it determines a physically realizable minimum-variance

currents Jmin(~s) that saturates Eq. (6).

Comparison to Past Uncertainty Relations. With a

blossoming variety uncertainty relations [30–47], it is nat-

ural to ask for direct comparisons. Reference [57] pro-

vides a thorough summary of previous TUR results, but

here we summarize the relevant comparisons. Barato and

Seifert [60] derived the ﬁrst TUR, ﬁnding that precision

could not be maximized without a corresponding increase

in the average entropy production:

2

J≥2

hΣi≡2

BS ,(7)

where the subscripts on the latter refer to the original

authors—the Barato-Seifert (BS) bound on scaled vari-

ance.

Further exploration found that detailed ﬂuctuation

theorems [54, 55] can be used to establish modiﬁed ther-

modynamic uncertainty relations. Hasegawa and Van Vu

[56] as well as Timparano, Guarnieri, Goold, and Landi

[62] used Eq. (4) to demonstrate that the scaled variance

is bounded below by:

2

J≥2

ehΣi−1≡2

HVV (8)

2

J≥csch2[g(hΣi/2)] ≡2

TGGL,(9)

where g(x) is the inverse of xtanh(x) and we have again

labeled the bounds for the authors. The 2

TGGL bound is

the tightest possible bound on scaled variance that can

be determined from the average entropy [62]. Comparing

these bounds independent of the TUT, one can see in

Figs. 1 and 2 that they are ordered:

2

BS(hΣi)> 2

TGGL(hΣi)> 2

HVV(hΣi).(10)

(See also Ref. [57] for a proof.) Note that, since 2

TGGL

and 2

HVV were derived from the TSCC DFT, which is

our starting point as well, the minimum scaled variance

is bounded below by these TURS, but not necessarily

2

BS.

With 2

Jmin’s exact form determined, though, a natu-

ral next question is how close the previous bounds, all

depending only on the average entropy production hΣi,

are to the actual minimum.

Fortunately, Timpanaro et al. [62] also showed that a

particular bimodal distribution:

min(Σ) ∝δ(Σ −a) + e−aδ(Σ + a)

achieves their lower bound 2

TGGL. This is the simplest

Figure 1. Thermodynamic Uncertainty Theorem and TURs:

Three dashed lines show previous TURS—2

BS,2

HVV, and

2

TGGL—all functions of average entropy production hΣi.

While they make nearly identical predictions for small en-

tropy production, they diverge as entropy increases, setting

very diﬀerent bounds for average entropy production as low

as hΣi= 2kB. In contrast, the minimum scaled variance 2

Jmin

is not strictly a function of average entropy. The entropy dis-

tribution Pr(Σ|µ, σ2) depends on the variance parameter σ2

and is displayed on a sliding scale from high to low variance

(from dark to light). σ2ranges from ≈8×10−3to 8 ×103.

While µis adjusted to keep hΣiﬁxed. This yields entropy dis-

tributions of the form shown on the right. The lowest values

of Σ2and var(Σ) closely match 2

TGGL. As σ2increases, the

variance of the entropy production increases and the curve

become lighter, achieving and surpassing the dashed line for

2

BS. Between these two extremes, there is a purple dashed

line that gives the minimum scaled variance of normal en-

tropy distributions.

distribution satisfying Pr(−Σ) = e−ΣPr(Σ). That is, it

consists of a delta function at entropy production Σ = a

and then contains a mirror of that entropy production

at Σ = −areduced by the exponential factor e−a. Any

other NESS entropy distribution can be constructed from

a superposition of such distributions.

We investigate our new minimum scaled variance by

exploring a variety of possible distributions. We take a

similar strategy, breaking the entropy distribution into

the piecewise function:

Pr(Σ|µ, σ2) = n(µ, σ2)





e(Σ−µ)2

2σ2if Σ ≥0

eΣe(−Σ−µ)2

2σ2if Σ <0

.(11)

Here, n(µ, σ2) = R∞

0e(Σ−µ)2

2σ2dΣ + R0

−∞ eΣe(−Σ−µ)2

2σ2dΣ is

the normalization factor. In essence, our probability dis-

tribution is a normal distribution with average µand

variance σ2over the positive interval. And, the TSCC

DFT deﬁnes the distribution to be Pr(Σ) = eΣPr(−Σ)

on the negative interval.

The variance σ2and average µof the positive en-

tropy portion of the distribution Pr(Σ|µ, σ2) deﬁne it.

In the limit σ2kBµ, the positive entropy distribu-

tion is nearly a delta function, and we recover roughly

the distribution Prmin(Σ) proposed by Timpanaro et al

[62]. We show this on the left side of Fig. 1 where

σ2≈10−3kBµ, corresponding to a two-peaked distri-

bution. In this case, 2

Jmin closely matches the bound

2

TGGL, as expected. However, Fig. 1 also shows that the

average entropy production is not the sole determinant

of the minimum scaled variance of the current.

As the variance of the positive normal distribution σ2

increases, Fig. 1 shows that the minimum scaled vari-

ance increases. Amidst that progression is a special dis-

tribution, where σ2= 2kBµ, for which Pr(Σ|µ, σ2) is a

normal distribution over the full range of entropy pro-

duction. It can be quickly shown that the variance for a

normal distribution that satisﬁes the TSCC DFT must

be σ2= 2kBµ. We highlight this special case with a pur-

ple dashed line in Fig. 1 and a blue dashed line in Fig.

2. NESSs, the most frequently studied subclass of TSCC

processes, approach a normal entropy production distri-

bution in the long time limit. Interestingly, the minimal

variance currents of the typical asymptotic behavior in

NESSs clearly violate the original TUR given by 2

BS in

the long-time limit.

If we continue beyond the normal distribution to higher

variances labeled in lighter colors in Fig. 1, the mini-

mum scaled variance 2

Jmin continues to increase, until

it surpasses the bound 2

BS [60]. Thus, by changing the

parameter σ2of the NESS distribution Pr(Σ|µ, σ2) and

its variance var(Σ) as well as other higher moments of

the entropy distribution Pr(Σ), the TUT interpolates be-

tween the TUR set by Timpanaro et al. and that set by

Barato and Seifert. Moreover, we can ﬁnd entropy distri-

butions that far exceed even Barato and Seifert’s bound.

Thermodynamic Simulations. The entropy produc-

tion distribution Pr(Σ|µ, σ2) is convenient to examine,

but it is not obvious how it can be physically generated.

We now describe two computational protocols that are

able to show similar breadth of behavior, but are ﬁrmly

rooted in dynamical models of physical processes. Both

are TSCCs in that they are implemented with time sym-

metric control of a potential energy landscape, where the

thermal inﬂuence of a bath is applied through Langevin

dynamics.

First, consider a simple reset protocol. A system con-

sisting of a single positional variable xin asymmetric

double well potential Ustore is initially set up in equilib-

rium with a thermal environment at temperature T. If

swap

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reset

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Figure 2. Bounds 2

BS > 2

TGGL > 2

HVV in solid lines (blue, red, and black respectively) and speciﬁc thermal processes with

dashed lines: The blue dashed line is the minimum scaled variance 2

Gaussian of any process that generates a Gaussian entropy

production distribution, which is achieved in the long-time limit of NESS processes. The red dashed line is the minimum

scaled variance 2

Discrete of an ideal discrete erasure. We compare two computational classes to the these bounds. (Left) we

plot 1366 diﬀerent time-symmetric erasures. As expected (see Ref. [57] ) they are bounded by the scaled variance of the ideal

discrete erasure 2

Discrete, which lies well above the bounds 2

TGGL and 2

HVV. A number of erasure operations are well above the

minimum 2

TGGL set by Barato and Seifert. (Right) we plot the result 1193 diﬀerent bit-ﬂips. As with the erasure protocol,

many computations are above the Barato-Seifert bound. However, many computations achieve a minimum scaled variance well

below the discrete erasure bound 2

Discrete. Many computations are quite close to the strongest possible TUR 2

TGGL, indicating

that this theoretical bound is indeed achievable with TSCCs.

the two metastable states are Aand B, then take Aas

the one with a deeper well and higher initial probability.

Then, the energy landscape is tilted and the energy bar-

rier is removed. This computational potential Ucomp is

held so that probability mass ﬂows from Ato B. This

is a “reset” in that it re-initializes the system to the B

state [59, 63]. Ref. [57] describes this protocol in detail.

The resulting entropy productions in minimum-

variance currents are shown in Fig. 2. Some computa-

tions are considerably less precise than speciﬁed by 2

BS,

the original TUR, but many lie well below this bound.

As expected, all computations are less precise than the

bounds 2

TGGL and 2

HVV, but none of them come very

close to the tightest theoretical bound given by 2

TGGL.

Instead, there is another curve that seems to bound

all of these time-symmetric erasures, shown in dashed

red. Reference [57] derives the bound 2

Discrete from

a discrete-state time-symmetric erasure, where the sys-

tem equilibrates to the tilted energy landscape before the

metastable information-storing potential is re-initialized

at the end of the computation.

Second, consider the same initial metastable states A

and B, which are stored in local equilibrium. Then, in-

stantaneously implement a harmonic potential and hold

the energy landscape for half a period of the oscillation.

If the coupling to the thermal environment is weak, then

this implements a reliable swap between Aand B, us-

ing momentum as memory to carry the distributions into

their new states. This is a slight modiﬁcation of momen-

tum computing protocols described in Refs. [64, 65]. The

entropy and scaled variance of these swaps are shown on

the right-hand side of Fig. 2. They span the same space

of possibilities shown for Pr(Σ|µ, σ2) in Fig. 1: some

lying above 2

BS and some TSCCs sitting just above the

minimum set by 2

TGGL.

For both the reset and swap, we see that the mini-

mum scaled variance tends to increase as the variance

of the entropy-production distribution increases. Higher

moments of entropy production are critically important

in predicting the precision of computations.

Conclusion. We introduced two equalities that pro-

vide an explicit expression for the most accurate current

in any entropy production distribution that satisﬁes Eq.

(4) and for the minimal scaled variance it achieves. Anal-

ogous to the transition from the Second Law inequality to

the ﬂuctuation relations of Crooks and Jarzynski, we de-

veloped a treatment of the entropy production that recog-

nizes it as a stochastic quantity with proper ﬂuctuations.

This treatment yields a result—the Thermodynamic Un-

certainty Theorem—that is (i) an equality rather than

an inequality and (ii) depends on the stochastic entropy

production’s ﬂuctuations rather than on only its average

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TheThermodynamicUncertaintyTheoremKyleJ.Ray,1,AlexanderB.Boyd,2,3,4,yGiacomoGuarnieri,5,zandJamesP.Crutcheld6,x1ComplexitySciencesCenterandPhysicsDepartment,UniversityofCaliforniaatDavis,OneShieldsAvenue,Davis,CA95616,USA2Correspondingauthor3TheDivisionofPhysics,MathematicsandAstronomy,CaliforniaI...

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