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Stochastic thermodynamic interpretation of information geometry

Sosuke ItoRIES, Hokkaido University, N20 W10, Kita-ku, Sapporo, Hokkaido 001-0020, Japan

(Dated: May 8, 2018)

In recent years, the unified theory of information and thermodynamics has been intensively dis-cussed in the context of stochastic thermodynamics. The unified theory reveals that informationtheory would be useful to understand non-stationary dynamics of systems far from equilibrium. Inthis letter, we have found a new link between stochastic thermodynamics and information theorywell known as information geometry. By applying this link, an information geometric inequalitycan be interpreted as a thermodynamic uncertainty relationship between speed and thermodynamiccost. We have numerically applied an information geometric inequality to a thermodynamic modelof biochemical enzyme reaction.

PACS numbers: 02.40.-k, 05.20.-y, 05.40.-a, 05.70.Ln, 89.70.-a

The crucial relationship between thermodynamicsand information theory has been well studied in lastdecades [1]. Historically, thermodynamic-informationallinks had been discussed in the context of the secondlaw of thermodynamics and the paradox of Maxwell’sdemon [2]. Recently, several studies have newly revealedthermodynamic interpretations of informational quanti-ties such as the Kullback-Leibler divergence [3], mutualinformation [4–6], the transfer entropy and informationflow [7–19]. The above interpretations of informationalquantities are based on the theory of stochastic thermo-dynamics [20, 21], which mainly focus on the entropyproduction in stochastic dynamics of small systems farfrom equilibrium.

Information thermodynamic relationship has been at-tracted not only in terms of Maxwell’s demon, but alsoin terms of geometry [22–28]. Indeed, differential geo-metric interpretations of thermodynamics have been dis-cussed especially in a near-equilibrium system [22, 29–34]. Moreover, the technique of differential geometry ininformation theory, well known as information geome-try [35], has received remarkable attention in the field ofneuroscience, signal processing, quantum mechanics, andmachine learning [36–38]. In spite of the deep link be-tween information and thermodynamics, the direct con-nection between thermodynamics and information geom-etry has been elusive especially for non-stationary andnon-equilibrium dynamics. For example, G. E. Crooksdiscovered a link between thermodynamics and informa-tion geometry [22, 32] based on the Gibbs ensemble, andthen his discussion is only valid for a near-equilibriumsystem.

In this letter, we discover a fundamental link be-tween information geometry and thermodynamics basedon stochastic thermodynamics for the master equation.We mainly report two inequalities derived thanks to in-formation geometry, and interpret them within the the-ory of stochastic thermodynamics. The first inequalityconnects the environmental entropy change rate to themean change of the local thermodynamic force rate. The

second inequality can be interpreted as a kind of ther-modynamic uncertainty relationships or thermodynamictrade-off relationships [39–51] between speed of a transi-tion from one state to another and thermodynamic costrelated to the entropy change of thermal baths in a near-equilibrium system. We numerically illustrate these twoinequalities on a model of biochemical enzyme reaction.Stochastic thermodynamics.– To clarify a link between

stochastic thermodynamics and information geometry,we here start with the formalism of stochastic thermo-dynamics for the master equation [20, 21], that is alsoknown as the Schnakenberg network theory [52, 53].We here consider a (n + 1)-states system. We assume

that transitions between states are induced by nbath-multiple thermal baths. The master equation for theprobability px (≥ 0,

∑nx=0 px = 1) to find the state at

x = {0, 1, . . . , n} is given by

d

dtpx =

nbath∑

ν=1

n∑

x′=0

W(ν)x′→xpx′ , (1)

where W(ν)x′→x is the transition rate from x′ to x induced

by ν-th thermal bath. We assume a non-zero value of

the transition rate W(ν)x′→x > 0 for any x 6= x′. We also

assume the condition

n∑

x=0

W(ν)x′→x = 0, (2)

or equivalently W(ν)x′→x′ = −∑x 6=x′ W

(ν)x′→x < 0, which

leads to the conservation of probability d(∑n

x=0 px)/dt =0. This equation (2) indicates that the master equationis then given by the thermodynamic flux from the statex′ to x [52],

J(ν)x′→x :=W

(ν)x′→xpx′ −W

(ν)x→x′px, (3)

d

dtpx =

nbath∑

ν=1

n∑

x′=0

J(ν)x′→x. (4)

2

If dynamics are reversible (i.e., J(ν)x′→x = 0 for any x, x′

and ν), the system is said to be in thermodynamic equi-librium. If we consider the conjugated thermodynamicforce

F(ν)x′→x := ln[W

(ν)x′→xpx′ ]− ln[W

(ν)x→x′px], (5)

thermodynamic equilibrium is equivalently given by

F(ν)x′→x = 0 for any x, x′ and ν.In stochastic thermodynamics [21], we treat the en-

tropy change of thermal bath and the system in a stochas-tic way. In the transition from x′ to x, the stochasticentropy change of ν-th thermal bath is defined as

∆σbath(ν)x′→x := ln

W(ν)x′→x

W(ν)x→x′

, (6)

and the stochastic entropy change of the system is definedas the stochastic Shannon entropy change

∆σsysx′→x := ln px′ − ln px, (7)

respectively. The thermodynamic force is then given bythe sum of entropy changes in the transition from x′ to

x induced by ν-th thermal bath F(ν)x′→x = ∆σ

bath(ν)x′→x +

∆σsysx′→x. This fact implies that the system is in equi-

librium if the sum of entropy changes is zero for anytransitions.The total entropy production rate Σtot is given by the

sum of the products of thermodynamic forces and fluxesover possible transitions. To simplify notations, we intro-duce the set of directed edges E = {(x′ → x, ν)|0 ≤ x′ <x ≤ n, 1 ≤ ν ≤ nbath} which denotes the set of all pos-sible transitions between two states. The total entropyproduction rate is then given by

Σtot :=∑

(x′→x,ν)∈E

J(ν)x′→xF

(ν)x′→x = 〈F 〉, (8)

where a parenthesis 〈· · · 〉 is defined as 〈A〉 :=∑

(x′→x,ν)∈E J(ν)x′→xA

(ν)x′→x for any function of edge A

(ν)x′→x.

Because signs of the thermodynamic force F(ν)x′→x and the

flux J(ν)x′→x are same, the total entropy production rate is

non-negative

〈F 〉 = 〈∆σbath〉+ 〈∆σsys〉 ≥ 0, (9)

that is well known as the second law of thermodynamics.Information geometry.– Next, we introduce informa-

tion theory well known as information geometry [35].In this letter, we only consider the discrete distributiongroup p = (p0, p1, . . . , pn), px ≥ 0, and

∑nx=0 px = 1.

This discrete distribution group gives the n-dimensionalmanifold Sn, because the discrete distribution is givenby n+1 parameters (p0, p1, . . . , pn) under the constraint∑n

x=0 px = 1. To introduce a geometry on the mani-fold Sn, we conventionally consider the Kullback-Leibler

FIG. 1: (color online). Schematic of information geometry onthe manifold S2. The manifold S2 leads to the sphere surfaceof radius 2 (see also SI). The statistical length L is boundedby the shortest length D = 2θ = 2 cos−1(rini · rfin).

divergence [55] between two distributions p and p′ =

(p′0, p′1, . . . , p

′n) defined as

DKL(p||p′) :=

n∑

x=0

px lnpxp′x. (10)

The square of the line element ds is defined as the second-order Taylor series of the Kullback-Leibler divergence

ds2 :=

n∑

x=0

(dpx)2

px= 2DKL(p||p+ dp), (11)

where dp = (dp0, dp1, . . . , dpn) is the infinitesimal dis-placement that satisfies

∑nx=0 dpx = 0. This square of

the line element is directly related to the Fisher infor-mation metric [54] (see also Supplementary Information(SI)).The manifold Sn leads to the geometry of the n-sphere

surface of radius 2 (see also Fig. 1), because the squareof the line element is also given by ds2 =

∑nx=0(2drx)

2

under the constraint r · r =∑

x(√px)

2 = 1 where r

is the unit vector defined as r = (r0, r1, . . . , rn) :=(√p0,

√p1, . . . ,

√pn) and · denotes the inner product.

The statistical length L [56, 57]

L :=

∫

ds =

∫

ds

dtdt, (12)

from the initial state rini to the final state rfin is thenbounded by

L ≥ 2 cos−1(rini · rfin) := D(rini; rfin), (13)

because D(rini; rfin) = 2θ is the shortest length betweenrini and rfin on the n-sphere surface of radius 2, whereθ is the angle between rini and rfin given by the innerproduct rini · rfin = cos θ.Stochastic thermodynamics of information geometry.–

We here discuss a relationship between the line element

3

and conventional observables of stochastic thermody-namics, which gives a stochastic thermodynamic inter-pretation of information geometric quantities.By using the master equation (1) and definitions of the

line element and thermodynamic quantities Eqs. (5), (6)and (11), we obtain stochastic thermodynamic expres-sions of ds2/dt2 (see also SI),

ds2

dt2=

n∑

x=0

pxd

dt

(

− 1

px

dpxdt

)

(14)

= −n∑

x=0

pxd

dt

(

nbath∑

ν=1

n∑

x′=0

W(ν)x→x′e

−F(ν)

x→x′

)

(15)

=

⟨

d∆σbath

dt

⟩

−⟨

dF

dt

⟩

. (16)

Equation (15) implies that geometric dynamics are driven

by the thermodynamic factor exp[−F (ν)x→x′ ], that is well

discussed in the context of stochastic thermodynamics(especially in the context of the fluctuation theorem [58–63]). The time evolution of the line element ds2/dt2

is directly related to the expected value of the timederivative of the rate-weighted thermodynamic factor

W(ν)x→x′e

−F(ν)

x→x′ .Another expression Eq. (16) gives a stochastic ther-

modynamic interpretation of information geometry, es-pecially in case of a near-equilibrium system. The con-

dition of an equilibrium system is given by F(ν)x′→x = 0

for any x′, x and ν. Then, the square of the line ele-ment is given by the entropy change in thermal bathsds2 ≃

⟨

d∆σbath⟩

dt in a near-equilibrium system.For example, in a near-equilibrium system, the

probability distribution is assumed to be the canon-ical distribution px = exp(β(φ − Hx)), where φ :=−β−1 ln[

∑nx=0 exp(−βHx))] is the Helmholtz free energy,

β is the inverse temperature andHx is the Hamiltonian ofthe system in the state x. To consider a near-equilibriumtransition, we assume that β andHx can depend on time.From ds2 = [

⟨

d∆σbath⟩

− 〈dF 〉]dt = −〈d∆σsys〉 dt, weobtain ds2 = −〈d∆σsys〉 dt = −〈d(β∆H)〉dt in a nearequilibrium system, where ∆Hx′→x := Hx − Hx′ is theHamiltonian change from the state x′ to x. Because−β∆H can be considered as the entropy change of ther-mal bath ∆σbath, an expression ds2 = −〈d(β∆H)〉dt forthe canonical distribution is consistent with a near equi-librium expression ds2 ≃

⟨

d∆σbath⟩

dt.We also discuss the second order expansion of ds2/dt2

for the thermodynamic force in SI, based on the linearirreversible thermodynamics [52]. Our discussion impliesthat the square of the line element (or the Fisher in-formation metric) for the thermodynamic forces is re-lated to the Onsager coefficients. Due to the Cramer-Rao bound [54, 55], the Onsager coefficients are directlyconnected to a lower bound of the variance of unbiasedestimator for parameters driven by the thermodynamicforce.

Due to the non-negativity of the square of line elementds2/dt2 ≥ 0, we have a thermodynamic inequality

⟨

d∆σbath

dt

⟩

≥⟨

dF

dt

⟩

. (17)

The equality holds if the system is in a stationary state,i.e., dpx/dt = 0 for any x. This result (17) implies thatthe change of the thermodynamic force rate is transferredto the environmental entropy change rate. The differ-ence 〈d∆σbath/dt〉 − 〈dF/dt〉 ≥ 0 can be interpreted asloss in the entropy change rate transfer due to the non-stationarity. If the environmental entropy change does

not change in time (i.e., d∆σbath(ν)x′→x /dt = 0 for any x′

and x), the thermodynamic force change tends to de-crease (i.e., 〈dF/dt〉 ≤ 0) in a transition. We stress thata mathematical property of the thermodynamic force inthis result is different from the second law of thermody-namics 〈F 〉 ≥ 0.From Eq. (16), the statistical length L =

∫ τ

0 dt(ds/dt)from time t = 0 to t = τ is given by

L =

∫ t=τ

t=0

dt

√

⟨

d∆σbath

dt

⟩

−⟨

dF

dt

⟩

. (18)

We then obtain the following thermodynamic inequalityfrom Eqs. (13) and (18),

∫ t=τ

t=0

dt

√

⟨

d∆σbath

dt

⟩

−⟨

dF

dt

⟩

≥ D(r(0); r(τ)). (19)

The equality holds if the path of transient dynamics is ageodesic line on the manifold Sn. This inequality gives ageometric constraint of the entropy change rate transferin a transition between two probability distributions p(0)and p(τ).Thermodynamic uncertainty.– We finally reach to a

thermodynamic uncertainty relationship between speedand thermodynamic cost. We here consider the actionC := (1/2)

∫ t=τ

t=0dt(ds2/dt2) from time t = 0 to t = τ .

From Eq. (16), the action C is given by

C =1

2

∫ t=τ

t=0

dt

[⟨

d∆σbath

dt

⟩

−⟨

dF

dt

⟩]

. (20)

Especillay in case of a near-equilibrium system, the ac-tion C is given by C ≃

∫ ⟨

d∆σbath⟩

/2. If we assumethe canonical distribution, we have C = −

∫

〈d(β∆H)〉/2.Even for a system far from equilibrium, we can considerthe action as a total amount of loss in the entropy changerate transfer. Therefore, the action can be interpreted asthermodynamic cost.Due to the Cauchy-Schwarz inequality

∫ τ

0dt∫ τ

0(ds/dt)2dt ≥ (

∫ τ

0(ds/dt)dt)2 [22], we obtain

a thermodynamic uncertainty relationship betweenspeed τ and thermodynamic cost C

τ ≥ L2

2C . (21)

4

FIG. 2: (color online). Numerical calculation of thermody-namic quantities in the three states model of enzyme reaction.We numerically shows the non-negativity of ds2/dt2 ≥ 0 andds2/dt2 = −〈dF/dt〉 + 〈d∆σbath/dt〉 in the graph. We alsoshow the total entropy change rate 〈F 〉 ≥ 0. We note thatd〈F 〉/dt is not equal to 〈dF/dt〉.

The equality holds if speed of dynamics ds2/dt2 does notdepend on time. By using the inequality (13), we alsohave a weaker bound

τ ≥ [D(r(0); r(τ))]2

2C . (22)

In a transition from r(0) to r(τ)(6= r(0)), thermody-namic cost C should be large if the transition time τis small. In case of a near-equilibrium system, we have2C =

∫ ⟨

d∆σbath⟩

(or 2C = −∫

〈d(β∆H)〉), and thenthe inequality is similar to the quantum speed limit thatis discussed in quantum mechanics [37]. We stress thatthis result is based on stochastic thermodynamics, noton quantum mechanics.The inequality (22) gives the ratio between time-

averaged thermodynamic cost 2C/τ and square of thevelocity on manifold ([D(r(0); r(τ))]/τ)2 . Then, this ra-tio

η :=[D(r(0); r(τ))]2

2τC . (23)

quantifies an efficiency for power to speed conversion.Due to the inequality (22) and its non-negativity, theefficiency η satisfies 0 ≤ η ≤ 1, where η = 1 (η = 0)implies high (low) efficiency.Three states model of enzyme reaction.– We numer-

ically illustrate thermodynamic inequalities of informa-tion geometry by using a thermodynamic model of bio-chemical reaction. We here consider a three states model(see also SI) that represents a chemical reaction A+B ⇋

FIG. 3: (color online). Numerical calculation of the thermo-dynamic uncertainty relationship in the three states modelof enzyme reaction. We numerically shows the geometric in-equality L ≥ D(r(0); r(τ )), the thermodynamic uncertaintyrelationship τ ≥ L2/(2C) ≥ [D(r(0); r(τ ))]2/(2C), and the ef-ficiency η in the graph.

AB with enzyme X ,

A+X ⇋ AX, (24)

A+B ⇋ AB, (25)

AX +B ⇋ AB +X. (26)

We here consider the probability distribution of statesx = A,AX,AB. We assume that the system is attachedto a single heat bath (nbath = 1) with inverse tempera-ture β. The master equation is given by Eq. (1), wherethe transition rates are supposed to be

W(1)A→AX = kAX+[X ], W

(1)AX→A = kAX+e

−β∆µAX ,

W(1)A→AB = kAB+[B], W

(1)AB→A = kAB+e

−β∆µAB ,

W(1)AX→AB = k+[B], W

(1)AB→AX = k+e

−β∆µ[X ],(27)

[X ] ([B]) is the concentration of X (B), kAX+, kAB+,and k+ are reaction rate constants, and ∆µAX , ∆µAB,and ∆µ are the chemical potential differences. In this

model, the entropy change of bath ∆σbath(ν)x′→x is given by

this chemical potential difference (see also SI) [64].In a numerical simulation, we set kAX+ = kAB+ =

k+ = 1, β∆µAX = 1, β∆µAB = 0.5, and β∆µ = 2.We assume that the time evolution of the concentrationsis given by [X ] = tan−1(ωXt), [B] = tan−1(ωBt) withωX = 1 and ωB = 2, which means that the concen-trations [X ] and [B] perform as control parameters. Attime t = 0, we set the initial probability distribution as(pA, pAX , pAB) = (0.9998, 0.0001, 0.0001).

5

In Fig. 2, we numerically show the inequality〈d∆σbath/dt〉 ≥ 〈dF/dt〉. We check that this inequal-ity does not coincide with the second law of thermody-namics 〈F 〉 ≥ 0. We also check the thermodynamic un-certainty relationship τ ≥ L2/(2C) in Fig. 3. Becausethe path from the initial distribution (pA, pAX , pAB) =(0.9998, 0.0001, 0.0001) to the final distribution is closeto the geodesic line, the thermodynamic uncertainty re-lationship gives a tight bound of the transition time τ .

Conclusion.– In this letter, we reveal a link be-tween stochastic thermodynamic quantities (J , F , ∆σsys,∆σbath) and information geometric quantities (ds2, L, D,C). Because the theory of information geometry is appli-cable to various fields of science such as neuroscience, sig-nal processing, machine learning and quantum mechan-ics, this link would help us to understand a thermody-namic aspect of such a topic. The trade-off relationshipbetween speed and thermodynamic cost Eq. (22) wouldbe helpful to understand biochemical reactions and givesa new insight into recent studies of the relationship be-tween information and thermodynamics in biochemicalprocesses [7, 42, 65–69].

ACKNOWLEDGEMENT

I am grateful to Shumpei Yamamoto for discussionsof stochastic thermodynamics for the master equation,to Naoto Shiraishi, Keiji Saito, Hal Tasaki, and Shin-Ichi Sasa for discussions of thermodynamic uncertaintyrelationships, to Schuyler B. Nicholson for discussionof information geometry and thermodynamics, and toPieter rein ten Wolde for discussions of thermodynam-ics in a chemical reaction. We also thank Tamiki Ko-matsuzaki to acknowledge my major contribution of thiswork and allow me to submit this manuscript alone. Imentioned that, after my submission of the first versionof this manuscript on arXiv [70], I heard that Schuyler B.Nicholson independently discovered a similar result suchas Eq. (16) [71]. I thank Sebastian Goldt, Matteo Polet-tini, Taro Toyoizumi, and Hiroyasu Tajima for valuablecomments on the manuscript. This research is supportedby JSPS KAKENHI Grant No. JP16K17780.

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7

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SUPPLEMENTARY INFORMATION

I. Intuitive proof of the fact that the manifold S2 gives the sphere surface of radius 2

We here intuitively show the fact that the manifold S2 gives the sphere surface of radius 2. The set of probabilityp = (p0, p1, p2) satisfies the normalization

∑2x=0 px = 1. The square of the line element ds is given by

ds2 =

2∑

x=0

(dpx)2

px. (28)

We here introduce the polar coordinate system (φ, ψ) where p0 = (cosψ)2, p1 = (sinψ)2(cosφ)2, p2 = (sinψ)2(sinφ)2.

We can check that the normalization∑2

x=0 px = 1 holds. By using the polar coordinate system, (dp0, dp1, dp2)is given by dp0 = −2(cosψ)(sinψ)dψ, dp1 = 2(cosψ)(sinψ)(cosφ)2dψ − 2(cosφ)(sinφ)(sinψ)2dφ, and dp2 =2(cosψ)(sinψ)(sin φ)2dψ + 2(cosφ)(sin φ)(sinψ)2dφ. From Eq. (28), we then obtain

ds2 = 4[(sinψ)2 + (cosψ)2(cosφ)2 + (cosψ)2(sinφ)2](dψ)2 + 0× (dφ)(dψ) + 4[(sinφ)2(sinψ)2 + (cosφ)2(sinψ)2](dφ)2

= 22[(dψ)2 + (sinψ)2(dφ)2]. (29)

Because the metric of the sphere surface of radius R is given by ds2 = R2[(dψ)2 + (sinψ)2(dφ)2], the manifold S2

gives the sphere surface of radius R = 2.

II. Detailed derivation of Eqs. (15) and (16) in the main text

We here discuss the detailed derivation of Eqs. (15) and (16) in the main text, and the relationship between thesquare of the line element and the Fisher information metric.

By using the definition of the thermodynamic force F(ν)x′→x := ln[W

(ν)x′→xpx′ ] − ln[W

(ν)x→x′px], the master equation is

given by

d

dtpx =

nbath∑

ν=1

n∑

x′=0

W(ν)x→x′pxe

−F(ν)

x→x′ . (30)

From Eqs. (28), (30) and∑n

x=0 d2px/dt

2 = 0, we obtain an expression Eq. (15) in the main text,

ds2

dt2=

n∑

x=0

1

px

(

dpxdt

)2

=

n∑

x=0

pxd

dt

(

− 1

px

)(

dpxdt

)

=

n∑

x=0

pxd

dt

(

− 1

px

)(

dpxdt

)

−n∑

x=0

(

d2pxdt2

)

=

n∑

x=0

pxd

dt

(

− 1

px

dpxdt

)

= −n∑

x=0

pxd

dt

(

nbath∑

ν=1

n∑

x′=0

W(ν)x→x′e

−F(ν)

x→x′

)

. (31)

8

Let E[A] :=∑n

x=0 pxA(x) be the expected value of any function A(x), and A(x) :=∑nbath

ν=1

∑nx′=0W

(ν)x→x′A

(ν)x→x′ be

the rate-weighted expected value of any function of edge A(ν)x→x′ with a fixed initial state x, respectively. By using

these notation, the result (31) can be rewritten as

ds2

dt2= −E

[

d

dte−F

]

. (32)

We here mention that a parenthesis in the main text is given by 〈A〉 = E[

A]

if A(ν)x→x′ is an anti-symmetric function

A(ν)x→x′ = −A(ν)

x′→x. Because the thermodynamic force is an anti-symmetric function F(ν)x→x′ = −F (ν)

x′→x, the total

entropy production rate is given by Σtot = E[

F]

. We also carefully mention that the expected value of e−F gives

E[e−F ] =∑nbath

ν=1

∑nx=0

∑nx′=0 px′W

(ν)x′→x = 0, compared to the integral fluctuation theorem 〈e−Ftraj〉traj = 1 with the

entropy production of trajectories Ftraj and the ensemble average of trajectories 〈· · · 〉traj [1, 2]. If the system is in astationary state, i.e., dpx/dt = 0 for any x, we have

E

[

d

dte−F

]

=d

dt

(

E

[

e−F

])

= 0. (33)

From Eq. (31), we also obtain

ds2

dt2=−

n∑

x=0

pxd

dt

(

nbath∑

ν=1

n∑

x′=0

W(ν)x→x′e

−F(ν)

x→x′

)

=−n∑

x=0

px

(

nbath∑

ν=1

n∑

x′=0

W(ν)x→x′

(

− d

dtF

(ν)x→x′

)

e−F(ν)

x→x′

)

−n∑

x=0

px

(

nbath∑

ν=1

n∑

x′=0

(

d

dtW

(ν)x→x′

)

e−F(ν)

x→x′

)

. (34)

The first term is calculated as follows

−n∑

x=0

px

(

nbath∑

ν=1

n∑

x′=0

W(ν)x→x′

(

− d

dtF

(ν)x→x′

)

e−F(ν)

x→x′

)

=−nbath∑

ν=1

n∑

x=0

n∑

x′=0

px′W(ν)x′→x

(

d

dtF

(ν)x′→x

)

=−nbath∑

ν=1

∑

x,x′|x>x′

px′W(ν)x′→x

(

d

dtF

(ν)x′→x

)

−nbath∑

ν=1

∑

x,x′|x′>x

px′W(ν)x′→x

(

d

dtF

(ν)x′→x

)

=−∑

(x′→x,ν)∈E

J(ν)x′→x

(

d

dtF

(ν)x′→x

)

= −⟨

dF

dt

⟩

, (35)

9

where we used F(ν)x′→x = −F (ν)

x→x′ and F(ν)x′→x′ = 0. The second term is also calculated as follows

−n∑

x=0

px

(

nbath∑

ν=1

n∑

x′=0

(

d

dtW

(ν)x→x′

)

e−F(ν)

x→x′

)

=−nbath∑

ν=1

n∑

x=0

n∑

x′=0

px′W(ν)x′→x

1

W(ν)x→x′

(

d

dtW

(ν)x→x′

)

=−nbath∑

ν=1

∑

x,x′|x′ 6=x

px′W(ν)x′→x

1

W(ν)x→x′

(

d

dtW

(ν)x→x′

)

−nbath∑

ν=1

n∑

x=0

pxW(ν)x→x

1

W(ν)x→x

(

d

dtW (ν)

x→x

)

=−nbath∑

ν=1

∑

x,x′|x>x′

px′W(ν)x′→x

1

W(ν)x→x′

(

d

dtW

(ν)x→x′

)

+

nbath∑

ν=1

n∑

x=0

px

∑

x′ 6=x

d

dtW

(ν)x→x′

=−nbath∑

ν=1

∑

x,x′|x 6=x′

px′W(ν)x′→x

(

d

dtln(W

(ν)x→x′)

)

+

nbath∑

ν=1

∑

x,x′|x′ 6=x

px′W(ν)x′→x

(

d

dtln(W

(ν)x′→x)

)

=

nbath∑

ν=1

∑

x,x′|x′ 6=x

px′W(ν)x′→x

(

d

dt∆σ

bath(ν)x′→x

)

=

nbath∑

ν=1

∑

x,x′|x>x′

px′W(ν)x′→x

(

d

dt∆σ

bath(ν)x′→x

)

−nbath∑

ν=1

∑

x,x′|x>x′

pxW(ν)x→x′

(

d

dt∆σ

bath(ν)x→x′

)

=∑

(x′→x,ν)∈E

J(ν)x′→x

(

d

dt∆σ

bath(ν)x′→x

)

=

⟨

d∆σbath

dt

⟩

, (36)

where we used W(ν)x′→x′ = −∑x 6=x′ W

(ν)x′→x, ∆σ

bath(ν)x′→x = −∆σ

bath(ν)x→x′ and ∆σ

bath(ν)x′→x′ = 0.

By using F(ν)x′→x = ∆σ

bath(ν)x′→x′ +∆σsys

x′→x′ , we obtain an expression

ds2

dt2=

⟨

d∆σbath

dt

⟩

−⟨

dF

dt

⟩

= −⟨

d∆σsys

dt

⟩

. (37)

Let (λ1, . . . , λn′) be the set of parameters such as control parameters. We also obtain the definition of the Fisherinformation metric [3]

gij = E

[(

∂ ln p

∂λi

)(

∂ ln p

∂λj

)]

=

n∑

x=0

px

[(

∂ ln px∂λi

)(

∂ ln px∂λj

)]

(38)

10

from the result (37),

ds2

dt2=

⟨

d∆σbath

dt

⟩

−⟨

dF

dt

⟩

=−∑

(x′→x,ν)∈E

J(ν)x′→x

[

1

px′

dpx′

dt− 1

px

dpxdt

]

=−nbath∑

ν=1

nbath∑

ν′=1

n∑

x=0

n∑

x′=0

n∑

x′′=0

px′W(ν)x′→x

[

1

px′

W(ν′)x′′→x′px′′ − 1

pxW

(ν′)x′′→xpx′′

]

=

n∑

x=0

px

[

nbath∑

ν=1

n∑

x′=0

px′W(ν)x′→x

px

nbath∑

ν′=1

n∑

x′′=0

px′′W(ν′)x′′→x

px

]

=

n∑

x=0

px

(

d ln pxdt

)2

=n∑

x=0

px

n′

∑

i=1

∂ ln px∂λi

dλidt

2

=

n′

∑

i=1

n′

∑

j=1

dλidtgijdλjdt

, (39)

where we used∑n

x=0W(ν)x′→x = 0 and the master equation dpx/dt =

∑nbath

ν=1

∑nx′=0 px′W

(ν)x′→x. This result is consistent

with the following calculation about the Fisher information metric

ds2 =

n∑

x=0

px(d ln px)2 =

n∑

x=0

px

n′

∑

i=1

(

∂ ln px∂λi

)

dλi

2

=

n′

∑

i=1

n′

∑

j=1

gijdλidλj . (40)

III. Linear irreversible thermodynamic interpretation of information geometry

We here discuss a stochastic thermodynamic interpretation of information geometry in a near-equilibirum sys-tem, where the entropy production rate is given by the second order expansion for the thermodynamic flow (or thethermodynamic force). This second order expansion is well known as linear irreversible thermodynamics [4].

If we assume F(ν)x′→x = 0, we have J

(ν)x′→x = 0. Thus, we have a linear expansion of thermodynamic force F

(ν)x′→x in

terms of the thermodynamic flow J(ν)x′→x for a near-equilibrium condition (i.e., F

(ν)x′→x ≃ 0 for any x and x′)

F(ν)x′→x = ln

(

1 +J(ν)x′→x

W(ν)x′→xpx

)

= α(ν)x′→xJ

(ν)x′→x + o(J

(ν)x′→x), (41)

α(ν)x′→x :=

1

W(ν)x′→xpx

∣

∣

∣

∣

∣

F(ν)

x′→x

=0

. (42)

We call this coefficient α(ν)x′→x as the Onsager coefficient of the edge (x′ → x, ν). The symmetry of the coefficient

α(ν)x′→x = α

(ν)x→x′ holds due to the condition F

(ν)x′→x = 0.

If we consider the Kirchhoff’s current law in a stationary state, the linear combination of the coefficient α(ν)x→x′ leads

to the Onsager coefficient [4]. Let {C1, . . . , Cm} be the cycle basis of the Markov network for the master equation.The thermodynamic force of the cycle F (Ci) is defined as

F (Ci) =∑

(x′→x,ν)∈E

S({x′ → x, ν}, Ci)F(ν)x′→x (43)

11

where

S({x′ → x, ν}, Ci) =

1 ({x′ → x, ν} ∈ Ci)

−1 ({x→ x′, ν} ∈ Ci)

0 (otherwise)

. (44)

The thermodynamic flow of the cycle J(Ci) is defined as

J(ν)x′→x =

m∑

i=1

S({x′ → x, ν}, Ci)J(Ci). (45)

We then obtain the linear relationship F (Cj) =∑m

i=1 LjiJ(Ci) (or J(Cj) =∑m

i=1 L−1ji F (Ci)) with the Onsager

coefficient

Lij =∑

(x′→x,ν)∈E

α(ν)x′→xS({x′ → x, ν}, Ci)S({x′ → x, ν}, Cj), (46)

for a near-equilibrium condition, the second law of thermodynamics

0 ≤ Σtot

=∑

(x′→x,ν)∈E

J(ν)x′→xF

(ν)x′→x

=∑

(x′→x,ν)∈E

m∑

i=1

S({x′ → x, ν}, Ci)J(Ci)F(ν)x′→x

=

m∑

i=1

J(Ci)F (Ci),

=

m∑

j=1

m∑

i=1

LijJ(Ci)J(Cj),

=

m∑

j=1

m∑

i=1

L−1ij F (Ci)F (Cj),

(47)

and the Onsager reciprocal relationship Lij = Lji. This result gives the second order expansion of the entropy

production rate Σtot for the thermodynamic flow J (or the thermodynamic force F ) in a stationary state. For m = 2,the second law of thermodynamics L11F (C1)

2+L22F (C2)2+2L12F (C1)F (C2) ≥ 0 is then given by L11 ≥ 0, L22 ≥ 0,

and L11L22 − L212 ≥ 0.

Here we newly consider the second order expansion of ds2 for the thermodynamic flow J (or the thermodynamicforce F ) in linear irreversible thermodynamics. In a near-equilibrium system, the square of line element ds is calculated

12

as follows

ds2 =−⟨

d∆σsys

dt

⟩

dt2

=−∑

(x′→x,ν)∈E

J(ν)x′→x

[

1

px′

dpx′

dt− 1

px

dpxdt

]

dt2

=−∑

(x′→x,ν)∈E

J(ν)x′→x

[

1

px′

nbath∑

ν′=1

n∑

x′′=0

J(ν′)x′′→x′ −

1

px

nbath∑

ν′=1

n∑

x′′=0

J(ν′)x′′→x

]

dt2

=∑

(x′→x,ν)∈E

nbath∑

ν′=1

n∑

x′′=0

J(ν)x′→x

[

1

pxJ(ν′)x′′→x − 1

px′

J(ν′)x′′→x′

]

dt2

=∑

(x′→x,ν)∈E

nbath∑

ν′=1

n∑

x′′=0

J(ν)x′→x

[

1

pxJ(ν′)x′′→x − 1

px′

J(ν′)x′′→x′

]

dt2

=

nbath∑

ν=1

nbath∑

ν′=1

n∑

x=0

n∑

x′=0

n∑

x′′=0

[

J(ν)x′→xJ

(ν′)x′′→x

px

]

dt2

=

nbath∑

ν=1

nbath∑

ν′=1

n∑

x=0

n∑

x′=0

n∑

x′′=0

[

F(ν)x′→xF

(ν′)x′′→x

α(ν)x′→xpxα

(ν′)x′′→x

]

dt2. (48)

We here consider the situation that the time evolution of control parameters λ(x′,x,νx) is driven by the thermodynamic

force F(νx)x′→x = dλ(x′,x,νx)/dt. The square of line element can be written by the following Fisher information metric

ds2 =

nbath∑

νx=1

n∑

x=0

n∑

x′=0

nbath∑

νy=1

n∑

y=0

n∑

y′=0

g(x′,x,νx)(y′,y,νy)dλ(x′,x,νx)dλ(y′,y,νy) (49)

g(x′,x,νx)(y′,y,νy) =δxy

α(νx)x′→xpxα

(νy)y′→y

. (50)

This result implies that the Fisher information metric for control parameters λ(x′,x,νx) driven by the thermodynamic

force F(νx)x′→x = dλ(x′,x,νx)/dt is related to the Onsager coefficients of the edge α

(νx)x′→x for a near-equilibrium condition.

Because the Cramer-Rao bound [3, 5] implies that the variance of unbiased estimator is bounded by the inverse of this

Fisher information metric, the Onsager coefficients of the edge α(νx)x′→x gives a lower bound of the variance of unbiased

estimator for control parameters driven by the thermodynamic forces in a near-equilibrium system.

IV. Detail of the three states model of enzyme reaction

Stochastic thermodynamics for the master equation is applicable to a model of chemical reaction [64]. We herediscuss the thermodynamic detail of the three states model of enzyme reaction discussed in the main text.

The master equation for Eq. (27) in the main text is given by

dpAdt

= −(kAX+[X ] + kAB+[B])pA + kAB−pAB + kAX−pAX ,

dpAB

dt= kAB+[B]pA − (kAB− + k−[X ])pAB + k+[B]pAX ,

dpAX

dt= kAX+[X ]pA + k−[X ]pAB − (kAX− + k+[B])pAX , (51)

13

where k−, kAB− and kAX− are given by the chemical potential differences

lnkAX+

kAX−= β∆µAX ,

lnkAB+

kAB−= β∆µAB,

lnk+k−

= β∆µ. (52)

We here assume that the sum of the concentrations [A] + [AB] + [AX ] = nA is constant. The probabilitiesdistributions pA, pAB, and pAX correspond to the fractions of pA = [A]/nA, pAB = [AB]/nA and pAX = [AX ]/nA,respectively. From the master equation (51), we obtain the rate equations of enzyme reaction

d[A]

dt= −(kAX+[X ] + kAB+[B])[A] + kAB−[AB] + kAX−[AX ],

d[AB]

dt= kAB+[B][A]− (kAB− + k−[X ])[AB] + k+[B][AX ],

d[AX ]

dt= kAX+[X ][A] + k−[X ][AB]− (kAX− + k+[B])[AX ]. (53)

which corresponds to the following enzyme reaction

A+X ⇋ AX,

A+B ⇋ AB,

AX +B ⇋ AB +X, (54)

where A is substrate, X is enzyme, AX is enzyme-substrate complex, and AB is product.In this model, the stochastic entropy changes of thermal bath are also calculated as

∆σbath(1)A→AB = β∆µAB + ln[B],

∆σbath(1)AB→AX = −β∆µ− ln[B] + ln[X ],

∆σbath(1)AX→A = −β∆µAX − ln[X ], (55)

which are the conventional definitions of the stochastic entropy changes of a thermal bath. In this model, the cyclebasis is given by one cycle {C1 = (A→ AB → AX → A)}. If the chemical potential change in a cycle C1 has non-zerovalue, i.e., ∆µcyc := ∆µAB −∆µ−∆µAX 6= 0, the system in a stationary state is driven by the thermodynamic force

of the cycle F (C1) = F(1)A→AB +F

(1)AB→AX +F

(1)AX→A = β∆µcyc. In a numerical calculation, we set β∆µcyc = −2.5 6= 0.

Then we consider non-equilibrium and non-stationary dynamics in a numerical calculation.

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