Nonequilibrium thermodynamics of uncertain stochastic processes Jan Korbel1 2and David H. Wolpert3 2 4 5 6 1Section for Science of Complex Systems CeMSIIS Medical University of Vienna Spitalgasse 23 1090 Vienna Austria

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Nonequilibrium thermodynamics of uncertain stochastic processes

Jan Korbel1, 2 and David H. Wolpert3, 2, 4, 5, 6

1Section for Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria∗

2Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria

3Santa Fe Institute, Santa Fe, NM, USA†

4Arizona State University, Tempe, AZ, USA

5International Center for Theoretical Physics, Trieste, Italy

6Albert Einstein Institute for Advanced Study, New York, USA

(Dated: May 23, 2023)

Stochastic thermodynamics is formulated under the assumption of perfect knowledge of all ther-

modynamic parameters. However, in any real-world experiment, there is non-zero uncertainty about

the precise value of temperatures, chemical potentials, energy spectrum, etc. Here we investigate

how this uncertainty modiﬁes the theorems of stochastic thermodynamics. We consider two sce-

narios: in the (called eﬀective) scenario we ﬁx the (unknown, randomly generated) experimental

apparatus and then repeatedly observe (stochastic) trajectories of the system for that ﬁxed appara-

tus. In contrast, in a (called phenomenological ) scenario the (unknown) apparatus is re-generated

for each trajectory. We derive expressions for thermodynamic quantities in both scenarios. We also

discuss the physical interpretation of eﬀective (scenario) entropy production (EP), derive the eﬀec-

tive mismatch cost, and provide a numerical analysis of the eﬀective thermodynamics of a quantum

dot implementing bit erasure with uncertain temperature. We then analyze the protocol for moving

between two state distributions that maximize eﬀective work extraction. Next, we investigate the

eﬀective thermodynamic value of information, focusing on the case where there is a delay between

the initialization of the system and the start of the protocol. Finally, we derive the detailed and

integrated ﬂuctuation theorems (FTs) for the phenomenological EP. In particular, we show how the

phenomenological FTs account for the fact that the longer a trajectory runs, the more information

it provides concerning the precise experimental apparatus, and therefore the less EP it generates.

I. INTRODUCTION

The microscopic laws of classical and quantum physics

are parameterized sets of equations that specify the evo-

lution of a closed system starting from a speciﬁc state. To

use those equations, we need to know that speciﬁc state,

we need to be sure the system is closed, and we need to

know the values of the parameters in the equations [1, 2].

Unfortunately, in many real-world scenarios, we are

uncertain about the precise state of the system, and very

often, the system is open rather than closed, subject

to uncertain interactions with the external environment.

Statistical physics accounts for these two types of un-

certainty by building on the microscopic laws of physics

in two ways. First, to capture uncertainty about the

state of the system, we replace the exact speciﬁcation

of the system’s state with a probability distribution over

states. Second, to capture uncertain interactions between

the system and the external environment, we add ran-

domness to the dynamics in a precisely parameterized

form [3].

In particular, in the sub-ﬁeld of classical stochastic

thermodynamics [1, 2] we model the system as a probabil-

ity distribution evolving under a continuous-time Markov

chain (CTMC) with a precisely speciﬁed rate matrix. Of-

ten in this work, we require that the CTMC obeys local

∗jan.korbel@meduniwien.ac.at

†david.h.wolpert@gmail.com; http://davidwolpert.weebly.com

detailed balance (LDB). This means that the rate ma-

trix of the CTMC has to obey certain restrictions, which

are parameterized by the energy spectrum of the system,

the number of thermodynamic reservoirs in the external

environment perturbing the system’s dynamics, and the

temperatures and chemical potentials of those reservoirs.

Often we also allow both the Hamiltonian of the system

and the rate matrix of the associated CTMC to change

in time in a deterministic manner, perhaps coupled by

LDB. That joint trajectory is referred to as a “protocol”.

However, in addition to uncertainty about the state of

the system and uncertainty about interactions with the

external environment, there is an additional unavoidable

type of uncertainty in all real-world systems: uncertainty

about the parameters in the equations governing the dy-

namics. In the context of stochastic thermodynamics,

this means that even if we impose LDB, we will never

know the reservoir temperatures and chemical potentials

to inﬁnite precision (often even being unsure about the

number of such reservoirs), we will never know the energy

spectrum to inﬁnite precision, and more generally, we will

never know the rate matrix and its time dependence to

inﬁnite precision.

At present, almost nothing is known about the thermo-

dynamic consequences of this third type of uncertainty

despite its unavoidability [4]. In this paper, we start

to ﬁll in this gap by considering how stochastic thermo-

dynamics (and non-equilibrium statistical physics more

generally) needs to be modiﬁed to account for this third

type of uncertainty, in addition to the two types of un-

certainty it already captures.

arXiv:2210.05249v2 [cond-mat.stat-mech] 22 May 2023

We deﬁne an apparatus α∈Ato be any speciﬁc set

of values of the thermodynamic parameters of an exper-

iment, including the number of reservoirs, their temper-

atures and chemical potentials, the precise initial distri-

bution over states (i.e., how the system was prepared)

the (deterministic trajectories of the) rate matrices, the

(deterministic trajectories of the) energy functions, etc.

Here and throughout, we assume that these thermody-

namic parameters are appropriately related by LDB for

any speciﬁc α. For simplicity, we also assume that for

all apparatuses, the system has the same state space, X.

Also for simplicity, we assume that all non-protocol com-

ponents of an apparatus (in particular the temperatures

and chemical potentials) do not change in time. In addi-

tion, we assume that for all α, the process takes place in

the same time interval, [ti, tf]. We write an element of

Xas x, and a trajectory of Xvalues across [ti, tf]as x

We suppose that αis not precisely known and write

its probability measure as dPα. Physically, it may be

that we have an inﬁnite set of apparatuses generated by

IID sampling dPα. Alternatively, dPαcould represent

Bayesian uncertainty or a detailed model of the noise in

the measuring instruments used to set the parameters in

α. (Below, we will often abuse notation/terminology and

refer to a “distribution” over apparatuses when properly

speaking, we should be couching the discussion in terms

of a probability measure.) Abusing notation, we will use

Ato denote both the random variable with values α, and

the event space of that random variable [5].

Concretely, we consider two kinds of experimental sce-

narios. Both start by sampling dPα, but they diﬀer after

that:

I) In the eﬀective scenario, we generate an appara-

tus by sampling dPα. For that ﬁxed apparatus

we then generate many stochastic trajectories x

After running all those trajectories for that ﬁxed

apparatus, we can, if we wish, rerun the scenario,

generating another sample of the distribution over

apparatuses, which we then use to generate a new

set of stochastic trajectories. We call this the ef-

fective scenario.

Experimentally, in the eﬀective scenario, one can

generate and then observe frequency counts of the

distribution p(x

x|α)for multiple random values of α,

but without ever directly observing α. For exam-

ple, the experimenter might construct a bit-eraser

experimental apparatus involving a single thermal

reservoir whose temperature is ﬁxed throughout

the experiment but only known to the ﬁnite pre-

cision of .1◦K. The experimenter then runs their

experiment many times using this ﬁxed apparatus

and collected statistics concerning the trajectories

across those experiments. They can then use those

estimates to make (perhaps Bayesian) estimates of

thermodynamic functions of a trajectory, like the

associated entropy production.

II) In the phenomenological scenario, we again gener-

ate an apparatus by sampling dPα, but the appa-

ratus cannot be ﬁxed while we generate multiple

trajectories. Instead, in order to generate a new

trajectory we must ﬁrst generate a new apparatus

by resampling dPα. We call this the phenomeno-

logical scenario.

To illustrate the phenomenological scenario we can

return to the example where the experimenter con-

structs a bit-eraser experimental apparatus involv-

ing a single thermal reservoir whose temperature

at the beginning of the experiment is only known

to some ﬁnite precision of .001◦K. Suppose though

that the temperature is very slowly drifting ran-

domly in time. Any given run of the experiment

is very fast on the timescale of that drift, so we

can treat the temperature as ﬁxed throughout the

run. However, after generating a trajectory by

running the experiment, it takes a long time for

the system to be reinitialized to rerun the experi-

ment, and during that time the temperature has

drifted to a new value that is statistically inde-

pendent of the value during the preceding run.

As in the eﬀective scenario, the experimenter runs

their experiment many times and collected statis-

tics concerning (functions of) the trajectories across

those experiments. However, in the phenomeno-

logical scenario, one can only observe frequency

counts of the α-averaged distribution over trajec-

tories, ¯p(x

x) := RdPαp(x

x|α).

Illustrations of both scenarios are depicted in Fig. 1, for

a simple three-state time-homogeneous system coupled

to one of the three possible apparatuses. In each case,

we measure the marginal probability distribution at the

ﬁnal time tf. Note that in neither scenario do we allow

any direct measurement of α. However, there may be

indirect information about αthat arises from the precise

trajectory of states that is generated once αis chosen.

Crucially, the ensemble-level thermodynamic quanti-

ties generated in these two scenarios can diﬀer, since the

two types of average involved (once over α, once over x)

do not necessarily commute. As an example, in the eﬀec-

tive scenario, since we can form an estimate of pt(x|α)by

running many iterations of a ﬁxed experimental appara-

tus, we can experimentally estimate the entropy deﬁned

S(Pt) = −ZdPαX

pt(x|α) ln pt(x|α)(1)

using empirical frequency counts.

This is not possible in the phenomenological scenario,

in which we can only experimentally estimate a more

“coarse-grained” version of entropy,

S(¯

Pt) = −X

xZdPαpt(x|α)ln ZdPαpt(x|α)(2)

E0E1

w12

w20

w10

w01

w21

w02

α1

p(α1)

α2

p(α2)

α3

p(α3)

(a)3-state system coupled to one of the apparatuses

Trajectories

ptf(E)=?

(b)distribution at time tf

tit

α1ptf(E|α1)

(c)Effective scenario: α1

tit

α2ptf(E|α2)

(d): Effective scenario: α2

tit

α3ptf(E|α3)

(e)Effective scenario: α3

FIG. 1. Comparison of the two uncertain apparatus scenarios considered in this paper. a) A simple, a three-state system that

can be coupled to one of the three apparatuses, with respective probabilities. b) We will plot trajectories of the values of a

quantity Ethat has three possible values across the time interval [ti, tf], along with the associated empirical estimate of the

relative probabilities that the system had each of those three values during [ti, tf]c),d),e) Plots for the eﬀective scenario, where

we can estimate the marginal distribution at time tfptf(E|α)for a ﬁxed apparatus α, shown for three separate instances of

the scenario corresponding to the three possible apparatuses. f) The phenomenological scenario, in which each trajectory is

sampled with a diﬀerent apparatus, and therefore only ¯ptf(E)can be estimated.

using empirical frequency counts.

Note also that if we are in the eﬀective scenario and

have the ability to force a new apparatus to be (ran-

domly) generated whenever we want, then we can imple-

ment the phenomenological scenario just by forcing a new

apparatus to be generated after every run. In this aug-

mented version of the eﬀective scenario, we could experi-

mentally estimate the quantity in Eq. (2) using empirical

frequency counts. However, if we are in the eﬀective sce-

nario and do not have this extra ability, then we cannot

estimate the quantity in Eq. (2), only the quantity in

Eq. (1). (In this paper, whenever we discuss the eﬀec-

tive scenario, we will assume we do not have this extra

ability.)

The diﬀerence between the thermodynamics of the two

scenarios will be a central focus of our analysis below.

Related research

It is important to distinguish between the focus of this

paper and some of the issues that have been investigated

in the recent literature. Some recent research has con-

sidered how to modify stochastic thermodynamics if the

experimentalist is not able to view all state transitions in

the system as it evolves [6, 7]. The uncertainty in these

papers concerns what is observed as the system evolves,

whereas we focus on uncertainty in the parameters gov-

erning that evolution. Similarly, some models consider

either spatial [8] or temporal [9] variation of temperature

and other parameters, but they assume that this evolu-

tion is known. In contrast, we assume that αis ﬁxed

throughout the interval, but to an unknown value.

Probably the closest research to what we consider in

this paper is sometimes called superstatistics. It has long

been known that an average over Gibbs distributions can-

not be written as some single Gibbs distribution (Thm. 1

in [10]). This means that even equilibrium statistical

physics must be modiﬁed when there is uncertainty in the

temperature of a system. The analysis of these modiﬁca-

tions was begun by Beck and Cohen [11], who developed

an eﬀective theory for thermodynamics with temperature

ﬂuctuating in time. They considered a system coupled

to a bath, which is in a local equilibrium under the slow

evolution of the temperature of the bath. The main as-

sumption they exploit is scale-separation: while for short

time scales, the distribution over states of the system is

an equilibrium, canonical distribution with inverse tem-

perature β, the long-scale behavior is determined by a su-

perposition of canonical distributions with some distribu-

tion of temperatures f(β). The resulting superstatistical

distribution p(E) = Rdβf(β) exp(−βE)/Z(β)was later

identiﬁed with the distribution corresponding to gener-

alized entropic functionals [12, 13] due to the fact that

particular generalized entropic functionals are maximized

by the same distribution that can be obtained the su-

perposition of the canonical distribution with given f(β)

[14, 15].

Later interpretations of superstatistics are not based

on the notion of local equilibria but rather on the

Bayesian approach to systems with uncertain tempera-

ture [16, 17]. These are conceptually closer to the focus

of this paper, which focuses on oﬀ-equilibrium systems

that are evolving quickly on the scale of the coupling

with the thermal reservoirs, and so cannot be modeled in

terms of time-scale separation.

Similar to the quasi-equilibrium scenarios considered

in superstatistics, other research has focused on deriving

an eﬀective description of the system in local equilibrium

averaged over uncertain thermodynamic parameters. In

particular, this is the basis of a very rich and well-studied

approach to analyzing spin glasses [18, 19], in which

the coupling constants Jij in the spin-glass Hamiltonian

H=−P(ij)Jij sisj, are random variables drawn from a

given distribution p(Jij ). Given such a distribution, the

famous replica trick ln Z= limn→0Zn−1

n[20] can be used

to calculate the Helmholtz free energy, averaged over all

Jij . Let us note that in the terminology used in disor-

dered systems, the annealed disorder corresponds to the

eﬀective scenario while quenched disorder corresponds to

the phenomenological scenario.

Finally, several authors [21–24] investigated the case

where the initial distribution diﬀers from the one that

would minimize EP. It describes the situation when the

system is designed by a scientist who was mistaken in

their assumption concerning the initial distribution. In

this case, the choice of the non-optimal solution gener-

ates the extra entropy production that can be described

by the so-called mismatch cost. These papers do not in-

volve a distribution diﬀerent initial distribution that is

re-sampled each time the experiment is re-run.

Roadmap

One of the major themes of our investigation is that

some of the details of how an experiment is conducted

that experimenters currently do not consider in fact have

major eﬀects on the precise forms of various thermody-

namic quantities. This is reﬂected in the diﬀerence be-

tween (the thermodynamics of) the eﬀective and phe-

nomenological scenarios, discussed above. Even within

the eﬀective scenario though, there are some important

distinctions between diﬀerent ways of running the exper-

iment (and so diﬀerent ways of deﬁning thermodynamic

quantities). In particular, there is a major eﬀect on the

thermodynamics itself that arises from whether the ex-

perimenter the protocol (time-dependent trajectory of

Hamiltonians of the system) changes from one run of an

experiment to the next, or instead is ﬁxed in all runs.

We call these the “unadapted” and “adapted” situations,

respectively. We start in Section II with a simple illustra-

tive example of these two situations, involving a moving

optical tweezer with uncertain stiﬀness parameter.

We then begin our more general analysis. First, in

Section III, we introduce the necessary notation and

brieﬂy recall the main results of traditional, full-certainty

stochastic thermodynamics. In Section IV, we present

the general form that stochastic thermodynamics takes in

the eﬀective scenario (recall the discussion of the eﬀective

and phenomenological scenarios in the introduction). We

begin by noting that the evolution of the eﬀective proba-

bility distribution is not Markovian. Then we derive the

forms of the ﬁrst and second laws of thermodynamics

for eﬀective thermodynamic quantities. Next we discuss

the relation between eﬀective EP and eﬀective dissipated

work. We illustrate this discussion with the numerical

example of a fermionic bit erasure with uncertain tem-

perature. We end this section by investigating the special

case where the only uncertainty concerns the initial dis-

tribution, calculating the associated eﬀective mismatch

cost.

In Section VI, we focus on (feedback) control proto-

cols for uncertain apparatuses. In contrast to the con-

ventional case where the apparatus is precisely known,

we assume we cannot tailor the protocol for each (un-

certain) apparatus separately, but instead must use the

same protocol for all apparatuses. We use this setting to

investigate how apparatus uncertainty aﬀects a founda-

tional concern of stochastic thermodynamics: How much

work can be extracted from a system during a process

that takes it from a given initial distribution to a given

target distribution.

First, we consider this issue when we are uncertain

both about the initial distribution (though not the ﬁ-

nal one) and about the temperature of the system as it

evolves. We focus on how that uncertainty changes the

results of the standard analysis of this issue, in which we

suppose a {quench; equilibrate; semi-statically-evolve}

process is applied to the system immediately after the

initial distribution is generated.

Next, we use this analysis to consider how uncertainty

aﬀects the “thermodynamic value of information” to a

feedback controller [25, 26]. We restrict attention to

the special case of the analysis where the temperature

is known exactly, so the only uncertainty is in the ini-

tial distribution. We also suppose that there is a (per-

fectly known) delay between when the initial distribution

is generated, ti, and the time τwhen the {quench; equi-

librate; semi-statically-evolve} process can begin, during

which time the system evolves according to a (perfectly

known) rate matrix. In particular, we derive expressions

for how the thermodynamic value of information varies

with the length of the delay.

In Section VII, we investigate the ensemble entropy

production calculated from eﬀective trajectory probabil-

ities, i.e., from trajectory probabilities given by averag-

ing over apparatuses. We call this the phenomenological

(ensemble) EP. We begin by proving that phenomeno-

logical ensemble EP is a lower bound on the average over

apparatuses of the eﬀective ensemble EP. So ﬁxing the

apparatus and averaging over the trajectories — though

without knowing what value the apparatus is ﬁxed to —

and then averaging over apparatuses increases EP, com-

pared to the case where we average apparatuses before

averaging trajectories.

The diﬀerence between eﬀective EP and phenomeno-

logical EP is called likelihood EP. It measures the diﬀer-

ence between log-likelihood functions estimated from the

forward and time-reversed trajectories.

Considering trajectory versions of all three EPs, we

establish three detailed ﬂuctuation theorems (DFT). In

addition to the well-studied DFT in the literature which

concerns a single, known apparatus, we establish the

DFT for the phenomenological EP and for likelihood EP.

The former represents the eﬀective irreversibility of the

system by coarse-graining all the apparatuses. The lat-

ter represents how irreversibility aﬀects the estimation of

the apparatus’ parameters when estimated by observing

the forward and time-reversed trajectories. These results

are illustrated by a simple example of a two-state system

coupled to one heat reservoir with uncertain tempera-

ture.

The paper ends with a discussion section in which we

describe just a few of the myriad directions for future

work.

II. ILLUSTRATIVE EXAMPLE

In this section we illustrate the importance of account-

ing for the uncertainty of the system parameters in an

experiment, with a simple example of a colloidal particle

in a moving laser trap. The dynamics of the particle is

given by the overdamped Langevin equation

˙x=−µ∂V

∂x +ξ

where ξis the white noise, and Vis the potential. Let us

consider that the particle is dragged by an optical tweezer

with the harmonic potential

Vk(x, t) = k

2(x−λ(t))2

where kis the stiﬀness parameter and λ(t)is the control

protocol. The average work is given by the Sekimoto

formula

W[λ(t)] = Ztf

dt˙

λVk(λ(t), x(t))

∂λ 

where h..iis the ensemble average. Let us consider µ= 1.

Our aim is to move the trap from λi= 0 at time ti= 0 to

λfat time tfsuch that the average work is minimal. Fol-

lowing [27], it is possible to express the optimal protocol

starting that minimizes the average work as

λ?

k=λf(1 + kt)

2 + ktf

and the corresponding optimal work as

k=kλ2

2 + ktf

(3)

The complete derivation is done in Appendix A.

We focus on the realistic situation where the experi-

menter has to measure the stiﬀness parameter to be able

to determine the optimal protocol. The estimation is typ-

ically done by repeated measurement of k, which leads to

a histogram of k. In practice, often an experimenter will

implicitly assume that the uncertainty in kis due to the

measurement and takes the average value of stiﬀness ¯

kas

the single possible value. However, often the uncertainty

in the parameters can have a physical reason, e.g., impre-

cise calibration of the laser. In those kinds of scenarios

the stiﬀness can change for each run of the experiment,

and so the experimenter’s implicit assumption is invalid.

Write the stiﬀness parameter that the experimenter

uses to set up the control protocol as k, with the real

stiﬀness parameter written as κ(which in general diﬀers

from k). In Appendix A we show that the work can then

be expressed as

Wκ[λk(t)] = W?

+λ2

(2 + ktf)2κ2−k2

κ+(k−κ)2

κe−κtf

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NonequilibriumthermodynamicsofuncertainstochasticprocessesJanKorbel1,2andDavidH.Wolpert3,2,4,5,61SectionforScienceofComplexSystems,CeMSIIS,MedicalUniversityofVienna,Spitalgasse23,1090Vienna,Austria*2ComplexityScienceHubVienna,JosefstädterStrasse39,1080Vienna,Austria3SantaFeInstitute,SantaFe,NM,USA4...

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Nonequilibrium thermodynamics of uncertain stochastic processes Jan Korbel1 2and David H. Wolpert3 2 4 5 6 1Section for Science of Complex Systems CeMSIIS Medical University of Vienna Spitalgasse 23 1090 Vienna Austria

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