The vanishing of excess heat for nonequilibrium processes reaching zero ambient temperature Faezeh Khodabandehlou1Christian Maes1Irene Maes2and Karel Neto cn y3

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The vanishing of excess heat for

nonequilibrium processes reaching zero ambient temperature

Faezeh Khodabandehlou,1Christian Maes,1Irene Maes,2and Karel Netoˇcn´y3

1Instituut voor Theoretische Fysica, KU Leuven, Belgium

2Departement Wiskunde, KU Leuven, Belgium

3Institute of Physics, Czech Academy of Sciences, Prague, Czech Republic

We present the mathematical ingredients for an extension of the Third Law of

Thermodynamics (Nernst heat postulate) to nonequilibrium processes. The central

quantity is the excess heat which measures the quasistatic addition to the steady

dissipative power when a parameter in the dynamics is changed slowly. We prove

for a class of driven Markov jump processes that it vanishes at zero environment

temperature. Furthermore, the nonequilibrium heat capacity goes to zero with tem-

perature as well. Main ingredients in the proof are the matrix-forest theorem for

the relaxation behavior of the heat ﬂux, and the matrix-tree theorem giving the low-

temperature asymptotics of the stationary probability. The main new condition for

the extended Third Law requires the absence of major (low-temperature induced)

delays in the relaxation to the steady dissipative structure.

Keywords: Nernst postulate, excess heat, nonequilibrium heat capacity, matrix-forest theo-

rem.

I. INTRODUCTION

The Nernst Postulate (1907) states that the change of entropy in any isothermal process

approaches zero as the temperature reaches absolute zero1. It evolved into the Planck

version of the Third Law of Thermodynamics, stating that the entropy of a perfect crystal

of a pure substance vanishes at absolute zero. That Third Law diﬀers from the First and

Second Laws, as it does not automatically follow from more microscopic considerations.

1Its historical origin lies in the variational principle of Thomsen and Berthelot, which was an empirical

precursor of the Gibbs variational principle, [1].

arXiv:2210.09858v2 [cond-mat.stat-mech] 13 Apr 2023

In fact, it sometimes fails, as exempliﬁed by ice models in [2]. Nevertheless, theorems on

the validity of the Third Law have been obtained for a large class of equilibria, cf. [3]: for

quantum and classical lattice systems, the entropy density at absolute zero temperature is

directly related to the degeneracy of the ground state corresponding to boundary conditions

with the highest degeneracy. The Third Law holds for example for any ferromagnetic Ising

system. Counterexamples include dimer systems and (spin) ice models, [2, 4–6].

It is natural to ask about thermal properties of nonequilibria as well. Then appears the

question of low-temperature asymptotics and a possible formulation of an extended Third

Law. Note that the temperature (which goes to zero) need not be physically associated

to the degrees of freedom of the system but rather to a thermal bath or large equilibrium

environment weakly coupled to the system. The main point to remember is that steady

nonequilibrium systems are open (and possibly small) and constantly dissipate heat into

the (much larger) environment. When parameters change, such as the temperature of that

heat bath or the volume of the system, the heat ﬂux may change. In other words, when

connecting two nonequilibrium conditions via a quasistatic transformation, an excess heat

may ﬂow. The present paper rigorously deﬁnes that notion of excess heat, and we prove,

under certain conditions, that this excess heat vanishes at zero ambient temperature. We

do that for irreducible (continuous time) Markov jump processes with ﬁnite state space. To

connect their mathematics with thermal properties, we require the interpretation of local

detailed balance, which identiﬁes the heat during a transition with the logarithmic ratio of

forward to backward rates.

In Section II we start with the setup, deﬁnitions and a presentation of the main results.

In particular, we specify the Markov jump processes with the physical interpretation of heat

and dissipated power in a thermal bath. The heat ﬂux is parameterized by the inverse

temperature βof the bath and by nother (external) real parameters α, summarized in

λ= (β−1, α)∈Rn+1. We show that when making a quasistatic transformation over a curve

Γ in that parameter space, the expected excess heat to the heat bath equals

Q(Γ) = ZΓ

dλ·DλDλ=h∇λVλis

λ(I.1)

in terms of the so-called quasipotential Vλ(x). The notation h·is

λindicates an average in the

stationary probability distribution ρs

λat parameter values λ.

Toward the end of Section II we already informally discuss our main result, that Dλ

vanishes as the temperature of the thermal bath tends to absolute zero. In fact, Dλ=

(−Cα(β), Dβ(α)) splits up in two components, depending on whether βis kept constant (in

Dβ(α)), or if αis kept constant (in Cα(β)) during the quasistatic transformation. Theorem

IV.1 gives conditions for Dβ(α) to vanish at absolute temperature. The heat capacity

Cα(β) = β2h∂

∂β Vλiλ(I.2)

is the variation with temperature of the excess heat toward the system. We show that

Cα(β) tends to zero as well for inverse temperature β↑ ∞ [Theorem IV.2].

The conditions for both Theorems are introduced at the end of Section II. We already

discuss the static condition in Section II C, to obtain ∇λρs

λ→0 when β↑ ∞.

Section III introduces the graph-theoretic elements needed for the proof of the boundedness

of the quasipotential. It can be stated in terms of the matrix-forest theorem. The suﬃcient

conditions are illustrated in Section C with simple examples, to show that they are on

target.

The actual proofs of the main theorems start in Section V. It ﬁrst concentrates on showing

the geometric result (I.1). Section V B shows the boundedness of Vλin β↑ ∞.

A summary of results and arguments is given in Section VI.

The Appendix makes the explicit links with the matrix-forest theorem.

We end this introduction by giving some additional context and background. Steady

state thermodynamics has been started in a number of papers like [7–9], where it attempts

to remove the condition of close-to-equilibrium that was prominent in much of irreversible

thermodynamics [10]. In particular, the idea of excess heat as used in the present paper has

been discussed in [8, 9]. Then, around 2011, nonequilibrium heat capacities were explicitly

introduced and examples were discussed in [11–14]. Other deﬁnitions of nonequilibrium heat

capacity have been proposed in various papers, including [15–17].

For more than a decade then, it remained very much an open question whether and when

those heat capacities would vanish at absolute zero. Only with the present paper, a precise

answer is given.

A physics presentation of the mathematical results contained here is found in [18]. In

particular, we discuss there how our setup covers certain quantum features and indeed ﬁts

the Nernst Postulate as usually presented in thermochemistry. We also give there a heuristic

argument to show that Third Law behavior follows when relaxation times do not exceed the

dissipation time. More illustrations and calculations of nonequilibrium heat capacities are

found in [19–21]. The present paper gives the rigorous version including all mathematical

details concerning the quasistatic limit of the excess heat, and the graphical-expression of the

quasipotential that leads to the proof of its boundedness. That has not appeared elsewhere.

II. SETUP AND MAIN RESULTS

A. Markov jump process

Consider a simple, connected and ﬁnite graph G= (V(G),E(G)) with vertex set V(G) =

Vand edge set E(G) = E. Vertices are written as x, y, . . . ; edges are denoted by e:= {x, y}

when unoriented and e:= (x, y) is an oriented (or, directed) edge which starts in xand ends

in y.

We consider a Markov jump process Xt=Xλ

t∈ V for times t≥0 and with rates kλ(x, y)>0

for the transition xto ywhen {x, y} ∈ E, and otherwise the rates are zero. The λrefers to

the dependence of the process on n+ 1 real parameters λ= (β−1, α), α = (αi, i = 1, . . . , n)

with β≥0 interpreted as the inverse temperature of a thermal bath, and other parameters

α∈ A for some open set A ⊂ Rn. The graph Gis ﬁxed throughout, and does not depend

on λ. We assume that all the transition rates kλ(x, y) are smooth in λ, which implies

smoothness of all derived quantities.

By irreducibility, there is a unique stationary probability distribution ρs

λ(x)>0, x ∈ V,

solution of the stationary Master Equations,

kλ(x, y)ρs

λ(x) = X

kλ(y, x)ρs

λ(y) (II.1)

Stationary expectations with respect to ρs

λare written as hfis

λ=Pxf(x)ρs

λ(x) for functions

f(x), x ∈ V.

The backward generator Lλis the |V| × |V|-matrix having elements Lλ(x, y) = kλ(x, y) and

Lλ(x, x) = −Pykλ(x, y). We ignore the λ−dependence in the notations if no confusion

can arise.

As physical orientation, we think of the vertices as states or conﬁgurations of an open

system. The transitions between states are possibly accompanied by exchanges of energy

or particles with the environment. The βis the inverse temperature of the environment

and the parameters αmay appear in (interaction or self-) energies, or can quantify external

parameters such as spatial volume or boundary conditions.

B. Excess heat

We assume that

βlog kλ(x, y)

kλ(y, x)=qα(x, y) (II.2)

does not depend on β. The qα(x, y) are obviously antisymmetric. Following the physical

condition of local detailed balance [22], qα(x, y) is interpreted as the heat to the thermal

bath (at inverse temperature β) in the transition xto y.

We then write

Pλ(x) := X

kλ(x, y)qα(x, y).(II.3)

for the expected instantaneous power when in x. By convexity hPλis

λ≥0, and hPλis

λis

called the stationary dissipated power. An important quantity will be the quasipotential Vλ,

a function on Vdeﬁned as

Vλ(x) := Z+∞

dt[hPλ(Xt)|X0=xiλ− hPλis

λ] (II.4)

=Z+∞

dt etL[Pλ− hPλis

λ](x) (II.5)

where in the last equality appears the semigroup S(t) = etL for which hg(Xt)|X0=xiλ=

S(t)g(x) for an arbitrary function g. The quasipotential will appear in the quasistatic

expression of excess heat, to which we turn next.

Given a smooth time-dependence λ(t),0≤t≤1, of the parameters, we call Γ its image

as a curve in parameter space R+× A. For a quasistatic process, we write λεand consider

a protocol where the system evolves under λε(t) := λ(εt),0≤t≤ε−1on Γ. The εis the

rate of change in the parameter protocol where λevolves. The limit where ε↓0 will make

the process to become quasistatic. For such a time-dependent process, at every moment t

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ThevanishingofexcessheatfornonequilibriumprocessesreachingzeroambienttemperatureFaezehKhodabandehlou,1ChristianMaes,1IreneMaes,2andKarelNetocny31InstituutvoorTheoretischeFysica,KULeuven,Belgium2DepartementWiskunde,KULeuven,Belgium3InstituteofPhysics,CzechAcademyofSciences,Prague,CzechRepublicWepre...

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The vanishing of excess heat for nonequilibrium processes reaching zero ambient temperature Faezeh Khodabandehlou1Christian Maes1Irene Maes2and Karel Neto cn y3

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