Stochastic thermodynamics of a finite quantum system coupled to a heat bath

1 Introduction

The study of nonequilibrium thermodynamics of systems in contact with a thermal reservoir (“heat bath”) of different temperatures has a long history. As an example of the approach via a Master equation and its weak coupling limit, we mention the work of Lebowitz and Spohn [1], which actually considers a finite number of reservoirs. During the last decades, new methods have been devised, in particular, the approach via fluctuation theorems [2]. The famous Jarzynski equation represents one of the rare exact results of nonequilibrium statistical mechanics. It is a statement about the expectation value of the exponential of the work e−βw performed on a system initially in thermal equilibrium with inverse temperature β, but possibly far from equilibrium after the work process. This equation was first formulated for classical systems [3] and subsequently proved for quantum systems [4–6]. Extensions for systems that are initially in local thermal equilibrium [5], micro-canonical ensembles [7], and grand-canonical ensembles [8–14] have been published. The literature on the Jarzynski equation and its applications is rich; a concise review is given in [14], focusing on the connection with other fluctuation theorems. The most common approach to the quantum Jarzynski equation is to consider sequential measurements. This approach is also followed in the present work. The general framework for such an approach was outlined in [15] and [16]. It is per se neither quantum mechanical nor classical and will be referred to as “stochastic thermodynamics” in the present work. At the end of Appendix C, we sketch realizations of the general framework in quantum mechanics and classical mechanics.

Interestingly, one can derive from the Jarzynski equation certain inequalities that resemble the second law, see, e.g., [14]. However, a closer inspection shows that these inequalities are not exactly statements about the non-decrease of entropy. But the entropy balance is not the problem: the total von Neumann entropy is constant during unitary time evolution and non-decreasing during projective measurements, see [17] or Theorem 11.9 in [18]. The problem is rather that in the quantum case, the entropy balance is not sufficient to cover all aspects of the second law.

To explain the latter, consider classical thermodynamics where there are several equivalent formulations of the second law. For example, from the non-decrease in total entropy, it can be deduced that heat (and entropy) always flows from the hotter to the colder body. The elementary argument goes as follows: If the hotter body with inverse temperature β transfers the infinitesimal heat δQ (of whatever sign) to the colder body with inverse temperature β₀ > β, then its entropy decrease will be dS₁ = βδQ, according to the Clausius equality. On the other hand, the colder body receives the heat δQ and its entropy increases by dS₂ = β₀δQ. The total entropy increase will be dS = dS₂ − dS₁ = (β₀ − β)δQ. Since (β₀ − β) > 0, we obtain dS ≥ 0 ⇔ δQ ≥ 0.

In the quantum case, this elementary argument breaks down. Between the two sequential measurements, the transferred heat ΔQ and entropy ΔS can no longer be considered as infinitesimal and the Clausius equality has to be replaced by two inequalities. Moreover, the energy increase of the first system is not exactly equal to the energy decrease of the second system. This holds only approximately if the interaction Hamiltonian can be neglected, which will not be the case for small ΔQ. (Strictly speaking, the latter objection already applies to the classical case.)

We will try here to modify the classical argument for the “correct” heat flow for quantum mechanics. To this end, we will adopt another approach to the problem of the direction of the heat flow that focuses on the N-level quantum system and describes the influence of the heat bath solely in terms of a transition matrix T. T is only a (left) stochastic matrix and can no longer be assumed to be bi-stochastic (in the strict or modified sense) and hence, the usual assumptions leading to a Jarzynski-type equation, see [15], are no longer satisfied. But it is possible to derive a more general J-equation that is only based on (left) stochasticity of T. Thus, we can find arguments for the “correct” flow of heat and entropy that only rely on the assumption that T leaves invariant some Gibbs state with inverse temperature β₀, see Section 2. β₀ is interpreted as the inverse temperature of the heat bath. In this sense, we derive the second law of thermodynamics from a form of the zeroth law. For this derivation, we combine known results that appear in various places in the literature, see [19–23], but are sometimes only proved under assumptions that are stronger than those we will assume in the present work. We will present the relationship of these references to our results in more detail in Section 4.

The structure of the paper is as follows. The general definitions and main results on the heat flow between the system and the heat bath are given in Section 2. These results are based on a Jarzynski-type equation (19), which is proved in a more general setting in Appendix C. Here, we also comment on the possibility to relax the usual assumption of an initial product state. Close to the equilibrium point, the entropy increase ΔS and the absorbed heat over temperature βΔQ have a common tangent, see Figure 1, with a positive slope, as proved in Appendix B. The analogous result on the entropy flow is formulated in Section 3. It depends on two “Clausius inequalities”, see Eq. (67), the second one of which again follows from the Jarzynski-type equation. The bi-stochastic limit is shortly considered in Section 4. The next Section 5 contains an analytically solvable example. We close with a summary and outlook in Section 6.

Figure 1:

Typical plot of the two “Clausius heat terms” β₀ΔQ, βΔQ, and the “entropy increase” ΔS as functions of the inverse temperature β of the NLS calculated for a randomly chosen transition matrix T and N = 4. Note that β₀ΔQ ≤ ΔS ≤ βΔQ holds in accordance with (67). The inverse temperature of the heat bath is β₀, where all three functions vanish and have a common tangent (dashed black line) with the slope a according to (56).

2 Main results on heat flow

We consider an N-level system (NLS) described by a finite index set N, energies E_n, and degeneracies d_n for n∈N. The NLS is assumed to be initially in a Gibbs state with probabilities

where the partition function Z is defined by

and β=1τ is the inverse temperature of the NLS. After an interaction with a heat bath a subsequent measurement of energy finds the NLS in the level m∈N with probability

Here, the “transition matrix” T is an N × N (left) stochastic matrix, i.e., satisfying

The entries of T will be sometimes written as conditional probabilities T_mn = P(m ← n) with self-explaining notation. We do not make any assumptions concerning thermalization and hence, the final probabilities q_m will, in general, not be of Gibbs type.

The usual Jarzynski equation is based on the property of T being a bi-stochastic matrix, i.e., additionally satisfying, in the non-degenerate case of d_n ≡ 1,

or, in the general case,

see Eq. (25) in [15]. This property holds if the system is closed and only subject to external forces performing work upon the system. But bi-stochasticity is no longer guaranteed for systems coupled to other ones (heat baths). However, if this is the case and if no external forces are applied, we may relax the (modified) bi-stochasticity of T to the following:

In this case, the transition matrix T, satisfying (4), (5) and (10), will be called a “Gibbs matrix” with temperature τ0=1β0. We will also refer to τ₀ as the “temperature of the heat bath”.

Recall that every (left) stochastic matrix T has an eigenvector p⁽⁰⁾ with non-negative entries corresponding to the eigenvalue 1, although p⁽⁰⁾ is generally not unique. If the entries pn(0) are different and positive, one can always define suitable energies En≔−logpn(0) such that (8) holds with β₀ = 1 and d_n ≡ 1. This has been used to generate numerical examples of Gibbs matrices, see Figures 1 and 2. Mathematically, T being bi-stochastic is a special case of being a Gibbs matrix, since (6) follows from (10) for β₀ = 0 and d_n = 1 for all n∈N. However, according to the above remarks, being a Gibbs matrix should rather be considered as a property of T relative to a given family of energies E_n, not as a property of T alone.

Figure 2:

The same plot as in Figure 1 but for the extended range of inverse temperatures −10β₀ ≤ β ≤ 10β₀ of the NLS. Note that also in this extended range β₀ΔQ ≤ ΔS ≤ βΔQ holds in accordance with (67). For negative temperatures, these “Clausius inequalities” are less restrictive since here β₀ΔQ and βΔQ have different signs. ΔS has a second zero for negative inverse temperatures that, in contrast to the first zero at β = β₀, does not correspond to a fixed point of the transition matrix T. For β → ±∞, the Gibbs state probability p(β) is concentrated on the level with the lowest/highest energy, respectively, and hence β₀ΔQ as well as ΔS approach constant values. Therefore, βΔQ is asymptotically linear at these limits.

Physically, the property (10) appears plausible if T represents the transition matrix due to the interaction with a heat bath of temperature τ₀. If the NLS already has the same temperature τ₀, its state should not change. This is not trivial since, in general, the Gibbs state of the combined system, NLS plus heat bath, with temperature τ₀ does not commute with the interaction Hamiltonian. However, it can be shown that (10) holds exactly for some analytically solvable examples [24], and in other cases, the real situation can be expected to be represented by (10) to an excellent approximation.

Next, we will recall some probabilistic framework concepts for the Jarzynski-type equation, see [15]. Let N×N be the set of “elementary events” such that one event (m,n)∈N×N represents the outcome of a sequential energy measurement at the NLS in the sense that the initial measurement yields the result E_n, and, after the interaction with the heat bath, the final measurement yields E_m. The probability function

defined for elementary events is given by

Analogously to the case of the ordinary Jarzynski equation, we consider random variables Y:N×N→R and their expectation value denoted by

An example is

defined by

This can be interpreted as the “heat” transferred to the NLS during the interaction with the heat bath since we have assumed that no external forces are active that could perform work on the NLS.

Another example is the random variable “entropy increase”

defined by

Its expectation value agrees with the familiar expression for entropy decrease, as will be shown in Section 3, see Eqs. (35)–(37).

Sometimes, instead of (13), we will also use the sloppy notation ⟨Y(m, n)⟩ for the expectation value.

Then we can state the following Jarzynski-type equation that follows from the general “J-equation” considered in Appendix C.

For the proof see Appendix C, where (19) is obtained as a special case.

The Jarzynski-type equation (19) is more of a template that can be used to generate further equations by choosing a special form of the general probability distribution p̃. As a particular choice, we will consider p̃=p. This yields

and further, using

and (15), the following equation:

This equation was also derived in [19] under stronger assumptions. As pointed out in [19], Eq. (23) implies that the probability of events where heat flows in the “wrong” direction, i.e., where β−β0ΔQ<0, must be exponentially suppressed. The reason is that in the case of a Jarzynski-type equation of the form eX=1 the contributions to the expectation value from large positive values of X must be counterbalanced by a large number of contributions from negative values of X in order to maintain the expectation value at 1.

As for the original Jarzynski equation, we may derive an inequality by invoking Jensen’s inequality (JI). Note that x ↦ −log x is a convex function. Hence

Thus, we have proven:

If the temperature τ of the NLS is lower than the temperature τ₀ of the heat bath, then β − β₀ ≥ 0 and hence, by means of (26), ΔQ≥0. It means that in this case the expectation value of the heat flowing into the NLS will be positive, and vice versa. In other words, heat will flow from the hotter body to the colder one, analogously to the result in classical thermodynamics, see Section 1.

3 Clausius inequalities

We adopt the notation of Section 2 but for the next steps will not need Assumption 1. Further define (sometimes skipping the expectation brackets ⟨…⟩ if no misunderstanding can occur):

Note that (1) implies the familiar identity

To show the consistency of (33) with the definition (17), we calculate the expectation value of the corresponding random variable

which agrees with (33).

Then, we can show the following:

This inequality has also be obtained in [20] by considering two systems with weak interaction and using the fact that the Gibbs state minimizes the free energy. This statement, in turn, can also be proven by the Gibbs inequality used below.

Interestingly, the first Clausius inequality can be sharpened to a Clausius equality in the weak coupling limit.

The proof can be found in Appendix A.

Next, we assume the situation of a “heat process” as in Section 2 together with Assumption 1 and hence, can interpret the energy difference ⟨ΔE⟩ as the heat ⟨ΔQ⟩ transferred to the NLS. We consider both sides of (38), β⟨ΔQ⟩ and ⟨ΔS⟩, as functions of the inverse temperature β. Both functions vanish at the inverse temperature β₀ of the heat bath and, due to (38), must have a common tangent at β = β₀, see Figure 1. We will calculate its slope a using the intermediate results

This yields

using Eq. (5) in (60). According to Theorem 1, it is clear that the slope of the tangent cannot be negative, a ≥ 0. Nevertheless, this will be checked independently, see Appendix B.

The linear part of the Taylor series of β⟨ΔQ⟩ w.r.t. (β − β₀) can be used to re-write ⟨ΔQ⟩ as a function of the dimensionless temperature τ=1β such that the zero of ⟨ΔQ⟩ at β = β₀ corresponds to the temperature τ0=1β0. The result

resembles the Fourier law or its precursor, Newton’s law of cooling [25], stating that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. We further remark that the explicit form (60) of the “heat conduction coefficient” a β₀ in (61) is reminiscent of the fluctuation–dissipation theorems mentioned in [7] in connection with the Jarzynski equation, see also Appendix B.2.

The above result that βΔQ=ΔS+O(β−β0)2 can be viewed as a confirmation of the Clausius identity in linear stochastic thermodynamics. The fact that the deviation to the Clausius identity is non-negative in the sense of Theorem 3 can be made plausible in the following way. Consider a state change of an NLS with a slightly lower temperature than the heat bath, τ < τ₀, consisting of two steps. In the first step, there is a limited contact with the heat bath such that only the heat Δ₁Q is flowing into the TLS leading, in linear approximation, to an increase of its entropy by Δ1S=Δ1Qτ. After this first step, the system, while being kept isolated, thermalizes and approximately assumes a Gibbs state with temperature τ₁ such that τ < τ₁ < τ₀. This can be reasonably expected if N is large enough (or if N = 2). In a second step, there is another contact with the heat bath leading to a further heat transfer of Δ₂Q and, in linear approximation, to an increase of its entropy by Δ2S=Δ2Qτ1<Δ2Qτ. The total heat transfer is ΔQ = Δ₁Q + Δ₂Q and the total increase of entropy is ΔS = Δ₁S + Δ₂S which is less than ΔQτ. An analogous reasoning applies to the case of τ > τ₀ and a cooling of the NLS in two steps. The “first Clausius inequality” β⟨ΔQ⟩ ≥ ⟨ΔS⟩ thus reflects the fact that β is the fixed initial inverse temperature of the NLS and possible changes of the NLS’s temperature during the interaction with the heat bath are ignored in the term β⟨ΔQ⟩ but would be relevant for the term ⟨ΔS⟩. On the other hand, the term β⟨ΔQ⟩ cannot be improved in a simple way, because after the interaction with the heat bath, the NLS may no longer be in a Gibbs state and thus has no temperature at all.

Next, we turn to a second Clausius inequality that can be obtained from the Jarzynski-type equation (19) by choosing p̃m=qm≔∑nTmnpn for all m∈N. This yields

As above, we may invoke Jensen’s inequality (JI) and the fact that x ↦ −log x is a convex function:

where we have suitably expanded the fraction (63) with the factors d_n and d_m. Together with (38), we have thus proven the following

The second Clausius inequality β₀⟨ΔQ⟩ ≤ ⟨ΔS⟩ has also been obtained in [21], Eq. (4.4), under the same conditions corresponding to our Assumption 1 and using the “monotonicity of the Kullback–Leibler (KL) divergence”. This proof is closely related to ours, since the said monotonicity is also a consequence of the general J-equation, see Appendix C.

Recall that according to Theorem 2, heat is always flowing from the hotter body to the colder one. According to the first Clausius inequality (38), the analogous statement for the entropy flow can only be shown in the case of 0 ≤ β ≤ β₀, i.e., where the NLS has initially a larger temperature than the heat bath. This follows since β − β₀ ≤ 0 implies ⟨ΔQ⟩ ≤ 0 by Theorem 2 and hence, ⟨ΔS⟩≤(67)β⟨ΔQ⟩≤0. The second Clausius inequality in (67) can now be used to extend the statement about the entropy flow to the case of β > β₀, i.e., where the NLS has initially a lower temperature than the heat bath. In this case, we always have ⟨ΔQ⟩ ≥ 0 by Theorem 2 and hence, ⟨ΔS⟩≥(67)β0⟨ΔQ⟩≥0. We thus have proven the following

4 Bi-stochastic limit case

As remarked in Section 2, in the limit case β₀ = 0 and if d_n = 1 for all n∈N, we obtain the special case of a bi-stochastic transition matrix T satisfying (6). For d_n ≡ 1, the entropy (31) can be identified with the Shannon entropy [26], up to the choice of units. Physically, this case can be realized by an NLS subject to external time-dependent forces but not coupled to a heat bath. Although this special case is actually outside the thematic scope of this article, it will be instructive to investigate it closer. The mathematics we used does not presuppose β₀ ≠ 0 and hence, this special case should be included in the preceding sections. According to the mentioned physical realization of the bi-stochastic limit case, we will refer to the random variable ΔE as “work” and denote it by the variable w.

In particular, we consider the “Clausius inequalities” (67) and re-write them as

For β ≥ 0, it implies ⟨w⟩ ≥ 0, a result that could also have been derived from the usual Jarzynski equation, see, e.g., [27].

Another consequence of (69) is ⟨ΔS⟩ ≥ 0 (in contrast to ⟨ΔS⟩ = 0 for classical adiabatic work processes). This result can be independently proven as follows: every bi-stochastic matrix T can be written as a convex sum of permutational matrices. This is the Birkhoff–von Neumann theorem, see [28], [29]. The Shannon entropy is invariant under permutations, but increases under a convex sum of probability distributions. The latter is due to the concavity of the Shannon entropy, see, e.g., Ex. 11.21 in [18].

5 Analytical example

As an example where the transition matrix T can be exactly calculated, we consider a single spin with spin quantum number s = 1 coupled to a harmonic oscillator that serves as a heat bath. Hence, we have a three-level system, N={1,0,−1} and N=N=3. The total Hamiltonian is

where

Here s∼x,s∼y,s∼z are the three spin operators and s∼±=s∼x±is∼y the two corresponding ladder operators. Similarly, A and A* are the lowering and raising operators, respectively, for the harmonic oscillator, such that H2=1spin⊗A*A+121HO and nn=0,1,2,… denotes the eigenbasis of A* A. λ is a real parameter. Further, let mm=1,0,−1 be the eigenbasis of s∼z such that m,nm∈N,n∈N is an orthonormal basis of the total Hilbert space.

The Hamiltonian (70)–(73) strongly resembles the Jaynes–Cummings model [30], which describes the interaction of a two-level system with a quantized radiation field. The extension to three-level systems has also been considered [31–33], but always assumes non-uniform level spacings and two radiation modes.

This system is analytically solvable since H₁₂ commutes with H₁ + H₂. The eigenspaces of H₁ + H₂ are, hence, left invariant under H. They are of the following form: either the singlet spanned by −1,0, or the doublet spanned by 0,0 and −1,1, or an infinite number of triplets spanned by 1,n−1, 0,n, and −1,n+1, where n = 1, 2, 3, …. Within the triplets H has the form

and the corresponding eigenvalues

Let π denote a Gibbs state of the harmonic oscillator, such that

We choose as an initial mixed state

with an arbitrary probability distribution p. After the time t, this state will evolve into

To simplify the following calculation, we will consider the time average

The time averaged probability of finally occupying the state r∈N will be

The latter equation follows since q_r is a linear function of p and defines the transition matrix T. From what has been said above it is clear that T will be obtained by a summation over all eigenspaces of H₁ + H₂. It proves to be independent of λ due to time averaging. After some computer-algebraic calculations, we obtain

Here Φ(z,s,a)≔∑k∈Nzk(k+a)s denotes the Lerch’s transcendent, see Section 25.14 in [34]. It can be shown that T is a left stochastic matrix and leaves the probability distribution

invariant that corresponds to a Gibbs state with inverse temperature β₀. It follows that Assumption 1 is satisfied and hence the results derived in Sections 2 and 3 hold for our example. We illustrate this by showing the three functions βΔQ, ΔS, and β₀ΔQ for −5β₀ ≤ β ≤ 5β₀ in Figure 3 satisfying the Clausius inequalities (67). For this example, we make the following observation, which seems to be typical for NLS coupled to a heat bath: If the three-level system is initially hotter than the heat bath, 0 < β < β₀, the maximal heat transfer |ΔQ| results for β → 0, as expected. In contrast, the entropy transfer |ΔS| takes its maximum at a positive inverse temperature β_max, see Figure 3. This is remarkable in the sense that one might naively think that in nonequilibrium thermodynamics the identity ΔS = β₀ΔQ, which holds for linear thermodynamics, could be weakened to a monotonic relation between ΔS and β₀ΔQ. However, this is not correct: if one increases the temperature above 1/β_max, the reduced heat transfer β0ΔQ also increases, while the entropy transfer ΔS decreases.

Figure 3:

Plot of the two “Clausius heat terms” β₀ΔQ, βΔQ, and the “entropy increase” ΔS as functions of the inverse temperature β of the three-level system considered in Section 5 and the corresponding transition matrix T according to (82)–(90). The range of the inverse temperature β is chosen as −5β₀ ≤ β ≤ 5β₀ and the calculations have been done for β₀ = 1. It is remarkable that for 0 < β < β₀, the absolute value |ΔS| has its maximum not for β = 0, as it is the case for |β₀ΔQ|, but for some positive inverse temperature of β_max ≈ 0.279896.

6 Summary and outlook

In this paper, we have presented an approach to the time-honored problem of the second law for a finite quantum system coupled to a heat bath. We have re-derived several known partial results, in part under weaker assumptions, and integrated them into a theory based on general J-equations. These resemble the famous Jarzynski equation and imply certain second law-like inequalities. It will be in order to provide a general survey that shows their logical dependencies, see Figure 4.

Figure 4:

Schematic representation of various forms of J-equations and inequalities and their logical dependencies. Detailed explanations are given in Section 4.

The most general J-equation is (C5), the central (red) equation of Figure 4. It holds for two sequential measurements under rather general assumptions, see Theorem 6, and contains two undetermined probability distributions p̃ and p⁽⁰⁾. There are two principal specialization paths that physically correspond to “work processes” (upward direction in Figure 4) and “heat processes” (downward direction in Figure 4).

For “work processes” performed on closed systems that are only marginally touched in this paper, the transition matrix T is bi-stochastic in the modified sense of Eq. (7). This entails the J-equation (27) in [15] (the upper blue equation in Figure 4) that can be further specialized according to the choice of p̃. The usual Jarzynski equations are derived for the choice of p̃=p, whereas the alternative p̃=q≔Tp leads to a scenario reminiscent of earlier work of W. Pauli and F. Klein, see [15], [16].

By contrast, the “heat processes” performed on systems without external forces but under contact with a heat bath are characterized by T p⁽⁰⁾ = p⁽⁰⁾ and lead to another special J-equation (19) (the lower green equation in Figure 4) that is of central importance for this work. Again, there are two further options. The choice p̃=p and the restriction to probability distribution given by Gibbs states leads to (23) and, by means of Jensen’s inequality (JI), to the relation (26). The latter states that, on statistical average, heat always flows from the hot system to the cold bath and vice versa.

For the analogous statement (68) about the average flow of entropy, we additionally required the two Clausius inequalities. The first one, ΔS ≤ βΔQ, is a simple consequence of the Gibbs inequality (GI). The second one, β₀ΔQ ≤ ΔS, follows via (JI) from the mentioned J-equation (19) and the choice p̃=q≔Tp.

In the following, we will relate our results to similar statements found in the relevant literature.

Jarzynski and Wójcik [19] consider two systems which initially have different temperatures, then interact weakly and finally (in the quantum case) are subjected to a separate energy measurement for both systems. They derive a Crooks-like equality, and from this a Jarzynski-type equation, Eq. (18) in [19] that corresponds to our Eq. (23). Important assumptions are: neglect of the interaction between both systems for the heat balance and microreversibility. In contrast, we will focus on the 1st system and describe the second one only in a general way by the transition matrix T. Our assumptions are weaker: T leaves invariant some Gibbs state without neglecting the interaction; time reversal invariance is not needed (and actually will be violated for the example presented in Section 5).

Jennings and Rudolph [20] also consider two systems, which, however, can also be initially entangled. The special case, which is interesting for our purposes, is that both systems are uncorrelated at the beginning and have different temperatures. A first Clausius inequality, corresponding to our (38), is derived from the property that the Gibbs state minimizes the free energy. From this directly, without using a Jarzynski-type equation, the heat flow inequality, Eq. (3) in [20], follows, which corresponds to our Eq. (26), but again assuming, as in [19], that the amount of heat emitted by the first system is exactly absorbed by the second one.

The second Clausius inequality that appears in our (67), can also be found in Chapter 4.1 of [21], and is proved there via the “monotonicity of the KL divergence”. This proof is closely related to ours, since the said monotonicity follows from the general J-equation, see Appendix C. The following statements in [21] interpreting the second Clausius inequality as a form of the second law of stochastic thermodynamics should be taken with some caution, see the discussion above.

Another related result has been proven already in 1978 by Spohn [22]: for an open quantum mechanical system described by a quantum dynamical semigroup that leaves invariant a certain state ρ⁰, the entropy production is non-negative. The latter is defined as the time derivative of the KL divergence between ρ_t and ρ⁰. This result has been recently reformulated in [23] in a way compatible with our approach; Eqs. (16) and (17) of that reference immediately imply the second Clausius inequality.

Summarizing the state of research, partial formulations of the second law for the coupling of an NLS to a heat bath can be found to a sufficient extent in recent years, but they have to be integrated into a unified theory and proved under conditions as weak as possible. This has been attempted in the present work.

Moreover, we have presented an analytically solvable example illustrating our approach. It consists of a three-level system (a spin with s = 1) coupled to a harmonic oscillator. For this example, our central Assumption 1, saying that the transition matrix has a fixed point of Gibbs type, is exactly satisfied. It would be a task for the future to investigate the conditions under which this assumption holds exactly or approximately.