Entropy production as correlation between system and reservoir

We derive an exact (classical and quantum) expression for the entropy production of a finite system placed in contact with one or several finite reservoirs each of which is initially described by a canonical equilibrium distribution. Whereas the total entropy of system plus reservoirs is conserved, we show that the system entropy production is always positive and is a direct measure of the system-reservoir correlations and/or entanglements. Using an exactly solvable quantum model, we illustrate our novel interpretation of the Second Law in a microscopically reversible finite-size setting, with strong coupling between system and reservoirs. With this model, we also explicitly show the approach of our exact formulation to the standard description of irreversibility in the limit of a large reservoir.

We derive an exact (classical and quantum) expression for the entropy production of a finite system placed in contact with one or several finite reservoirs each of which is initially described by a canonical equilibrium distribution. Whereas the total entropy of system plus reservoirs is conserved, we show that the system entropy production is always positive and is a direct measure of the system-reservoir correlations and/or entanglements. Using an exactly solvable quantum model, we illustrate our novel interpretation of the Second Law in a microscopically reversible finite-size setting, with strong coupling between system and reservoirs. With this model, we also explicitly show the approach of our exact formulation to the standard description of irreversibility in the limit of a large reservoir. Starting with the groundbreaking work of Boltzmann, there have been numerous attempts to construct a microscopic derivation of the Second Law. The main difficulty is that the prime microscopic candidate for the entropy, namely, the von Neumann entropy S = −Trρ ln ρ with ρ the density matrix of the total or compound system, is a constant in time by virtue of Liouville's theorem. Related difficulties are the time-reversibility of the microscopic laws and the recurrences of the micro-states. A common way to bypass these difficulties is to introduce irreversibility in an ad hoc way, for example by reasoning that the system is in contact with idealized infinitely large heat reservoirs. Nevertheless, as was realized early on by Onsager, a consistent description of the resulting irreversible behavior still carries the undiluted imprint of the underlying time-reversibility and Liouville's theorem for the system. Examples are the symmetry of the Onsager coefficients and the fluctuation dissipation theorem. As examples of more recent discussions we cite results on work theorems and fluctuation theorems [1,2,3]. Even more relevant to the question pursued here, we cite the microscopic expression for the entropy production as the breaking, in a statistical sense, of the arrow of time [4,5,6,7,8,9,10]. We also mention that significant effort has been devoted to a detailed description and understanding of the interaction with the heat reservoirs, in particular the difficulties of dealing with the case of strong coupling [11,12].
In this letter we show that the problem of entropy production can be addressed within a microscopically exact description of a finite system, without resorting to infinitely large heat reservoirs and without any assumption of weak coupling. Whereas the von Neumann entropy of system plus reservoirs is conserved, the entropy production of the system is always positive, even though it displays oscillations and recurrences typical of the finite total system. Interestingly, this entropy production is expressed in terms of the correlations and/or entanglement between system and reservoirs, so that its positivity can be explained by a corresponding negative entropy contribution contained in the correlations and/or entanglement with the reservoirs. As the size of the reservoirs increases, the recurrences die out, the negative entropy contribution is diluted in an intricate way over the increasing number of correlations with reservoir degrees of freedom, and the entropy production of the system itself approaches the standard thermodynamic form. We will illustrate this novel interpretation of the Second Law on an exactly solvable model, namely, a spin interacting with an N -level quantum system via a random matrix coupling. We focus on the derivation for the quantum case, but the analogous treatment for the classical system is straightforward.
The set-up is as follows. We consider one or several finite quantum systems r which play the role of finitesize heat reservoirs. Accordingly, their density matrices ρ r (t) at the initial time t = 0 are assumed to be of the canonical equilibrium form, Here β r , H r and Z r are the corresponding inverse temperature at t = 0 (Boltzmann's constant k B is set equal to 1), the Hamiltonian, and the partition function at t = 0. Being reservoir systems, it is further natural to assume that their Hamiltonians H r are time-independent. At time t = 0 we connect a finite quantum system s, characterized by Hamiltonian H s (t) and density matrix ρ s (t), to the reservoirs by switching on an interaction Hamiltonian V (t). The initial state of the compound system, characterized by the density matrix ρ(t), does not display any entanglement or correlation, Correlations and/or entanglements do develop in the subsequent time evolution of ρ(t), which obeys Liouville's equation for the total Hamiltonian Note that in addition to the issue of relaxation of a system in contact with a reservoir, this scenario includes the ingredients for the study a driven system, cf. the timedependence of the system's Hamiltonian, as well as that of a nonequilibrium steady state, which can be realized in view of the presence of several heat reservoirs. In fact, the above construct can easily be generalized to include particle reservoirs described via grand-canonical distributions. This would allow the consideration of particle flows in addition to heat flows. We are primarily interested in the occurrence and characterization of irreversible behavior in the system, and we thus focus our attention on the entropy S(t) of the system, where ρ s (t) is the trace of ρ(t) over the degree of freedom of all the reservoirs. Contrary to the total von Neumann entropy, the entropy of the system is in general a function of time, technically speaking because the dynamics of ρ s (t) is not unitary. More to the point for the ensuing discussion, we note that from the thermodynamic point of view we are dealing with an energetically open system. We now show that it is precisely the time invariance of the total von Neumann entropy which induces a natural separation of the entropy change of the system into separate contributions from an entropy flow and an entropy production. Using −Trρ(t) ln ρ(t) = −Trρ(0) ln ρ(0) = −Tr s ρ s (0) ln ρ s (0) − r Tr r ρ eq r ln ρ eq r , we find for the entropy change of the system We conclude that the change in the entropy of the system can be written in the standard thermodynamic form [13] ∆S(t) = ∆ i S(t) + ∆ e S(t).
The entropy flow, representing the reversible contribution to the system entropy change due to heat exchanges, is identified as the last term in (5). After some manipulation using the explicit form of ρ eq r , it can be written as where Of particular interest is the resulting expression for the entropy production, which represents the irreversible contribution to the entropy change of the system. Here, D[ρ||ρ ′ ] is the quantum relative entropy between two density matrices ρ and ρ ′ , It has the following important properties [14,15]. The relative entropy is positive, and equal to zero only when the two matrices are identical. We thus conclude that the entropy production introduced above is indeed a positive quantity, ∆ i S(t) ≥ 0, and vanishes only when the system and the reservoirs are totally decorrelated. Furthermore, the relative entropy is a measure of the "distance" between two density matrices. Hence, as announced earlier, the entropy production explicitly expresses how "far" the actual state ρ(t) of the total system is from the decorrelated/disentangled product state ρ s (t) r ρ eq r . To further clarify the significance of our central result (6), we make a number of additional comments. First, starting with Eq. (6) we can rewrite the entropy production as ∆ i S(t) = ∆S(t) − ∆ e S(t). ∆S(t) is the exact entropy change of the system. If one assumes that the entropy change in each heat reservoir is given by ∆S r (t) = −β r Q r (t), and if one further erroneously supposes that the total entropy is simply the sum of the system and reservoir entropy, one concludes that the positive entropy production ∆ i S(t) is the entropy increase in the total system. This is of course in flagrant contradiction with the premise that led to the identification of ∆ i S(t), namely, that the entropy of the total system remains unchanged. The error resides in disregarding a contribution −∆ i S(t) to the total entropy, which is precisely the negative entropy contribution contained in the correlations and entanglement between system and reservoir. The argument may on the surface appear circular, but the neglect of the negative entropy contribution is actually quite natural from an operational point of view: while one has full microscopic access to the system's properties, one only controls or measures the energy and no other properties of the reservoir. In this sense, the above procedure leading to an apparent total positive entropy change can be viewed as a coarse graining operation that retains the full microscopic description of the system but reduces the reservoirs plus correlations to an idealized heat reservoir description. In the limit of large reservoirs it is likely that this latter description deviates very little from the canonical distribution. Concerning the correlations, one expects that they will be diluted over the exponentially many higher-order correlations, becoming in effect irretrievable. Furthermore, this will happen exponentially fast in time if the reservoirs display nonintegrable, chaotic properties.
Second, while ∆ i S(t) is a positive quantity, it does not increase monotonically in time. In fact, oscillations are bound to arise in view of the recurrences in the state of the finite total system. In this respect, it is important to stress that we consider the entropy change starting from the natural but specific initial condition (2). The transient decreases of ∆ i S(t) can be interpreted as the reappearance of the negative entropy, hidden in the correlations, as system and reservoir transiently return to states close to this decoupled initial state. In the limit of large reservoirs, recurrences will become less and less likely, and ∆ i S(t) is expected to converge to a convex monotonically increasing function of t.
Third, we make the connection with a recent discussion [11,12] concerning the appropriate definition of work and free energy in a driven system strongly coupled to a heat reservoir. We consider the case of a single reservoir at temperature T = β −1 , for convenience dropping the subscript r. Using the fact that TrH(t)ρ(t) = 0, the work done on the total system can be written as The change of the energy of the system, including the contribution of the interaction term, reads Using Tr H s (t) + V (t) ρ(t) = −TrH rρ (t), we find that this energy change can be written, in accordance with the First Law, as the sum of work and heat, Next, introducing the nonequilibrium free energy we can rewrite the expression (6) for the entropy production in the standard thermodynamic form for a driven system in contact with a heat reservoir, This expression is exact. If we assume that the total system relaxes to a final canonical equilibrium at temperature β −1 , this nonequilibrium free energy difference reduces to the equilibrium expression identified in the context of the work theorem in both the weak coupling [2,4,16] and strong coupling regimes [11,12]. Finally, we discuss the connection with the following alternative definition for the irreversible entropy change, proposed in open quantum system theory [15]: The entropy flow is now defined as ∆ eS (t) ≡ β( H s t − H s 0 ) and ρ eq s = exp (−βH s )/Z s . To compare this expression with our definition (8) for the entropy production, we note that total energy is conserved by the dynamics, H t = H 0 , and hence The two definitions thus differ by the interaction term, which vanishes in the limit of either weak coupling or high temperature. The definition (15) has the obvious advantage of being exclusively expressed in terms of the system density matrix, while our definition (8) requires the total density matrix. However, we will show that contrary to our expression, (15) is not always a positive quantity. The positivity of (15) can be proven when ρ eq s is the stationary solution of the reduced dynamics [15]. This will generically be the case in the weak-interaction large-reservoir limit, i.e., precisely when the interaction term in (16) can be neglected and (15) becomes identical to (8). An even stronger statement can be made when, in the same limit, the system dynamics can be described by a Markovian quantum master equation of the forṁ ρ s (t) = Lρ s (t), where L is a Redfield superoperator satisfying Lρ eq s = 0 [15,17]. Under these conditions, it is known that the entropy production is a convex functional of the system density matrix [15], with a positive rate of entropy production d dt ∆ i S(t) ≈ d dt ∆ iS (t) ≥ 0. We will now illustrate the above findings in a twolevel quantum spin coupled to an N -level reservoir via a random matrix. The total Hamiltonian reads σ x,z are the well known Pauli matrices. The reservoir Hamiltonian H r is a diagonal matrix with N equally spaced eigenvalues between −0.5 and 0.5. The coupling matrix is R = X/ √ 8N , where X is a Gaussian orthogonal random matrix of size N with probability density proportional to exp(− 1 4 TrX 2 ) [18]. This model is similar to the spin-GORM model of Ref. [19]. The system is initially assumed to be in the pure lower energy state ρ s (0) = |0 0|, where σ z |0 = −|0 , and the reservoir is initially in a canonical equilibrium state at the temperature β −1 . In the weak-coupling large-reservoir limit, the resulting Redfield equation leads to the following closed relaxation equation for the z-component of the spin ( = 1): Hereα(ω) is the Fourier transform of the reservoir correlation function α(t) = Tr r ρ eq r exp [iH r t]R exp [−iH r t]R, α(|ω|) = 1 16 e −β/2 e β|ω| − e β/2 e −β/2 + e β/2 = e β|ω|α (−|ω|). (19) The x and y components of the spin evolve independently of the z component, and are zero for our initial condition. We are now in a position to compare the definition (15) of the entropy change with the irreversible entropy change (8) that follows from Redfield theory. This is accomplished through an exact numerical solution of our model for finite N . The results are summarized in Fig.  1. Note that we show single realizations of the random matrix. For small N , we observe a pronounced oscillatory behavior and even near-recurrences very close to zero of our entropy change (8). While the latter always remains positive, the entropy change (15) can be negative for small values of N , which is clearly not acceptable. In the limit of a large reservoir (N → ∞ ), both expressions converge to one another and coincide with the positive and convex irreversible entropy change predicted by the Redfield equation.
We conclude that (8) is a proper definition for the entropy change, one that remains valid for a small system strongly coupled to small reservoirs. In the limit of large reservoirs, it converges to a convex irreversible entropy, coinciding with the familiar definition of entropy production [15] for the quantum master equation. Our identification of the entropy production (8) within an exact microscopic framework vindicates the description of irreversibility as a property of open systems, with the entropy production rather than the entropy of the total system playing the central role. The microscopic origin of the entropy production, explained in terms of correlations established between the system and its reservoirs, is reminiscent of Boltzmann's Stosszahlansatz. However, our analysis of the micro-dynamics and the identification of the entropy production of the system are exact. The appearance of irreversibility as the omission, in the reservoirs, of its correlations with the system, provides a natural, precise and transparent interpretation of the Second Law.
M. E. is supported by the FNRS Belgium (chargé de recherches) and by the Luxembourgish government (Bourse de formation recherches). K.L. and C. VdB. gratefully acknowledge the support of the US National Science Foundation through Grant No. XXXXX.