Entropy production as correlation between system and reservoir

Abstract

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.PACS numbers: 05.70. Ln, 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...

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“…This relation may be established both for quantum and classical systems [20] as a consequence of the microscopic dynamics [21][22][23][24][25] or as a general thermodynamic result [13]. Here, Q is the heat flow into the thermal reservoir, and we have identified the irreversible entropy production ∆ i S. Any process that saturates the inequality in Eq.…”

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“…4 The above derivation hinges on the non-negativity of relative entropy. A similar approach has recently been taken to obtain inequalities related to the second law of thermodynamics [21][22][23][24], in situations when the system of interest does not necessarily begin (or end) in states of thermal equilibrium. (See also Ref.…”

Entropy Production in the Expanding Universe

“…In this set-up, it has been assumed that the total von Neumann entropy induces a natural separation of the entropy change of the system into separate contributions from an entropy flow and an entropy production and further erroneously suppose that the total entropy production is simply the sum of the system and reservoir entropy. Therefore, the change in the entropy of the system, which is a positive quantity, can be written in the standard thermodynamic form [25,47] …”

“…The positive entropy production introduced above represents the irreversible contribution to the entropy change of the system, which indeed vanishes only when the system and the reservoir are totally decorrelated. In other words, the entropy production explicitly expresses how far the actual state of the total system is from the decorrelated product state [25]. In sec.…”

“…An exact expression has been obtained for deriving the entropy production of a system that is in contact with a thermal reservoir with strong coupling between system and reservoirs [25]. The system entropy production has been recognised as a direct measure of the system-reservoir correlations and/or entanglements.…”