The Entropy Production of a Nonequilibrium Open System

In the study of nonequilibrium systems different projection operators are introduced to present a macroscopic description of the system in order to simplify the problem [1-6]. In this approach the macroscopic state of the system is determined by expectation values of a set of basis macro variables, and equations of motions for these expectation values, the transport equations, are derived in the projection operator formalism.


Introduction
In the study of nonequilibrium systems different projection operators are introduced to present a macroscopic description of the system in order to simplify the problem [1][2][3][4][5][6]. In this approach the macroscopic state of the system is determined by expectation values of a set of basis macro variables, and equations of motions for these expectation values, the transport equations, are derived in the projection operator formalism.
When studying a nonequilibrium open system, the influence of the environment upon the open system is one of the important topics in such studies. It has been shown [7] that the influence from the environment comes from two parts: one is the time-rate of the averaged macro variables resulting from the interaction Hamiltonian H SR and the other from an additional influence term, therefore, the influence of the environment can be completely separated from the corresponding closed system.
When the relevant statistical operator of the system is of a generalized canonical statistical operator (GCSO) by which the entropy of the open system is defined, if the environment is a reservoir, then the memory and influence terms in the transport equation can be given in terms of correlation functions of fluctuations of random forces and interacting random forces, and they can be cast into the Volterra equation formalism.
The purpose of the present paper is to generalize the results to the case that the environment is not a reservoir which may linearly deviate from its initial state under the reaction from the open system. We will show that the memory and influence terms can still be expressed in terms of correlation functions of fluctuations of random forces and interaction random forces, but no longer be able to cast into the Volterra equation formalism, so is the entropy production rate of the open system.
The results obtained in this paper are compared with approaches in linear thermodynamics and statistical mechanics, focusing on the entropy production of a nonequilibrium open system, which is local in both space and time. In contract, the entropy generation [8] is also important in the study of nonequilibrium systems, which is global in space and time, being especially useful in cases involving effects of irreversibility. In addition, another important development in physics today is the so-called quantum thermodynamics [9][10][11][12][13][14] which has extended the thermodynamics study from the macroscopic scale to the nanometer scale, and even down to the single atom and single photon scale. In Section 2, transport equations of the system are briefly reviewed. In Section 3, a GCSO is introduced. The entropy production rate is derived in Section 4. The influence term and its contribution to the entropy production is studied in Section 5. Comparison of the results with well-known approaches is presented in Section 6 and conclusions are drawn in Section 7.

Transport equations
Consider an open system S under the influence of its environment R. The total system S ⊕ R is characterized by Hamiltonian H=H S +H R +λH SR and statistical operator (so) W(t) . The open system s is described describing the influence of R upon S, where L S X= i [HS, X], h=1.
Suppose we are satisfied with the description of system S at the macroscopic level by expectation values (EVs) of a set of basis macrovariables { A j , j=1,…,m} of S, such macroscopic description can be realized by a relevant so ρ r (t) which is picked up by a time-dependent projection operator ρ(t) from ρ(t): ρ r (t)= ρ(t) ρ(t). We may choose the following projection operator as ρ(t) [3]: Introduce q(t) =1-p(t) , p(t)q(t) = 0 , we have [6] is a time-ordered evolution operator satisfying

The transport equation for EV < A j (t)>=tr S [ρ(t)A j ]=tr s [ρ(t)A j takes the form [6]
In Heisenberg picture, here the first term gives the organized motion, the second term the initial condition and the third term the disorganized motion or the memory term [4] and is an additional term describing the external influence from the environment upon the open system; ( ) ( ) is an anti-time-ordered evolution operator defined by is the transport equation of the corresponding closed system, i.e. the time rate of EV resulting from H S , the Hamiltonian of the system S itself; and is the time rate resulting from the interaction H SR .
The meaning of (2.9) is clear and simple: The transport equation of an open system is the sum of transport equation of the corresponding closed system, the time-rate of the EV due to the interaction Hamiltonian H SR and the additional influence term Y j ( t ).
The influence term (2.6) can be written as (j=1,2…,m) (j=1,2…,m) , (2.11) denotes the random force, which may be split into two: being respectively the random force and interaction random force associated with the time rate of the basis variable A j due to H S and H SR , respectively. Since the average of the random force over given In the rest of the paper we will no longer distinguish ( ) Introducing the generalized quantum correlation function and making use of (3.2), the integrand in (2.5) may be written as here the memory term is expressed in terms of quantum correlation function of fluctuations of random forces. The influence term Y j (t) will be further analysed in Section 5.

Entropy production rate
Now define the entropy of the noequilibrium open system through its relevant statistical operator [1,15,16] where k B is the Boltzmann constant. The entropy production rate reads [7] ( ) which is the sum of products of transport equations and the conjugate parameters. If assume that the initial state of the system is a GCSO: ρ (0)=ρ r (0) then the initial term in (3.6) vanishes. Combining (4.2) with (2.4) given by (3.6) and (2.11), we obtain because of (3.5); and the third term resulting from the influence term (2.11) is These expressions represent the contributions of each term in the transport equation to the entropy production, respectively. Besides, Eq.(4.4) does not involve iL S A j , indicating that in the organized motion term H S contributes nothing to the rate.

Non-reservoir environment
Now we further analyze the contribution of the influence term Y j (t). Suppose that the environment R is not a reservoir and may linearly deviate from its initial state under the reaction from S. For simplicity, we assume SR , k A and k B respectively pertain to S and R, and they are initially independent: By ( ) ( ) ( ) For weak interaction, keeping only the linear term in λ , we obtain ( ) being the zeroth and first order terms of the EV of B K when R linearly deviates from its initial state under the weak reaction from S. By (5.1) and (5.2), the integrand in (2.11) takes the form and its contribution to the entropy production in Shrodinger and Heisenberg pictures: Now consider the case that the initial state of S is given by a GCSO: is the averaged interaction random force. Thus we obtain Besides, as a special case of (5.9) and (5.10), we have Finally we obtain the transport equation for the only basis variable H S : Eqs.(5.9), (5.10) and (5.14), (5.15) involve the averaged interaction random force (5.8) which has incorporated the linear deviation of the environment from its initial state. Now consider the case without a given initial condition. By (3.1) and the Kubo identity, we have Eq.(5.19) is similar to Eq.(22) in [17] where the environment is a reservoir and 0 < > SR R L is time-independent. In the following, we will follow the argument in [17], however, take into consideration that