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- Ming Bao Yu

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**Received Date:** April 15, 2014; **Accepted Date:** January 05, 2015; **Published Date:** January 10, 2015

**Citation:** Yu MB (2015) The Entropy Production of a Nonequilibrium Open System. J Phys Math 6:128. doi: 10.41722090-0902.1000128

**Copyright:** © 2015 Yu MB. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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A nonequilibrium open system is studied in the projection operator formalism. The environment may linearly deviate from its initial state under the reaction from the open system. If the relevant statistical operator of the system is a generalized canonical one, the transport equation, the second kind of fluctuation-dissipation theorem and the entropy production rate of the open system can be derived and expressed in terms of correlation functions of fluctuations of random forces and interaction random forces.

Non equilibrium open system; Projection operator; Entropy production rate

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.

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 [9-14] 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)

*η(t)=-iλtr _{R}[L_{SR}W(t)]* (2.1)

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]:

(2.2)

Introduce q(t) =1-p(t) , p(t)q(t) = 0 , we have [6]

(2.3)

with is a time-ordered evolution operator satisfying

*∂ g(t,u)/∂ u=ig(t, u) q(u)L _{S} and g(t, t)=1.*

The transport equation for EV < A_{j}(t)>=tr_{S}[ρ(t)A_{j}]=trs[ρ(t)A_{j} takes
the form [6]

(2.4)

In Heisenberg picture,

(2.5)

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

(2.6)

is an additional term describing the external influence from the environment upon the open system; (t>u) is an anti-time-ordered evolution operator defined by and G(t, t)=1; Q( t )=1- P( t ) , P( t ) is the transposed projection operator of P(t) [4],

(2.7)

Satisfying

(2.8)

Since , (2.4) may be written as [7]

(j = 1,2,..m)(2.9)

Where

(2.10a) 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

(2.10b)

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)

Where denotes the random force, which may be split into two:

(2.12a)

(2.12b)

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
ensembles vanishes, so In the rest of the paper we
will no longer distinguish from its fluctuation

**Generalized canonical statistical operator In order to go
steps further**

let us assume to be a GCSO:

(3.1)

where λ_{t} (t) (l=1,…,m) are conjugate parameters of the basis
macrovariables {A_{l}}

Making use of the Kubo identity we have

(3.2a)

(3.2b)

Introducing the generalized quantum correlation function

(3.3)

and making use of (3.2), the integrand in (2.5) may be written as

(3.4)

Since f_{j}(ut) is in the irregular space because of Q(u) while is in the
regular space, their correlation is zero, thus

(3.5)

Where is given by (2.12b). Same argument will apply to similar cases later.

Therefore Equation.(2.5) takes the form

(3.6)

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].

(4.1)

where k_{B} is the Boltzmann constant. The entropy production rate reads
[7].

(4.2)

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

(4.3)

the first term resulting from the organized motion in (3.6) reads [7]

(4.4)

the second term resulting from the disorganized motion in (3.6) takes the form

(4.5)

because of (3.5); and the third term resulting from the influence term (2.11) is

(4.6)

These expressions represent the contributions of each term in the
transport equation to the entropy production, respectively. Besides,
Equation.(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 Yj(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 and B_{k} respectively pertain to S and R,
and they are initially independent:

where

By , we have , here and . By (2.1), ƞ(u) may be written as

(5.1)

(5.2)

determines the evolution of the EV of macrovariable B_{k}

Since thus For weak interaction, keeping only the linear term in λ , we obtain

(5.3a)

(5.3b)

(5.3c)

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

(5.4a)

(5.4b)

in Schrodinger and Heisenberg pictures, respectively; where . Hence we have the influence term

(5.5a)

(5.5b)

and its contribution to the entropy production in Shrodinger and Heisenberg pictures:

(5.6a)

(5.6b)

Now consider the case that the initial state of S is given by a GCSO:

(5.7)

By the Kubo identity and the initial condition we have , Thus (5.4) may be written as

here we have conducted argument similar to that leads (3.4) to (3.5), and

(5.8)

is the averaged interaction random force. Thus we obtain

(5.9)

(5.10)

With

If the open system is initially in an equilibrium state

(5.11)

is the initial inverse temperature of the system,
then ρ_{eq} is a special case of (5.7) in which , and . Since A_{1} is the only basic variable, so and is the inverse
temperature of S. Because , thus and

(5.12)

the memory term in (3.6) becomes and vanishes, so

(5.13)

(5.14)

(5.15)

(5.16)

Finally we obtain the transport equation for the only basis variable
H_{S}:

(5.17)

Equations.(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

(5.18)

With (5.1) and (2.3),

(5.19)

Equation.(5.19) is similar to Equation.(22) in [17] where the environment is a reservoir and is time-independent. In the following, we will follow the argument in [17], however, take into consideration that is time-dependent. Making use of (5.18) and (3.3), leads to

(5.20)

(5.21)

(5.22)

(5.23)

(5.24)

Substituting (2.3) into (5.23) and repeating the above arguments, we have

(5.25)

which many be written as

(5.26)

(5.27)

here we have had argument similar to that leads (3.4) to (3.5). Thus we obtain the influence term and its contribution to the entropy production:

(5.28)

(5.29)

If we are satisfied with keeping the linear term of λ in *M _{l}(u,t)*, then

(5.30)

(5.31)

(5.32)

Therefore we have the approximate expressions

(5.33)

(5.34)

they are up to λ^{2} by (2.12a). Comparing (5.34) with (4.5), we see clearly that plays the role of in the case of corresponding isolate
system.

Now we rewrite the results obtained above in the form of special dependent. For simplicity, we focus on the simpler expression (5.34). The entropy production of the open system reads

(5.35)

Where

(5.36a)

results from the organized motion in the transport equation due to
H_{SR};

(5.36b)

from the disorganized motion and

(5.36c)

from the influence term, respectively.

**Comparison**

In this section, we compare the results obtained in the proceeding sections with the well known approaches in the linear nonequilibrium thermodynamics and statistical mechanics. The time rate of the entropy density s(x, t) of a nonequilibrium system takes the form [8,18,19]:

(6.1)

Where (6.2)

is the entropy production density occurring inside the system which is given in terms of the sum of products of thermodynamic fluxes and the conjugate thermodynamic forces ; and is the density of entropy flux through the border into the system.

Onsager proposed a linear relationship between the fluxes and forces

(6.3)

with reciprocity relations

(6.4)

Thus we have

(6.5)

For the special case considered in Sect.5, the interaction between open system S and its environment R takes the form , for example, S and R are composed of different kinds of harmonic oscillator [7]. Such interaction implies no obvious border separating S and R, leading to absence of the divergence term on the right hand side of (6.5). Thus the variation of entropy density results from inner entropy production only:

(6.6)

Besides, in the Green-Kubo formalism, the transport coefficients
L_{ik} can be expressed in terms of time correlation functions of the time
rate of corresponding variables [19,20].

(6.7)

where the average is taken over an equilibrium ensemble and the Markovian effect is taken into account.

In this paper, we study a nonequilibrium open system whose
transport equations (2.9)-(2.11) are nonlinear differential-integral
ones. Now let us compare (5.36b) with (6.6). We notice that (5.36b)
share the same structure as (6.6) because of the facts : (1) parameters ( j=1,…m) play the role Thermodynamic forces since they
may involve spacial gradients of, e.g., temperature, velocity, chemical
potential or electric, magnetic fields, etc,; (2) the random forces (2.12b)
involve the time rates of variables because of using projection operator
technique and (3) the average is taken over GCSO (3.1) instead of an
equilibrium ensemble. As for (5.36c), the contribution of to the
entropy production, in which the free term in the Volterra
equation is indeed a generalization of in (5.26b), hence (5.36c)
possesses the same structure as (6.6) also. Acoordingly, we see clearly
that the entropy production rate (5.35) is a natural generalization of
(6.6) where the non-linearity and the non-Markovian effect have been
taken into consideration. In addition to the entropy production, the *entropy generation* is another useful tool in the study of nonequilibrium
systems [8] and especially useful in the analysis of a process occurring
in the system during a period of time τ . It is worth noticing the
major differences between the two: the entropy production needs the
hypothesis of local equilibrium but the entropy generation does not;
the former does not consider the time but the latter introduces the
lifetime τ of the process [8]. The two different approaches are closely
related and complementary one to another.

In the present paper we have studied a nonequilibrium open system
in interaction with its environment which may linearly deviate from its
initial state under the reaction of the open system. We have shown that
if the relevant statistical operator of the system is of the form of GCSO,
then the transport equation is given by (3.6) and (5.28) or (5.33). The
memory term in (3.6) and the influence term (5.28) or (5.33) can be
expressed in terms of quantum correlation functions of fluctuations
of random forces and interaction random forces, giving the second
kind of fluctuation-dissipation theorem for this nonequilibrium open
system. We have also shown that the entropy production rate is given
by the sum of products of transport equations and the corresponding
parameters. In the organized motion term, H_{S} contributes nothing to
the rate, but H_{SR} does; the contributions of the memory and influence
terms are expressed in terms of quantum correlation functions of
fluctuations of random forces and interaction random forces. The total
entropy production rate is given by the sum of contributions resulting
from each term in the transport equation, given respectively by (4.4),
(4.5) and (5.29) or (5.34). They are natural generalizations of those for a
linear nonequilibrium closed system to a nonlinear open system.

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