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The Entropy Production of a Nonequilibrium Open System | OMICS International
ISSN: 2090-0902
Journal of Physical Mathematics
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The Entropy Production of a Nonequilibrium Open System

Ming Bao Yu*

407 Oak Tree Square, Athens, Georgia 30606, USA

*Corresponding Author:
Ming Bao Yu
407 Oak Tree Square, Athens
Georgia 30606, USA
Tel: 706-410-2137
E-mail: [email protected]

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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Abstract

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.

Keywords

Non equilibrium open system; Projection operator; Entropy production rate

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 [1-6]. 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=HS+HR +λHSR and statistical operator (so) W(t) . The open system s is described by a reduced statistical operator ρ(t)=trRW(t) satisfying ∂ ρ(t)/ ∂ t=-iLS ρ(t)+ƞ(t) with

η(t)=-iλtrR[LSRW(t)] (2.1)

describing the influence of R upon S, where LSX= 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 { Aj, j=1,…,m} of S, such macroscopic description can be realized by a relevant so ρr(t) which is picked up by a time-dependent projection operator ρ(t) from ρ(t): ρr(t)= ρ(t) ρ(t). We may choose the following projection operator as ρ(t) [3]:

equation(2.2)

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

equation(2.3)

with equation is a time-ordered evolution operator satisfying

∂ g(t,u)/∂ u=ig(t, u) q(u)LS and g(t, t)=1.

The transport equation for EV < Aj(t)>=trS[ρ(t)Aj]=trs[ρ(t)Aj takes the form [6]

equation(2.4)

In Heisenberg picture,

equation(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

equation(2.6)

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

equation(2.7)

Satisfying

equation(2.8)

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

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

Where

equation(2.10a) is the transport equation of the corresponding closed system, i.e. the time rate of EV resulting from HS, the Hamiltonian of the system S itself; and

equation(2.10b)

is the time rate resulting from the interaction HSR.

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 HSR and the additional influence term Yj (t).

The influence term (2.6) can be written as (j=1,2…,m) (j=1,2…,m) , (2.11)

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

equation(2.12a)

equation(2.12b)

being respectively the random force and interaction random force associated with the time rate of the basis variable Aj due to HS and HSR, respectively. Since the average of the random force over given ensembles vanishes, so equation In the rest of the paper we will no longer distinguish equation from its fluctuation equation

Generalized canonical statistical operator In order to go steps further

let us assume equation to be a GCSO:

equation(3.1)

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

Making use of the Kubo identity equation we have

equation(3.2a)

equation(3.2b)

Introducing the generalized quantum correlation function

equation(3.3)

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

equation(3.4)

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

equation(3.5)

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

Therefore Equation.(2.5) takes the form

equation(3.6)

here the memory term is expressed in terms of quantum correlation function of fluctuations of random forces. The influence term Yj (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].

equation(4.1)

where kB is the Boltzmann constant. The entropy production rate reads [7].

equation(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)=ρr (0) then the initial term in (3.6) vanishes. Combining (4.2) with (2.4) given by (3.6) and (2.11), we obtain

equation(4.3)

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

equation(4.4)

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

equation(4.5)

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

equation(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 iLSAj, indicating that in the organized motion term HS 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 equation and Bk respectively pertain to S and R, and they are initially independent:

equation where equation

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

equation(5.1)

equation(5.2)

equation determines the evolution of the EV of macrovariable Bk

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

equation(5.3a)

equation(5.3b)

equation(5.3c)

being the zeroth and first order terms of the EV of BK 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

equation(5.4a)

equation(5.4b)

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

equation(5.5a)

equation(5.5b)

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

equation(5.6a)

equation(5.6b)

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

equation(5.7)

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

equation

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

equation(5.8)

is the averaged interaction random force. Thus we obtain

equation(5.9)

equation(5.10)

With equation

If the open system is initially in an equilibrium state

equation(5.11)

equation is the initial inverse temperature of the system, then ρeq is a special case of (5.7) in which equation, equation and equation. Since A1 is the only basic variable, so equation and equation is the inverse temperature of S. Because equation, thus equation and

equation(5.12)

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

equation(5.13)

equation(5.14)

equation(5.15)

equation(5.16)

Finally we obtain the transport equation for the only basis variable HS:

equation(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

equation(5.18)

With (5.1) and (2.3),

equation(5.19)

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

equation(5.20)

equation(5.21)

equation(5.22)

equation(5.23)

equation(5.24)

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

equation(5.25)

which many be written as

equation(5.26)

equation(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:

equation(5.28)

equation(5.29)

If we are satisfied with keeping the linear term of λ in Ml(u,t), then

equation(5.30)

equation(5.31)

equation(5.32)

Therefore we have the approximate expressions

equation(5.33)

equation(5.34)

they are up to λ2 by (2.12a). Comparing (5.34) with (4.5), we see clearly that equation plays the role of equation 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

equation(5.35)

Where

equation(5.36a)

results from the organized motion in the transport equation due to HSR;

equation(5.36b)

from the disorganized motion and

equation(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]:

equation(6.1)

Where equation(6.2)

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

Onsager proposed a linear relationship between the fluxes and forces

equation(6.3)

with reciprocity relations

equation(6.4)

Thus we have

equation(6.5)

For the special case considered in Sect.5, the interaction between open system S and its environment R takes the form equation, 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 equation only:

equation(6.6)

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

equation(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 equation ( 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 equation to the entropy production, in which the free term equation in the Volterra equation is indeed a generalization of equation 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.

Conclusion

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, HS contributes nothing to the rate, but HSR 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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