Entropy and entropy production in some applications

Abstract

By using entropy and entropy production, we calculate the steady flux of some phenomena. The method we use is a competition method, S S /τ + σ = maximum, where S S is system entropy, σ is entropy production and τ is microscopic interaction time. System entropy is calculated from the equilibrium state by studying the flux fluctuations. The phenomena we study include ionic conduction, atomic diffusion, thermal conduction and viscosity of a dilute gas.PACS numbers: 05.70. Ln, 51.30.+i, 51.20.+d Keywords: entropy; entropy production; ionic conductivity; thermal conductivity; viscosity; atomic diffusion INTRODUCTIONTransport phenomena, including electrical conduction, thermal conduction, viscosity and diffusion, are all about non-equilibrium states. A non-equilibrium state always tends to relax to the equilibrium state. But an opposite tendency can also exist if there is an external force, because an external force always induces its conjugate flux to tend to become greater. The two tendencies compete with each other and the compromise is a steady flux. However, their strengths seem measured in different quantities: one is system entropy, the other entropy production. In order to make them comparable, a time parameter τ must come in.τ relates to microscopic interactions. For instance, it can be the mean molecular collision time, τ = λ/v. During such a τ , a molecule averagely collides once. The collision result is non-deterministic. So randomness arises, which is then described by system entropy. A system entropy is thus associated with a τ as well as a series of discrete processes. By using τ , system entropy and entropy production can be compared. There are three methods to do so, see Tab I.tendency for a flux to relax tendency for a flux to increase system entropy S S entropy production σ S S − S S0 ∝ −J 2 σ ∝ J method 1 relaxation entropy production usual entropy production= σ method 2 system entropy environment entropy S S S E = S E0 + τ σ S S + S E = maximum method 3 system entropy and τ entropy production S S /τ σ S S /τ + σ = maximum The first method is an entropy production method [1][2][3]. In this method, one constructs 2 a new entropy production from the system entropy, by considering that a given flux relaxes at the maximum rate. The corresponding relaxation time is τ which now is interpreted as the shortest possible relaxation time. We can do this because the lower limit of the relaxation time should be the microscopic interaction time. Thus, a new entropy production is constructed, which is then compared with the other. When they are equal, the steady flux is found. Note that this method needs to be used together with two other principles: the steepest entropy ascent principle (SEA) [4], and the maximum entropy production principle (MEP) [5][6][7][8].The second method is an entropy method [3]. In this method, one constructs a new entropy from the entropy production, by considering that the entropy producing process is a series of discrete processes. Each process lasts a time τ . The newly constructed e...

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