Nonequilibrium thermodynamics. II. Application to inhomogeneous systems (original) (raw)
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Entropy Principle and Recent Results in Non-Equilibrium Theories
Entropy, 2014
We present the state of the art on the modern mathematical methods of exploiting the entropy principle in thermomechanics of continuous media. A survey of recent results and conceptual discussions of this topic in some well-known non-equilibrium theories (Classical irreversible thermodynamics CIT, Rational thermodynamics RT, Thermodynamics of irreversible processes TIP, Extended irreversible thermodynamics EIT, Rational Extended thermodynamics RET) is also summarized.
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The paper discusses how the two thermodynamic properties, energy (U) and exergy (E), can be used to solve the problem of quantifying the entropy of non-equilibrium systems. Both energy and exergy are a priori concepts, and their formal dependence on thermodynamic state variables at equilibrium is known. Exploiting the results of a previous study, we first calculate the non-equilibrium exergy E n-eq can be calculated for an arbitrary temperature distributions across a macroscopic body with an accuracy that depends only on the available information about the initial distribution: the analytical results confirm that E n-eq exponentially relaxes to its equilibrium value. Using the Gyftopoulos-Beretta formalism, a non-equilibrium entropy S n-eq (x,t) is then derived from E n-eq (x,t) and U(x,t). It is finally shown that the non-equilibrium entropy generation between two states is always larger than its equilibrium (herein referred to as "classical") counterpart. We conclude that every iso-energetic non-equilibrium state corresponds to an infinite set of non-equivalent states that can be ranked in terms of increasing entropy. Therefore, each point of the Gibbs plane corresponds therefore to a set of possible initial distributions: the non-equilibrium entropy is a multi-valued function that depends on the initial mass and energy distribution within the body. Though the concept cannot be directly extended to microscopic systems, it is argued that the present formulation is compatible with a possible reinterpretation of the existing non-equilibrium formulations, namely those of Tsallis and Grmela, and answers at least in part one of the objections set forth by Lieb and Yngvason. A systematic application of this paradigm is very convenient from a theoretical point of view and may be beneficial for meaningful future applications in the fields of nano-engineering and biological sciences.
The status of heat and work in nonequilibrium thermodynamics is quite confusing and nonunique at present with conflicting interpretations even after a long history of the first law dE(t) = d e Q(t) − dW e (t) in terms of exchange heat and work, and is far from settled. Moreover, the exchange quantities lack certain symmetry (see text). By generalizing the traditional concept to also include their time-dependent irreversible components d i Q(t) and d i W (t) allows us to express the first law in a symmetric form dE(t) = dQ(t) − dW (t) in which dQ(t) and work dW (t) appear on equal footing and possess the symmetry. We prove that d i Q(t) ≡ d i W (t); as a consequence, irreversible work turns into irreversible heat. Statistical analysis in terms of microstate probabilities p i (t) uniquely identifies dW (t) as isentropic and dQ(t) as isometric (see text) change in dE(t), a result known in equilibrium. We show that such a clear separation does not occur for d e Q(t) and dW e (t). Hence, our new formulation of the first law provides tremendous advantages and results in an extremely useful formulation of non-equilibrium thermodynamics, as we have shown recently [Phys. Rev. E 81, 051130 (2010); ibid 85, 041128 and 041129 (2012)]. We prove that an adiabatic process does not alter p i. All these results remain valid no matter how far the system is out of equilibrium. When the system is in internal equilibrium, dQ(t) ≡ T (t)dS(t) in terms of the instantaneous temperature T (t) of the system, which is reminiscent of equilibrium, even though, neither d e Q(t) ≡ T (t)d e S(t) nor d i Q(t) ≡ T (t)d i S(t). Indeed, d i Q(t) and d i S(t) have very different physics. We express these quantities in terms of d e p i (t) and d i p i (t), and demonstrate that p i (t) has a form very different from that in equilibrium. The first and second laws are no longer independent so that we need only one law, which is again reminiscent of equilibrium. The traditional formulas like the Clausius inequality d e Q(t)/T 0 < 0, ∆ e W < −∆ [E(t − T 0 S(t))], etc. become equalities dQ(t)/T (t) ≡ 0, ∆W = −∆ [E(t − T (t)S(t)], etc, a quite remarkable but unexpected result in view of the fact that ∆ i S(t) > 0. We identify the uncompensated transformation N (t, τ) during a cycle. We determine the irreversible components in two simple cases to show the usefulness of our approach; here, the traditional formulation is of no use. Our extension bring about a very strong parallel between equilibrium and non-equilibrium thermodynamics, except that one has irreversible entropy generation d i S(t) > 0 in the latter.
A Derivation of the Main Relations of Nonequilibrium Thermodynamics
ISRN Thermodynamics, 2013
The principles of nonequilibrium thermodynamics are discussed, using the concept of internal variables that describe deviations of a thermodynamic system from the equilibrium state. While considering the first law of thermodynamics, work of internal variables is taken into account. It is shown that the requirement that the thermodynamic system cannot fulfil any work via internal variables is equivalent to the conventional formulation of the second law of thermodynamics. These statements, in line with the axioms introducing internal variables can be considered as basic principles of nonequilibrium thermodynamics. While considering stationary nonequilibrium situations close to equilibrium, it is shown that known linear parities between thermodynamic forces and fluxes and also the production of entropy, as a sum of products of thermodynamic forces and fluxes, are consequences of fundamental principles of thermodynamics.
Nonequilibrium entropy and the second law of thermodynamics: A simple illustration
International Journal of Thermophysics, 1993
The objective of this paper is twofold: first, to examine how the concepts of extended irreversible thermodynamics are related to the notion of accompanying equilibrium state introduced by Kestin; second, to compare the behavior of both the classical local equilibrium entropy and that used in extended irreversible thermodynamics. Whereas the former does not show a monotonic increase, the latter exhibits a steady increase during the heat transfer process; therefore it is more suitable than the former one to cope with the approach to equilibrium in the presence of thermal waves.
On the Problem of Formulating Principles in Nonequilibrium Thermodynamics
Entropy, 2010
In this work, we consider the choice of a system suitable for the formulation of principles in nonequilibrium thermodynamics. It is argued that an isolated system is a much better candidate than a system in contact with a bath. In other words, relaxation processes rather than stationary processes are more appropriate for the formulation of principles in nonequilibrium thermodynamics. Arguing that slow varying relaxation can be described with quasi-stationary process, it is shown for two special cases, linear nonequilibrium thermodynamics and linearized Boltzmann equation, that solutions of these problems are in accordance with the maximum entropy production principle.
On the (Boltzmann) entropy of non-equilibrium systems
Physica D: Nonlinear Phenomena, 2004
Boltzmann defined the entropy of a macroscopic system in a macrostate M as the log of the volume of phase space (number of microstates) corresponding to M . This agrees with the thermodynamic entropy of Clausius when M specifies the locally conserved quantities of a system in local thermal equilibrium (LTE). Here we discuss Boltzmann's entropy, involving an appropriate choice of macro-variables, for systems not in LTE. We generalize the formulas of Boltzmann for dilute gases and of Resibois for hard sphere fluids and show that for macro-variables satisfying any deterministic autonomous evolution equation arising from the microscopic dynamics the corresponding Boltzmann entropy must satisfy an H-theorem.
Phenomenological thermodynamics and entropy principles
2003
There is no unified approach to irreversible thermodynamics in the phenomenological theories of continuum thermodynamics. The different approaches are based on different forms of the second law. Depending upon which basic underlying principles are postulated, the entropy principle yields different implications on the nonequilibrium quantities for these to fulfil the irreversibility requirements.