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Register a new userGeneralized Non-equilibrium Heat and Work and the Fate of the Clausius Inequality
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P. D. Gujrati
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By generalizing the traditional concept of heat dQ and work dW to also include their time-dependent irreversible components d_{i}Q and d_{i}W allows us to express them in terms of the instantaneous internal temperature T(t) and pressure P(t), whereas the conventional form uses the constant values T_{0} and P_{0} of the medium. This results in an extremely useful formulation of non-equilibrium thermodynamics so that the first law turns into the Gibbs fundamental relation and the Clausius inequality becomes an equality ointdQ(t)/T(t)equiv0 in all cases, a quite remarkable but unexpected result. We determine the irreversible components d_{i}Qequivd_{i}W and discuss how they can be determined to obtain the generalized dW(t) and dQ(t).
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The status of heat and work in nonequilibrium thermodynamics is quite confusing and non-unique at present with conflicting interpretations even after a long history of the first law in terms of exchange heat and work, and is far from settled. Moreover, the exchange quantities lack certain symmetry. By generalizing the traditional concept to also include their time-dependent irreversible components 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 an equal footing and possess the symmetry. We prove that 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); such a clear separation does not occur for exchange quantities. 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. 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)equivT(t)dS(t) in terms of the instantaneous temperature T(t) of the system, which is reminiscent of equilibrium. We 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 {oint}d_{e}Q(t)/T_{0}<0, etc. become equalities {oint}dQ(t)/T(t)equiv0, etc, a quite remarkable but unexpected result in view of irreversibility. We determine the irreversible components in two simple cases to show the usefulness of our approach; here, the traditional formulation is of no use.
We extend certain basic and general concepts of thermodynamics to discrete Markov systems exchanging work and heat with reservoirs. In this framework we show that the celebrated Clausius inequality can be generalized and becomes an equality, significantly extending several recent results. We further show that achieving zero dissipation in a system implies that detailed balance obtains, and as a consequence there is zero power production. We obtain inequalities for power production under more general circumstances and show that near equilibrium obtaining maximum power production requires dissipation to be of the same order of magnitude.
Specific heat of dipolar glasses does not obey Debye law. It is of interest to know if the non-Debye specific heat can be accounted for in terms of Schottky-type specific heat arising from rotational tunneling states of the dipoles. This paper deals with rotational tunneling spectra of NH$_{4}^{+}$ ions and the non-Debye specific heat of mixed salts (e.g. (NH$_{4})_{x}$Rb$_{1-x}$Br) of ammonium and alkali halides which are known to exhibit dipolar glass phase. We have measured specific heat of above mixed salts at low temperatures (1.5 K $< T <$ 15 K). It is seen that while the specific heat of pure salts obeys Debye law, the specific heat of mixed salts does not obey Debye law. We have studied the effect of the NH$_{4}^{+}$ ion concentration, first neighbor environment of NH$_{4}^{+}$ ion and the lattice strain field on the non-Debye specific heat by carrying out measurements on suitably chosen mixed salts. Independent of above, we have measured the rotational tunneling spectra, $f(omega $), of the NH$_{4}^{+}$ ions in above salts using technique of neutron incoherent inelastic scattering. The above studies show that both the non-Debye specific heat and the tunneling spectra of the NH$_{4}^{+}$ ions depend on the NH$_{4}^{+}$ ion concentration, first neighbor environment of NH$_{4}^{+}$ ions and the lattice strain field. We have further shown that the temperature dependence of the measured specific heat can be explained for all the samples in terms of a model that takes account of contributions to the specific heat from the Debye phonons and the rotational tunneling states of the NH$_{4}^{+}$ ions. To the best of our knowledge, this is a first study where it is shown that measured specific heat of (NH$_{4})_{x}$Rb$_{1-x}$Br can be quantitatively explained in terms of an experimentally measured rotational tunneling spectra $f(omega $) of the NH$_{4}^{+}$ ions.
We show how Jarzynski relation can be exploited to analyze the nature of order-disorder and a bifurcation type dynamical transition in terms of a response function derived on the basis of work distribution over non-equilibrium paths between two thermalized states. The validity of the response function extends over linear as well as nonlinear regime and far from equilibrium situations.
The fluctuation-dissipation relation is usually formulated for a system interacting with a heat bath at finite temperature in the context of linear response theory, where only small deviations from the mean are considered. We show that for an open quantum system interacting with a non-equilibrium environment, where temperature is no longer a valid notion, a fluctuation-dissipation inequality exists. Clearly stated, quantum fluctuations are bounded below by quantum dissipation, whereas classically the fluctuations can be made to vanish. The lower bound of this inequality is exactly satisfied by (zero-temperature) quantum noise and is in accord with the Heisenberg uncertainty principle, both in its microscopic origins and its influence upon systems. Moreover, it is shown that the non-equilibrium fluctuation-dissipation relation determines the non-equilibrium uncertainty relation in the weak-damping limit.
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