A Rigorous Derivation of the Entropy Bound and the Nature of Entropy Variation for Non-equilibrium Systems during Cooling

We use rigorous non-equilibrium thermodynamic arguments to prove (i) the residual entropy of any system is bounded below by the experimentally (calorimetrically) determined absolute temperature entropy, which itself is bounded below by the entropy of the corresponding equilibrium (metastable supercooled liquid) state, and (ii) the instantaneous entropy cannot drop below that of the equilibrium state. The theorems follow from the second law and the existence of internal equilibrium and refer to the thermodynamic entropy. They go beyond the calorimetric observations by Johari and Khouri [J. Chem. Phys. 134, 034515 (2011)] and others by extending them to all non-equilibrium systems regardless of how far they are from their equilibrium states. We also discuss the statistical interpretation of the thermodynamic entropy and show that the conventional Gibbs or Boltzmann interpretation gives the correct thermodynamic entropy even for a single sample regardless of the duration of measurements.Comment: 25 pages; 1 figure; new result

Complexity Thermodynamics, Equiprobability Principle, Percolation, and Goldstein's Conjectures

The configurational states as introduced by Goldstein represent the system's basins and are characterized by their free energies $\varphi(T,V)$ as we show here. We find that the energies of some of the special points (termed basin identifiers here) like the basin minima, maxima, lowest energy barriers, etc. cannot be used to characterize the configurational states of the system in all cases due to their possible non-monotonic behavior as we explain. The complexity $\mathcal{S(}\varphi,T,V\mathcal{)},$ represents the configurational state entropy. We prove that $S(T,V) \equiv S(\varphi_b,T,V) + S_b(T,V)$, where $S_{\text{b}},$ and $\varphi_{\text{b}}$ are the basin entropy and free energy, respectively. We further prove that all basins at equilibrium have the same equilibrium basin energy $E(T,V)$ and entropy S_{\text{b}% }(T,V). Here, $\varphi$\ and $E$ are measured with respect to the zero of the potential energy. The Boltzmann equiprobability principle is shown to apply to the basins in that each equilibrium basin has an equal probability $\mathcal{P=}\exp(-\mathcal{S})$ to be explored. This principle allow us to draw some useful conclusions about the time-dependence in the system. We discuss the percolation due to basin connectivity and its relevance for the dynamic transition. Our analysis validates modified Goldstein's conjectures. All the above results are shown to be valid at all temperatures, and not just low temperatures as originally propsed by Goldstein.Comment: 14 pages; no figur

Consequences of the Detailed Balance for the Crooks Fluctuation Theorem

We show that the assumptions of the detailed balance and of the initial equilibrium macrostate, which are central to the Crooks fluctuation theorem (CFT), lead to all microstates along a trajectory to have equilibrium probabilities. We also point out that the Crooks's definition of the backward trajectory does not return the system back to its initial microstate. Once corrected, the detailed balance assumption makes the CFT a theorem only about reversible processes involving reversible trajectories that satisfy Kolmogorov's criterion. As there is no dissipation, the CFT cannot cover irreversible processes, which is contrary to the common belief. This is consistent with our recent result that the JE is also a result only for reversible processes.Comment: 15 page

Correcting the Mistaken Identification of Nonequilibrium Microscopic Work

The energy change dE_k for the kth microstate is erroneously equated with the external work done on the microstate. It ignores the ubiquitous internal energy change d_iW_k due to force imbalance between the internal and external forces. We show that this contribution is present even in a reversible process, which is a surprise. We show that the correct identification is dE_k=-dW_k, where dW_k is the generalized work done by the microstate. We prove that the thermodynamic average of the internal work gives dissipation and is not captured by the external work. The latter effectively sets d_iW_k =0 and results in no dissipation. Using dW_k to account for irreversibility, we obtain a new work relation that works even for free expansion, where the Jarzynski equality fails. In the new work relation, dW_k depends only on the energies of the initial and final states and not on the actual process. This makes the new relation very different from the Jarzynski equality. The correction has far-reaching consequences and requires reassessment of current applications of external work in theoretical physics.Comment: 20 pages, 4 Figures. Some overlap with arXiv:1702.0045

Nonequilibrium Thermodynamics. Symmetric and Unique Formulation of the First Law, Statistical Definition of Heat and Work, Adiabatic Theorem and the Fate of the Clausius Inequality: A Microscopic View

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.Comment: 39 pages, 1 figur

Stationary Metastability in an Exact Non-Mean Field Calculation for a Model without Long-Range Interactions

We introduce the concept of stationary metastable states (SMS's) in the presence of another more stable state. The stationary nature allows us to study SMS's by using a restricted partition function formalism as advocated by Penrose and Lebowitz and requires continuing the free energy. The formalism ensures that SMS free energy satisfies the requirement of thermodynamic stability everywhere including T=0, but need not represent a pysically observable metastable state over the range where the entropy under continuation becomes negative. We consider a 1-dimensional m-component axis-spin model involving only nearest-neighbor interactions, which is solved exactly. The high-temperature expansion of the model representys a polymer problem in which m acts as the activity of a loop formation. We follow deGennes and trerat m as a real variable. A thermodynamic phase transition occurs in the model for m<1. The analytic continuation of the high-temperature disordered phase free energy below the transition represents the free energy of the metastable state. The calculation shows that the notion of SMS is not necessaily a consequence of only mean-field analysis or requires long-range interactions.Comment: 9 pages, 2 figure

General Mechanism for a Positive Temperature Entropy Crisis in Stationary Metastable States: Thermodynamic Necessity and Confirmation by Exact calculations

We study stationary metastable states(SMS's)using a restricted partition function formalism. The formalism ensures that SMS free energy exists all the way to T=0, and remains stable. We introduce the concept of the reality condition, according to which the entropy $S(T)$ of a set of coupled degrees of freedom must be non-negative. The entropy crisis, which does not affect stability, is identified as the violation of the reality condition. We identify and validate rigorously, using general thermodynamic arguments, the following general thermodynamic mechanism behind the entropy crisis in SMS. The free energy $F_{\text{dis}}(T)$ of any SMS must be equal to the T=0 crystal free energy $E_{0}$ at two different temperatures $T=0,$ and $T=T_{\text{eq}}>0$. Thus, the stability requires $F_{\text{dis}}(T)$ to possess a maximum at an intermediate but a strictly positive temperature $T_{\text{K}},$ where the energy is $E=E_{\text{K}}.$ The SMS branch below $T_{\text{K}}$ gives the entropy crisis and must be replaced by hand by an ideal glass free energy of constant energy $E_{\text{K}},$ and vanishing entropy. Hence, $T_{\text{K}}>0$ represents the Kauzmann temperature. The ideal glass energy $E_{\text{K}}$ is higher than the crystal energy $E_{0}$ at absolute zero, which is in agreement with the experimenatal fact that the extrapolated energy of a real glass at T=0 is higher than its T=0 crystal energy. We confirm the general predictions by two exact calculations, one of which is not mean-field. The calculations clearly show that the notion of SMS is not only not vaccuous, but also not a consequence of a mean-field analysis. They also show that certain folklore cannot be substantiated.Comment: 31 pages, 2 figure

Nonequilibrium Entropy

We consider an isolated system in an arbitrary state and provide a general formulation using first principles for an additive and non-negative statistical quantity that is shown to reproduce the equilibrium thermodynamic entropy of the isolated system. We further show that the statistical quantity represents the nonequilibrium thermodynamic entropy when the latter is a state function of nonequilibrium state variables; see text. We consider an isolated 1-d ideal gas and determine its non-equilibrium statistical entropy as a function of the box size as the gas expands freely isoenergetically, and compare it with the equilibrium thermodynamic entropy S_{0eq}. We find that the statistical entropy is less than S_{0eq} in accordance with the second law, as expected. To understand how the statistical entropy is different from thermodynamic entropy of classical continuum models that is known to become negative under certain conditions, we calculate it for a 1-d lattice model and discover that it can be related to the thermodynamic entropy of the continuum 1-d Tonks gas by taking the lattice spacing {\delta} go to zero, but only if the latter is state-independent. We discuss the semi-classical approximation of our entropy and show that the standard quantity S_{f}(t) in the Boltzmann's H-theorem does not directly correspond to the statistical entropy.Comment: 13 pages, 2 figure

Comment on "Comment on: On the reality of residual entropies of glasses and disordered crystals" [J. Chem. Phys. 129, 067101 (2008)]

By using very general arguments, we show that the entropy loss conjecture at the glass transition violates the second law of thermodynamics and must be rejected.Comment: 6 pages, 2 figure

Entropy Crisis, Defects and the Role of Competition in Monatomic Glass Formers

We establish the existence of an entropy crisis in monatomic glass formers. The work finally shows that the entropy crisis is ubiqutous in all supercooled liquids. We also study the roles of defects and energetic competition on the ideal glass.Comment: 4 pages, 3 figure