Starting from the second law of thermodynamics applied to an isolated system consisting of the system surrounded by an extremely large medium, we formulate a general non-equilibrium thermodynamic description of the system when it is out of equilibrium. We then apply it to study the structural relaxation in glasses and establish the phenomenology behind the concept of the fictive temperature and of the empirical Tool-Narayanaswamy equation on firmer theoretical foundation.

We provide an extension of a recent approach to study non-equilibrium thermodynamics [Phys.Rev. E 81, 051130 (2010), to be denoted by I in this work] to inhomogeneous systems by considering the latter to be composed of quasi-independent subsystems. The system Σ along with the (macroscopically extremely large) medium Σ forms an isolated system Σ 0 . Starting from the Gibbsian formulation of the entropy for Σ 0 , which is valid even when Σ 0 is out of equilibrium, we derive the Gibbsian formulation of the entropy of Σ, which need not be in equilibrium. We show that the additivity of entropy requires quasi-independence of the subsystems, which requires that the interaction energies between different subsystems must be negligible so that the energy also becomes additive. The thermodynamic potentials of subsystems such as the Gibbs free energy that continuously decrease during approach to equilibrium are determined by the field parameters (temperature, pressure, etc.) of the medium and exist no matter how far the subsystems are out of equilibrium so that their field variables may not even exist. This and the requirement of quasiindependence make our approach different from the conventional approach due to de Groot and others as discussed in the text. As the energy depends on the frame of reference, the thermodynamic potentials and Gibbs fundamental relation, but not the entropy, depend on the frame of reference. The possibility of relative motion between subsystems described by their net linear and angular momenta gives rise to viscous dissipation. The concept of internal equilibrium introduced in I is developed further here and its important consequences are discussed for inhomogeneous systems. The concept of internal variables (various examples are given in the text) as variables that cannot be controlled by the observer for non-equilibrium evolution is also discussed. They are important because the internal equilibrium in the presence of internal variables is lost if internal variables are not used in thermodynamics. It is argued that their affinity vanishes only in equilibrium. Gibbs fundamental relation, thermodynamic potentials and irreversible entropy generation are expressed in terms of observables and internal variables. We use these relations to eventually formulate the non-equilibrium thermodynamics of inhomogeneous systems. We also briefly discuss the case when bodies form an isolated system without any medium to obtain their irreversible contributions and show that this case is no different than when bodies are in an extremely large medium.

A detailed analysis of deterministic (one-to-one) and stochastic (one-to-many) dynamics establishes that dS/dt > 0 is only consistent with the latter, which contains violation of temporal symmetry and homogeneity. We observe that the former only supports dS/dt = 0 and cannot give rise to Boltzmann's molecular chaos assumption. The ensemble average is more meaningful than the temporal average, especially in non-equilibrium statistical mechanics of systems confined to disjoint phase space components, which commonly occurs at low temperatures. We propose that the stochasticity arises from extra degrees of freedom, which are not part of the system. We provide a simple resolution of the recurrence and irreversibility paradoxes.

The mechanism behind the steep slowing down of molecular motions upon approaching the glass transition remains a great puzzle. Most of the theories relate this mechanism to the cooperativity in molecular motion. In this work, we estimate the length scale of molecular cooperativity xi for many glass-forming systems from the collective vibrations (the so-called boson peak). The obtained values agree well with the dynamic heterogeneity length scale estimated using four-dimensional NMR. We demonstrate that xi directly correlates to the dependence of the structural relaxation on volume. This dependence presents only one part of the mechanism of slowing down the structural relaxation. Our analysis reveals that another part, the purely thermal variation in the structural relaxation (at constant volume), does not have a direct correlation with molecular cooperativity. These results call for a conceptually new approach to the analysis of the mechanism of the glass transition and to the role of molecular cooperativity.

We identify the mechanism behind a rapid entropy drop in the metastable (ML) polymer liquid and clarify the significance of the Kauzmann paradox. We also establish a thermodynamic basis for an apparent critical mode-coupling transition between supercooled (SCL) and ML polymer liquids, and for the ideal glass transition but only in ML. The latter need not ever form an equilibrium phase. The crystal can have higher entropy than ML or SCL polymer liquids.

A lattice model of semiflexible linear chains (with equilibrium polydispersity) containing free volume is solved exactly on a Husimi cactus. A metastable liquid (ML) is discovered to exist only at low temperatures and is distinct (and may be disjoint) from the supercooled liquid (SCL) that exists only at high temperatures. The free volume plays a significant role in that the spinodals of the ML and SCL merge and then disappear as the free volume is reduced. The Kauzmann temperature T(K) occurs in the ML without any singularity. At T(MC)>T(K), the ML specific heat has a peak. For infinitely long polymers, the peak height diverges and the free volume vanishes at T(MC), resulting in a continuous liquid-liquid transition. Contrary to the conventional wisdom, both T(K) and T(MC) occur in the ML and not in the SCL.

Abstract:We review the concept of nonequilibrium thermodynamic entropy and observables and internal variables as state variables, introduced recently by us, and provide a simple first principle derivation of additive statistical entropy, applicable to all nonequilibrium states by treating thermodynamics as an experimental science. We establish their numerical equivalence in several cases, which includes the most important case when the thermodynamic entropy is a state function. We discuss various interesting aspects of the two entropies and show that the number of microstates in the Boltzmann entropy includes all possible microstates of non-zero probabilities even if the system is trapped in a disjoint component of the microstate space. We show that negative thermodynamic entropy can appear from nonnegative statistical entropy.

We discuss the need for discretization to evaluate the configurational entropy in a general model. We also discuss the prescription using restricted partition function formalism to study the stationary limit of metastable states where a more stable equilibrium state exists. We introduce a lattice model of dimers as a paradigm of molecular fluid and study stationary metastability in it to investigate the root cause of glassy behavior. We demonstrate the existence of entropy crisis in metastable states, from which it follows that the entropy crisis is the root cause underlying the ideal glass transition in systems with particles of all sizes. The orientational interactions in the model control the nature of the liquid-liquid transition observed in recent years in molecular glasses.