Temperature of systems out of thermodynamic equilibrium

Nonequilibrium Temperature: An Approach from Irreversibility

Nonequilibrium temperature is a topic of research with continuously growing interest because of recent improvements in and applications of nonequilibrium thermodynamics, with particular regard to information theory, kinetic theory, nonequilibrium molecular dynamics, superfluids, radiative systems, etc. All studies on nonequilibrium temperature have pointed out that the definition of nonequilibrium temperature must be related to different aspects of the system, to the energy of the system, and to the energy fluxes between the system and its environment. In this paper, we introduce a definition of nonequilibrium temperature based on the Gouy-Stodola and Carnot theorems in order to satisfy all these theoretical requirements. The result obtained links nonequilibrium temperature to the electromagnetic outflow, generated by irreversibility during microscopic interaction in the system; to the environmental temperature; to the mean energy; and to the geometrical and physical characteristics of the system.

Shannon entropic temperature and its lower and upper bounds for non-Markovian stochastic dynamics

In this article we have studied Shannon entropic nonequilibrium temperature (NET) extensively for a system which is coupled to a thermal bath that may be Markovian or non-Markovian in nature. Using the phase-space distribution function, i.e., the solution of the generalized Fokker Planck equation, we have calculated the entropy production, NET, and their bounds. Other thermodynamic properties like internal energy of the system, heat, and work, etc. are also measured to study their relations with NET. The present study reveals that the heat flux is proportional to the difference between the temperature of the thermal bath and the nonequilibrium temperature of the system. It also reveals that heat capacity at nonequilibrium state is independent of both NET and time. Furthermore, we have demonstrated the time variations of the above-mentioned and related quantities to differentiate between the equilibration processes for the coupling of the system with the Markovian and the non-Markovian thermal baths, respectively. It implies that in contrast to the Markovian case, a certain time is required to develop maximum interaction between the system and the non-Markovian thermal bath (NMTB). It also implies that longer relaxation time is needed for a NMTB compared to a Markovian one. Quasidynamical behavior of the NMTB introduces an oscillation in the variation of properties with time. Finally, we have demonstrated how the nonequilibrium state is affected by the memory time of the thermal bath.

Nonequilibrium entropic temperature and its lower bound for quantum stochastic processes

In this paper, we have studied the Shannon "entropic" nonequilibrium temperature (NET) of quantum Brownian systems. The Brownian particle is attached to either a bosonic or fermionic bath. Based on the Fokker-Planck description of the c-number quantum Langevin equation, we have calculated entropy production, NET, and their bounds. Entropy production (EP), the upper bound of entropy production (UBEP), and the deviation of the UBEP from EP monotonically decrease as functions of time to equilibrium value for both of the thermal baths. The deviation decreases with increase of temperature of the bosonic thermal bath, but it becomes larger as the temperature of the fermionic bath grows. We also observe that nonequilibrium temperature and its lower bound monotonically increase to equilibrium value with the progression of time. But their difference as a function of time shows an optimum behavior in most cases. Finally, we have observed that at long time, the entropic temperature (for a bosonic thermal bath) first increases nonlinearly as a function of thermodynamic temperature (TT) and, if the TT is appreciably large, then it grows linearly. But for the fermionic thermal bath, the entropic temperature decreases monotonically as a nonlinear function of thermodynamic temperature to zero value.

Affinity and its derivatives in the glass transition process

The thermodynamic treatment of the glass transition remains an issue of intense debate. When associated with the formalism of non-equilibrium thermodynamics, the lattice-hole theory of liquids can provide new insight in this direction, as has been shown by Schmelzer and Gutzow [J. Chem. Phys. 125, 184511 (2006)], by Möller et al. [J. Chem. Phys. 125, 094505 (2006)], and more recently by Tropin et al. [J. Non-Cryst. Solids 357, 1291 (2011); ibid. 357, 1303 (2011)]. Here, we employ a similar approach. We include pressure as an additional variable, in order to account for the freezing-in of structural degrees of freedom upon pressure increase. Second, we demonstrate that important terms concerning first order derivatives of the affinity-driving-force with respect to temperature and pressure have been previously neglected. We show that these are of crucial importance in the approach. Macroscopic non-equilibrium thermodynamics is used to enlighten these contributions in the derivation of C(p),κ(T), and α(p). The coefficients are calculated as a function of pressure and temperature following different theoretical protocols, revealing classical aspects of vitrification and structural recovery processes. Finally, we demonstrate that a simple minimalist model such as the lattice-hole theory of liquids, when being associated with rigorous use of macroscopic non-equilibrium thermodynamics, is able to account for the primary features of the glass transition phenomenology. Notwithstanding its simplicity and its limits, this approach can be used as a very pedagogical tool to provide a physical understanding on the underlying thermodynamics which governs the glass transition process.

Shannon-entropy-based nonequilibrium "entropic" temperature of a general distribution

The concept of temperature is one of the key ideas in describing the thermodynamical properties of systems. In classical statistical mechanics of ideal gases, the notion of temperature can be described in at least two different ways: the kinetic temperature (related to the average kinetic energy of the particles) and the thermodynamic temperature (related to the ratio between infinitesimal changes in entropy and energy). For the Boltzmann distribution, the two notions lead to the same result. However, for nonequilibrium phenomena, while the kinetic temperature has been commonly used both for theoretical and simulation purposes, there appears to be no corresponding general definition of thermodynamic or entropic temperature. In this paper, we consider the statistical or Shannon entropy of a system and use the "de Bruijn identity" from information theory (see Appendix A 2 for a derivation of this identity) to show that it is possible to define a "Shannon temperature" or "entropic temperature" T for a nonequilibrium system as the ratio between the average curvature of the Hamiltonian function associated with the system and the trace of the Fisher information matrix of the nonequilibrium probability distribution (see Appendix A 1 for a definition of the Fisher information). We show that this definition subsumes many other attempts at defining entropic temperatures for nonequilibrium systems and is not restricted to equilibrium or near equilibrium systems. Intuitively, the gist of our approach is to use the Shannon or Gibbs entropy of a system and make use of the relation dS=dQ(rev)/T as a definition of temperature. We achieve this by positing a statistical notion of infinitesimal heating as the addition of uncorrelated random variables (in a special way). As an example of the utility of such a definition, we obtain the nonequilibrium entropic temperature for a system satisfying the Langevin equations. For such a system, we show that while the kinetic temperature is related to the changes in the energy of the system, the entropic or Shannon temperature is related to the changes in the entropy of the system. We show that this notion, together with the well known Cramer-Rao inequality in statistics demonstrates the validity of the second law of thermodynamics for such a nonequilibrium system.

The potential energy landscape contribution to the dynamic heat capacity

The dynamic heat capacity of a simple polymeric, model glassformer was computed using molecular dynamics simulations by sinusoidally driving the temperature and recording the resultant energy. The underlying potential energy landscape of the system was probed by taking a time series of particle positions and quenching them. The resulting dynamic heat capacity demonstrates that the long time relaxation is the direct result of dynamics resulting from the potential energy landscape. Moreover, the equilibrium (low frequency) portion of the potential energy landscape contribution to the heat capacity is found to increase rapidly at low temperatures and at high packing fractions. This increase in the heat capacity is explained by a statistical mechanical model based on the distribution of minima in the potential energy landscape.

Nonequilibrium thermodynamics: structural relaxation, fictive temperature, and Tool-Narayanaswamy phenomenology in glasses

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 nonequilibrium thermodynamic description of the system when it is out of thermal and mechanical equilibrium with the medium. Our approach allows us to identify the correct form of the Gibbs free energy and enthalpy. We also obtain an extension of the classical nonequilibrium thermodynamics due to de Donder in which one normally assumes thermal and mechanical equilibrium with the medium; see text. We find that the temperature and pressure differences between the system and the medium act as thermodynamic forces, which are normally neglected in the classical nonequilibrium thermodynamics. The Prigogine-Defay ratio is found to be greater than 1 merely due to the lack of equilibrium with the medium, even though we do not consider any internal order parameters. This shows that these forces should play an important role in relaxation processes. We then apply our approach to study the general trend during 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.