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P. D. Gujrati

Publications and source records attributed to P. D. Gujrati.

At least 37 records · Page 2Linked to original sources
arXiv2016

Determination of Nonequilibrium Temperature and Pressure using Clausius Equality in a State with Memory: A Simple Model Calculation

Use of the extended definition of heat dQ=deQ+diQ converts the Clausius inequality dS greater than or equal to deQ/T0 into an equality dS=dQ/T involving the nonequilibrium temperature T of the system having the conventional interpretation that heat flows from hot to cold. The equality is applied to the exact quantum evolution of a 1-dimensional ideal gas free expansion. In a first ever calculation of its kind in an expansion which retains the memory of initial state, we determine the nonequilibrium temperature T and pressure P, which are then compared with the ratio P/T obtained by an independent method to show the consistency of the nonequilibrium formulation. We find that the quantum evolution by itself cannot eliminate the memory effect.cannot eliminate the memory effect; hence, it cannot thermalize the system.

cond-mat.stat-mech↗
arXiv2013

Some Rigorous Results Relating Nonequilibrium, Equilibrium, Calorimetrically Measured and Residual Entropies during Cooling

We use rigorous nonequilibrium thermodynamic arguments to establish that (i) the nonequilibrium entropy S(T_{0}) of any system is bounded below by the experimentally (calorimetrically) determined entropy S_{expt}(T_{0}), (ii) S_{expt}(T_{0}) is bounded below by the equilibrium or stationary state (such as the supercooled liquid) entropy S_{SCL}(T_{0}) and consequently (iii) S(T_{0}) cannot drop below S_{SCL}(T_{0}). It then follows that the residual entropy S_{R} is bounded below by the extrapolated S_{expt}(0)>S_{SCL}(0) at absolute zero. These results are very general and applicable to all nonequilibrium systems regardless of how far they are from their stationary states.

physics.chem-ph↗
arXiv2013

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 δ 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.

physics.chem-ph↗
arXiv2012

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.

physics.chem-ph↗
arXiv2011

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.

cond-mat.stat-mech↗
arXiv2011

Generalized Non-equilibrium Heat and Work and the Fate of the Clausius Inequality

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}Q\equivd_{i}W and discuss how they can be determined to obtain the generalized dW(t) and dQ(t).

physics.chem-ph↗
arXiv2011

Is Structural Relaxation During Vitrification the Inverse of the Glass Transition?

We have recently applied the second law to an isolated system, consisting of a system Σ such as a glass surrounded by an extremely large medium Σ, to show that the instantaneous temperature T(t), thermodynamic entropy S(T_0,t) and enthalpy H(T_0,t) of Σ decrease in any isothermal relaxation towards their respective equilibrium values under isobaric cooling. The Gibbs statistical entropy also conforms to the above relaxation behavior in a glass, which however is contrary to the conjecture by Gupta, Mauro and coworkers that the glass transition and the structural relaxation are inverse processes. However, they do not establish that the entropy loss during the glass transition is accompanied by a concomitant entropy gain of the medium to maintain the second law. They use a novel statistical formulation of entropy based on several conjectures such as it being zero for a microstate, but do not compare its behavior with the thermodynamic entropy. The formulation is designed to show the entropy loss. Its subsequent rise not only contradicts our result but also implies that the glass during relaxation must have a negative absolute temperature. To understand these puzzling results and the above conjecture, we have carried out a critical evaluation of their unconventional approach. We find that the inverse conjecture is neither supported by their approach nor by the second law. The zero-entropy microstate conjecture is only consistent with S(T_0,t)\equiv 0 at all temperatures, not just at absolute zero and is found to have no scientific merit. We show that the maximum entropy gain of the medium during the glass transition invalidates the entropy loss conjecture. After pointing out other misleading, confusing and highly exaggerated statements in their work, we finally conclude that their unconventional statistical approach and computational scheme are not appropriate for glasses.

cond-mat.stat-mech↗
arXiv2011

Non-equilibrium thermodynamics.III. Thermodynamic Principles, Entropy Continuity during Component Confinement, Energy Gap and the Residual Entropy

To investigate the consequences of component confinement such as at a glass transition and the well-known energy or enthalpy gap (between the glass and the perfect crystal at absolute zero, see text), we follow our previous approach [Phys. Rev. E 81, 051130 (2010)] of using the second law applied to an isolated system Σ_0 consisting of the homogeneous system Σ and the medium Σ. We establish on general grounds the continuity of the Gibbs free energy G(t) of Σ as a function of time at fixed temperature and pressure of the medium. It immediately follows from this and the observed continuity of the enthalpy during component confinement that the entropy S of the open system Σ must remain continuous during a component confinement such as at a glass transition. We use these continuity properties and the recently developed non-equilibrium thermodynamics to formulate thermodynamic principles of additivity, reproducibility, continuity and stability that must also apply to non-equilibrium systems in internal equilibrium. We find that the irreversibility during a glass transition only justifies the residual entropy S_{R} to be at least as much as that determined by disregarding the irreversibility, a common practice in the field. This justifies a non-zero residual entropy S_{R} in glasses, which is also in accordance with the energy or enthalpy gap at absolute zero. We develop a statistical formulation of the entropy of a non-equilibrium system, which results in the continuity of entropy during component confinement in accordance with the second law and sheds light on the mystery behind the residual entropy, which is consistent with the recent conclusion [Symmetry 2, 1201 (2010)] drawn by us.

cond-mat.stat-mech↗
arXiv2011

Non-equilibrium thermodynamics. II: Application to inhomogeneous systems

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. The additivity of entropy requires quasi-independence of the subsystems, which limits the size of various subsystems. The thermodynamic potentials of subsystems such as the Gibbs free energy are determined by the field parameters (temperature, pressure, etc.) of the medium even if the latter may not exist for the subsystems. This and the requirement of quasi-independence make our approach different from the conventional approach due to de Groot and others. 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. Important consequences of internal equilibrium are discussed. Internal variables as variables that cannot be controlled by the observer for non-equilibrium evolution are also discussed. We finally 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.

cond-mat.stat-mech↗
arXiv2011

Non-equilibrium thermodynamics. IV: Generalization of Maxwell, Claussius-Clapeyron and Response Functions Relations, and the Prigogine-Defay Ratio for Systems in Internal Equilibrium

We follow the consequences of internal equilibrium in non-equilibrium systems that has been introduced recently [Phys. Rev. E 81, 051130 (2010)] to obtain the generalization of Maxwell's relation and the Clausius-Clapeyron relation that are normally given for equilibrium systems. The use of Jacobians allow for a more compact way to address the generalized Maxwell relations; the latter are available for any number of internal variables. The Clausius-Clapeyron relation in the subspace of observables show not only the non-equilibrium modification but also the modification due to internal variables that play a dominant role in glasses. Real systems do not directly turn into glasses (GL) that are frozen structures from the supercooled liquid state L; there is an intermediate state (gL) where the internal variables are not frozen. Thus, there is no single glass transition. A system possess several kinds of glass transitions, some conventional (L \rightarrow gL; gL\rightarrow GL) in which the state change continuously and the transition mimics a continuous or second order transition, and some apparent (L\rightarrow gL; L\rightarrow GL) in which the free energies are discontinuous so that the transition appears as a zeroth order transition, as discussed in the text. We evaluate the Prigogine-Defay ratio Π in the subspace of the observables at these transitions. We find that it is normally different from 1, except at the conventional transition L\rightarrow gL, where Π=1 regardless of the number of internal variables.

cond-mat.stat-mech↗
arXiv2009

Energy Gap and the Ideal Glass as a Defective Crystal: A Lattice Model of Monatomic Systems

We use the cell model to justify the use of a lattice model to study the ideal glass transition. Based on empirical evidence and several previous exact calculations, we hypothesize that there exists an energy gap between the lowest possible energy of a glass (the ideal glass IG) and the crystal (CR). The gap is due to the presence of strongly correlated excitations with respect to the ideal CR; thus, one can treat IG as a highly defective crystal. We argue that an excitation in IG requires energy that increases logarithmically with the size of the system; as a consequence, we prove that IG must emerge at a positive temperature T_{K}. We propose an antiferromagnetic Ising model on a lattice to model liquid-crystal transition in a simple fluid or a binary mixture, which is then solved exactly on a recursive (Husimi) lattice to investigate the ideal glass transition, the nature of defects in the supercooled liquid and CR analytically, and the effects of competing interactions on the glass transition. The calculation establishes the gap. The lattice entropy of the supercooled liquid vanishes at a positive temperature T_{K}>0, where IG emerges but where CR has a positive entropy. The macrostate IG is in a particular and unique disordered microstate at T_{K}, just as the ideal CR is in a perfectly ordered microstate at absolute zero. This explains why it is possible for CR to have a higher entropy at T_{K} than IG. The demonstration here of an entropy crisis in monatomic systems along with previously known results strongly suggests that the entropy crisis first noted by Kauzmann and demonstrated by Gibbs and DiMarzio in long polymers appears to be ubiquitous in all supercooled liquids.

cond-mat.mtrl-sci↗
arXiv2009

Non-equilibrium 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 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.

cond-mat.stat-mech↗
arXiv2009

Comment on "Non-equilibrium entropy of glasses formed by continuous cooling" [J. Non-Cryst. Solids 355 (2009) 600]

We use the general statement of the second law applied to an isolated system, the glass in an extremely large medium, to prove that the entropy of the glass must decrease with time during its relaxation towards the supercooled liquid state. This result contradicts the claim of Mauro et al and their computational result of the entropy, according to which their entropy of the glass increases. Our approach using the isolated system completely bypasses the issue of ergodicity loss in glasses, as discussed in the comment.

cond-mat.dis-nn↗
arXiv2009

Where is the residual entropy of a glass hiding?

We revisit the controversy, discussed recently by Goldstein in this journal[J. Chem. Phys. 128,154510 (2008)], whether the residual entropy is real or fictional. It is shown that the residual entropy loss conjecture (ELC) at the glass transition, which results in a discontinuous entropy violates many fundamental principles of classical thermodynamics, and also contradicts some experimental facts. Assuming, as is common in the field, that glasses are in internal equilibrium, we show that the continuity of enthalpy and volume at the glass transition require the continuity of the Gibbs free energy and the entropy, which contradicts ELC. It is then argued that ELC is founded on an incorrect understanding of what it means for a glass to be kinetically trapped in a basin and of the concept of probability and entropy. Once this misunderstanding is corrected in our approach by the proper identification of entropy as the ensemble entropy, which is in accordance with the principle of reproducibility (see Sect. II), it follows immediately that the residual entropy does not disappear in a kinetically frozen glassy state and all the violations of thermodynamics disappear. We show that the temporal definition of entropy over finite times does not make sense for glasses as it is not unique. There is no loss of ergodicity and causality, contrary to some recent claims.

cond-mat.mtrl-sci↗
arXiv2008

Irreversibility, Molecular Chaos, and A Simple Proof of the Second Law

The irreversibility in a statistical system is traced to its probabilistic evolution, and the molecular chaos assumption is not its unique consequence as is commonly believed. Under the assumption that the rate of change of the each microstate probability vanishes only as time diverges, we prove that the entropy of a system at constant energy cannot decrease with time.

cond-mat.stat-mech↗
arXiv2008

Compressible or incompressible blend of interacting monodisperse star and linear polymers near a surface

We consider a lattice model of a mixture of repulsive, attractive, or neutral monodisperse star (species A) and linear (species B) polymers with a third monomeric species C, which may represent free volume. The mixture is next to a hard, infinite plate whose interactions with A and C can be attractive, repulsive, or neutral. These two interactions are the only parameters necessary to specify the effect of the surface on all three components. We numerically study monomer density profiles using the method of Gujrati and Chhajer that has already been previously applied to study polydisperse and monodisperse linear-linear blends next to surfaces. The resulting density profiles always show an enrichment of linear polymers in the immediate vicinity of the surface, due to entropic repulsion of the star core. However, the integrated surface excess of star monomers is sometimes positive, indicating an overall enrichment of stars. This excess increases with the number of star arms only up to a certain critical number and decreases thereafter. The critical arm number increases with compressibility (bulk concentration of C). The method of Gujrati and Chhajer is computationally ultrafast and can be carried out on a PC, even in the incompressible case, when simulations are unfeasible. Calculations of density profiles usually take less than 20 minutes on PCs.

cond-mat.soft↗