__P. D. Gujrati__
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.

__P. D. Gujrati__
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.

__P. D. Gujrati__
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.

__P. D. Gujrati__
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.

__P. D. Gujrati__,
__P. P. Aung__
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 {\Pi} 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
{\Pi}=1 regardless of the number of internal variables.

__P. D. Gujrati__
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 {\Sigma}_0 consisting of the homogeneous system {\Sigma} and the medium
{\Sigma}. We establish on general grounds the continuity of the Gibbs free
energy G(t) of {\Sigma} 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
{\Sigma} 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.

__P. D. Gujrati__,
__Bradley P. Lambeth Jr__,
__Andrea Corsi__,
__Evan Askanazi__
We consider a general incompressible finite model protein of size M in its
environment, which we represent by a semiflexible copolymer consisting of amino
acid residues classified into only two species (H and P, see text) following
Lau and Dill. We allow various interactions between chemically unbonded
residues in a given sequence and the solvent (water), and exactly enumerate the
number of conformations W(E) as a function of the energy E on an infinite
lattice under two different conditions: (i) we allow conformations that are
restricted to be compact (known as Hamilton walk conformations), and (ii) we
allow unrestricted conformations that can also be non-compact. It is easily
demonstrated using plausible arguments that our model does not possess any
energy gap even though it is supposed to exhibit a sharp folding transition in
the thermodynamic limit. The enumeration allows us to investigate exactly the
effects of energetics on the native state(s), and the effect of small size on
protein thermodynamics and, in particular, on the differences between the
microcanonical and canonical ensembles. We find that the canonical entropy is
much larger than the microcanonical entropy for finite systems. We investigate
the property of self-averaging and conclude that small proteins do not
self-average. We also present results that (i) provide some understanding of
the energy landscape, and (ii) shed light on the free energy landscape at
different temperatures.

__Sagar S. Rane__,
__P. D. Gujrati__
The bare chi characterizing polymer blends plays a significant role in their
macroscopic description. Therefore, its experimental determination, especially
from small-angle-neutron-scattering experiments on isotopic blends, is of prime
importance in thermodynamic investigations. Experimentally extracted quantity,
commonly known as the effective chi is affected by thermodynamics, in
particular by polymer connectivity, and density and composition fluctuations.
The present work is primarily concerned with studying four possible effective
chi's, one of which is closely related to the conventionally defined effective
chi, to see which one plays the role of a reliable estimator of the bare chi.
We show that the conventionally extracted effective chi is not a good measure
of the bare chi in most blends. A related quantity that does not contain any
density fluctuations, and one which can be easily extracted, is a good
estimator of the bare chi in all blends except weakly interacting asymmetric
blends (see text for definition). The density fluctuation contribution is given
by (Delta v^bar)**2/2TK_T, where Delta v^bar is the difference of the partial
monomer volumes and K_T is the compressibility. Our effective chi's are
theory-independent. From our calculations and by explicitly treating
experimental data, we show that the effective chi's, as defined here, have weak
composition dependence and do not diverge in the composition wings. We
elucidate the impact of compressibility and interactions on the behavior of the
effective chi's and their relationship with the bare chi.