Response to “Comment on ‘Heat capacity, enthalpy fluctuations, and configurational entropy in broken ergodic systems’” [J. Chem. Phys. 134, 147101 (2011)]

In his comment on our recent manuscript,¹ Johari² argues that “a state is either ergodic or it is not. There is no varying degree of (continuously) broken-ergodicity.” He argues that ergodicity breaking happens only once when the liquid vitrifies, e.g., when a liquid observed at a high frequency is cooled below its normal glass transition temperature, and “there should be no further ‘ergodicity breaking.’” This comment by Johari reflects a misunderstanding of ergodic theory and the statistical mechanics of glass-forming systems, a misunderstanding that is repeated in a series of recent publications by Johari on the same topic.^3–5 In each case Johari arrives at the erroneous conclusion that glass has a finite residual entropy at absolute zero, a result in violation of the third law of thermodynamics and in contradiction to our recent proof¹ that only a time-average formalism of statistical mechanics can correctly account for the well established experimental results that configurational degrees of freedom are frozen at low temperatures and/or high frequencies.

To address this misinterpretation, we should begin with the definition of ergodicity. An ergodic system is the one in which the time-average of every measurable property gives an equivalent result as the corresponding ensemble average.⁶ In condensed matter systems, ergodicity implies that the system can sample a sufficiently large representative region of phase space within the relevant experimental time scale.

The importance of time scale in determining the ergodicity of a system is highlighted by Palmer⁷ in his classic and pedagogical work on broken ergodicity. Using several examples, Palmer stresses that for any experiment there are two relevant time scales: an internal time scale τ_int on which the dynamics of the system occur and an external scale τ_ext on which properties are measured. The internal time scale is a result of the intrinsic relaxation dynamics of the system; the external time scale defines a measurement window over which the system is observed.

The significance of time scale was also emphasized by Reiner⁸ in his definition of the Deborah number,

The scientific implication of this equation is that systems that appear static on a laboratory (external) time scale, i.e., τ_int ≫ τ_ext or D ≫ 1, may in fact relax on a time scale inaccessible in a laboratory, i.e., when τ_int < τ_ext or D < 1. In condensed matter physics, the liquid-to-glass transition is an archetypical example of ergodic breakdown,^7,9 since the liquid state is ergodic (having τ_int < τ_ext or D < 1) and the glassy state is nonergodic (having τ_int > τ_ext or D > 1). Since any experimental glass transition is continuous rather than discontinuous in nature,¹⁰ the breakdown of ergodicity is a continuous process involving a successive partitioning of the phase space into so-called components or metabasins, each one satisfying the condition of internal ergodicity.^11,12 As we demonstrate below, this successive partitioning of the phase space is a key to the concept of continuously broken ergodicity.

Typical laboratory glass formation involves the cooling of a liquid from high temperature at a fixed observation time scale. In other words, τ_ext is fixed while τ_int increases exponentially during the cooling process. In the high temperature liquid state, τ_int < τ_ext and the system is ergodic. As the system is cooled, τ_int increases continuously. The glass transition occurs when τ_int ∼ τ_ext, i.e., the initial breakdown of ergodicity. Further ergodic breakdown may occur upon continued cooling of the glass, since faster atomic relaxation processes can also become frozen. In the limit of absolute zero temperature, all classical (i.e., nonquantum) relaxation processes are frozen, so the macrostate of a glass is determined by a single microstate. This, in words, is why the entropy of a glass is zero at absolute zero, in agreement with the third law of thermodynamics and the principle of causality.^13,14 The formal proof for this is given in Ref. 1.

Cooling from a melt is just one example of a laboratory glass transition, in which τ_ext is fixed and τ_int increases. From the definition of the Deborah number above, it is clear that a glass transition can also occur with fixed τ_int but decreasing τ_ext. This is what occurs during specific heat spectroscopy¹⁵ and other relaxation spectroscopy experiments¹⁶ by varying the measurement frequency (ν_ext = 1/τ_ext) at a fixed temperature (i.e., a constant τ_int). Such an experiment also leads to a continuous loss of ergodicity as the Deborah number becomes less than unity. For this case of an isothermal glass transition, it is the denominator (τ_ext) of Eq. (1) that changes to induce this breakdown of ergodicity, whereas glass formation by cooling from a melt corresponds to changing the numerator (τ_int). In either case, the Deborah number becomes less than unity, and there is a breakdown of ergodicity. The example given by Johari² of cooling a system measured at high frequency corresponds to changing first the denominator (τ_ext) and then the numerator (τ_int). As in any other case, the breakdown of ergodicity will be continuous and will depend on the particular distribution of relaxation modes in the system.

In relaxation spectroscopy experiments, “the high-frequency limit reflects only the contribution of the fast modes,” to quote Birge and Nagel.¹⁵ Similar to the low-temperature limit of traditional glass formation, this high-frequency limit in relaxation spectroscopy reflects the nonergodic, solid glassy state. In the extreme limit of infinite frequency, the measured macrostate is the result of just a single microstate, and thus the entropy is zero.¹ Additional experimental proof for the isothermal glass transition is provided in the classic review article by Litovitz and Davis,¹⁷ who show that the high frequency limit in the liquid state at a temperature T corresponds to a glassy state having a fictive temperature also equal to T. The value for glass changes with time at a fixed bath temperature due to aging (relaxation), which also changes its fictive temperature.

In fact Johari² concedes that “at high frequencies, C_p^′ and G^′ values may be mostly nonconfigurational.” However, he misinterprets the physics of these experiments by trying to argue that “they do not change with time and hence should be seen as a liquidlike, or ergodic, response.” Here the concept of ergodicity is incorrectly treated by Johari to be equivalent to that of the thermodynamic equilibrium. If one observes an equilibrated liquid at a sufficiently high frequency, the response is not liquidlike but solidlike since configurational changes are forbidden at such high frequency. This solidlike response is the result of a breakdown of ergodicity, as described above. The steady-state nature of this response is due to two factors: (a) the liquid is in thermal equilibrium and hence the microstate occupation probabilities follow an equilibrium Boltzmann distribution; and (b) for a fixed measurement frequency the observation time is constant. If one decreases the measurement frequency, then the observation time increases and one obtains a different response as various configurational modes are activated. At sufficiently low frequencies, the liquidlike response of the system is recovered as ergodicity is restored.

In our work,^1,9 this difference between equilibrium and ergodicity is reflected in the use of two different types of probabilities: basin occupation probabilities (p_i) and conditional probabilities (f_ij). The p_i values give the probability of occupying a given microstate i at any point in time. For equilibrated systems, the distribution of p_i values follows a Boltzmann distribution according to equilibrium statistical mechanics. The possibility of broken ergodicity is included through use of the conditional probabilities, f_ij, which give the probability of occupying some final microstate j after starting in an initial microstate i and allowing the dynamics of the system to proceed over some observation time period, τ_ext. In the limit of a fully ergodic system, the conditional probabilities reduce to the equilibrium basin occupation probabilities, i.e., f_ij = p_j^eq, and equilibrium statistical mechanics is recovered. In the limit of a fully confined nonergodic system, the conditional probabilities reduce to a Kronecker delta function, f_ij = δ_ij, reflecting this total confinement. The continuous breakdown of ergodicity occurs with everything between these two extremes. Full details on this formalism have been provided in our previous papers.^1,9

Finally, we wish to underscore the importance of having a physically rigorous model for the entropy of nonequilibrium systems, such as glass. While entropy itself is not a directly measurable property, it has important practical implications for other properties, such as heat capacity, elastic moduli, and viscosity. The theory of continuously broken ergodicity has already proved essential to the development of a computational framework for solving the dynamics of glass-forming systems on a laboratory time scale.^11,18 This, in turn, provided the framework for the first-ever comprehensive study of the temperature and thermal history dependence of glass viscosity during the glass transition—where there is a continuous loss of configurational entropy—and in the isostructural regime at even lower temperatures.¹⁹ This approach was also essential for the recent discovery of the inherent nonmonotonic relaxation behavior of density fluctuations in glass-forming systems, which is confirmed by in situ small-angle x-ray scattering experiments.²⁰ Finally, the description of configurational entropy described in our work is the only one consistent with a topological view of glass structure,²¹ in which the so-called “floppy modes” are shown to be the dominant contributors to configurational entropy.^22,23 At low temperatures when thermal energy is lost, all of the floppy modes become rigid, leading to vanishing of configurational entropy.²⁴