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Abstract
We apply microcanonical ensemble considerations to suggest that, whenever it may thermalise, a general disorder-free many-body Hamiltonian of a typical atomic system has solid-like eigenstates at low energies and fluid-type (and gaseous, plasma) eigenstates associated with energy densities exceeding those present in the melting (and, respectively, higher energy) transition(s). In particular, the lowest energy density at which the eigenstates of such a clean many body atomic system undergo a non-analytic change is that of the melting (or freezing) transition. We invoke this observation to analyse the evolution of a liquid upon supercooling (i.e. cooling rapidly enough to avoid solidification below the freezing temperature). Expanding the wavefunction of a supercooled liquid in the complete eigenbasis of the many-body Hamiltonian, only the higher energy liquid-type eigenstates contribute significantly to measurable hydrodynamic relaxations (e.g. those probed by viscosity) while static thermodynamic observables become weighted averages over both solid- and liquid-type eigenstates. Consequently, when extrapolated to low temperatures, hydrodynamic relaxation times of deeply supercooled liquids (i.e. glasses) may seem to diverge at nearly the same temperature at which the extrapolated entropy of the supercooled liquid becomes that of the solid. In this formal quantum framework, the increasingly sluggish (and spatially heterogeneous) dynamics in supercooled liquids as their temperature is lowered stems from the existence of the single non-analytic change of the eigenstates of the clean many-body Hamiltonian at the equilibrium melting transition present in low energy solid-type eigenstates. We derive a single (possibly computable) dimensionless parameter fit to the viscosity and suggest other testable predictions of our approach.
Acknowledgements
I am extremely grateful to N. Weingartner for help with the numerical fits and gratefully acknowledge work with N. B. Weingartner, F. S. Nogueira, K. F. Kelton, M. Blodgett, E. Altman, S. Banerjee, and L. Rademaker on related problems. In particular, in [Citation62,Citation63,Citation75] we tested and extended many of the ideas that were first introduced in the current work. I wish to thank G. Biroli, G. Parisi, and especially to T. Schaefer for prompting me to write up my approach and am appreciative of positive remarks by T. Egami and questions by G. Tarjus and F. Zamponi.
Notes
No potential conflict of interest was reported by the author.
This article was originally published with errors. This version has been amended. Please see Addendum (http://dx.doi.org/10.1080/14786435.2017.1317431).
1 Commonly, the glass transition temperature is defined to be the temperature at which the viscosity is
Pa
sec (or
Poise).
2 Rather explicitly, for a Hamiltonian which is the sum of a kinetic energy (of particles of mass m) and a potential energy V, by the Ehrenfest theorem (or the expectation values of the equivalent Heisenberg or Schrödinger equations)(1)
When the uncertainties in the particle positions in the state
are small (as they typically are in semiclassical systems), then we may substitute
(2)
With this, Equations (Equation42(42) ) reduce to the classical equations of motion. In the quantum setting, the time evolution (from time t to time
) of the expectation values is given by
. Similarly, if the dynamics in the classical system are subject to external random (thermal or other) forces
then Equation (Equation42
(42) ) is replaced by
(3)
If at time the classical and quantum expectation values are identical and if the noise
in the classical system coincides with the difference in the forces as measured in the quantum and classical systems, i.e. if
(4)
then the solutions and
to the classical equations of motion with external noise (Equations (Equation44
(44) )) will coincide with the measured observables of the corresponding position and momentum operators in the quantum state
. The noise
in Equation (Equation44
(44) ) may indeed be thermal noise that assists a classical supercooled liquid as it evolves according to the equations of motion to explore the ‘energy landscape’. One may anticipate the magnitude of the ‘uncertainties’ in position (triggered by external thermal noise) to become larger as the system is at higher temperature T. Since the Hamiltonian H is a sum of position- and momentum-dependent terms, this anticipation of a larger uncertainty is in line with the larger effective broadening
that we will find for the energy eigenvalues in the state
at higher energy densities or, equivalently, higher temperatures (see Equation (Equation27
(27) ) for a fixed
and effective average heat capacity C). For a classical system governed by Equation (Equation44
(44) ), the energy uncertainty formally introduced by the random thermal agitation
can be substantial; typically, this uncertainty has a scale set by the temperature.
3 For any function F that (away from any phase space regions of localized support) is a piecewise linear (or constant) function in the each of the phase coordinate variables or
, we have that
(5)
Thus, for general functions or
that are indeed linear in the phase space coordinates, Hamilton’s classical equations of motion (which are those in
in which
(with
or
) are the classical phase space coordinates) are identical to those implied by Ehrenfest’s theorem in the quantum arena. In such circumstances, even if there are quantum uncertainties in any phase space variables, the classical equations of motion will still be obeyed (following the replacement of the classical variables x and p by their quantum expectations values
and
) whenever the quantum Hamiltonian H is quadratic in these phase space variables. Whenever the wavefunction is localised about particular phase space coordinates, the corresponding position or momentum operators z exhibit no fluctuations about their average values and the relation
will, similarly, be ensured. That is, in such instances, Ehrenfest’s theorem automatically implies the classical (i.e. Hamilton’s) equations of motion are satisfied. Solving any such system quantum mechanically or analysing it classically will lead to identical results.
4 In drawing conclusions (A) and (B), we assumed that the standard equilibrium averages of Equations (Equation2(2) ) and (Equation3
(3) ) indeed describe the long-time behaviour of the disorder free system of Equation (Equation1
(1) ) that is known to obey equilibrium statistical mechanics when it is not supercooled. In Section 5, we discuss in more detail (see, e.g. Equations (Equation14
(14) ) and (Equation17
(17) )), the averages over restricted subsets of eigenstates with specific quantum numbers (different from the average over all eigenstates of specific energies as in Equation (Equation2
(2) )). The many body localised eigenstates of Refs. [Citation45–Citation50] (and of Appendix 4) correspond to limiting cases of such restricted sets of eigenstates. If the average over all such restricted subsets leads to the same value of any observable O then conclusions (A) and (B) will hold over all special subsets (if and when they exist). That is, insofar as that any measurement of the observable O can attest, the system eigenstates will be liquid or solid like depending on whether their associated energy densities will be greater than or smaller than
.
5 Although we will not consider it in the current work, instead of Equation (Equation4(4) ), one may also examine a single step process whereupon the initiation of cooling
(i.e. at all times
the Hamiltonian will be given by
). Similarly, the evolution operator
may embody a ‘rejuvenation’ process wherein the system is heated from the supercooled state.
6 The equalities of Equations (Equation9(9) ) and (Equation10
(10) ) concerning
are not merely asymptotic statements. In systems that do thermalise, the temporal averages
might be directly measured in all but the shortest times. This assertion may be demonstrated by several approaches (that largely emulate standard steps in the derivation of the time-energy uncertainly relation). In the context of Equation (Equation9
(9) ), to illustrate this point, one may, e.g. consider
to be a Gaussian of width
with the off-diagonal matrix elements of a local operator
set to a fixed constant between the eigenstates that may differ in energy with the aforementioned Gaussian set by
. In such a case, a straightforward integration demonstrates that already for times
, rapid off-diagonal phase cancellations will occur and the temporal average
will, essentially, be given by its long time average value. In typical equilibrated systems, the fluctuation associated with the local
is
. At room temperature (
, the ensuing very short equilibration time scale (set by
femtoseconds) may, in principle, be observable by luminescence measurements.
7 A priori, the expectation value of an observable such as the velocity operator () discussed in Section 6,
and other similar quantities, might not depend only on the energy
of the eigenstate
. Thus, as in Equation (Equation14
(14) ), the average in Equation (D3) may, theoretically, depend on quantum numbers
other than the energy. Such a dependence may, in principle, distinguish between candidate localised and delocalised states that share the same energy density.
8 By the ‘off-diagonal terms’ that have been omitted in the second equality of Equation (Equation19(19) ), we allude to the sum
(25)
Similar to the relations of Equations (Equation9(9) ) and (Equation10
(10) ) yet weaker, individual
terms summed over in Equation (Equation47
(47) ) may each have a vanishing long time average.
9 More exact than Equation (Equation22(22) ) is the statement that the average over a system size-independent energy interval
defining the microcanonical ensemble, the average
(28)
Here, as in Equation (Equation2(2) ),
is the number of the eigenstates whose energy
satisfies
. Taken together with Equation (Equation48
(48) ), the semi-positive definiteness of the velocity of the dropping sphere (with the z axis oriented downwards along the direction of the external gravitational field), i.e.
(29)
for all eigenstates implies that the expectation value of the velocity operator must vanish for all eigenstates
(Equation (Equation22
(22) )). Equation (Equation49
(49) ) may be demonstrated by, e.g. noting that given any state
that evolves with H, in the long time limit the average velocity of the sphere is, trivially, non-negative. Since this semi-positive definiteness applies to any state
, that the expectation value of the velocity must be non-negative in any eigenstate
. Equivalently, if the sphere is confined to the half space
(as it essentially is) then its velocity must satisfy Equation (Equation49
(49) ). As an aside, we remark that even without Equation (Equation22
(22) ), the equality of Equation (Equation48
(48) ) suffices to establish our approximate expression of Equation (Equation23
(23) ) for the viscosity.
10 The motion (or ‘flow’) of defects in solids (e.g. dislocations or grain boundaries) is a fascinating topic. Illuminating analysis [Citation95] suggests that in the properly defined limit of vanishing external stress (or small mass of the sphere dropped into the supercooled liquid), the time scale for flow in equilibrium solids (if these are crystalline) indeed does diverge. For finite applied stress, the relaxation time is finite (and does not diverge in the thermodynamic limit).
11 Rather explicitly, any distribution may generally be expressed as a weighted sum of numerous rectangular distributions of varying widths
at locations
. That is, we may normally write
(32)
Here, the weights are normalised (
) and
is the Heaviside function. If the distribution
is narrow then, for any observable
, the expectation value
(33)
where(34)
In a narrow distribution , the pertinent energies
are very close to each other. In such a case, all microcanonical ensemble averages
in Equations (Equation51
(51) ) and Equation52
(52) ) will obtain the same value (
). Thus, from Equation (Equation51
(51) ), the expectation value of any operator
computed with a distribution
of vanishing width in the energy density will be equal to the average of
when calculated within the micro-canonical ensemble. As it is important, we briefly expand on this conclusion. Because all equilibrium thermodynamic observables
depend uniquely on temperature or, equivalently, on the energy density (and may similarly be set by all other intensive state variables in an ensemble containing other intensive state variables such as pressure or chemical potential), barring special quantum effects (see Section (4)), a deviation from equilibrium will imply that the standard deviation of the energy divided by the energy itself must be finite. That is, in order to experimentally observe deviations between the long time averages
from the equilibrium averages
, the ratio
must vanish in the thermodynamic (large N) limit. This is so as If the ratio
tends to zero in this limit then, as made explicit in Equations (Equation50
(50) ), (Equation51
(51) ), and (Equation52
(52) ), averaging the equilibrium expectation values of
over systems with the same energy density will trivially lead to the expectation values of
in an equilibrium system at the same energy density. So long as (
in the
limit, the average of any observable will coincide with its equilibrium expectation value regardless of any possible weak dependence of the width
on the particle number N. This conclusion applies (yet is certainly not limited) to the usual situation (realised in, e.g. the canonical ensemble) in which the standard deviation
scales as
. We remark that unusual behaviours, including the inequivalence of ensembles, arise in systems with sufficiently long-range interactions such that dissimilar energy densities (and associated entropies and free energies) appear in different ensembles [Citation96–Citation99].
12 The derivation of this well-known relation is sketched here for the benefit of the reader. If we enforce a fixed variance with a Lagrange multiplier
while maximising the Shannon entropy of Equation (Equation28
(28) ) (and the conditions of normalisation
enforced with a Lagrange multiplier
along with the demand that the quantity E appearing in the definition of the variance
is, indeed, the average energy, viz.,
enforced via a third Lagrange multiplier
), then the corresponding variational equation trivially reads
. Imposing normalisation, and the given values of the standard deviation
, and average energy E fixes the three Lagrange multipliers and leads to Equation (Equation24
(24) ). In the standard canonical ensemble, of the latter three constraints, that of a fixed standard deviation is not fixed (i.e.
is not present) and the standard exponential (or Boltzmann) distribution results.
13 In physical systems with a bounded spectrum, the probability density strictly vanishes at sufficiently low energies where
is the ground state energy. Albeit extremely small, the Gaussians of Equation (Equation24
(24) ) (for equilibrated finite size systems) and of Equation (Equation29
(29) ) (for the supercooled fluid) do not strictly vanish at arbitrarily low energies
. For the energy densities of interest, we expect Equation (Equation24
(24) ) to hold.
14 The equilibrium Boltzmann distribution does not maximise the entropy with a fixed finite (i.e. non-zero) standard deviation of the energy density. Rather, the Boltzmann distribution maximises the entropy with the constraint of a fixed energy mean. Here, the sum is performed over all eigenstates n with their respective probabilities (and not the cumulative probability
). In Section 10.2.3, we return to this distribution and explain how a Boltzmann distribution may be found for the expectation value of the energy even when when the Hamiltonian H cannot be simultaneously diagonalised with the density matrix.
15 In this brief comment, we return to the discussion of the equilibrium system and highlight a simple conceptual point concerning intuition that cannot be blindly borrowed from standard equilibrium statistical mechanics. For any Hamiltonian that is a sum of local terms that have their support in local regions R (i.e. for Hamiltonians of the type
with
operators
) all relative fluctuations must vanish in the thermodynamic limit. This is because in such equilibrated systems, the connected correlation functions between far separated local terms vanish,
. Here. the average
is performed relative to an equilibrium state defined by H. In such a case, only connected correlation functions involving geometrically close pairs of regions R and
contribute to
. In the aftermath, only
connected correlation functions will substantially contribute to
(in accordance with the earlier stated known results concerning fluctuations in equilibrated systems). However, when the state
in which the energy fluctuations are computed is not generated by H (i.e.
does not represent an equilibrium state formed by a superstition of eigenstates
that share the same energy density), such locality arguments become void. To clarify this point, one may consider the extreme case of random state
. Relative to any such random state that may be formed coupling to a rapidly evolving external heat bath, the fluctuations
may be sizable.
16 For completeness, we list the remaining non-vanishing off-diagonal matrix elements.(43)
The above matrix elements and those of higher powers of will appear when evaluating higher order cumulants beyond the variance. If the probability distribution
is close to being a Gaussian then these higher order cumulants will be small. Although we introduced the harmonic oscillator example for pedagogical purposes only, the general scaling of the variance with the square of the energy holds for general systems with first- (or second-) quantised Hamiltonians
and
. Indeed, as we explained in the main text (Section 10.2.2), Entropy Maximisation suggests that the most likely probability distributions will be Gaussian.
17 To elucidate this point, one may divide the energy () domain into regions of width
. There are exponentially many states of the form of Equation (Equation5
(5) ) that share fixed set of values
over a wide interval of energies
. The number of such states is given by the product of the number of possible superpositions of eigenstates in a given energy interval
that give rise to the same value of
. As the ratio
, the number of such states that have the same probability distribution is exponentially large in the system size N (and the corresponding configurational entropy is extensive).
18 We remark that if the contributions in Equation (Equation18
(18) ) cannot be neglected and, consequently, lead to ratio
that is significantly smaller than two then there is a theoretically appealing possibility that we may be tested. Along these lines, one may replace the temperature
in Equation (Equation33
(33) ) by a new temperature
and similarly replace the viscosities in Equation (Equation33
(33) ) (i.e. exchange
and
with
and
). Here, the temperature
is defined by the implicit equation
where
is given by extrapolating the viscosity of the equilibrium high-temperature liquid above melting to the temperature
corresponding to an energy density of the supercooled liquid that lies within the
region. Below this energy density, at the lower energy within the
, the eigenstates are more solid-like and little hydrodynamic transport occurs. In the fits discussed in this work, we avoided introducing such a new defined temperatures and employed the bare experimentally measured melting temperature
.
19 It is conceivable that distributions more complex than that of Equation (Equation29(29) ) may appear when the supercooling is achieved in a very special way. Furthermore, the amended variant of Equation (Equation10
(10) ) (Equation (Equation14
(14) ) for general observables
) may be of use in analysing rare yet notable situations, see, e.g. [Citation100], wherein different spatially inhomogeneous preparations of the supercooled state generally trigger significantly different behaviors.
21 In a similar vein, if we define the cumulative probability of being in the mixed coexistence () region by
(where, as throughout,
) then the combined sum of the liquid like and the mixed solid–liquid state weights will be trivially bounded,
.