Fluctuation-dissipation theorem for frequency-dependent specific heat

A derivation of the fluctuation-dissipation ~FD! theorem for the frequency-dependent specific heat of a system described by a master equation is presented. The FD theorem is illustrated by a number of simple examples, including a system described by a linear Langevin equation, a two-level system, and a system described by the energy master equation. It is shown that for two quite different models with low-energy cutoffs—a collection of two-level systems and a system described by the energy master equation—the frequency-dependent specific heat in dimensionless units becomes universal at low temperatures, i.e., independent of both energy distribution and temperature. These two models give almost the same universal frequencydependent specific heat, which compares favorably to experiments on supercooled alcohols. @S0163-1829 ~96!06446-6#


I. INTRODUCTION
ac calorimetry was introduced in the 1960's as an accurate method for measuring the ordinary dc specific heat. 1,2he use of ac calorimetry for measuring the frequencydependent specific heat was pioneered in 1985 by Birge and Nagel 3 and by Christensen. 4Frequency-dependence of the specific heat occurs, e.g., in liquids close to the glass transition, 5 where structural relaxations take place on a time scale depending strongly on temperature. 6Other systems with slow enthalpy relaxations like unfolding proteins 7 also exhibit frequency dependence of the specific heat.For measurements of the frequency-dependent specific heat, the accessible time scale typically involves times longer than one second.
][10][11] The study of enthalpy relaxation upon a small change of temperature, which is a linear phenomenon, is conceptually simpler but experimentally more difficult.This is probably why the first measurements of frequencydependent specific heat appeared just 11 years ago.From a theoretical point of view, however, a linear response is much easier to model and understand than a nonlinear response.][14] The main reason for the simplicity of a linear response is the fluctuation-dissipation ͑FD͒ theorem, which expresses the response of a system upon a slight time-dependent change of external parameters in terms of equilibrium fluctuations.In effect, a linear response probes the equilibrium dynamics ͑i.e., the dynamics in absence of any external dis-turbance͒.The standard proof of the FD theorem considers an external field that perturbs the Hamiltonian. 157][18][19] This, however, is inaccurate since the specific heat is not a response to an external field perturbing the Hamiltonian.
1][22][23][24] The frequency-dependent specific heat was viewed in the context of generalized hydrodynamics for liquids with structural relaxation. 20,21It was suggested by Zwanzig that the frequency-dependent specific heat is directly related to the frequency-dependent longitudinal viscosity. 23Ja ¨ckle 24 discussed heat conduction and frequency-dependent specific heat in one-component liquids near the glass transition and showed that a frequency dependence of the thermal conductivity is connected to a wave vector and frequency dependence of the specific heat.Despite these theoretical developments, a proof of the FD theorem for frequency-dependent specific heat does not seem to exist in the literature.This is ironic, since one of the very earliest indications of the existence of FD theorems is Einstein's well-known result that the ordinary dc specific heat c p is proportional to the mean-square enthalpy fluctuation in the canonical ensemble ͑here and henceforth the subscript 0 denotes an equilibrium average͒:

͑1͒
The outline of the paper is as follows.Section II briefly reviews the stochastic framework within which the FD theorem for frequency-dependent specific heat is derived in Sec.IV.This is done by analogy with the derivation of the ordinary FD theorem, which is reviewed in Sec.III.Section V illustrates the FD theorem for specific heat by working out a number of simple examples.In Sec.VI we discuss ''quasiuniversality'' of the frequency-dependent specific heat in certain systems and compare the quasiuniversal frequencydependent specific heat to data for six supercooled alcohols.Section VII contains the conclusion.

II. STOCHASTIC MODEL
The system is described by probabilities to be in some ''states'' and the dynamics is given by a master equation.If P i is the probability to be in state i and W i j is the probability per unit time for a transition from state j to state i, the master equation 25 is the following first order differential equation: The first term on the right-hand side represents the probability per unit time of leaving state i while the second term represents the probability per unit time of jumping into state i.Equation ͑2͒ is linear.The Green's function G 0 (i,t͉ j,tЈ) is the probability to be in state i at time t given that the system is in state j at time tЈϽt; in terms of the Green's function the linearity of Eq. ͑2͒ implies 25 The equilibrium probability P eq,i is time independent and thus P eq,i ϭ ͚ j G 0 ͑ i,t͉ j,tЈ͒P eq, j ͑ tϾtЈ͒.͑4͒

III. THE ORDINARY FLUCTUATION-DISSIPATION THEOREM
Consider a system which is in thermal equilibrium at time tϭϪϱ.Applying a time-dependent external field f (t), the average deviation from equilibrium of some physical quantity, ␦B, depends linearly on f (t) if the field is sufficiently small.The general linear expression for ␦B(t) reflecting causality and time homogeneity is In the frequency domain, one considers a small oscillating perturbation, f (t)ϭRe͕ f 0 e st ͖, where s is the imaginary fre- In terms of the Green's function the correlation function is given by ͗⌬A͑0͒⌬B͑t͒͘ 0 ϭ ͚ i, j G 0 ͑ i,t͉ j,0͒P eq, j ⌬A j ⌬B i .͑10͒ Equation ͑10͒ comes from considering the actual ''path'' of the system in state space due to equilibrium thermal fluctuations.The probability to be in state j at tϭ0 is P eq, j and the probability of subsequently being in state i at time t is given by the Green's function.Note that the correlation function refers to equilibrium fluctuations; the point of the FD theorem is precisely that the linear response to an external field is determined by fluctuations in thermal equilibrium with no external field.Following Doi and Edwards 26 the derivation of the ordinary FD theorem proceeds in the following way.Suppose the system is equilibrated at a small constant time-independent external field f 0 , which is turned off at time tϭ0.For tϾ0 the system relaxes towards equilibrium corresponding to the zero field situation.The approach of the variable B towards equilibrium is via Eq.͑5͒ given by If the unperturbed energy of state j is E j and the value of A is A j , the initial state probability at tϭ0, P j (tϭ0), is the equilibrium probability in the external field f 0 ͓HϭH 0 Ϫ f 0 A implies that the energy of state j is E j Ϫ f 0 A j ͔: where ␤ϭ1/(k B T) and Zϭ ͚ j e Ϫ␤(E j Ϫ f 0 A j ) .It is straightforward to show that Eq. ͑13͒ implies To first order in f 0 one therefore has Using Eqs.͑3͒, ͑4͒, ͑10͒, ͑15͒, and the fact that ͗⌬B͘ 0 ϭ0, to first order in the external field

͑17͒
Via Eq. ͑12͒ one now arrives at the ordinary FD theorem Eq. ͑8͒.

IV. FLUCTUATION-DISSIPATION THEOREM FOR SPECIFIC HEAT
In measurements of the specific heat two different experimental situations may be realized.The most common is keeping the pressure constant ͑isobaric conditions͒; alternatively the volume may be kept constant ͑isochoric condi-tions͒.For both corresponding canonical ensembles, the equilibrium distribution is Boltzmann-like with an appropriate quantity Y j in the exponential: where Zϭ ͚ j e Ϫ␤Y j .In the canonical isobaric ensemble Y j is the enthalpy (Y j ϭE j ϩpV j ), in the canonical isochoric ensemble Y j is the energy. 27onsider now a system exposed to a temperature ''field'' T(t) given by T͑t ͒ϭ ͭ Tϩ␦T, tϽ0 T, tϾ0.͑19͒ To first order in ␦T, the distribution P j (tϭ0)ϭP eq, j (T ϩ␦T) may via Eq.͑20͒ be expressed in terms of the equilibrium distribution at T: The time dependence of the average deviation of a physical quantity B from equilibrium is for tϾ0 via Eqs.͑3͒, ͑4͒, and ͑21͒ given by

͑23͒
The frequency-dependent response function is given by Eqs.
͑7͒ and ͑12͒, leading to In particular, the case where B is the energy ͑in the constant volume case͒ or the enthalpy ͑in the constant pressure case͒ gives the FD theorem for frequency-dependent specific heat.

V. FREQUENCY-DEPENDENT SPECIFIC HEAT IN SOME SIMPLE MODELS
In the isobaric case Y is the enthalpy H and the frequency-dependent specific heat is given by After a partial integration this becomes

͑26͒
For sϭ0 we recover the standard expression for the dc specific heat Eq.͑1͒.It is sometimes convenient to rewrite the FD theorem by substituting the identity ͗⌬H(0)⌬H(t)͘ 0 ϭ͗H(0)H(t)͘ 0 Ϫ͗H͘ 0 2 into Eq.͑26͒, leading to If the pressure is zero, the enthalpy is equal to the energy.In this case, one calculates the frequency-dependent isobaric specific heat by considering equilibrium energy fluctuations, as done below for some simple models.Note that c p c v even when pϭ0, although both quantities are given by formally identical expressions involving energy autocorrelation functions.This is because the dynamics in the two situations are different ͑when the pressure is constant the volume fluctuates and more states are allowed than in the constant volume case͒.

͑29͒
For the energy autocorrelation function one finds As is well known, the linear Langevin equation Eq. ͑28͒ has the property that x(t) is distributed according to a Gauss- Gaussian variables the following identity holds: ͗x is substituted into Eq.͑30͒ the following expression for the equilibrium energy autocorrelation function is obtained ͑using the fact that in a harmonic potential ͗E͘ 0 ϭ 1 2 k B T)

͑31͒
For the frequency-dependent specific heat one finally obtains from Eq. ͑25͒ This is the Debye expression for a linear response function, 28 first derived for dielectric relaxation ͑remember that sϭi).At high frequencies the specific heat goes to zero.This is because for a fast small periodic temperature variation the average energy only has time to change insignificantly.

B. Two-level system
Consider next a system which can be in two different states iϭ0,1 with energies E i .It is assumed that there is an energy maximum E max which the system has to pass to go from one state to the other.In thermal equilibrium the probability flow between the two states is zero ͑principle of detailed balance͒, 25 implying that ␥ 0 P eq,0 ϭ␥ 1 P eq,1 , ͑33͒ where ␥ i is the rate of jumps out of state i.This is fulfilled if the rates are given by those of transition state theory ͑where 0 is the inverse of the ''attempt frequency''͒: The differential equation for the probability of being in state number 1 is ͑utilizing the fact that P 0 ϩ P 1 ϭ1) ‫ץ‬P 1 ‫ץ‬t ϭϪ␥ 1 P 1 ϩ␥ 0 P 0 ϭϪ͑␥ 0 ϩ␥ 1 ͒P 1 ϩ␥ 0 .͑35͒ The solution is The energy autocorrelation function is given by

͑37͒
It straightforward to evaluate this sum using Eq.͑36͒ from which all four conditional probabilities G 0 (i,t͉ j,0) may be found.However, it is easier to note that, since Eq.͑36͒ implies that each of the conditional probabilities is a linear function of e Ϫ(␥ 0 ϩ␥ 1 )t , the entire energy autocorrelation function also depends linearly on this quantity.Thus, for some constants A and B we have ͗⌬E͑0͒⌬E͑t͒͘ 0 ϭAe Ϫ͑␥ 0 ϩ␥ 1 ͒t ϩB.

͑39͒
When substituted into the FD theorem Eq. ͑25͒ we thus find For sϭ0 Eq. ͑40͒ reduces to Eq. ͑1͒.
Equation ͑40͒ is formally indistinguishable from the frequency-dependent specific heat of the Langevin equation for a harmonic potential Eq. ͑32͒, since for a suitable choice of ⌫ Eqs.͑32͒ and ͑40͒ may both be written c(s) ϭc(0)⌫/(sϩ⌫).Thus the frequency-dependent specific heat is not able to distinguish between the continuous Langevin model and a discrete two-level system.
It is straightforward to show that for a two-level system

͑41͒
Combining Eqs.͑34͒, ͑40͒, and ͑41͒ and introducing the dimensionless imaginary frequency the following expression is arrived at

C. Collection of two-level systems
Consider a collection of two-level systems, each with the same E 0 and E max , but with a distribution E 1 's, f (E 1 ), where E 0 ϽE 1 ϽE max .The frequency-dependent specific heat of this system is just the sum of all the single two-level system specific heats given by Eq. ͑43͒; thus c͑s ˜͒ϭ 1

͑44͒
A discussion of the low-temperature limit of this expression is given in Sec.VI.

D. Energy master equation
0][31][32] The idea is the following.The supercooled liquid is thought of as temporarily trapped in some configuration.To jump into another configuration, a number of molecules ͑a ''region''͒ collectively have to pass an energy barrier given by ⌬EϭE max ϪE, where E is the energy of the initial state.The maximum to be overcome, E max , is assumed to be independent of the initial state.A further assumption is that, once excited into the transition state, the region has forgotten which state it came from and ends up in a randomly chosen state.If n(E) is the normalized density of states and one defines ⌫͑E ͒ϭ⌫ 0 e Ϫ␤͑E max ϪE ͒ , ͑45͒ the above assumptions lead to the energy master equation 31,32 ͑EME͒ for the energy probability distribution P(E,t):

͑46͒
The first term on the right-hand side represents the probability per unit time for jumping away from a state of energy E, the second term is the probability per unit time for having a region changing its energy into E. To make sense physically, the energy integration in Eq. ͑46͒ should only extend up to E max ; however, at not too high temperatures the integration may be extended to infinity without changing the relaxation properties of the equation around the main parts of the relaxation spectrum.If G 0 (E͉EЈ)(s) is the Laplace transform of the Green's function, Eq. ͑27͒ implies ͑47͒ To find G 0 (E͉EЈ)(s) one refers to the general solution of the EME ͓Eq.͑A10͒ in Ref. 32͔, according to which the Laplace transform of P(E,t) is given by G 0 (E͉EЈ)(s) is found by substituting P(EЉ,tϭ0) ϭ␦(EЉϪEЈ) into Eq.͑48͒.After some elementary reductions utilizing the fact that P eq (E)ϰn(E)/⌫(E), one finally finds 33 from Eq. ͑47͒

VI. QUASIUNIVERSALITY OF THE FREQUENCY-DEPENDENT SPECIFIC HEAT AT LOW TEMPERATURES
This section discusses quasiuniversality at low temperatures for two models.The one model is a collection of twolevel systems ͑Sec.V C͒, the other is defined by the EME ͑Sec.V D͒.Each model has a common value of the energy maximum to be overcome in transitions between any two states.As shown below for both models, if the density of states has a sharp low-energy cutoff, the frequencydependent specific heat at low temperatures in dimensionless units becomes independent of both temperature and density of states.It turns out that the two universal curves are very similar, justifying the term ''quasiuniversality.''Consider first a collection of two-level systems, each with ground state energy E 0 .If the distribution of excited states, f (E 1 ), obeys f (E 0 ) 0, this model results in a normalized specific heat which is universal.This is because, in the lowtemperature limit, f (E 1 ) in the integral of Eq. ͑44͒ may be replaced by the constant f (E 0 ) and the integration may be extended to infinity.Doing this one finds that the normalized specific heat, c ˜(s ˜)ϭc(s ˜)/c(0), is given by where s ˜is given by Eq. ͑42͒ and F͑a,b ͒ϭ ͵ 0 ϱ x 2 e x ϩae Ϫx ϩb dx.͑51͒ Consider now the energy master equation.In this case, whenever there is a sharp low-energy cutoff at EϭE 0 for the density of states ͓i.e., n(E)ϭ0 for EϽE 0 and n(E 0 ) 0͔, n(E) may be replaced by n(E 0 ) at low temperatures.Thus, as shown in Appendix A, Eq. ͑49͒ implies ͓where s ˜ϭs/⌫(E 0 ) and Li m (z) is the mth polylogarithm 34 ͔ c ˜͑s ˜͒ϭ Ϫ2 Figure 1 shows a Cole-Cole plot of the two universal frequency-dependent specific heats and data for six supercooled alcohols obtained by the ''adiabatic'' method. 4,35The two universal curves are almost identical.Data for four of the alcohols compare favorably to the universality prediction; however, two alcohols deviate considerably.At present no clear conclusion can be reached regarding whether the simple models behind the quasiuniversality prediction reflect the actual behavior of supercooled alcohols.If this is the case, one may speculate that the two alcohols deviating from universality both have additional ␤ relaxation above the dominant ␣ relaxation.

VII. CONCLUSION
We have derived the fluctuation-dissipation theorem for frequency-dependent specific heat.In this case the ''external field'' does not enter into the Hamiltonian but is the temperature-a statistical concept.Consequently, the suitable framework for discussing frequency-dependent specific heat is to consider systems described by a master equation.
The FD theorem was illustrated by simple examples, including a system described by a linear Langevin equation and a two-level system.These models both give rise to Debye frequency dependence of the specific heat, so specific heat spectroscopy cannot distinguish between these conceptually very different models ͑of course, the models can be distinguished by the temperature dependence of the loss peak frequency͒.This is analogous to what happens for dielectric relaxation described by the same two models; it reflects the fact that both models have a single relaxation time.
In the low-temperature limit, a distribution of two-level systems-each with the same energy difference between ground state and barrier to be overcome-gives rise to a frequency-dependent specific heat, that in dimensionless units is independent of the distribution of level spacings.Interestingly, almost the same universal frequencydependent specific heat arises from a quite different approach, that of the energy master equation.In real units the frequency-dependent specific heat is usually strongly temperature dependent; the prediction of universality just means that the shape of the specific heat loss peak is temperature independent.Thus, universality implies that these systems at low temperatures have frequency-dependent specific heats obeying the time-temperature superposition principle.The two universal curves give a good fit to four of the six alcohols studied in Refs. 4 and 35.Whether this is a pure coincidence or a result of general significance remains to be seen.It is thus an open question whether or not supercooled liquids close to the laboratory glass transition ͑where the measurements of Fig. 1 were made͒ are at ''low temperatures'' in the sense of Sec.VI, meaning that the molecular configuration is close to the ground state ͑excitations of which give rise to the universal frequency-dependent specific heat͒.Further work is needed to clarify this point.

APPENDIX: DERIVATION OF UNIVERSAL LOW-TEMPERATURE FREQUENCY-DEPENDENT SPECIFIC HEAT IN THE EME WITH A LOW-ENERGY CUTOFF
We here derive Eq. ͑52͒, the universal low-temperature frequency-dependent specific heat for the EME with a lowenergy cutoff in the density of states at EϭE 0 .Introducing the quantity Eq. ͑49͒ becomes FIG. 1. Cole-Cole plot of normalized frequency-dependent specific heat data for six supercooled alcohols ͑symbols͒ 4,35 compared to the two universality predictions valid at low temperatures for systems with sharp low-energy cutoffs "Eq.͑50͒ for a collection of two-level systems ͓full curve͔ and Eq.͑52͒ for the energy master equation ͓dashed curve͔….The horizontal axis gives the real part and the vertical axis gives the negative imaginary part of the normalized frequency-dependent specific heat ͑with the frequency as parameter tracing out the curve͒.The figure shows that the two universality predictions are very similar, thus giving rise to a hypothesis of ''quasiuniversality'' at low temperatures among different models with sharp low-energy cutoffs.The quasiuniversal specific heat compares favorably to the data for four of the alcohols.The remaining two alcohols give rather poor fits, possibly due to additional ␤ relaxations above the dominant ␣ relaxation.

͑A2͒
The equilibrium energy probability distribution is given by P eq (E)ϭn(E)e Ϫ␤E /Z where Zϭ͐ E 0 ϱ n(E)e Ϫ␤E dE.At low temperatures the density of states n(E) may be replaced by the number n(E 0 ) and the equilibrium energy probability is simply P eq (E)ϭ␤e Ϫ␤(EϪE 0 ) .Without loss of generality it may be assumed that E 0 ϭ0 and Eq.͑A1͒ becomes While Eq. ͑A9͒ was derived assuming ͉s ˜͉Ͻ1 the result is valid for all frequencies by analytic continuation.The analytic continuation of the polylogarithm may be evaluated from the following relation 36 valid for mϭ2,3, . . .:

͑A10͒
Here B m (z) is the mth Bernoulli polynomial and the argument for the complex logarithm is chosen between 0 and 2.
For s ˜→0 Eq. ͑A9͒ implies c ˜ϭ1Ϫs ˜/4.For s ˜→ϱ Eq. ͑A9͒ implies for the real part of the frequency-dependent specific heat ͓s ˜ϭi ˜͔ c ˜Јϭln 2 ( ˜)/(8 ˜) and for the imaginary part c ˜ЉϭϪln 3 ( ˜)/(12 ˜).The loss peak frequency is given by ˜mϭ4.51 and the maximum of the negative imaginary part is given by c ˜max Љ ϭϪ0.313.