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Glass Thermodynamics: Clausius Theorem and a New Tensorial Definition of Temperature

Glass Thermodynamics: Clausius Theorem and a New Tensorial Definition of Temperature

Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B, February 2021, 62 (1), 8–18

Glass : Clausius theorem and a new tensorial definition of

Akira Takada,a,b,* Reinhard Conradtc & Pascal Richetd a University College London, Gower Street, London W1E6BT, UK b Ehime University, 10-13 Dogo-Himata, Matsuyama, Ehime 790-8577, Japan c UniglassAC GmbH, Nizzaallee 75, D-52072, Aachen, Germany d Institut de Physique du Globe de Paris, 1 rue Jussieu, 75005 Paris, France

Manuscript received 23 July 2020 Accepted 23 September 2020

This paper aims at unifying the and thermodynamics of glass through a new extended definition of temperature. Along with new insights into the Clausius theorem, this definition has been derived from a three-level model system. As previously shown in our study of a two-level system, introduction of an additional parameter, i.e. an internal temperature, is useful to describe the relation between and changes in non-equilibrium states. Because a further extension is indispensable to deal with more realistic glass models, we define here temperature as a tensor whose diagonal components, the statistical , constitute a bridge between thermodynamic variables and a partition function in the framework of non-equilibrium statistical mechanics. On the other hand, the non-diagonal components, named thermodynamic temperatures, are used to extend the Clausius theorem within the framework of non-equilibrium thermodynamics. The new formalism is applied to a three-level model as the simplest example of multi- level system. It succeeds in describing the relation between energy changes and entropy in non-equilibrium states. In addition, this study sheds a new light on the interpretation of the second law of thermodynamics. As a result, Clausius’ definition of entropy, and its famous inequality for non-equilibrium cases can be merged into a single equality without the introduction of any additional terms. In addition, this study introduces a more general definition of the zeroth law of thermodynamics for non-equilibrium conditions, and extends the concept of fictive temperature as an order parameter to distinguish non-equilibrium glasses.

1. Introduction Much effort has thus been made to extend such a Historically, thermodynamics and statistical mechan- framework to non-equilibrium states, e.g. Ref. 11. As ics have been the two theoretical frameworks relevant an example, the standard formalism of spin glasses to the investigation of the thermal properties of gases, has made it possible to estimate the regions of exist- liquids and solids (i.e. crystals and glasses). Within ence of spin-glass phases and the hierarchy of spin the former, one correlates energy with temperature, clusters.(12–14) But the applicability of the second and , , entropy and other thermodynamic third to non-equilibrium variables of the macroscopic world.(1–3) For glasses, systems in general remains debated so that an issue the relation between elementary structural entities as important as the definition of partition functions and thermodynamics can be studied in this way,(4,5) for non-equilibrium states has not been settled yet. for example by calorimetric methods with which Within the framework of statistical thermodynamics, thermodynamic variables can be determined even for two main approaches have been followed to solve non-equilibrium states.(6,7) In statistical mechanics,(8,9) these problems. In the first approach, one introduces Boltzmann established his famous link between en- the concept of ensemble and investigates all possible tropy and the number of microstates whereas Gibbs microstates, the replicas of the spin-glass community. introduced the concept of ensemble, which made it In the second approach, one investigates the temporal possible to calculate partition functions and then all response of correlation function by using the fluctua- thermodynamics variables. tion–dissipation (FD) theorem. Because no decisive As a fundamental problem, however, the conven- results have been obtained in both cases and none tional framework of both thermodynamics and statis- seems easily applicable to structural glasses, the tical mechanics is valid for equilibrium states only.(10) purpose of the present study is to explore new ground with a third kind of approach.

*Corresponding author. Email [email protected] The basic point here is to rely only on instantane- DOI: 10.13036/17533562.62.1.011 ous atomic coordinates, velocities and . As

8 Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE generally accepted, two extensive variables such as the system by internal or external effects. volume and the number of particles can be calculated The experimental or theoretical determination of as states variables at each instant without requiring this uncompensated , as it was called by Clausius, either the concept of ensemble or the FD theorem. has always been a matter of concern. In one of the It follows that entropy can also be calculated as most popular approaches, one introduces a general- an extensive variable without requiring replicas or ized thermodynamic force F and thermodynamic temporal responses. This statement directly leads flow X; to our new “spatial sampling” framework in which d S=FdX (4) all thermodynamic variables can be defined for each i microstate, contrary to the commonly held opinion For chemical reactions, the force F is de Donder’s affin- that they can be defined only for macrostates. Build- ity and the fluxX then denotes reaction coordinates.(21) ing on this starting point, we make in addition use In other situations, the combination of force F and of a tensorial form of temperature that originates in fluxX may represent a temperature gradient and the an analogy with the non-equilibrium deformation resulting heat flow, or a concentration gradient and of materials under force. Whereas a unique value a mass flow. This formulation has thus been applied of pressure specifies the mechanical state of a body to thermal conduction, diffusion, chemical reaction under an isotropic force, pressure must be expressed and many other non-equilibrium phenomena.(22) instead in a tensorial form under an anisotropic stress. For thermal conduction, the time change of entropy

Consequently, the problem is to define a new diS/dτ is statistical temperature T for non-equilibrium states sta d S/dτ=F(dX/dτ) (5) by a combination of thermodynamic and statistical i mechanical methods. To solve it, we will first briefly F=−grad(lnT)/T (6) review the Clausius theorem, which underlies calo- rimetric entropy determinations, and its relationship dX/dτ=dQ/dτ (7) with the concept of fictive temperature. We will then turn to the concept of partition function, which is a where τ is time and Q is heat flow. An important as- core feature of statistical mechanics. As a starting sumption is that the formulation of Equations (4) or point toward more complex systems, we will then (5)–(7) relies on “local equilibrium.” For example, T extend our previous (15–20) with a three-level in Equation (6) must be an equilibrium temperature model to illustrate how one can unify the two differ- with respect to spatial coordinates. For this reason, ent frameworks of thermodynamics, as represented applications of Equations (5)–(7) are limited to by the Clausius theorem, and statistical mechanics, as phenomena close to equilibrium or to steady-state represented by partition function, through a tensorial non-equilibrium as expressed by a linear response definition of temperature. function (i.e. a linear force–flow relation). Independently of any non-equilibrium considera- 2. Clausius theorem tions, the concept of “fictive temperature” is used to characterize the structural and thermodynamic As is well known, Clausius theorem is one expres- states of a glass.(23,24) Since the fictive temperature sion of the second law of thermodynamics. Along a differs from the temperature of a heat bath or that of reversible cycle, the environment, it represents an additional internal temperature characterizing the extent of departure dS=dQ/T (1) ext from equilibrium. From a practical point of view, where dS, dQ and Text are the entropy and energy structural relaxation models based on this concept changes and the external temperature (i.e. heat bath have long proven useful to analyze residual stresses or environmental temperature), respectively. On the and tempering processes. On the theoretical side, an other hand, in an irreversible cycle, investigation of a two-level system(15,16) demonstrated that such an internal temperature T unrelated to dS>dQ/T (2) int ext either Equation (2) or (3) can be uniquely defined The entropy appearing in Equation (2) is usually split so as to satisfy Clausius theorem even under non- into two contributions, equilibrium conditions; dS=deS+diS (3) dS=dQ/Tint (8)

In the absence of mass transfer between the system We call Tint thermodynamic temperature because it is at and its environment, the first, ed S, is the entropy as- the roots of the fundamentally important thermody- sociated with the heat dQ transferred from or to the namic Equation (4). As already stated, however, this external heat bath under the reversible conditions of concept of a single additional temperature cannot be

Equation (1). The second, diS, is necessarily positive applied to the general case of a multi-level systems because it is the entropy irreversibly created within such a real glass in which we are interested here.

Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 9 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE

Regarding non-equilibrium temperatures,(25) a ensemble for a system composed of N particles, the simple thermodynamic approach is to extend their single-particle partition function per particle z is definition through the use of the fluctuation–dis- defined as: sipation theorem.(26,27) For dynamics near thermal z=Σp(i)=Σexp[−e(i)/(k T )] (11) equilibrium, this theorem provides the following B ext (26) relation: where p(i) is the probability of the energy level e(i), Text an external temperature and k Boltzmann’s constant. C(ω)=(2T(ω)/ω)ImR(ω) (9) B The index i is varied over all states that a particle may where C and R are correlation and response func- occupy at equilibrium. Fortunately, in the case of a tions of a general physical quantity, ω and T(ω) are model of independent and distinguishable particles, frequency and temperature, respectively. In other the energy distribution summed over all the atoms words, a generalized susceptibility (i.e. a response can replace the probability distribution of energy per function) caused by external forces can be linked to atom. The partition function for the whole system correlations of fluctuations. If one can calculateT (ω) composed of n particles Z is: as a function of ω from Equation (9), an effective tem- Z=Σexp[−e(i)/(kBT)] (12) perature Te can thereby be defined as a limiting value; Once Z is calculated, one easily derives the Helmholtz TTe = lim w (10) wÆ0 ( ) free energy A, E, and entropy S by This formalism based on a linear response theory using Z as follows: seems to work well in a near-equilibrium linear A=−k Tln(Z) (13) region. Efforts have also been made to extend the B formalism to the statistical mechanics of a many- E=Σe(i)exp[−e(i)/(k T)/Z] (14) particle system.(28) B Alternatively, a generic formalism(29–31) relies on S=E/[T+k ln(Z)] (15) the equations governing a Hamilton-like evolution B of a nonequilibrium , which Because a theoretical framework expressed as Equa- includes T as one of its variables. A third formalism tions (11)–(15) is only applicable to equilibrium, we involves entropy calculations with the steepest-ascent introduce an extended formulation. As a first step method(32) whereby the equations of nonequilibrium in that direction, we have recently proposed an ex- dynamics are solved to determine how the system tended partition function relying on a vectorial form moves to achieve a maximum entropy production. of temperature. The starting point of this approach is A fourth approach introduces the concept of exergy, that, according to Boltzmann factor, the probability which represents here the extra ideal work that could of the energy level p(e(i)) of state i is completely be extracted from the body when the system tends determined by Equation (11): to its equilibrium state. Entropy and nonequilibrium p(e(i))=exp[−e(i)/(k T )]/Z (16) temperature are then calculable from energy and B ext exergy as indicated in Ref. 33. In non-equilibrium states, any arbitrary energy Although it remains to determine which approach distribution can in contrast exist so that the single should be preferred, all of them share the common temperature Text must be replaced by a vectorial point of using a sort of effective scalar value of pseudo-temperature.(14) This parameter was also termed temperature as a function of space and time. The statistical temperature because it controls the statistical significant difference with the present approach is probability of each energy level and thus differs from thus that we attempt to construct a tensorial form of the thermodynamic temperature defined by Equations temperature from the use of only spatial information. (1) or (8). The number of vectorial components is equal to 3. Partition function that of energy levels. By using the statistical tempera- ture (or pseudo-temperature) Tsta, we then define an Thermodynamic variables in non-equilibrium states extended partition function as: have been discussed from a variety of macroscopic Z=Σexp[−e(i)/(k T (i))] (17) standpoints. As far as we are aware, however, refer- B sta ences to microscopic pictures are lacking although From this formula we derive the probability p(e(i)) it is extremely important to propose an extended of state i, the energy per particle E, and the entropy definition of partition function consistent with S as follows: Clausius equation and applicable to non-equilibrium. p(e(i))=exp[−e(i)/(k T (i))]/Z (18) To simplify the relation between the macroscopic B sta and microscopic world, the system investigated E=Σe(i)[exp(−e(i)/(kBTsta(i))/Z] (19) throughout this paper is a model of independent and distinguishable particles. In the traditional canonical S=Σ[e(i)/(kBTsta(i))p(e(i))]+kBln(Z) (20)

10 Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE

Three important points may be stressed here. First, than confi gurational rearrangements in glasses and the statistical temperature represents a distribution liquids, Tkin is supposed to equilibrate immediately not in the real three-dimensional space, but in the with the heat-bath temperature Text. In other words, energy space. Second, each component of statistical each particle has a kinetic energy that is instantane- temperature refl ects a deviation from the normal ously determined by the heat-bath temperature. By Boltz mann distribution rather than an amount of heat contrast, a confi gurational rearrangement takes much as typifi ed by the concept of “hot” or “cold.” In other longer because it is determined by considerably lower words, diff erences between the values of statistical hopping frequencies. The relative ease of hopping temperature refl ects the extent of non-equilibrium. depends on energy barriers e(i) and temperature

Third, all components of the statistical temperature Tkin. A matrix representation of Equation (21) reads tend to merge into a single uniform value Text when as follows: the system approaches equilibrium. But the diff erence Èdp (1) / dt ˘ È- a11 º ap1m ˘È(1) ˘ between the statistical temperature used in Equations Í ˙ Í ˙Í ˙ ··=  (23) (17)–(20) and the thermodynamic temperature used in Í ˙ Í ˙Í ˙ Í ˙ Í ˙Í ˙ Equations (1) or (8) remains to be clarifi ed. Îd()/dpmt ˚ Î am1 º- amm ˚Î pm() ˚ All p(i) are normalized, hence Σp(i)=1 (24) When the heat-bath temperature changes in a quasi-static manner, a partition function z and thermodynamic variables E and S can be derived from Equations (11)–(15) each time a new set of p(i) is calculated. As explained below in the Discussion, when the system is not in equilibrium, in contrast,

one must fi rst estimateT kin before following the route from Equation (17) to (20) so as to keep the formula- Figure 1. Schematic diagram of a multi-level system [Col- tion of statistical mechanics. our available online] For simplicity, the master equation is employed 4. Modelling of a multi-level system here. The formalism embodied by Equations (11)–(15) implicitly relies on the assumption of a canonical The multi-level system sketched in Figure 1 is an ex- ensemble. On the other hand, no a priori assump- tension of our previous two-level model(15) to simulate tions are made for Equations (17)–(20), in terms of phenomena in which particles hop between sites by fi xed type of ensemble, equipartition of energy or thermal excitation. If each site is regarded as a par- symmetry breaking. As investigated in the following ticular local structure, each hopping represents a local section, once all the distributions of p(e(i)) are calcu- structural transformation whereas the probability lated at each instant, a unique distribution of Tsta(i) is distribution of the particles refl ects their structural calculated without the assumption of any particular disorder. Similar models in which excitations from ensemble. As an important relation, there is a one- one state to the other states have been used to model to-one correspondence between the distributions of phenomenologically the thermodynamics and dy- p(e(i)) and Tsta(i). This is one form of expression of namics of real glasses, e.g. Ref. 40. Before discussing ‘internal temperature’ refl ecting the distribution of new defi nitions of temperature, we must calculate internal energy. the occupation probabilities p(i). A master equation is commonly used to calculate the time evolution of 5. Results the probability. The equation used here is: A three-level model with dimensionless parameters dp(i)/dτ=−Σaijp(i)+Σajip(j) (21) was used with values of e1, e2, e3, bij and kB set to where τ is time and aij the parameter that determines −2·0, −1·7, −1·0, 0·01, 1·0, respectively. The heat-bath the probability of hopping from site i to j, and vice temperature (i.e. Text or Tkin) with which the system versa for aji. This parameter is itself calculated from was equilibrated was initially 5·0. The heat-bath temperature was then decreased down to zero during a =b exp(−e(i))/(k T )) (22) ij ij B kin 50,000 time-steps. Three cases were considered. In where bij is a constant and Tkin a kinetic temperature the fi rst (case-1), equilibrium calculations were made refl ecting the magnitude of particle movement. The such that dp(i)/dτ was set to be zero at each step and choice of bij and e(i) will be addressed in Section p(i), abbreviated instead of p(e(i)), was calculated with

5. In general, bij≠bji and e(i)≠e(j). In the equilibrium Equation (16). The second (case-2) and the third (case- state, the right-hand side of Equation (21) is zero. 3) were a relatively slow and a rapid quenching with a Because vibrational energy transfer is much faster time-step increment of 10 and 0·1, respectively. Since

Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 11 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE

Text Energy

Figure 2. Quasi-static profi le of heat bath or external Figure 3. Quasi-static profi le of energy [Colour available temperature online] this study is not intended to simulate a specifi c real system, all the dimension parameters were chosen so that stable numerical solutions could be obtained near absolute zero. Note that a state at 0 K is not only a physically unreachable and mathematically singular point, but would require a quantum-mechanical formulation if phenomena around the region were dealt with. To apply this model to real phenomena at low temperatures, further investigations on the formulation of a particle hopping model, parameter identifi cation and numerical analysis techniques will be required.

For case-1, the temperature Text, the site-occupancy Figure 4. Quasi-static profi le of site-occupancy probability. probability p(i), internal energy E, and entropy S are Three values of p(1), p(2) and p(3) denote their site-occu- shown in Figures 2–5. As expected, the calculated pancy probability at site-1, site-2 and site3, respectively entropy decreases monotonously and reaches zero [Colour available online] at the end. For case-2 and case-3, the site-occupancy prob- ability p(i), internal energy E, and entropy S are shown in Figures 6–8 and 9–11, respectively. All curves look similar in the fi rst half before deviating more and more from each other in the second one. As also expected, a more rapid heating/cooling leads S to larger deviations from equilibrium. All curves are fl att ened at the fi nal stage, which indicates that trans- fer between sites is rapidly slowing down and then nearly freezes in as observed for a two-level model.(15) The time evolution of the site-occupancy has an Arrhenian activation energy so that the relaxation phenomena are thought to be those involved in the Figure 5. Quasi-static profi le of entropy ageing in glasses.(34)

adopted the constraint that all Tsta(i) converge at the 6. Discussion uniquely defi ned heat-bath temperature when the system is equilibrated under adiabatic conditions. 6.1 New defi nition of the statistical temperature The practical calculation procedure then is as follows:

The statistical temperature Tsta(i) out of equilibrium (1) Once each p(e(i)) is calculated, the energy E is is primarily defi ned with Equation (18) where the derived from number of parameters i involved is m for an m-level E=Σp(e(i))e(i) (25) system. Although the number of individual relations in Equation (18) is also m, the normalization Equa- (2) This energy E/N is that of the non-equilibrium tion (24) reduces the eff ective number of equations state. On the other hand, an eff ective temperature to m−1 so that an additional constraint is needed. We Te ff is calculated such that the state has the same

12 Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE

Energy Energy

Figure 6. Time profi le of energy in case-2 (slow quenching) Figure 9. Time profi le of energy in case-3 (rapid quenching) [Colour available online] [Colour available online]

Figure 7. Time profi le of site-occupancy probability in Figure 10. Time profi le of site-occupancy probability in case-2 (slow quenching). Three curves of p(1), p(2) and case-3 (slow quenching). Three curves of p(1), p(2) and p(3) denote their site-occupancy probability at site-1, site-2 p(3) denote their site-occupancy probability at site-1, site-2 and site3, respectively [Colour available online] and site3, respectively [Colour available online]

S S

Figure 8. Time profi le of entropy in case-2 (slow quench- Figure 11. Time profi le of entropy in case-3 (rapid quench- ing) ing)

energy as it would have at a Teff equilibrium its magnitude is thought to be Te ff . This defi nition

temperature. In other words, if the present non- signifi es that Tsta moves around on a sphere of

equilibrium system is equilibrated under adiabatic radius Te ff /N, which is determined by the system

conditions, its temperature will become Te ff . energy E.

(3) The introduction of the concept “norm” (length (4) All Tsta(i) are calculated numerically from m equa- of a vector) used in mathematics leads to the tions, namely, Equations (17) and (26).

following constraint linking Tsta(i) and Te ff ; Calculations for the three-level system are shown

2 for case-2 and case-3 in Figures 12 and 13, respective- Te ff =√[Σ(Tsta) ]/N (26) ly, where Tsta(i) is expressed as T(ii) to distinguish this

When Tsta(i) is regarded as a vector in an m- element of statistical temperature from T(ij) defi ned dimensional temperature space, its distance or later as the thermodynamic temperature. It is recalled

Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 13 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE

(a) (a)

(a) (a) Figure 12. Time profi le of statistical temperature in case- Figure 13. Time profi le of statistical temperature in case-3 2 (slow quenching). T(11), T(22) and T(33) is statistical (rapid quenching). T(11), T(22) and T(33) is statistical temperature defi ned at site-1, site-2 and site-3, respectively. temperature defi ned at site-1, site-2 and site-3, respectively. (a) Whole time span. (b) Time span between 45000 and (a) Whole time span. (b) Time span between 45000 and 50000 time-steps [Colour available online] 50000 time-steps [Colour available online]

that all Tsta(i) are always equal to the heat-bath tem- interference from the other sites or from the heat bath, perature Text in equilibrium. All Tsta(i) greatly deviate one fi nds that Equations (11) or (17) lead to: from the heat-bath temperature T at the fi nal stage ext p(i)/p(j)=exp[−e(i)/(k T (ij))]/exp[−e(j)/k T (ij))] (27) because the changes in site-occupancy probability B the B the are nearly frozen in. To describe the state accurately, ln(p(i)/p(j))=(e(j)−e(i))/(kBTthe(ij)) (28) the set of Tsta represents essential parameters whose number is the same as that of energy levels sites. The contribution of the two sites to the entropy then It is interesting to note that no formulation based is calculated with Gibbs formula: on linear-response theory (see Equations (9) and (10)) s(ij)=−k [p(i)ln(p(i))+p(j)ln(p(j))] (29) is used here. In other words, the statistical tempera- B ture Tsta can be defi ned exclusively from information If an energy transfer of probability Δp occurs from obtained at each instant in time regardless of the site i to site j, the occupancy changes to p(i)−Δp and extent of non-equilibrium. p(j)+Δp for the former and latt er, respectively. The value of energy transfer ΔQ(ij) between two sites is: 6.2 New defi nition of the thermodynamic ΔQ(ij)=e(i)*(−Δp)+e(j)*(+Δp)=(e(j)−e(i))*Δp (30) temperature If Δp is small, As outlined above, the introduction of a new, plural ln(p+Δp)≈ln(p)+Δp/p (31) notion of temperatures seems a promising way to develop a new general defi nition of thermodynamic From Equations (29) and (31), the change of entropy temperature Tthe(i) in non-equilibrium. Like for an ΔS(ij) is: equilibrium system, Equation (8) is valid for a two- ΔS(ij)=Δpln((p(i)/p(j)) (32) level non-equilibrium system where energy transfer is determined by the temperature Tint. In this case, From Equations (28), (30), and (32), the entropy consider the situation where two sites i and j are change ΔS is alternatively picked up and all others are frozen in. With the as- ΔS(ij)=ΔQ(ij)/(e(j)−e(i))*(e(j)−e(i))/(k T (ij)) sumption that the two sites are thermally equilibrated B the (33) =ΔQ(ij)/(kBTthe(ij)) under the same temperature Tthe(ij), without any

14 Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE

Figure 14. Time profi le of thermodynamic temperature in Figure 15. Time profi le of thermodynamic temperature in case-2 (slow quenching). T(12), T(23), and T(31) is ther- case-3 (rapid quenching). T(12), T(23), and T(31) is ther- modynamic temperature defi ned by correlation between modynamic temperature defi ned by correlation between site-1 and site-2, between site-2 and site-3, and between site-1 and site-2, between site-2 and site-3, and between site-3 and site-2, respectively. (a) Whole time span. (b) site-3 and site-2, respectively. (a) Whole time span. (b) Time span between 45000 and 50000 time-steps [Colour Time span between 45000 and 50000 time-steps [Colour available online] available online]

Let us assume that energy transfer in the whole probability are nearly frozen in. system is the sum of contributions of all such in- In the statistical physics of spin glasses, not only dependent pairs. The total entropy change in the eff ective temperatures(35,36) but also a similar tensorial (37) whole system then is the sum of these individual form of temperature Tαβ are introduced to express contributions: the relaxation time between two valleys Vα and Vβ in the hierarchical tree structure of replicas. In this ΔS(tot)=Σ(ΔQ(ij)/(k T (ij)) (34) B the study, we have not used any information assuming We now need to calculate ΔQ(ij) to apply Equation such a replica structure but derived the tensorial (34) to the selected two-level system. The changes in form only from an extension of the Clausius theorem. site-occupancy p(i) are calculated at each time step The plural temperatures used in spin glasses seem with useful to analyze the cascade of diff erent timescale of relaxation phenomena. In contrast, the plural Δp(ij)=(Δp(j)−Δp(i))/m (35) temperatures introduced in this study will be useful and to analyze the structural origin of non-equilibrium, because each value of temperature can be att ributed ΔQ(ij)=(e(j)*Δp(j)−e(i)*Δp(i))/m (36) to a pair of atoms. where the factor m (number of sites) in the denomina- Next, the entropy profi les calculated from the tor is introduced to keep ∑Δp(ij)=0. three diff erent formulae (37), (1) and (34) are shown For cases-2 and -3 with the three-level system, the in Figure 16. Type-1 is calculated with Gibbs formula: calculated thermodynamic temperatures T (ij) are the s(type−1)=−k Σ[p(i)ln(p(i))] (37) shown in Figures 14 and 15, respectively, where all B

Tthe(ij), (which remain equal to the heat-bath tempera- Case-2 is calculated from an integration of Equa- ture Text in equilibrium) are expressed as T(ij). In a tion (1) where Text is taken as Tint. Finally, case-3 is similar manner with the curves of Tsta(i), Tthe(ij) greatly calculated from an integration of Equation (34). The deviate from the heat-bath temperature Text at the remarkable conclusion is that the calculated curve fi nal stage, again because changes of site-occupancy of case-3 is always the same as that of case-1 even in

Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 15 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE

equilibrium, all matrix components have diff erent values. If the system concerned assumes equilibrium, all components merge into a unique value, which is typical of temperature used in the framework of statistical mechanics and thermodynamics. Entropy can be calculated by two routes; one uses Equation (20) and the diagonal terms of T, the other uses Equation (34) and the non-diagonal terms of T. Since we started from the same distribution of occupancy probability p(i) and constructed the two routes of Sections 3 and 6.2, the two values of entropy are always the same. We can express the relation between the diagonal and non-diagonal terms by rewriting Equation (28);

ln(p(i)/p(j))=(e(j)−e(i))/(kBTsta(ij)) (39)

=e(j)/kBTthe(ii)−e(i)/kBTthe(jj) Hence, the above relation is always preserved so that the distribution p(i) and entropy are also preserved.

There is one-to-one correspondence between Tsta

and Tthe. Finally, a few words may be useful on the relation of the new temperature concept with the second law of thermodynamics. This law is normally expressed Figure 16. Time profi le of entropy calculated by three in terms of the Clausius theorem: different methods. Three curves of entropy (Type-1, Type-2, and Type-3) signify value of entropy estimated dS=dQ/Text in equilibrium (40) by the Gibbs formula, that by the Clausius formula and and heat bath temperature, and that by the Clausius formula and thermodynamic temperature, respectively. (a) Case-2 dS>dQ/Text in non-equilibrium (41) (slow quenching). (b) Case-3 (rapid quenching) [Colour If a tensorial form of temperature Tten(i,j) and energy available online] change Q(i,j) is used, the single equality dS=Σ(ΔQ(ij)/(k T (ij)) (42) non-equilibrium. In addition, in all cases, there is a B ten fi nal non-zero residual entropy. The case-2 calcula- is in contrast valid for both equilibrium and non- tion represents the calorimetric entropy. It is lower equilibrium. In the long history of thermodynamics, than the values obtained by the two other methods, non-equilibrium states have been primarily dealt which agrees well with the theoretical conclusion with Equation (3) where diS, which denotes entropy that it is always a lower bound of the true entropy.(38) generation, is a relaxation term vanishing at equi- librium. In contrast to the various att empts made 6.3 New integrated formula of temperature and previously at quantifying and modeling diS, our goal relation to the second law of thermodynamics has been instead to deal with non-equilibrium by extending the concept of temperature as a tensorial The statistical and thermodynamic temperatures are quantity, which nevertheless remains the conjugate both important because the former enables parti- variable of entropy. tion functions to be defi ned even out of equilibrium At this point it is interesting to note that the whereas the latt er makes Clausius theorem valid tensorial form of pressure expresses the anisotropic without extra terms under non-equilibrium condi- deformation of materials in real space. In contrast, tions. As a further step, these two temperatures can the tensorial form of temperature expresses the ani- be integrated into one single tensorial expressions sotropic deformation away from the state determined as: by Boltz mann factor in the energy space. A remaining problem to be considered in our ȺTT11 1m ˘ Í˙ next study is the relation between time’s arrow,  (38) Í˙ entropy and the tensorial form of temperature in Í˙ Î˚TTm1 º mm more complex system. To validate our approach, where the diagonal terms Tii and non-diagonal terms other nonequilibrium methods such as the steepest-

Tij denote the statistical and thermodynamic tempera- entropy-ascent will be very useful. For doing so, ture, respectively. In the most general case of non- some conversion equations between scalar eff ective

16 Physics and Chemistry of Glasses: European Journal of Glass Science and Technology Part B Volume 62 Number 1 February 2021 A. TAKADA ET AL: GLASS THERMODYNAMICS: CLAUSIUS THEOREM AND A NEW TENSORIAL TEMPERATURE

7. Conclusions In this paper, two new defi nitions of temperature and Clausius theorem have been investigated with a three-level system. To extend the traditional concept of temperature, which is valid only for equilibrium, a new tensorial formula of temperature has been devel- oped. It allows one to describe non-equilibrium states in general, including those of glasses. The diagonal components of the temperature tensor, named statisti- Figure 17. Schematic diagram of instantaneous thermal cal temperature, is used to construct a bridge between equilibrium. The label of e(i) or e(j) indicates energy level thermodynamic variables and a partition function in at i-site or j-site. Number of disks and arrows shows site- the framework of non-equilibrium statistical mechan- occupancies and heat transfer between A and B [Colour ics. On the other hand, the non-diagonal components, available online] named thermodynamic temperature, are used to extend the Clausius theorem in the framework of non- equilibrium thermodynamics. The new formula has temperatures used in other studies and the tensorial been applied to a three-level system as the simplest form of temperature might be found in the future. example of multi-level systems mimicking a model glass. The new formula succeeds in describing the 6.4 Extended defi nition of the zeroth law of relation between changes in energy and entropy in thermodynamics non-equilibrium states. In addition, this study sheds new light on the interpretation of the second law of It is often said that the zeroth law of thermodynam- thermodynamics. As a result, Clausius’ defi nition of ics ensures the existence of the temperature concept. entropy, and its famous inequality for non-equilibri- Planck stated, “If a body C, be in thermal equilibrium um cases can be merged into a single equality without with two other bodies, A and B, then A and B are in introducing any additional terms. Finally, this study thermal equilibrium with one another.”(39) The statement proposes an extended defi nition of the zeroth law of suggests three scalar variables of thermodynamic thermodynamics in non-equilibrium. This study can potential, which drive heat fl ows among three bodies, be regarded as a rational extension of traditionally balance each other. The above statement is only valid useful concept of fi ctive temperature and a new order for equilibrium states. parameter to distinguish non-equilibrium glasses. If “instantaneous thermal equilibrium (i.e. bal- We plan to apply this theoretical framework to real ance)” characterizes a state in which two bodies glasses. have the same thermodynamic potentials, i.e. the same tensorial value of temperature, the net heat References fl ow at any instant is zero even in non-equilibrium. 1. Gutz ow, I. & Schmelzer, J. W. P. The Vitreous State: Thermodynamics, A schematic diagram of heat fl ow between A and B Structure, Rheology, and Crystallization. 2013. Springer, Berlin/Heidel- bodies is shown Figure 17. If A and B have tensorial berg. A B 2. Nemilov, S. V. Thermodynamic and Kinetic Aspects of the Vitreous State. temperature [Tij ] and [Tij ], for any combination of i 1994. CRC Press, London. and j, the driving forces for heat transfer at the i-site 3. Gupta, P. K. J. Non-Cryst. Solids,1988, 102, 231. of A to j-site of A and to j-site of B are equal at any 4. Shakhmatkin, B. A., Vedishcheva, N. M, Schultz , M. M. & Wright, A. C. instant. 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