Formulation of Genuine Thermodynamic Variables from Special Microscopic States
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Formulation of Genuine Thermodynamic Variables from Special Microscopic States
Koretaka Yuge and Shouno Ohta Department of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan
For classical discrete systems under constant composition, it has been considered that genuine thermody-namic variables such as free energy cannot be generally determined from information about a single or a fewselected microscopic states. Despite this fact, we here show that Helmholtz free energy for any given composi-tion for disordered states can be well characterized by information about a few ( R +3, where R denotes numberof components) specially selected microscopic states, whose structure can be known a priori without requiringany thermodynamic information. The present study is a non-trivial extension of our recently-developed theo-retical approach for special microscopic states in canonical ensemble to semi-grand canonical ensemble, whichadditionally enables to characterize temperature dependence of other thermodynamic variables such as internalenergy and entropy. I. INTRODUCTION
In classical discrete systems, Helmholtz free energy is typ-ically given by F = − β − ln Z , (1)where β denotes inverse temperature ( k B T ) − and Z denotespartition function given by Z = ∑ I exp ( − β E I ) , (2)where summation is taken over all possible microscopic stateson configuration space. Since number of possible statesshould astronomically increase with increase of system size, itis generally impossible to exactly determine the value of freeenergy for practical systems. Therefore, several theoreticaltechniques have been developed to effectively sample impor-tant microscopic states, such as thermodynamic integration and Wang-Landau algorism. The reason for performing suchnumerical simulation comes from the fact that a set of mi-croscopic states having dominant contribution to F should inprinciple depends on temperature and system (i.e., potentialenergy landscape), clearly stated by classical statistical me-chanics.Despite these facts, we recently find a special micro-scopic state, called projection state, which can character-ize macroscopic structure in thermodynamically equilibriumstate, where its structure is independent of temperature and ofinteractions. This strongly indicates the importance of ge-ometry on configuration space, i.e., the landscape of configu-rational density of states (CDOS) for non-interacting system,which can break the curse of dimensionality lies under thecurrent classical statistical mechanics. The present study per-form non-trivial extention of our previous work to semi-grandcanonical ensemble, in order to figure out a set of special mi-croscopic states that always exhibit dominant contribution to genuine thermodynamic variables such as Helmholtz free en-ergy. The details are shown below.
II. DERIVATION AND CONCEPT
Since marginal distribution of a fixed composition can bewell characterized by multidimensional gaussian and addi-tional odd (especially, the third) order genelarized moments,we can reasonally start from expanding semi-grand canonicalaverage of composition q = x − h q i S ( β ) = h q i − β f ∑ α = h q q α i h I | q i h I | q α i + β f ∑ α , β = (cid:10) q q α q β (cid:11) h I | q i h I | q α i (cid:10) I (cid:12)(cid:12) q β (cid:11) + · · · , (3)where I = E − ∆µ N ( q + ) ∆µ denotes difference in chemical potential, ∆µ = µ A − µ B . Here, h i represents semi-grand canonical average for non-interacting system, h | i denotes inner product onconfiguration space, i.e., trace over possible states for wholecomposition, and subscript for q denotes dimension of the fig-ure. Based on the symmetric definition of spin variable σ ± h q i S ( β ) ≃ h q i − β (cid:10) q (cid:11) h I | q i + β · ∑ m (cid:10) q q m (cid:11) h I | q i h I | q m i + · ∑ r ≥ ∑ m , n D q q r m q ( r + ) n E h I | q r m i D I (cid:12)(cid:12)(cid:12) q ( r + ) n E! − β ∑ r ≥ ∑ m , n ∑ a = , K · (cid:16)D q q r m q ( r + a ) n E − (cid:10) q (cid:11) D q r m q ( r + a ) n E − h q q r m i D q q ( r + a ) n E(cid:17) h I | q i h I | q r m i D I (cid:12)(cid:12)(cid:12) q ( r + a ) n E − β ∑ s · (cid:0)(cid:10) q q s (cid:11) − (cid:10) q (cid:11) h q q s i (cid:1) h I | q i h I | q s i (5)where the subsubscripts denote the class of figure in a givendimension. K = a = m = n , and K = h q i S ( β ) ≃ h q i − β (cid:10) q (cid:11) h I | q i + β h I | q i ∑ m (cid:10) q q m (cid:11) h I | q m i + f ′ ∑ c = t X λ c ( U c ⊗ V c ) X ! − β · h I | q i f ′′ ∑ d = t Y η d ( T d ⊗ W d ) Y − β · h I | q i ∑ s (cid:0)(cid:10) q q s (cid:11) − (cid:10) q (cid:11) h q q s i (cid:1) h I | q s i , (6)where λ and η denote singular values, and U c , V c , T d and W d are respectively c -th and d -th column of matrix U , V , T and W given by the SVD: A = UDV T B = TD ′ W T (7)and A i j = (cid:10) q q i q j (cid:11)(cid:0) i ∈ r m , j ∈ r ′ n , r ≥ , r ′ ≥ (cid:1) B i ′ j ′ = (cid:10) q q i ′ q j ′ (cid:11) − (cid:10) q (cid:11) (cid:10) q i ′ q j ′ (cid:11) − h q q i ′ i (cid:10) q q j ′ (cid:11)(cid:0) i ′ ∈ r m , j ′ ∈ r ′ n , r ≥ , r ′ ≥ (cid:1) . (8) X and Y are f ′ − and f ′′ − dimensional vectors whose com-ponents consist of a set of (cid:10) I (cid:12)(cid:12) q g (cid:11) , where figure g is includedin individual summations. When we define that λ M and η M are the largest singular values for respective SVD, we can ex-press semi-grand canonical average for composition only byenergy (and given chemical potential) of ”7” special micro-scopic states: h q i S ( β ) ≃ h q i − β (cid:10) q (cid:11) I + β ( I · I + I · I ) − β (cid:0) I · I · I − I · I (cid:1) . (9) The corresponding microscopic structures are explicitly givenby str1 : { , , , · · · , } str2 : (cid:8) , (cid:10) q q (cid:11) , · · · , (cid:10) q q e (cid:11) , , · · · , (cid:9) str3 : n , p λ M U M , · · · , p λ M U f ′ M o str4 : n , p λ M V M , · · · , p λ M V f ′ M o str5 : (cid:8) , √ η M T M , · · · , √ η M T f ′′ M (cid:9) str6 : (cid:8) , √ η M W M , · · · , √ η M W f ′′ M (cid:9) str7 : (cid:8) , , · · · , Q , · · · , Q e ′ , , · · · , (cid:9) , (10)where Q s = (cid:10) q q s (cid:11) − (cid:10) q (cid:11) h q q s i . (11)It is now clear from Eq. (10) that structure of the 7 special mi-croscopic states can be known a priori without any thermo-dynamic information, since the corresponding values can beobtained from information about CDOS for non-interacting system. Therefore, by using the standard relationship in ther-modynamics of ∆µ = ∂ F ∂ ( Nx ) , (12)we can quantitatively determine the value of Helmholtz freeenergy F at any given temperature and composition.We finally demonstrate how to estimate the values for spe-cial states. For instance, non-zero contribution of (cid:10) q · q i (cid:11) isgiven by (cid:10) q · q i (cid:11) = N ( DN ) ∑ k F i ( k ) h q k i = N ( DN ) F i ( ) h q i = N ( DN ) · DN · = N , (13)where the factor 2 in numerator comes from the permutationof two 1-body figure. Other non-zero contribution to the 3-order moment should always consist of one 1-body, and m -and m + (cid:10) q · q i · q p (cid:11) = N · DN · T N ∑ k F i p ( k ) h q k i = N · DN · T N F i p ( ) h q i = JN D , (14)where J = − N p ( N p is the number of pair 2 i included in a single triplet figure 3 p ) if the triplet 3 p includes at least onepair 2 i , and J = III. CONCLUSIONS
Based on geometry of configurational density of states, wederive 7 special microscopic states that always exhibit domi-nant contribution to Helmholtz free energy, which is invariantfor the choice of constituent elements, temperature and po-tential energy landscape. Significant information about non-interacting system should be re-emphasized in statistical me-chanics to break the curse of dimensionality.
IV. ACKNOWLEDGEMENT
This work is supported by a Grant-in-Aid for Scientific Re-search on Innovative Areas (18H05453) and a Grant-in-Aidfor Scientific Research (16K06704) from the MEXT of Japan,Research Grant from Hitachi Metals · Materials Science Foun-dation, and Advanced Low Carbon Technology Research andDevelopment Program of the Japan Science and TechnologyAgency (JST). A. van de Walle and M. Asta, Modell. Simul. Mater. Sci. Eng. ,521 (2002). A. van de Walle and G. Ceder, J. Phase Equilib. , 348 (2002). A. van de Walle, M. Asta, and G. Ceder, Calphad , 539 (2002). A. van de Walle, Calphad , 266 (2009). F. Wang and D.P. Landau, Phys. Rev. Lett. , 2050 (2001). K. Yuge, arXiv:1704.07725 [cond-mat.dis-nn]. K. Yuge, J. Phys. Soc. Jpn. , 084801 (2015). T. Taikei, T. Kishimoto, K. Takeuchi and K. Yuge, J. Phys. Soc.Jpn. , 114802 (2017). K. Yuge, T. Taikei and K. Takeuchi, arXiv:1706.08796 [cond-mat.dis-nn]. K. Yuge, J. Phys. Soc. Jpn.85