Thermodynamics from First Principles: Correlations and Nonextensivity

Thermodynamics from ﬁrst principles: correlations and nonextensivity

S. N. Saadatmand,1, ∗ Tim Gould,2 E. G. Cavalcanti,3 and J. A. Vaccaro1 1Centre for Quantum Dynamics, Griffith University, Nathan, QLD 4111, Australia. 2Qld Micro- and Nanotechnology Centre, Griffith University, Nathan, QLD 4111, Australia. 3Centre for Quantum Dynamics, Griffith University, Gold Coast, QLD 4222, Australia. (Dated: March 17, 2020) The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this letter, we use the fundamental principles of ergodicity (via Liouville’s theorem), the self-similarity of correlations, and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long-range-interacting systems. Significantly, our formalism provides a justification for the well-studied nonextensive thermostatistics characterized by the Tsallis distribution, which it includes as a special case. We also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the Boltzmann-Gibbs-Shannon entropy. The consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics.

Introduction. The ability to describe the statistical tion [30] or the structure of the microstates [27]. Another state of a macroscopic system is central to many ar- widely-used approach is to generalize the MaxEnt prin- eas of physics [1–4]. In thermostatistics, the statistical ciple to apply to a different entropy functional in place of BGS state of a system of N particles in equilibrium is de- S ({wz}). At the forefront of this effort is the so-called Ts scribed by the distribution function wz over z where q-thermostatistics based on Tsallis’ entropy Sq ({wz}) ∝ P q z = ({q1, ··· , qN }, {p1, ··· , pN }) defines a point (the (1− z wz)/(q −1) and expressing the constraints as av- q microstate) in the concomitant 6N-dimensional phase erages with respect to escort probabilities {wz} [4,8]. space. The central question addressed in this letter is, q-thermostatistics is known to describe a wide range of what is the generalized form of wz for a composite sys- physical scenarios [3, 13, 20, 23, 31–46], including high-Tc tem at thermodynamic equilibrium that features corre- superconductivity, long-range-interacting Ising magnets, lated subsystems? turbulent pure-electron plasmas, N-body self-gravitating This question has been the subject of intense research stellar systems, high-energy hadronic collisions, and low- for more than a century [3–14, 16–23]. Correlations dimensional chaotic maps. The approach has also been and nonextensive energies are associated with long-range- refined and extended [3, 47–50]. interacting systems, which are at the focus of much of A contentious issue, however, is that the Tsallis en- the effort (in particular see [13, 18–20, 22]). The most tropy does not satisfy Shore and Johnson’s system- widespread approach to finding wz is the maximum en- independence axiom [26–29]. Although Jizba et al. [22] tropy (MaxEnt) principle introduced by Jaynes [6,7] recently made some headway towards a resolution, objec- on the basis of information theory. The principle en- tions remain [51], and the generalization of the MaxEnt tails making the least-biased statistical inferences about principle continues to be controversial. This brings into a physical system consistent with prior expected values focus the need for an independent approach to our central of a set of its quantities {f¯(1), f¯(2),..., f¯(l)}. It requires question. the distribution wz to maximize the Gibbs-Shannon We propose an answer by introducing a general for- GS (GS) logarithmic entropy functional S ({wz}) subject malism based on ergodicity [1] for deriving equilibrium P (i) ¯(i) GS to constraints fz wz = f . Here, S ({wz}) = distributions, including ones for correlated systems. Pre- P z −k z wz ln(wz) for constant k > 0. Despite outstanding viously this derivation was thought impossible as corre- success [24, 25] in capturing thermodynamics of weakly- lations have been linked with nonergodicity (see e.g. [4], interacting gases, the principle – in its original form – p. 68 and p. 320). However, we circumvent these diffi- does not describe correlated systems. Attempts have culties by showing how the self-similarity of correlations been made to generalize the principle, however, as there can be invoked to derive the distribution wz under well- is no accepted method for doing so, controversy has en- defined criteria. Then we show how to employ the Max- sued [26–28]. Ent principle consistently with correlations encoded as a self-similarity constraint function. After comparing our arXiv:1907.01855v4 [cond-mat.stat-mech] 16 Mar 2020 One approach is based on the extension of the MaxEnt principle by Shore and Johnson [29], and entails gener- results with previous works, we present a numerical ex- alizing the way knowledge of the system is represented ample for completeness and end with a conclusion. by constraints [25, 29]. Information about correlations Key ideas. Our approach rests on two key ideas. are incorporated, e.g. by modifying the partition func- (i) Liouville’s theorem for equilibrium systems. Con- 2 sider a generic, classical, dynamical system described by B, where wA and wB are not the marginals of wAB for zA zB zAB Hamiltonian H and phase-space distribution w(z; t). Be- q 6= 1. As this result has previously been regarded [4] as ing a Hamiltonian system guarantees the incompressibil- incompatible with Eq. (2), it shows that Liouville’s the- ity of phase-space flows, which is represented via Liou- orem has an underappreciated application for describing ville’s equation [1,5] by w being a constant of motion highly correlated systems. along a trajectory, i.e. Finding a generalized distribution. With these ideas in mind, we derive our main results for a composite, ∂w ∂w dw + z˙ · ∇w = + {w, H} = = 0, (1) self-similar, classical Hamiltonian system in thermody- ∂t ∂t dt namic equilibrium. For brevity, we explicitly treat a com- where { , } denotes the Poisson bracket. Imposing the posite system AB composed of two subsystems A and equilibrium condition ∂w/∂t = 0 implies B, although our results are easily extendable to com- positions involving an arbitrary number of macroscopic ∂w/∂t = −{w, H} = 0, (2) subsystems. Let the tuples (wAB ,HAB ), (wA ,HA ), zAB zAB zA zA (wB ,HB ) denote the composite and isolated equilib- zB zB i.e. the existence of a steady-state, w(z; t) = wz ∀ t. Any rium distributions and Hamiltonians of the composite Hamiltonian ergodic system at equilibrium will obey this AB, and separate A, B subsystems, respectively; wX zX condition. A possible solution of Eq. (2) is given by is the equilibrium probability that system X is in phase w = af(bH + c), where f(·) is any differentiable func- z z space point zX . The following criteria encapsulate prop- tion, for macrostate-defining and normalisation constants erties of the system required for subsequent work. They a, b and c (see e.g. [1, 56]). We only consider solutions of immediately lead to two key Theorems, which generalize this form, which is equivalent to invoking the fundamen- thermostatistics. tal postulate of equal a priori probabilities for accessible Criterion I – Thermodynamic limit: Consider a se- microstates [1]. For brevity, we shall write the solution quence of systems A1,A2, ··· for which the solution as (n) Eq. (3) for the nth term is given by wAn = G (HAn ). zAn An zAn A sequence that increases in size is said to have a ther- wz = GX (Hz) (3) modynamic limit if G(n) attains a limiting parametrized An and leave the dependence on the parameters a, b and c form as A becomes macroscopic, i.e. if G(n) → G as n An A as being implicit in the label X. n → ∞. The distribution wA = G (HA ), where the de- zA A zA (ii) Deriving equilibrium distributions. Consider the pendence on system, macrostate, and normalisation con- equilibrium distributions wA, wB and Hamiltonians HA, B stants is implicit in the label A on GA, is said to represent H of two isolated, conservative, short-range-interacting the thermostatistical properties of the physical material systems labelled A and B where comprising A in the thermodynamic limit. AB A B Examples of limiting forms include the BG distri- Hz = Hz + Hz , (4) AB A B bution G(H ) = ae−bHz and the Tsallis distribution AB A B z w = w w (5) −bHz zAB zA zB G(Hz) = aeq for macrostate-dependent parameter b and normalization constant a. are the total Hamiltonian and joint distributions, respec- Criterion II – Compositional self-similarity: We define tively, for the isolated, composite system AB at equi- a system as having compositional self-similarity if there librium, and z ≡ (z , z ). From (3), each distribu- AB A B exist mapping functions and such that the composite tion is a function of its respective Hamiltonian. Taken C H equilibrium distribution and energy of macroscopic AB together, Eqs. (3)-(5) imply the general solution wX zX are related to the isolated equilibrium distribution and is the Boltzmann-Gibbs (BG) distribution G (HX ) = X zX bH energy of macroscopic A and B by the following relations ae zX for macrostate-dependent constants a, b and X = A, B and AB. This well-known result can easily be generalised. For AB A B Hz = H(Hz ,Hz ) (7a) example, replacing Eq. (5) with AB A B wAB = (wA , wB ), 0 ≤ ≤ 1 , (7b) zAB C zA zB C wAB = wA ⊗ wB , (6) zAB zA q zB for all zAB, where H embodies the nature of the interac- where ⊗q is the q-product [3], correspondingly implies tions, and C embodies the nature of the correlations. For that the general solution is given by the Tsallis distribu- example, short-range interactions are well approximated bH tion GX (H ) = ae zX where ex is the q-exponential of by HAB = HA + HB and wAB = wA wB , whereas zX q q zAB zA zB zAB zA zB x provided due care is taken in respect of applying the the Tsallis distribution in Eq. (6) has been applied to a q-algebra [3, 55] and normalisation [56]. Note that each wide range of physical situations [3, 13, 20, 23, 31–46] ex- wX is the equilibrium distribution for system X in iso- hibiting strong correlations and long-range interactions. zX lation, and Eq. (6) represents a correlated state of A and Other relations hold in general, as shown in TableI[56]. 3

For brevity we will henceforth use “self-similar” to refer P wAB HAB − H¯ AB = 0, respectively, where H¯ AB zAB zAB zAB to compositional self-similarity. is the average energy. The prior knowledge of the self- Theorem I : For systems satisfying compositional self- similar correlations is represented by Eq. (7b) as a func- similarity in the thermodynamic limit, the equilibrium tional constraint over the phase space. Thus, the con- distribution is given by wX = G (HX ) where the func- strained maximization of SBGS({wAB})/k leads to zX X zX z tion GX satisfies ∂ X − ln wAB +aI({wAB })+bE({wAB }) (G (HA ), G (HB )) = G ( (HA ,HB )). (8) AB zAB zAB zAB C A zA B zB AB H zA zB ∂w 0 zAB zAB X + c (wAB − (wA , wB )) = 0 (12) Proof : This follows directly from Criteria I and II . zAB zAB C zA zB Hence, finding a G that satisfies Eq. (8) allows one zAB to calculate the equilibrium distribution in Eq. (3). See with Lagrange multipliers a, b, and {c }, where c Supplementary Material [56] for a simple example. In zAB zAB general, finding G is difficult, however, the next theorem is a function over the phase space. supplies a solution for an important class of situations. In [56], we show Eq. (12) yields Theorem II : Given single-variable invertible maps F 1 and H satisfying the following functional equations [ln (wA , wB )−aAB −cAB ] (13) bAB C zA zB zAB A B A B 1 A A A 1 B B B FAB( (w , w )) = FA(w ) + FB(w ) (9a) = ( [ln w −a −c ], [ln w −a −c ]), C H A zA zA B zB zB A B A B b b HAB(H(H ,H )) = HA(H ) + HB(H ) (9b) where equilibrium distributions are given by ln wX = zX then there exists a family of equilibrium distributions aX + bX HX + cX for phase space functions cAB , zX zX zAB given by cA , and cB that satisfy the above equation. Setting zA zB G−1(wX ) = 1 [ln wX −aX −cX ] shows that Eq. (13) is X X −1 X X X zX bX zX zX wz ≡ GX (Hz ) = FX (a H(Hz ) + b ) ∀z (10) equivalent to Eq. (8), and so the solutions found here are

X X equivalent to those given by the solutions of Eqs. (3) and where a and b are constants obeying the system com- (8) for corresponding values of the Lagrange multipliers position rules aX and bX . aAB = aA = aB, bAB = bA + bB. (11) Relationship with previously-studied thermostatistic classes. TableI compares the forms of F and H, and Note that aX and bX are generalisations of a common limitations of various classes of distributions. inverse-temperature-like quantity β = aX and an exten- An interesting result is that, although the Tsallis dis- 1 −βq H X X tribution, eq , is known to exhibit nonadditive av- sive average-energy-like quantity Ho = −b /β in the Zq more familiar form of Eq. (10), wX = F −1(β(H(HX ) − erage energy [4], our formalism shows that it corresponds X Ho )). to systems with additive Hamiltonians as demonstrated Proof : We defer the proof and a nontrivial example to in the table. Evidently, the nonadditivity of the average Supplementary Material [56]. energy is due to correlations forming between subsystems In our generalized thermostatistic formalism, the solu- (see also [2]). Nevertheless, our results effectively rule out tions to Eq. (8) give the most general form of the equi- the validity of the Tsallis distribution for systems with an librium distribution and Eq. (10) provides a recipe for interaction term in the Hamiltonian and satisfying Cri- finding it for the cases satisfying Eq. (9). Solutions to teria I and II. This is also true for the examples of mul- Eq. (9) can be guessed for a number of cases of practical tifractal and φ-exponential-class thermostatistics [3,4] interest, as shown below. However, the analytical forms that are characterized by the Tsallis distribution. of F and H are expected to be difficult to find, in gen- Moreover, our formalism covers thermostatistics of ex- eral. Nevertheless, we demonstrate below a systematic treme cases of correlations, most notably the well-studied numerical method that can find F and H for a given C case of one-dimensional Ising ferromagnets at vanishing and H, and thus determine the corresponding equilibrium temperature. See [56] for details. Aside from such triv- thermostatistics in the general case. ial maximally-correlated cases and the long-range Ising MaxEnt principle with correlations. We now show models [23, 40, 44–46, 58] (corresponding to the third row that the MaxEnt principle for SGS gives an indepen- of Table I as long as subsystems are macroscopic), we are dent derivation of Eq. (8) when the self-similar corre- not aware of any previous thermostatistic formalism that lations are treated as prior data along with the nor- can describe nontrivial long-range-interacting systems, as malization and mean energy conditions [25, 29]. For in the last row, by finding the equilibrium distribution. composite system AB, the constraints for the normal- Numerical example. The well-studied examples dis- ization and mean energy are the conventional ones, i.e. cussed above all have analytic solutions. Next, we I({wAB }) = P wAB − 1 = 0 and E({wAB }) = demonstrate the versatility of our approach by numer- zAB zAB zAB zAB 4

TABLE I. A summary of appropriate choices for {F, H} to reproduce well-established classes of thermostatistics, which allowed us to also indicate their potential limitations. We have included the conventional partition-funtion-type normalization constants in some cases for completeness. Note, however, that such constants can be re-expressed as a and b or β and Ho— i.e. ZBG = −1 GS −βHo+k S −βq Ho −1 e and Zq = eq (where βq = β[1 + (1 − q)βHo] ).

Type of Fails to Correlations Hamiltonian F(w) H(H) Distribution thermostatistics describe: (w , w ) (H ,H ) systems failing this work C 1 2 H 1 2 - - Eq. (10) (self-similar) (arbitrary) Criteria I and II 1 −βH correlations, conventional w1w2 H1 + H2 e ln(w) H ZBG nonadditive thermostatistics [5–7] (independent) (noninteracting) (exponential class) Hamiltonians 1 −βq H Tsallis’ (q-) w1 ⊗q w2 H1 + H2 eq nonadditive lnq(w) H Zq thermostatistics [4,8] (correlated) (noninteracting) (q-deformed class) Hamiltonians an exactly-solvable H example exhibiting w1 ⊗q w2 H1 ⊕p H2 H expq − β ln(ep ) lnq(w) ln(ep ) - both correlations and (correlated) (interacting) +Ho nonextensive energies

ically evaluating the statistics of a complex long-range- 15 15 interacting model (an extreme case of correlations and ) ) nonadditivity). To demonstrate how our approach might 10 H 10 w ( e handle a practical problem, we intentionally choose com- ( position rules, 5 5 (3.3 − wA)(3.3 − wB) (wA, wB) =wAwB , (14) C 2 0 0 2.3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (HA,HB) =0.7(HA + HB) , (15) w e H H 1.0 which have no known analytical solution for w within our BG formalism, and which would require extremely long-range 0.8 Our mapping interactions for the energy composition rule. 0.6 )

−H H

In Fig. (1), we show F and Q, where H(H) = Q(e ) (

w 0.4 [in BG, F ∝ Q ∝ ln(x)], for Eqs. (14) and (15), and w (w , w ) = w w (3.3 w1)(3.3 w2) −1 1 2 1 2 2.32 versus H, where w(H) = F (β(H(H) − Ho)). Here, Ho 0.2 (H , H ) = 0.7(H + H ) ensures normalization R w(H)dH = 1¯ = 1 and β = −1 1 2 1 2 R ensures Hw¯(H)dH = H¯ = 1, which corresponds, in 0.0 0 2 4 6 8 10 BG thermostatistics, to having an inverse temperature H of βBG = −1 in unitless parameters. It is interesting to see the significant differences between the generalized dis- BG −H FIG. 1. The top two panels show the mapping functions F tributionw ¯(H) and the normalized w = e (bottom and Q forw ¯ and q = e−H , respectively (solid lines), and plot), being flatter for small energies and decaying more their BG equivalents (dotted lines). The bottom panel shows −1 rapidly for larger energies. Full details of the numeri- the distribution w(H) = F (β(H(H) − Ho)) (solid line) for cal implementation are discussed in the Supplementary the mappings in Eqs. (14) and (15), and the normalized BG Material [56]. distribution with the same average energy (dotted line). Conclusions. We employed an approach based on Li- ouville’s theorem for equilibrium conditions obeying a thermodynamic limit and self-similarity criterion, to pro- Tsallis q-thermostatistics as demonstrated in TableI. In- vide an alternative derivation of consistent generalized terestingly, our formalism implies that, for systems sat- thermostatistics for systems with correlations and non- isfying our criteria, the latter family of thermostatistics additive Hamiltonians (this is in comparison to the con- can only capture the thermodynamics of systems with ventional MaxEnt formulations [3,4,6,7]). In our for- additive Hamiltonians. malism, the equilibrium distributions of such systems are Our extension of the MaxEnt principle with SGS to fully characterized by G in Eq. (3) or by {F, H} maps in include self-similar correlations as priors, gives an in- Eq. 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Supplemental material for “Thermodynamics from ﬁrst principles: correlations and nonextensivity”

S. N. Saadatmand, Tim Gould, E. G. Cavalcanti, and J. A. Vaccaro

In this supplemental material, we first discuss the restrictions on the normalization of Tsallis distribution and when it can be considered as a valid probability. We then present a simple example of finding the distribution function, G, satisfying Eq. (8) of the main text. The proof of Theorem II of the main text and an associated nontrivial example is given in Sec. III. Later, in Sec. IV, we discuss how to find Eq. (13) from Eq. (12). In Sec. V, we demonstrate how our formalism recovers wBG for the trivial case of one-dimensional Ising ferromagnets at vanishing temperatures. The details of our numerical approach to find F and H maps in Eq. (9) is presented in the last section.

RESTRICTIONS ON THE NORMALIZATION OF TSALLIS DISTRIBUTION

bHz Care needs to be taken with the normalization of the Tsallis distribution, wz ∝ eq , that is introduced following Eq. (6) in the main text [S1]. For example, it cannot be normalized and interpreted as a valid probability distribution when q > 1, b < 0 and the Hamiltonian involves unbounded kinetic energy terms. It can, however, be normalised for b > 0 for q >2. Also, it is important to note that b is not generally the negative of the inverse temperature and has a nontrivial connection to the physical temperature; a consistent equilibrium q-thermostatistics is presented in [S2].

RECOVERING wBG USING EQUATION (8)

In the main text, we argued that ﬁnding G to satisfy Eq. (8) of the main text results in the equilibrium distribution in Eq. (3). Here is a simple example: consider a conventional short-range-interacting system where the relations wAB = wAwB and HAB = HA + HB hold to a very a good approximation. In this case, C(wA, wB) = wAwB and H(HA,HB) = HA +HB. Equation (8) becomes wAwB = G(HA +HB), which is trivially satisﬁed by any G satisfying −b/a Hz/a fz = G(a ln(fz) + b) (or equivalently G(Hz) = e e ); here fz is a bounded, otherwise arbitrary, phase-space function, {a, b} are constants of integration, and we have bAB = bA + bB, where bX is proportional to the size of BG −βHz BG the system X as before. This gives the Boltzmann-Gibbs (BG) exponential class of distributions wz = e /Z ¯ −1 GS BG P −βHz −βH+k S BG ¯ GS with Z = z e = e , as expected. Here, we have β ≡ −1/a, b ≡ − ln(Z )/β = H − S /(kβ), GS P ¯ BG S ({wz}) = −k z wz ln(wz) and, therefore, it is clear that the constants β, H, and Z can be generally interpreted as the inverse temperature of the equilibrium, internal mean energy, and partition function respectively (as illustrated in Table I of the main text).

PROOF OF THEOREM II

We prove Theorem II by combining FX (·) with the function GX (·) in Eq. (3) to give the composite function X X A B AB FX (wz ) = FGX00 (Hz ) where FGX00 ≡ FX ◦ GX0 . This allows Eq. (9a), after C(w , w ) is replaced by w according to Eq. (7b), to be written as

AB A B FGAB(H ) = FGA(H ) + FGB(H ). (S1)

In comparison Eq. (9b), with H(HA,HB) replaced by HAB according to Eq. (7a), is

AB A B HAB(H ) = HA(H ) + HB(H ), (S2)

X X which suggests that FGX (·) and HX (·) are related functions. Indeed, the equality FGX (H ) = HX (H ) satisﬁes Eqs. (S1) and (S2) as does the linear relationship

X X X X FGX (H ) = a HX (H ) + b (S3) for constants aX and bX provided we adopt the system composition rules

aAB = aA = aB, bAB = bA + bB. (S4) 2

Applying the inverse function F −1 to both sides of Eq. (S3) and making use of Eq. (3) then yields the desired result, Eq. (10) with Eq. (S4) as condition Eq. (11). While other relationships may hold, the linear relationship in Eq. (S3) and its corresponding equilibrium distribution in Eq. (10) are suﬃcient for our purposes here. As an example of the application of Theorem II, consider a nontrivial correlated and interacting system satisfying AB A B A B AB A B A B w = C(w , w ) = w ⊗q w and H = H(H ,H ) = H ⊕p H , where ⊗q and ⊕p are the generalized product A B and sum of the q-algebra respectively [S3, S4]. It is clear that choosing F = lnq will result in F(C(w , w )) = A B Hz A B A B F(w ) + F(w ), while setting H(Hz) = ln(ep ) gives H(H(H ,H )) = H(H ) + H(H ). Therefore, Eq. (7) tells Hz us that the equilibrium distribution is simply wz = expq a ln(ep ) + b . This example appears in Table I in the main text.

FINDING EQUATION (13) GIVEN EQUATION (12)

In the main text, we argued that the constrained maximization of SGS({wAB})/k leads to ∂ [− P ln wAB + z ∂wAB zAB zAB z0 AB aI({wAB })+bE({wAB }) + P c (wAB − (wA , wB ))] = 0 with Lagrange multipliers a, b, and {c }, where zAB zAB zAB zAB zAB C zA zB zAB czAB is a function over the phase space. As (wA , wB ) has no explicit dependence on wAB , the above equation results in −1−ln wAB +a+bHAB +c = 0. C zA zB zAB zAB zAB zAB From Axiom II, this gives

ln wX = aX + bX HX + cX (S5) zX zX zX for X = A, B, and AB, where we have redeﬁned aX as 1 + aX for convenience. Taking Eq. (S5) with X = AB, and substituting for wAB and HAB using Eq. (6) gives ln (wA , wB ) = aAB + bAB (HA ,HB ) + cAB . Taking this zAB zAB C zA zB H zA zB zAB and substituting for HA and HB using Eq. (S5) with X = A and B, respectively, then yields 1 [ln (wA , wB )− zA zB bAB C zA zB aAB −cAB ] = ( 1 [ln wA −aA −cA ], 1 [ln wB −aB −cB ]) as required. zAB H bA zA zA bB zB zB

RECOVERING wBG FOR ONE-DIMENSIONAL ISING FERROMAGNETS

P Consider macroscopic steady-state ground states of the nearest-neighbor Ising model, HIsing = −J i SiSi+1, J > 0,Si = ±1 ∀i (e.g. see [S5] for a review). It can be easily shown that this system is describable, with a good AB A B A B AB AB A B A B approximation, by w(i,j) = C(wi , wj ) = wi wj exp[−γ(1 − |m(i,j)|)] and H(i,j) = H(Hi ,Hj ) = Hi + Hj (neglecting the interaction term and boundary effects due to the large size of the systems) — here, i denotes a collective spin state, γ → ∞ (playing the role of the diverging inverse temperature and, therefore, the exponential acts effectively as X P X X a δ A B -function). Also, we use m = Si/N , where N is the number of sites, to denote the magnetization mi ,mj i per site for systems X = A, B, and their composition AB. (Notice that the self-similarity rules already imply that, for all single systems, there are two highly likely, equiprobable, and degenerate states with mX = ±1, as expected from the spontaneous magnetization.) Similar to our previous BG-type example in Sec. I above, it is easy to check Hi = G(a ln(Hi)+γ(1−|mi|)+b) satisfies Eq. (8) for some constant a and additive parameter b (the middle term in G argument always vanishes for single systems); therefore, the equilibrium distribution is of the wBG-form as expected.

DETAILED DESCRIPTION OF THE NUMERICAL PROCEDURE

Here, we detail the numerical procedure used to calculate F and H for arbitrary mappings C and H. These procedures were used to generate Fig. (1) in the main text. Of relevance here are three primary points: 1) that F can be found accurately in most cases; 2) that H can A B be found similarly by transforming to q = e−H , to get Q(qA, qB) = e−H(− log(q ),− log(q )), H(H) = Q(e−H ), and H−1 = − log(Q−1); 3) that for the BG case, we have F BG ∝ − log(w), QBG ∝ − log(w) and GBG = e−H in dimensionless units with βBG = 1. 3

−1 Finding F and F given C

To calculate F, we use an iterative procedure over the mapping C. Specifically, we exploit the fact that the mapping C(wA, wB) has attractors for wA/B = 0, and a non-attractive fixed point wA = wB = 1, and that these are the only fixed points in [0, 1]2). Thus, we can use the following procedure:

1 Choose an initial weight value w0 = 0.99 < 1, and set F(w0) = f0 = 0.01.

2 Choose a secondary value w1 = C(w0, w0), so that F(w1) = F(w0) + F(w0) = 2f0.

3 Iterate wn>1 = C(wn−1, wn) until wn < 1E − 5, and evaluate F(wn>1) = F(wn−1) + F(wn−2) using existing values.

This gives a set of pairs of values (wi, F(wi)) over i, where we got O(10) pairs in all our tests. A B Our next step is to generate a continuous function F(w). We recognize that, for the BG mapping CBG(w , w ) = A B i w w , we get FBG(w) ∝ − log(w). Running the BG case through the distribution gives wi = w0, and F(wi) = (i + 1)f0. Clearly, this gives evenly distributed pairs (log(wi), F(wi)). We assume that this behaviour is approximately preserved in general mappings. We thus evaluate F(w) for general w by interpolating (using a cubic spline) the pairs we obtained by iteration on the logarithm of the weights, i.e., we interpolate F(wn) versus log(wn). This method is exact for the BG case. Without loss of generality, we finally R 1 normalize F so that 0 xF(x)dx = 1. As a final step, we recognise that F is monotone. This means we can similarly find the inverse function F −1(z) by interpolating log(xn) versus F(xn), which is again exact for the BG case.

−1 Finding H and H given H

We note that this iterative approach does not work for H, which does not have any ﬁxed point. But it does work for q = e−H , giving

A B − (− log(qA),− log(qB )) Q(q , q ) =e H , (S6) with H(H) = Q(e−H ) and H−1 = − log(Q−1). We can thus use the above approach to calculate mappings for H, by going via Q. Note that in the BG case, we ﬁnd QBG(q) ∝ − log(q) and see that the method is once again exact.

∗ n.saadatmand@griﬃth.edu.au [S1] J. F. Lutsko, J. P. Boon, EPL 95, 20006 (2011). [S2] F. Tanjia, S. N. Saadatmand, and J. A. Vaccaro, In Preparation. [S3] H. Suyari, Physica A: Statistical Mechanics and its Applications 368, 63 (2006). [S4] C. Tsallis, Introduction to Nonextensive Statistical Mechanics (Springer, 2009). [S5] R. Baxter, Exactly Solved Models in Statistical Mechanics, Dover books on physics (Dover Publications, 2007).