Information Geometry, Phase Transitions, and Widom Lines : Magnetic and Liquid Systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Information Geometry, Phase Transitions, andWidom Lines : Magnetic and Liquid Systems
Anshuman Dey, Pratim Roy, Tapobrata Sarkar ∗ Department of Physics,Indian Institute of Technology,Kanpur 208016,India
Abstract
We study information geometry of the thermodynamics of first and secondorder phase transitions, and beyond criticality, in magnetic and liquid sys-tems. We establish a universal microscopic characterization of such phasetransitions via the equality of correlation lengths ξ in coexisting phases,where ξ is related to the scalar curvature of the equilibrium thermodynamicstate space. The 1-D Ising model, and the mean-field Curie-Weiss modelare discussed, and we show that information geometry correctly describesthe phase behavior for the latter. The Widom lines for these systems arealso established. We further study a simple model for the thermodynamicsof liquid-liquid phase co-existence, and show that our method provides asimple and direct way to obtain its phase behavior and the locations of theWidom lines. Our analysis points towards multiple Widom lines in liquidsystems. ∗ E-mail: deyanshu, proy, tapo @iitk.ac.in Introduction
The physics of phase transitions [1] has been a fascinating area of research formore than a century, starting from the celebrated van der Waals equation forliquid-gas systems. Application of information geometric methods to such stud-ies are relatively recent, but have yielded several useful insights especially inthe context of liquid-gas co-existence [2]. This method involves the underlyingRiemannian geometry of the equilibrium thermodynamic state space, and wasmainly initiated through the work of Weinhold [3] and Ruppeiner [4]. The keyidea here is to utilise the positivity condition on the Hessian of the entropy fora thermodynamic system in equilibrium, so as to define a Riemannian metric onthe thermodynamic state space. In a striking conjecture, Ruppeiner proposed,via the theory of Gaussian fluctuations, that the scalar curvature R (or, moreappropriately, | R | ), arising out of such a metric is related to the correlation length ξ of the system, an idea that has since been tested in a variety of models. Indeed,it has been established by now that the scalar curvature of the thermodynamicstate space diverges at critical points, for a wide variety of systems that exhibitsecond order phase transitions. Most of the analysis done in the past dealt with the geometry of thermody-namics at or near criticality. However, it has recently been established [6] thatmethods of information geometry can be used to study discontinuous first orderphase transitions, via the equality of the correlation lengths of the coexistingphases, in liquid-gas systems. In [6], such equality has, in fact, been verified withexperimental data from NIST [7]. Apart from providing an alternative routeto study phase transitions that bypasses several long-standing problems in stan-dard thermodynamics, this approach has the potential of analytically predictingthe “Widom line” [8], [9], a conjectured continuation of phase co-existence thatextends beyond criticality, and distinguishes between supercritical “phases” ofa system. The presence of the Widom line, which is usually defined as the lo-cus of maxima of the correlation length has recently been experimentally estab-lished [10], and certainly puts information theoretic studies of phase transitionsin perspective. Note that very close to criticality, all response coefficients scale aspowers of the correlation length, and hence the loci of maxima of these serve asequivalent definitions of the Widom line [8]. Slightly away from criticality, how-ever, this is not the case and our method provides a way of locating the Widomline as per its actual definition, even away from criticality.Whereas methods of information geometry in liquid-gas systems have beenfirmly established by now, magnetic and liquid-liquid systems have been muchless studied. Although it was known that the thermodynamic scalar curvaturecan be calculated for some magnetic systems, and these show the standard di-vergences associated with a second order critical point, it is not known howRiemannian geometry captures first order transitions in these systems. Evenless is known about the information geometry of liquid systems and its role in Information geometry in the context of thermodynamics is also called “thermodynamicgeometry.” Information geometry of quantum phase transitions has also been an area of extensiveresearch, see e.g. [5].
We begin with a very brief overview of information geometry. It will be enough forus to consider this in the context of equilibrium thermodynamics, a formulationdue to Ruppeiner [4]. More details can be found in the excellent review of Brodyand Hook [2]. The main idea is to consider the line element due to a positivedefinite Riemannian metric which is defined by the Hessian of the entropy perunit volume s , dl = g ij da i da j , g ij = − k B (cid:18) ∂ s∂a i ∂a j (cid:19) (1)where a and a denote the internal energy and the particle number per unit vol-ume (for magnetic systems, one has to appropriately use thermodynamic quanti-ties per unit spin). As shown by Ruppeiner [4], this introduces the concept of adistance in the space of equilibrium thermodynamic states, i.e, a large distancebetween two such equilibrium states is interpreted as a small probability thatthese are related by a thermal fluctuation.One can consider various other forms of the metric related to the one givenin eq.(1) by Legendre transforms. Indeed, a choice of the thermodynamic poten-tial in appropriate coordinates might render the metric diagonal, and simplifyalgebraic calculations. For example, for single component fluids and magnetic2ystems, a particularly simple diagonal form of the metric can be used [4] dl = 1 k B T (cid:18) ∂s∂T (cid:19) ρ dT + 1 k B T (cid:18) ∂µ∂ρ (cid:19) T dρ (2)where µ = (cid:16) ∂f∂ρ (cid:17) T , f being the Helmholtz free energy per unit volume (per unitspin for magnetic systems), and ρ is the inverse of the volume. For magneticsystems, the magnetization per unit volume plays the role of ρ .For such diagonal metrics, the scalar curvature of the manifold takes a wellknown simple form, R = 1 √ g (cid:20) ∂∂T (cid:18) √ g ∂g ρρ ∂T (cid:19) + ∂∂ρ (cid:18) √ g ∂g T T ∂ρ (cid:19)(cid:21) (3)where g T T and g ρρ are the coefficients multiplying dT and dρ respectively, ineq.(2), and g = g T T .g ρρ . In passing, we note that for positivity of the line element, g T T and g ρρ should be positive definite in the regions of interest. Non-diagonalforms of the metric (as in eq.(1)) can also be used, and give equivalent results for R . By augmenting this geometric picture with Gaussian fluctuation theory, Rup-peiner conjectured that such a scalar curvature can, in fact, be related to thecorrelation length of the system [4], | R | ∼ ξ d , where d is the system dimension.We also mention that the geometry described here becomes trivial in the presenceof a single fluctuating variable, and meaningful results can only be obtained fortwo or more fluctuating thermodynamic quantities.In the context of the 1-D Ising model, information geometry has been studiedby Ruppeiner [12] and later by Janyszek and Mrugala, [13]. We briefly recalltheir results. We start with the thermodynamic potential per unit spin in theferromagnetic case (with unit coupling constant), which is given byΦ = − β ln h e β cosh ( α ) + (cid:0) e β sinh ( α ) + e − β (cid:1) / i (4)where H and T denote the applied magnetic field and the temperature respec-tively, and α = H/T , β = 1 /T (the Boltzmann’s constant has been set to beunity). From eq.(4), the authors of [13] showed that in the ( α, β ) coordinates,the scalar curvature of the thermodynamic state space is given by the simpleanalytical expression | R | = cosh ( α ) (cid:0) sinh ( α ) + e − β (cid:1) / + 1 (5)This also agrees with the result of [12]. In the zero field limit, the curvaturediverges at T = 0 indicating the sole critical point of this theory. Also, it can bechecked that ξ | H =0 = 1 | ln (tanh ( β )) | ≃ | R | H =0 (6)3 Figure 1: “Widom lines” for the 1-D Ising model on the H − T plane. The redcurve is the locus of maxima of the scalar curvature R . The blue and the greencurves are the maxima of the specific heats C M and C H respectively.In order to understand the Widom line, we study information geometry of the1-D Ising model away from criticality for non-zero T and H . We compute themaxima of the correlation length (equivalently, | R | ) in the supercritical region,and contrast it with the maxima of other response functions. We first record thestandard expression for the magnetization per spin m = sinh ( α ) (cid:0) sinh ( α ) + e − β (cid:1) (7)The specific heat at constant magnetization is then obtained by differentiatingthe entropy obtained from eq.(4), while keeping m fixed, C m = 4 e α T (cid:0) e β sinh ( α ) + 1 (cid:1) / (cid:16) e β cosh ( α ) + (cid:0) e β sinh ( α ) + 1 (cid:1) / (cid:17) cosh ( α )(8)with a lengthy expression for the specific heat C H at constant applied field H which is not reproduced here. These of course reduce to the standard expression C m,H | H =0 = β sech β in the zero field case. Now, we calculate the maxima of | R | as a function of temperature. From eq.(5), it can be seen that such maximaoccur for a given value of H when the the temperature T satisfies the equationtanh (cid:18) HT (cid:19) = 2 H ( e /T −
1) (9)and a similar analysis can be carried out to determine the maxima of the specificheats C m and C H . Numerical solutions of these have been plotted in fig.(1), We will loosely refer to the locus of maxima of the correlation length and the other responsecoefficients as the “Widom lines.” | R | , C H , and C m respectively. We observe that near the critical point, i.e for very small valuesof the applied magnetic field H , the maxima of the specific heat do not asymptoteto T = 0, but reaches a limiting temperature T ≃ .
8. The maxima of | R | (i.ethe correlation length ξ ) for such small values of H also does not asymptote tozero, but goes to a limit T ≃ .
18 for H = 10 − . It is interesting to note that thelocus of maxima of the correlation length does not begin from the critical pointfor the 1-D Ising model. As we will see, this feature will not be present in any ofthe mean-field models that we will analyse.Before concluding this section, let us summarize the main results here. Wehave considered the 1-D Ising model from the point of view of information ge-ometry. Analysing of the same away from the critical point, we have located thelocus of maxima of the correlation length ξ and the specific heats C m and C H ,in the T − H plane. Our analysis points to the fact that in this simple example,the Widom lines associated with the locus of maxima of R and also the specificheats do not seem to converge to the critical point ( T = 0) in the limit H → In this section, we study the information geometry of the classical mean-fieldCurie-Weiss (CW) ferromagnetic model. This model has been studied extensivelyin the past, and details can be found in standard textbooks [14], [15]. We willadhere to the notations of [14]. In the context of information geometry, theCW model was first studied by Janyszek and Mrugala in [13]. These authorsestablished the divergence of the scalar curvature close to the critical point. Weseek to understand this model away from criticality, and study first order phasetransitions, via the geometry of the thermodynamic state space.As is well known, in the thermodynamic limit, the free energy of the CWmodel is given by G = − T (ln2 + max f ( m )) (10)where the Boltzmann constant has been set to unity, and m is the magnetizationper spin, which solves the equation m = tanh (cid:18) T c T m + HT (cid:19) (11)Here, T is the temperature, T c its critical value, and H is the applied magneticfield. In order to write this in terms of the temperature and the thermodynamicextensive variable, m , we perform a Legendre transform and write the free energyas [13] f = − T ln2 − T c m + T (cid:0) − m (cid:1) + T m tanh − m (12)with (cid:0) ∂f∂m (cid:1) T = H . This can be used to calculate the scalar curvature of theequilibrium thermodynamic state space, but note that the entropy per spin is5iven by eq.(12) as s = ln2 −
12 ln (cid:0) − m (cid:1) − m tanh − m (13)As can be immediately seen from eq.(13) and eq.(1), information geometry be-comes trivial in this context, since the entropy is a function of a single variable,and is not amenable to a Riemannian geometric analysis. In order to remedy thesituation, the authors of [13] proposed to modify the Hamiltonian, which in itsoriginal form for N spins, is given by H = − T c N X i 4, 0 . . T c = 1. - - Figure 3: Isothermal R vs m for theCurie-Weiss model, beyond criticality.The green line shows a diverging curva-ture at the origin for T = T c = 1. Themagenta and red curves correspond to T = 1 . . T c = 1 and a power law, C L ( T ) = 1 + T + T . This is onlyfor illustration, and as can be checked, any other value of the critical temperatureor any other regular functional form of C L will not alter our discussion below.With these assumptions, the curvature simplifies to R = 12 (1 − m ) [ m (1 + 2 T + 3 T ) + 4 T − − m − T ) (1 + T + T ) (18)We also calculate the specific heat at constant applied field H , using H = T tanh − m − T c m (19)and obtain from eq.(13), for T c = 1, C H = T (cid:18) ∂s∂T (cid:19) H = T (cid:0) tanh − m (cid:1) ( m − − m − T (20)We observe that the curvature scalar blows up wherever C H does, as expected,for T < T c . (However, C H does not diverge at the critical point). Further, thedenominator of R behaves as the product of the square of C L and the square ofthe denominator of C H . We keep this in mind, as this seems to be a universalfeature in mean-field theories.Now, we study the behavior of R as a function of m . This is shown in fig.(2)for temperatures below criticality and in fig.(3) for temperatures at and beyondcriticality. For T < T c , we find that R diverges symmetrically on the m -axis fora given value of temperature. This value of m can, in fact, be identified with7 Figure 4: R vs H for isotherms theCurie-Weiss model. The green curvescorrespond to | R | in the two physical re-gions at T = 0 . 6, and the blue curvesare for T = 0 . 8. The red curve corre-sponds to the isotherm T = 1 . 35. Cross-ing of the physical branches of | R | indi-cate phase transition, always at H = 0. –0.3–0.2–0.100.10.20.3H 0.95 1 1.05 1.1 1.15 1.2T Figure 5: Loci of maxima for the CWmodel. The green line along the H -axis denotes phase co-existence and ter-minates at T c = 1. The red line, along H = 0 are the maxima of | R | (and χ T ),the Widom line for various values of T .The two blue lines are the symmetricmaxima of C H as a function of T .the turning point of the magnetic isotherms in the H − m plane. We have notshown these isotherms here, but we note that for values of m that lies betweenthe two divergences, the specific heat C H is negative, i.e, the system becomsunstable. The two divergence on the m -axis get closer as we approach T c = 1, atwhich point they merge into a single divergence at m = 0. Beyond criticality, thesituation is depicted in fig.(3). Here we see that isothermal R has a maximumat m = 0, with its value at the maximum decreasing as we move away fromcriticality.It is more useful to consider the behavior of R as a function of the appliedmagnetic field, H , with m being treated as a parameter. We can guess the resultby exploiting the symmetry of the situation. From the expression for R of eq.(18),we see that R ( m ) = R ( − m ). For two physical branches (where C H is positive),the R ’s should thus “cross” where m = 0. From eq.(19), this occurs when H = 0.We show this in fig.(4), for the isotherms T = 0 . T = 0 . T = 1 . 35 (red), which we now explain this in details. Take, for example, theisotherm T = 0 . R in the the regions where C H is positive, with thecurve that asymptotes to infinity for positive values of H are for negative valuesof m and the other one is for corresponding positive values of m . Specifically,from fig.(2), the T = 0 . R -isotherm diverges at m = ± . H is aparametric plot for m < − . H is for m > . H = ± . T c , there is no R -crossing.8e thus see that below the critical temperature, R has two physical branches,which cross at H = 0. Indeed, as alluded to in the introduction, this impliesequality of correlation lengths at H = 0 which we interpret as the phase tran-sition, with the residual magnetization being the values of m where R divergesfor this temperature. Of course beyond T c , there is no residual magnetization, ascan be seen from the behavior of R in fig.(3). This is the information geomet-ric description of discontinuous transitions in magnetic systems, and agrees withstandard results.As we move closer to T c , the value of R at which the physical branches cross,increases. This can be seen by comparing the green and blue curves in fig.(4),which denote physical branches of R at T = 0 . T = 0 . T c =1, the crossing point is pushed to infinity. Beyond T c , R shows a maxima, whichis always at H = 0, which can again be justified using symmetry arguments. Thelocation of the Widom line is hence along the H -axis, evan away from criticality.This is depicted in fig.(5), where the green line ( H =0), is the first order line thatculminates at T c = 1, and the locus of maxima of R is its continuation, along thered line.To contrast the behavior of the maxima of R with other response coefficients,we have also plotted, in fig.(5), the locus of maxima of the specific heat C H . Wefind that unlike R (and the susceptibility χ T , whose locus of maxima is the sameas that of R ), C H shows two symmetric maxima for a given temperature on the H -axis. All maxima converge to the critical point for very small values of H .To conclude this section, we summarize the main results. Here, we haveprovided a novel characterisation of first order processes in a simple mean-fieldmagnetic model, the CW model of ferromagnetism, via information geometry.We have seen that equality of the correlation length ξ of co-existing phases accu-rately predicts known behavior of this model, near or away from criticality. Theinterpretation of the scalar curvature of the thermodynamic state space R furtherallows us to calculate the Widom line as the locus of maxima of the correlationlength ξ (via R ∼ ξ d ) which is shown to lie on the H -axis, as an expected contin-uation of the phase co-existence line, in the T − H plane. Such loci of maximafor the specific heat C H however do not lie on the H -axis. We further note thatour information geometric study of first order processes in the CW model hasessentially similar features as mean-field liquid-gas systems [6], as expected, andindicate the wide applicability of this method in studying phase transitions. Asan illustration of this, we now proceed to study liquid systems. We now apply our information geometric method to a simplified toy example inthe thermodynamics of liquid-liquid phase transitions. Although the literatureon the topic is vast (see, for example, [16] for a recent analysis in silica) analyticinsight in such systems is often difficult to provide in practice. We choose a simplemodel that nevertheless captures the essential physics. The model that we studyhas internal energy U = A ( V ) + B ( V ) T / + kT (21)9 Figure 6: Isotherms in the V − P planefor various values of temperature. Figure 7: Meta-stability region in the T − V plane.where k is related to Boltzmann’s constant. This form of the energy is motivatedfrom the fact that the potential energy U ∼ T / for some simple liquids [17],[18]. Here, A ( V ) and B ( V ) are functions of the volume V , which can be takento be polynomial fits, determined from simulation data. The exact nature ofthese functions for liquid silica has been dealt with extensively in [18] (wherea similar form of the internal energy was used), and they were fitted to fourthorder polynomials in the volume. For analytical tractability, it is enough for usto choose a simpler situation, and we assume A ( V ) = A + A V + A V B ( V ) = B + B V (22)where A i , i = 0 · · · B j , j = 0 , B ( V ) does not allow for analytic handling of the model, and we havedropped this for the time being. Indeed, this simple choice of the functions A and B will serve to illustrate the main features for any physical system with a T / dependence of the internal energy, and it can be checked that the inclusion of aquadritic term in B ( V ) will not alter the qualitative aspects of our discussion.We will also assume throughout that the units are properly chosen, as in [18].Given the internal energy U , one can compute the entropy of the system by text-book methods [1]. In particular, starting from a reference entropy S ( T , V ) at areference temperature T , and a reference volume V , the entropy at arbitrary val-ues of the temperature and volume can be calculated by adding its change alongand isotherm and along an isochore. This requires the expression for the pressurefor the reference temperature, which can be again obtained as a polynomial fitin the density [18]. In our example, we choose P | T = T = C + C V + C V (23)Where the C s are coefficients that can be calculated from molecular dynamicssimulations. The calculation for the entropy has been pedagogically explained10n [18], and we obtain S = 12 T / V (cid:2) T / (2 V ln T + 5 V B ( V ) ++ 2 V A ( V ) + 2 (cid:0) V + C V + C V ln V − C (cid:1)(cid:1) − V B ( V ) (cid:3) (24)where we have set the constant k in eq.(21), and the reference temperature T tounity, without loss of generality. The Helmholtz free energy for this model canbe calculated from F = U − T S , and taking its derivative yields the pressure P = T P − B (cid:0) T / − T (cid:1) + ( T − 1) ( A + 2 A V ) (25)The curve of metastability can be obtained by setting (cid:18) ∂P∂V (cid:19) T = 2 A ( T − − TV ( C V + 2 C ) = 0 (26)The critical volume is calculated by solving for T from the above equation andthen using (cid:0) ∂T m ∂V (cid:1) = 0, where T m is the solution for the temperature obtainedfrom eq.(26). These finally yield the critical temperature and volume : T c = 54 A C A C − C , V c = − C C (27)The constant volume specific heat for this model can be calculated to be C V = 5 T / + 3 ( B + B V )5 T / (28)And the isothermal compressibility is given by K T = − V (cid:18) ∂V∂P (cid:19) T = V T ( C V + 2 C ) − A V ( T − 1) (29)The expression for C P , the specific heat at constant pressure is easy to obtainby differentiating eq.(24) while keeping the pressure in eq.(25) fixed, but gives alengthy expressions, which we shall not present fully here, but we note here thatit has the same denominator as eq.(29), as expected.Our main interest here is in the scalar curvature of the equilibrium thermo-dynamic state space of this system. This can be calculated with the help ofeither eq.(1) or eq.(2), and we obtain an expression for R in terms of T and V .Using eq.(2) for example, we obtain a diagonal form of the metric, given by thecomponents g T T = C V V T , g ρρ = V T K T (30)where ρ = 1 /V . As required [4], these are certainly positive in the domain ofinterest, as is the quantity √ g T T g ρρ . The expression for R is too lengthy toreproduce here, but for the moment, let us note that its denominator is R den = (cid:2)(cid:0) T / + 3 ( B + B V ) (cid:1) (cid:0) T ( C V + 2 C ) − A V ( T − (cid:1)(cid:3) (31)11 Figure 8: R as a function of volume forvarious isotherms. The red, green andblue curves correspond to T = 1 . 6, 1 . . | R | shows two local maxima. Figure 9: Maxima of | R | (black andbrown curves), C P (red) and K T (blue)in the V − T plane, beginning fromthe critical point in the spinodal (green)curve in the V − T plane.implying that the curvature scalar diverges along the divergence of K T or C P .Also note that the denominator of R is equal to the product of the square of thenumerator of C V and the square of the denominator of C P (or K T ). This is afeature that we noticed in the Curie-Weiss model of the previous section, andseems to be an universal property for mean-field systems.In order to make our results more tractable in general, we now make a choiceof constants, and set A = − B = 5 × , C = 10, C = − 3, with all theother constants set to unity. One can check that this ensures the positivity ofphysical quantities like temperature, volume, isothermal compressibility etc. insome domain, with our analysis being valid in this domain. With this choice ofparameters, the metastability condition of eq.(26) yields, as a solution for thetemperature, T m = 2 V V − V + 2 (32)with the system being stable outside the region defined by the above equation,plotted in the T − V plane. The critical temperature for our model is, from eq.(27), T c = 2, with V c = 1 and P c = 16 . 21 (in appropriate units). In fig.(6), we haveshown some isotherms on the P − V plane, along with the region of metastabilityfor our model. The latter region is plotted in the T − V plane in fig.(7). Wenow present our results on the scalar curvature graphically. We find that below T c , isothermal R , as a function of the volume, diverges at the boundaries of themetastability curve of fig.(7). This is shown in fig.(8). Consider, for example,the isotherm at T = 1 . 6, plotted in red in fig.(8). For this temperature, C P diverges for V = 0 . V = 1 . V forwhich R diverges, and as can be seen from fig.(7) (or, equivalently, from eq.(26)after putting in the constants), these are the values of V between which thesystem becomes unstable. As we look at isotherms with higher temperature, for12 Figure 10: Isothermal R-crossing as afunction of P . Two curves of the samecolor denote | R | in the physical regionsfor a given isotherm. The red curvescorrespond to T = 1 . 6, the green to T = 1 . 65 and the blue to T = 1 . 7. Thephysical regions of the P − V isothermshave been used for the plots. Figure 11: Widom lines for the modelliquid-liquid system. The green andblack curves correspond to the two max-ima of | R | . The red and blue curves arealmost indistinguishable and correspondto the maxima of C P and K T respec-tively. Near the critical pressure, thesemerge into a single line.example the T = 1 . R come closer to each other, and at T c = 2, they merge at V = 1. Beyond T = 1, | R | ( ∼ ξ ) shows two maxima, as can be seen from the blue curve infig.(7) which corresponds to an isotherm T = 2 . 1. The Widom line therefore hasto be understood in more details here.For this, beyond T c , we study the maxima of | R | by the equations (cid:18) ∂ | R | ∂T (cid:19) V = 0 , (cid:18) ∂ | R | ∂V (cid:19) P = 0 , (cid:18) ∂ | R | ∂T (cid:19) P = 0 (33)We find that whereas the maxima of | R | obtained from the first of these relations(i.e isochoric maxima of | R | with respect to the volume) is virtually indistin-guishable from the K T maxima line, those obtained from the second and thirdrelations of eq.(33) indicate two local maxima for | R | near criticality. The valuesof V obtained from both these relations (where | R | maximises) are indistinguish-able. Fig.(9) summarizes our results. Here, the red curve indicate the locus ofmaxima of C P , and the blue one is the corresponding quantity for K T . The so-lution of the first relation of eq.(33) is identical to the blue curve. The solutionsof the other two relations in eq.(33) give the brown and black curves.As in the case of the CW model of the previous section, instead of lookingat isochores, it is more useful to look at the behavior of | R | with respect toa thermodynamic intensive variable, such as the temperature or the pressure.In fig.(10), we have shown the behavior of isothermal | R | as a function of thepressure, for various values of the temperature, with volume being the parameter13n the plot. Consider, for example, the red curves in fig.(10). These are isothermalplots of | R | as a function of the pressure, for T = 1 . 6, in the two physical domainsof V (as we have discussed) and correspond to the red curve in fig.(8). One of thered curves here (the one diverging at P ∼ . 8) corresponds to V < . V > . | R | for the physical branches is thus seen tocross at P ∼ . 85. This is thus the value of P for which the correlation lengthsof the coexisting phases become equal, and we interpret this as the pressure forwhich a first-order liquid-liquid phase transition occurs at T = 1 . 6. Similar valuesof the pressure can be calculated using isotherms for different values of T . Thecollection of all such points in the T − P plane is the phase co-existence curvefor the model.As we approach criticality, the crossing point of R is pushed to infinity. Be-yond criticality, there is no crossing, i.e a single phase exists, but | R | shows twomaxima with respect to the pressure, i.e there are two Widom lines that orig-inate from the critical point. In fig.(11), we have plotted the Widom lines forthis system in the P − T plane. Projected on this plane, the maxima of | R | areshown in the green and black curves and the other curves, which are the maximaof C P and K T become almost indistinguishable. All these curves converge at thecritical point.Before we end, we summarize the main results in this section. Here, wehave shown that information geometry can be effectively used to establish phasebehavior in liquid systems, in a simple way. The main inputs that went intoour calculation is the power law behavior of the internal energy of eq.(21) andthe coefficients appearing in eqs.(22) and (23). We stress here that althoughfor illustrative purpose we have chosen a toy model, once these quantities arecalculated in any liquid system, our method offers an easy way to determinephase behavior, and very importantly, the Widom lines, and may be used forexperimental predictions. We have also seen an important qualitative differencebetween magnetic and liquid systems, namely that in the latter, there might bemultiple isothermal Widom lines. We note here that this situation is differentfrom liquid-gas systems as well, where there is a single locus of maxima of thecorrelation length ξ for isotherms [6]. In this paper, we have studied in details information geometry of magnetic andliquid systems. We first established the idea that equality of correlation lengthsof co-existing phases, calculated via the scalar curvature of the equilibrium ther-modynamic state space, is indicative of first order phase behavior in mean-fieldmagnetic systems, where the curvature diverges appropriately at criticality. Wethen applied this to liquid systems and predicted liquid-liquid phase transition ina simple model. Our main conclusion here is that geometric techniques provide auniversal new method of characterizing first and second order phase transitions,and can also be used to predict the behavior of the system beyond criticality, viathe Widom line. Our results show that the definition of the latter as the locus ofmaxima of the correlation length is somewhat ambiguous in liquid systems, and14an lead to multiple lines, which originate from criticality. This is due to the T / behavior of the internal energy. This is to be contrasted with liquid-gas systemswhere the prediction of isobaric or isothermal Widom lines are unique [6]. (Formagnetic systems, such multiple locus of maxima occur instead for the specificheats). For the 1-D Ising model, we have seen that the Widom line does notoriginate from the critical point. This however, can be attributed to the some-what unphysical nature of the model, and this feature is not seen for any of themean-field theories analysed in this paper. It is an important question whetherthe Widom line is unique, and future experiments will probably indicate which ofthe multiple lines is chosen by the system to distinguish between phases, beyondcriticality.In liquid-gas systems, prediction of first order co-existence via the equality ofthe correlation length originates from the idea of Widom [11] that near a first or-der transition, density fluctuations in one phase of a fluid results in the formationof a second phase. The thickness of the interface between the two phases is theninterpreted as the correlation length, which therefore should be equal measuredin either phase [6]. Our results indicate that a similar phenomenon happens inmagnetic and liquid systems as well, and this should be studied further.Compared to the more sophisticated computer simulation techniques to studyliquid-liquid phase transitions advocated over the last two decades, ours is asimple method which can be applied to any theoretical equation of state, orequivalently, experimental data. Admittedly, we have chosen a simple scenarioto illustrate our method, but this can be easily generalised to more realisticsituations, and the qualitative details should remain unchanged.It will be of interest to apply our technique to phenomenological models ofliquid-liquid phase transitions. We note here that the predictions using the scalarcurvature remain valid as long as the numerical value of | R | is more than typicalmolecular volumes. For liquid-gas systems, these restrict the use of geometricmethods beyond a certain range of temperature and pressure, though these aretypically away from the scaling region. A similar analysis in liquid systems wouldinvolve analysing data from molecular dynamics simulations. We leave this for afuture publication. Acknowledgements We wish to sincerely thank Amit Dutta, V. Subrahmanyam and K. P. Rajeev forextremely useful discussions. References [1] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics ,John Wiley & Sons, New York, 1985.[2] D. C. Brody, D. W. Hook, “Information geometry in vapor-liquidequilibrium,” J. Phys. A42 (2008) 023001, arXiv : 0809.1166[cond-mat.stat-mech] . 153] F. Weinhold, “Metric geometry of equilibrium thermodynamics,” J. Chem.Phys. (1975) 2479.[4] G. Ruppeiner, “Riemannian geometry in thermodynamic fluctuation the-ory,” Rev. Mod. Phys. , 605 (1995), erratum ibid , 313 (1996).[5] P. Zanardi, P. Giorda, M. 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