Soft Report - Dipartimento di Fisica - Sapienza

Phase Transitions and TopologyPhase transitions are a very well understood subjectin statistical mechanics and a huge amount of workhas been done in the last century. Recently,however, a novel characterization of phasetransitions has been proposed [1]: the singularbehaviour of thermodynamic observables at a phasetransition is attributed to major topology changes inphase space or, equivalently, in configuration space.More precisely, the conjecture is that for a systemdefined by a continuous potential energy functionV(q) – q denotes the N-dimensional vector of thegeneralized coordinates – a thermodynamic phasetransition occurring at T c (corresponding to energyV c) is the manifestation of a topological discontinuitytaking place at the specific value V c of the potentialenergy function V (strong topological hypothesis).The most striking consequence of this hypothesis isthat the signature of a phase transition is present inthe topology of the configuration spaceindependently on the statistical measure defined onit. Through Morse theory, topological changes arerelated to the presence of stationary points of V,and, more specifically, to the discontinuousbehaviour of invariant quantity defined on them, asthe Euler characteristic χ.However, there are many open questions. Amongthem: what are the necessary and/or sufficiencyconditions for the topological hypothesis? What is theexact correspondence between the thermodynamicand the topological transition points? What about theorder of the phase transition?On the one hand, there is a theorem [1] assertingthat, for smooth, finite-range and confininginteraction potentials a topology change at some V θis a necessary condition for a phase transition totake place at the corresponding energy value. On theother hand, there are numerical studies of variousmodels for which a variety of results has beenobtained.Our main contributions to this topic can besummarised as follow.1. By studying an analytically solvable mean-fieldmodel with k-body interaction (k-trigonometricmodel [2]), that, according to the value of k,undergoes no (k=1), second-order (k=2) or firstorder(k>2) phase transition, we were able todirectly relate the different thermodynamicbehaviours to the Euler characteristic (see Figure 1),finding that the discontinuity of χ signals thepresence of the phase transition and the concavityits order.2. There is a non-trivial relation between the phasetransition critical energy (V c) and the topologicalcritical energy (V θ). By studying different modelsystems (mean-field Φ 4 [3], two one-dimensionalmodels [4]), we found that the relevant topologicalpoints are the “underlying stationary points”(saddles), defined through a map V s=M(V) fromenergy level V to stationary point energy V s. If thereis a topological singularity at V θ a phase transition isFig. 1: Logarithmic Euler characteristic as afunction of potential energy V for the k-trigonometric model with k=1,2,3.also present if and only if there is a temperature T csuch that V s(T c)= V θ (weak topological hypothesis).Our findings seem to indicate that a sufficiencycriterion for the phase transition to take placerequires the introduction of a statistical measure, asthe map M is defined through an average over thestatistical measure. Moreover, the introduced weaktopological hypothesis, based on the concept ofunderlying saddles (well known from glassysystems), appears as a possible framework to fit theresults of a variety of model systems.We hope that this approach can be of interest from atheoretical point of view, in elucidating the deepunderlying relationship between phase transitionsand topology, and from a practical reason, for theinvestigation of mesoscopic systems (e.g. proteinsand large molecules) where the number of degreesof freedom is small enough to detect the presence ofa phase transition using standard techniques.References[1] L. Casetti, M. Pettini, and E.G.D. Cohen, Phys.Rep. 337, 237 (2000); R. Franzosi and M. Pettini,Phys. Rev. Lett. 92, 060601 (2004).[2] L. Angelani, L. Casetti, M. Pettini, G. Ruocco, andF. Zamponi, Europhys. Lett. 62, 775 (2003) ; Phys.Rev. E 71, 036152 (2005).[3] A. Andronico, L. Angelani, G. Ruocco, and F.Zamponi, Phys. Rev. E 70, 041101 (2004).[4] L. Angelani, G. Ruocco, and F. Zamponi, Phys.Rev. E 72, 016122 (2005).AuthorsL. Angelani (a), G. Ruocco (b,c), F. Zamponi (b,d)(a) CRS SMC-INFM-CNR, Roma, Italy(b) CRS SOFT-INFM-CNR, Roma, Italy(c) Dip. di Fisica, Univ. di Roma La Sapienza, Roma,I(d) Ecole Normale Superieure, Paris, France.51SOFT Scientific Report 2004-06