Theorem on the Origin of Phase Transitions

Franzosi, Roberto; Pettini, Marco

doi:10.1103/physrevlett.92.060601

Cited by 108 publications

(197 citation statements)

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“…Several results strongly support this Topological Hypothesis and suggest that a phase transition might well be the consequence of an abrupt transition between different rates of change in the topology above and below the critical point. More details can be found in the review paper [36] and in the subsequent papers: [58] where the topology of the M v is analytically studied for the mean-field XY model; [59,60] where the topology of the M v is analytically studied for a trigonometric model undergoing also a first-order phase transition; [60,61] where an analytic relationship between topology and thermodynamic entropy is given among other results; [62,63] where a preliminary account of a general theorem on topology and phase transitions is given.…”

Section: Hamiltonian Dynamics Phase Transitions and Topologymentioning

confidence: 99%

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Pettini

Casetti²,

Cerruti-Sola

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. A first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: i) a Stochasticity Threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. ii) a Strong Stochasticity Threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weakly and strongly chaotic regimes. It is stable with N . A second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. The starting of this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N , or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, a third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions. "...Fermi expressed often the belief that future fundamental theories in physics may involve non-linear operators and equations, and that it would be useful to attempt practice in the mathematics needed for the understanding of nonlinear systems. The plan was then to start with the possibly simplest such physical model and to study the results of the calculation of its long-time behavior.... The motivation then was to observe the rates of mixing and thermalization with the hope that the computational results would provide hints for a future theory. One could venture a guess that one motive in the selection of problems could be traced to Fermi's early interest in the ergodic theory..."Actually, Fermi's early interest in ergodic theory is witnessed by his contribution to a theorem due to Poincaré and thenceforth known as the Poincaré-Fermi theorem. This asserts that neither analytic (Poincaré) nor smooth (Fermi) integrals of motion besides the energy can survive a generic perturbation of an integrable system with three or more degrees of freedom, thus, in the absence of other isolating integrals of motion, any constant energy surface of these generic systems is expected to be everywhere accessible to the phase space trajectory. At this level, no hindrance to ergodicity seems to be possible. Whence the surprise of Fermi, Pasta and Ulam (FPU) when no apparent...

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Section: Hamiltonian Dynamics Phase Transitions and Topologymentioning

confidence: 99%

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Pettini

Casetti²,

Cerruti-Sola

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

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“…It has been conjectured [3,6] that some of these topology changes are the "deep" cause of the presence of a phase transition. The correspondence between major topology changes of the M v 's and Σ v 's and phase transitions has been checked in two particular models [5,7]; more recently, it has been proved [8] that a topology change is a necessary condition for a phase transition under rather general assumptions [28], but the sufficiency conditions are still lacking. A natural way to characterize topology changes involves the computation of some topological invariants of the manifolds under investigation.…”

Section: Topology Of Configuration Spacementioning

confidence: 99%

Topology and phase transitions: From an exactly solvable model to a relation between topology and thermodynamics

et al. 2005

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The elsewhere surmised topological origin of phase transitions is given here new important evidence through the analytic study of an exactly solvable model for which both topology and thermodynamics are worked out. The model is a mean-field one with a k-body interaction. It undergoes a second order phase transition for k = 2 and a first order one for k > 2. This opens a completely new perspective for the understanding of the deep origin of first and second order phase transitions, respectively. In particular, a remarkable theoretical result consists of a new mathematical characterization of first order transitions. Moreover, we show that a "reduced" configuration space can be defined in terms of collective variables, such that the correspondence between phase transitions and topology changes becomes one-to-one, for this model. Finally, an unusual relationship is worked out between the microscopic description of a classical N -body system and its macroscopic thermodynamic behaviour. This consists of a functional dependence of thermodynamic entropy upon the Morse indexes of the critical points (saddles) of the constant energy hypersurfaces of the microscopic 2N -dimensional phase space. Thus phase space (and configuration space) topology is directly related to thermodynamics.

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“…Indeed, similar considerations have become central in the study of phase changes for physical systems. Franzosi and Pettini (2004), for instance, write "The standard way of [studying phase transition in physical systems] is to consider how the values of thermodynamic observables, obtained in laboratory experiments, vary with temperature, volume, or an external field, and then to associate the experimentally observed discontinuities at a PT [phase transition] to the appearance of some kind of singularity entailing a loss of analyticity... However, we can wonder whether this is the ultimate level of mathematical understanding of PT phenomena, or if some reduction to a more basic level is possible... [Our] new theorem says that nonanalyticity is the 'shadow' of a more fundamental phenomenon occurring in configuration space: a topology change...…”

Section: Dynamic Manifoldsmentioning

confidence: 99%

“…Next we briefly recapitulate part of the standard treatment of large fluctuations (Onsager and Machlup, 1953;Fredlin and Wentzell, 1998).…”

Section: Higher Order Coevolutionmentioning

confidence: 99%

Punctuated Equilibrium in Statistical Models of Generalized Coevolutionary Resilience: How Sudden Ecosystem Transitions Can Entrain Both Phenotype Expression and Darwinian Selection

Wallace¹,

Wallace

2008

Lecture Notes in Computer Science

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We argue that mesoscale ecosystem resilience shifts akin to sudden phase transitions in physical systems can entrain similarly punctuated events of gene expression on more rapid time scales, and, in part through such means, slower changes induced by selection pressure, triggering punctuated equilibrium Darwinian evolutionary transitions on geologic time scales. The approach reduces ecosystem, gene expression, and Darwinian genetic dynamics to a least common denominator of information sources interacting by crosstalk at markedly differing rates. Pettini's 'topological hypothesis', via a homology between information source uncertainty and free energy density, generates a statistical model of sudden coevolutionary phase transition based on the Rate Distortion and Shannon-McMillan Theorems of information theory which links all three levels. Holling's (1992) extended keystone hypothesis regarding the particular role of mesoscale phenomena in entraining both slower and faster dynamical structures produces the result. A main theme is the necessity of a cognitive paradigm for gene expression, mirroring I. Cohen's cognitive approach to immune function. Invocation of the necessary conditions imposed by the asymptotic limit theorems of communication theory enables us to penetrate one layer more deeply before needing to impose a phenomenological system of 'Onsager relation' recursive coevolutionary stochastic differential equations. Extending the development to second order via a large deviations argument may permit modeling the influence of human cultural structures on ecosystems.

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Theorem on the Origin of Phase Transitions

Cited by 108 publications

References 15 publications

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Topology and phase transitions: From an exactly solvable model to a relation between topology and thermodynamics

Punctuated Equilibrium in Statistical Models of Generalized Coevolutionary Resilience: How Sudden Ecosystem Transitions Can Entrain Both Phenotype Expression and Darwinian Selection

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