Phase Transitions Detached from Stationary Points of the Energy Landscape

Abstract

The stationary points of the potential energy function V are studied for the φ 4 model on a two-dimensional square lattice with nearest-neighbor interactions. On the basis of analytical and numerical results, we explore the relation of stationary points to the occurrence of thermodynamic phase transitions. We find that the phase transition potential energy of the φ 4 model does in general not coincide with the potential energy of any of the stationary points of V . This disproves earlier, allegedly rigorous, claims in the literature on necessary conditions for the existence of phase transitions. Moreover, we find evidence that the indices of stationary points scale extensively with the system size, and therefore the index density can be used to characterize features of the energy landscape in the infinite-system limit. We conclude that the finite-system stationary points provide one possible mechanism of how a phase transition can arise, but not the only one. The stationary points of the potential energy function or other classical energy functions can be employed to calculate or estimate physical quantities. Well-known examples include transition state theory or Kramers's reaction rate theory for the thermally activated escape from metastable states, where the barrier height (corresponding to the difference between potential energies at certain stationary points of the potential energy function) plays an essential role. More recently, a large variety of related techniques has become popular under the name of energy landscape methods [1], with applications to many-body systems as diverse as metallic clusters, or biomolecules and their folding transitions. While the mentioned applications focus mostly on the numerical investigation of finite systems, the analysis of stationary points has also proved useful for analytical studies of N -body systems in the thermodynamic limit. One field of research where such methods have been fruitfully applied is disordered systems undergoing a dynamical glass transition [2].Another line of research based on stationary points but focusing on equilibrium phase transitions in the thermodynamic limit N → ∞, dates back to about the same time [3]. This approach, originally formulated in terms of topology changes of configuration space submanifolds, can be rephrased in terms of stationary points of the potential energy function V , i.e. configuration space points q s satisfying ∇V (q s ) = 0. The underlying idea can be understood as follows [4]: Thermodynamic equilibrium properties are encoded in the thermodynamic limit value of the microcanonical configurational entropywhere Γ denotes configuration space and dx its volume measure, Σ v ⊂ Γ is the hypersurface of constantpotential energy V = N v, and dΣ stands for the (N − 1)-dimensional Hausdorff measure on Σ v . At a stationary point, we have ∇V = 0, the integrand on the righthand side of (1) diverges, and we may expect the stationary point to give an important contribution to the integral. Indeed, it has been shown that, for finite N...

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