Tunneling Catastrophes of the Partition Function

Carvalho, C. A. A. de; Cavalcanti, R. M.

doi:10.1590/s0103-97331997000300006

Cited by 11 publications

(25 citation statements)

References 13 publications

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“…Lowering the temperature, we go from a single-solution to a three-solution regime. Lowering it even further, we reach a five-solution regime and so on [23]. 2 It is essential for this piece of the argument that the maxima of the inverted potential are degenerate.…”

Section: Coalescence Of Solutionsmentioning

confidence: 94%

“…The curves defining the frontiers between two such regions are named caustics, for they are analogous to the optical phenomenon. In our case, the classical solutions play the role of the light rays and the action replaces the optical distance [23].…”

Section: Coalescence Of Solutionsmentioning

confidence: 99%

“…Among the numerous solutions, we use the framework of catastrophe theory to identify the only two relevant ones, following Refs. [23,24]. We then compute the partition function and specific heat, obtaining the correct limits at both high and low temperatures, and a regular behavior where the usual quadratic approximation diverges.…”

Section: Introductionmentioning

confidence: 99%

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Semiclassical partition function for the double-well potential

et al. 2014

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We compute the partition function and specific heat for a quantum-mechanical particle under the influence of a quartic double-well potential nonperturbatively, using the semiclassical method. Near the region of bounded motion in the inverted potential, the usual quadratic approximation fails due to the existence of multiple classical solutions and caustics. Using the tools of catastrophe theory, we identify the relevant classical solutions, showing that at most two have to be considered. This corresponds to the first step towards the study of spontaneous symmetry breaking and thermal phase transitions in the nonperturbative framework of the boundary effective theory.

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Section: Coalescence Of Solutionsmentioning

confidence: 94%

Section: Coalescence Of Solutionsmentioning

confidence: 99%

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Semiclassical partition function for the double-well potential

et al. 2014

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“…This was already recognized in the semiclassical calculation of partition functions for quantum mechanical systems, where the need to consider a varying number of classical solutions in different temperature regimes became evident [4].…”

Section: Semiclassical Physicsmentioning

confidence: 96%

“…In Section 2. 4 we define what we mean by a renormalized focal decomposition, state a new result concerning the convergence of the renormalized focal decompositions to the asymptotic universal focal decomposition and give an idea of the main step towards the proof of this result. In Sections 2.5 and 2.6 we state new results concerning the Hamiltonian character of the asymptotic trajectories and the corresponding actions.…”

Section: Introductionmentioning

confidence: 99%

Focal decomposition, renormalization and semiclassical physics

Carvalho

Peixoto

Pinheiro³

et al. 2011

Journal of Difference Equations and Applications

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We review some recent results concerning a connection between focal decomposition, renormalization and semiclassical physics. The dynamical behaviour of a family of mechanical systems which includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time can be characterized through a renormalization scheme acting on the dynamics of this family. We have proved that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence, we have obtained an asymptotic universal focal decomposition for this family of mechanical systems which can now be used to compute estimates for propagators in semiclassical physics.

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Semiclassical Series at Finite Temperature

et al. 1999

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We derive the semiclassical series for the partition function of a one-dimensional quantum-mechanical system consisting of a particle in a single-well potential. We do this by applying the method of steepest descent to the path-integral representation of the partition function, and we present a systematic procedure to generate the terms of the series using the minima of the Euclidean action as the only input. For the particular case of a quartic anharmonic oscillator, we compute the first two terms of the series, and investigate their high and low temperature limits. We also exhibit the nonperturbative character of the terms, as each corresponds to sums over infinite subsets of perturbative graphs. We illustrate the power of such resummations by extracting from the first term an accurate nonperturbative estimate of the ground-state energy of the system and a curve for the specific heat. We conclude by pointing out possible extensions of our results which include field theories with spherically symmetric classical solutions.Comment: 25 pages + 5 figure

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Tunneling Catastrophes of the Partition Function

Cited by 11 publications

References 13 publications

Semiclassical partition function for the double-well potential

Semiclassical partition function for the double-well potential

Focal decomposition, renormalization and semiclassical physics

Semiclassical Series at Finite Temperature

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