Abstract
Motivated by the study of the metastable stochastic Ising model at subcritical temperature and in the limit of a vanishing magnetic field, we extend the notion of \((\kappa , \lambda )\)-capacities between sets, as well as the associated notion of soft-measures, to the case of overlapping sets. We recover their essential properties, sometimes in a stronger form or in a simpler way, relying on weaker hypotheses. These properties allow to write the main quantities associated with reversible metastable dynamics, e.g. asymptotic transition and relaxation times, in terms of objects that are associated with two-sided variational principles. We also clarify the connection with the classical “pathwise approach” by referring to temporal means on the appropriate time scale.
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Notes
This is because [18] sticks to the conventions of its main reference [24], where the authors introduced a rather unusual factor 1/2 in the Hamiltonian. Since Fig. 1 illustrates a dynamics ran at inverse temperature \(\beta = 2 / 3\) without such a convention, it would correspond to a trajectory sampled at inverse temperature \(\beta ' = 4 / 3\), i.e., to a temperature twice as small, with the convention of [24] and [18].
In the case of the kinetic Ising model, one can instead think of two opposite strong drifts: one towards the metastable state and one towards the stable state. The nucleation of the supercritical droplet occurs by fluctuation against the first drift and the system follows the second drift afterwards. This is coherent with the fast relaxation to local equilibria.
Note that for the kinetic Ising model, despite the strong drift towards equilibrium at the appearance of the first supercritical droplet, there is still some time to wait before the relaxation is achieved: the pictures on the second line of Fig. 1 show atypical configurations with respect to the equilibrium measure. The exit from \(\mathcal R\) does not coincide with global thermalization.
In this framework, we would like to describe \(A_\theta \) as a continuous process with a discontinuous limit. This is not compatible with Skorohod topology. Without this modification one should introduce an ad hoc topology. The modification allows to work with the standard Skorohod topology.
In the case of our kinetic Ising model this amounts to describing the shape of subcritical, critical and subcritical droplets. The techniques presented in [18] allow for proving that the critical droplet is Wulff-shaped and this question is still largely open for subcritical and supercritical droplets.
This is however an asymptotic formula and there is no equality.
In [11] metastable states are associated with disjoint subsets \(\mathcal R_i\), \(i \in I\), of \(\mathcal X\) and each of them is associated with a parameter \(\kappa _i\) used to defined the soft capacities \(C_{\kappa _i}^{\kappa _j}(\mathcal R_i, \mathcal R_j)\) introduced in [8]. We do not see obstructions in extending the results of the present paper to more than two overlapping sets \(\mathcal R_i\) covering \(\mathcal X\). In this framework the relevant times to consider instead of \(T_{\lambda _{\mathcal S}}\) and \(T_{\kappa _{\mathcal R}}\) would be the \(\min _{j \ne i} T_j\), \(i \in I\), where \(T_j\) is the killing time of X absorbed in \(\mathcal R_j\) at rate \(\kappa _j\).
This is an alternative construction that does not use the exponential variable \(\sigma _\lambda \) of Sect. 1.1 and allows for less homogeneous killing rates.
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Acknowledgements
A. G. and P. M. thank Maria Eulalia Vares, the Universidade federal do Rio de Janeiro and the Università di Padova for the kind hospitality which gave us the opportunity to lay the foundations of this work.
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Communicated by Yvan Velenik.
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Bianchi, A., Gaudillière, A. & Milanesi, P. On Soft Capacities, Quasi-stationary Distributions and the Pathwise Approach to Metastability. J Stat Phys 181, 1052–1086 (2020). https://doi.org/10.1007/s10955-020-02618-9
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DOI: https://doi.org/10.1007/s10955-020-02618-9
Keywords
- Soft capacity
- Soft measure
- Quasi-stationary measure
- Restricted ensemble
- Metastability
- Potential theory
- Relaxation time