Conference Agenda

We prove a variational characterisation of the free energy of the interacting Bose gas in the thermodynamic limit. The formula minimises the sum of entropy and energy over point processes of loops and interlacements. The starting point is a Poisson-point process representation of the gas that is based on a by now classical path-integral representation via the Feynman--Kac formula in terms of the Brownian loop soup. The new contribution is a clear separation between short and long loops, the latter yield the interlacement part. On the way, we introduce two new notions of specific relative entropy densities.

This talk introduces a computational method for generating digital twins of the 3D morphology of (functional) materials through stochastic geometry models, calibrated by means of 2D image data. By systematic variations of model parameters a wide spectrum of structural scenarios can be investigated, such that the corresponding digital twins can be exploited as geometry input for numerical simulations of macroscopic effective properties [1,2]. For calibrating models that can generate virtual 3D microstructures by stochastic simulation, generative adversarial networks (GANs) have gained an increased popularity [3]. While classical (non-parametric) GANs offer a data-driven approach for modeling complex 3D morphologies, the systematic variation of their model parameters for generating diverse, not yet measured structural scenarios can be difficult. In contrast, parametric models of stochastic geometry (e.g., Gaussian random fields) allow for the generation of realistic, yet unobserved, structures through systematic parameter variation. However, as model complexity increases---such as for excursion sets of more general random fields or random tessellations induced by marked point processes, which are necessary to capture more intricate microstructures---the required number of model parameters increases substantially. This makes classical model calibration by means of interpretable descriptors impractical. Combining GANs with advanced stochastic geometry models can overcome these limitations and, in addition, allows for the calibration of model parameters solely based on 2D image data of planar sections through the 3D structure [4]. These parametric hybrid models are flexible enough to stochastically model complex 3D morphologies, enabling the systematic exploration of different structures. Moreover, by combining stochastic and numerical simulations, the impact of morphological descriptors on macroscopic effective properties can be investigated and quantitative structure-property relationships can be established. Thus, the presented method allows for the generation of a wide spectrum of virtual 3D morphologies, that can be used for identifying structures (e.g., cathodes in Li-ion batteries) with optimized functional properties.

References

[1] B. Prifling, M. Röding, P. Townsend, M. Neumann and V. Schmidt, Large-scale statistical learning for mass transport prediction in porous materials using 90,000 artificially generated microstructures. Frontiers in Materials 8 (2021) 786502.

[2] O. Furat, L. Petrich, D. Finegan, D. Diercks, F. Usseglio-Viretta, K. Smith and V. Schmidt, Artificial generation of representative single Li-ion electrode particle architectures from microscopy data. npj Computational Materials 7 (2021) 105.

[3] S. Kench and S.J. Cooper, Generating three-dimensional structures from a two-dimensional slice with generative adversarial network-based dimensionality expansion. Nature Machine Intelligence 3 (2021) 299-305.

[4] L. Fuchs, O. Furat, D.P. Finegan, J. Allen, F.L.E. Usseglio-Viretta, B. Ozdogru, P.J. Weddle, K. Smith and V. Schmidt, Generating multi-scale NMC particles with radial grain architectures using spatial stochastics and GANs. arXiv preprint arXiv:2407.05333 (2024).

The Spin-Boson model describes the interaction between a quantum mechanical two-level system and a bosonic field. Its Hamiltonian, a self-adjoint and lower-bounded operator, is said to have a ground state if the infimum of its spectrum is an eigenvalue. Using the Feynman-Kac formula, one can express matrix elements of the semigroup generated by the Hamiltonian in terms of a self-attractive jump process. Associated with this process is a continuum FK-percolation model which randomly partitions the half-axis into intervals, with distinct intervals possibly being connected by bonds. Applying this representation, we show that, in the infrared-critical case, a phase transition occurs: as the coupling strength increases, the system transitions from having a ground state to having none. Based on recent work with Volker Betz, Benjamin Hinrichs and Mino Nicola Kraft.

A quantum particle coupled to a quantised field behaves as if it were effectively heavier than its actual mass. Enhanced binding refers to the phenomenon that due to this effective mass of the particle, the system admits a ground state, even if this is not the case for the uncoupled system. Feynman-Kac formulas allow a probabilistic interpretation of the problem: one studies Brownian motion perturbed by two attractive potentials, $$\hat{\mathbb{P}}_{\delta,\alpha,T}(\mathrm{d}x) \propto \mathrm{e}^{ \delta \int_0 ^T \mathrm{d}s V(x_s) + \alpha \int_0 ^T \int_0 ^T \mathrm{d}s \mathrm{d}t W( x_t - x_s, t-s)} 1_{B(0,R)}(x_T) \mathbb{P}(\mathrm{d}x).$$ Here, $V$ can be thought of as rewarding the path for being close to the origin and $W$ as giving rewards to the path if it stays locally where it came from. $\alpha > 0$ is a parameter which determines how much Brownian motion is "slowed down" by the pair potential $W$. The main challenge is then showing that $$\liminf\limits_{T \rightarrow \infty}\hat{\mathbb{P}}_{\delta,\alpha,T} \left( \Vert x_{T/2} \Vert \le R \right) > 0,$$ which can be interpreted as the particle localising. This can be seen to imply the existence of a ground state in the quantum system. This is joint work with Volker Betz and Mark Sellke.