Bashiri, Kaveh: Gradient Flows, Metastability and Interacting Particle Systems. - Bonn, 2020. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.

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@phdthesis{handle:20.500.11811/8415,

urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-58455,

author = {{Kaveh Bashiri}},

title = {Gradient Flows, Metastability and Interacting Particle Systems},

school = {Rheinische Friedrich-Wilhelms-Universität Bonn},

year = 2020,

month = jun,

note = {Many stochastic models exhibit a phenomenon called metastability. The first goal of this thesis is to study this phenomenon for certain classes of interacting particle systems. The second goal of this thesis is the following. Many models that are expected to exhibit metastable behaviour consist of a large number of particles. Thus, their dynamics takes place in a high-dimensional configuration space. It is then a typical idea to describe the system on the macroscopic level by introducing a macroscopic order parameter. In the case of high-dimensional diffusion systems, the empirical distribution turns out to be a suitable order parameter. Hence, the macroscopic level is given by the infinite-dimensional space of probability measures. Therefore, in order to study the macroscopic behaviour, it is useful to have the structure of a Riemannian manifold on the space of probability measure. It is known that the so-called Wasserstein formalism provides such a structure. The second goal of this thesis is to extend this Wasserstein formalism to a certain class of diffusion equations, and to use this formalism to build a rigorous bridge between the microscopic and the macroscopic level in the case of local mean-field interacting diffusions.

The outline of this thesis is as follows.

In Chapter 2 we study the metastable behaviour of three modifications of the standard, two-dimensional Ising model. The first model is an anisotropic version of the Ising model, where the interaction energy takes different values on vertical and horizontal bonds. The second model adds next-nearest-neighbour attraction to the standard Ising model. In the third model, the magnetic field is assumed to have different alternating signs on even and on odd rows.

In Chapter 3 we first establish a gradient flow representation for evolution equations that depend on a non-evolving parameter. These equations are connected to a local mean-field interacting spin system. We then use the gradient flow representation to prove a large deviation principle and a law of large numbers for the empirical process associated to this system. This is done by using the so-called Fathi-Sandier-Serfaty approach.

In Chapter 4 we consider a system of N mean-field interacting diffusions that are driven by a single-site potential of the form z↦z^4/4-z^2/2. The strength of the noise is measured by ε>0, and the strength of the interaction by J>1. Choosing the empirical mean, P, as the macroscopic order parameter, we show that the resulting macroscopic Hamiltonian admits two global minima, one at -m^*<0, and one at m^*>0. We are interested in the transition time to the hyperplane P^{-1}(m^*), when the initial configuration is close to P^{-1}(-m^*). The main result is a formula for this transition time, which is reminiscent of the celebrated Eyring-Kramers formula up to a multiplicative error term that tends to 1 as N→∞ and ε↘0. We also provide some estimates on this transition time in the case ε=1 and for a large class of single-site potentials.

In Chapter 5 we again consider the system of Chapter 4 in the case ε=1 and for a large class of single-site potentials. This time, instead of the empirical mean, we choose the empirical distribution as the order parameter. We then prove some results about the ergodicity and the basins of attraction in the macroscopic energy landscape.},

url = {http://hdl.handle.net/20.500.11811/8415}

}

urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-58455,

author = {{Kaveh Bashiri}},

title = {Gradient Flows, Metastability and Interacting Particle Systems},

school = {Rheinische Friedrich-Wilhelms-Universität Bonn},

year = 2020,

month = jun,

note = {Many stochastic models exhibit a phenomenon called metastability. The first goal of this thesis is to study this phenomenon for certain classes of interacting particle systems. The second goal of this thesis is the following. Many models that are expected to exhibit metastable behaviour consist of a large number of particles. Thus, their dynamics takes place in a high-dimensional configuration space. It is then a typical idea to describe the system on the macroscopic level by introducing a macroscopic order parameter. In the case of high-dimensional diffusion systems, the empirical distribution turns out to be a suitable order parameter. Hence, the macroscopic level is given by the infinite-dimensional space of probability measures. Therefore, in order to study the macroscopic behaviour, it is useful to have the structure of a Riemannian manifold on the space of probability measure. It is known that the so-called Wasserstein formalism provides such a structure. The second goal of this thesis is to extend this Wasserstein formalism to a certain class of diffusion equations, and to use this formalism to build a rigorous bridge between the microscopic and the macroscopic level in the case of local mean-field interacting diffusions.

The outline of this thesis is as follows.

In Chapter 2 we study the metastable behaviour of three modifications of the standard, two-dimensional Ising model. The first model is an anisotropic version of the Ising model, where the interaction energy takes different values on vertical and horizontal bonds. The second model adds next-nearest-neighbour attraction to the standard Ising model. In the third model, the magnetic field is assumed to have different alternating signs on even and on odd rows.

In Chapter 3 we first establish a gradient flow representation for evolution equations that depend on a non-evolving parameter. These equations are connected to a local mean-field interacting spin system. We then use the gradient flow representation to prove a large deviation principle and a law of large numbers for the empirical process associated to this system. This is done by using the so-called Fathi-Sandier-Serfaty approach.

In Chapter 4 we consider a system of N mean-field interacting diffusions that are driven by a single-site potential of the form z↦z^4/4-z^2/2. The strength of the noise is measured by ε>0, and the strength of the interaction by J>1. Choosing the empirical mean, P, as the macroscopic order parameter, we show that the resulting macroscopic Hamiltonian admits two global minima, one at -m^*<0, and one at m^*>0. We are interested in the transition time to the hyperplane P^{-1}(m^*), when the initial configuration is close to P^{-1}(-m^*). The main result is a formula for this transition time, which is reminiscent of the celebrated Eyring-Kramers formula up to a multiplicative error term that tends to 1 as N→∞ and ε↘0. We also provide some estimates on this transition time in the case ε=1 and for a large class of single-site potentials.

In Chapter 5 we again consider the system of Chapter 4 in the case ε=1 and for a large class of single-site potentials. This time, instead of the empirical mean, we choose the empirical distribution as the order parameter. We then prove some results about the ergodicity and the basins of attraction in the macroscopic energy landscape.},

url = {http://hdl.handle.net/20.500.11811/8415}

}