Samenvatting
The discovery of Wasserstein gradient flows provided a mathematically precise formulation for the way in which thermodynamic systems are driven by entropy. Since the entropy of a system in equilibrium can be related to the large deviations of stochastic particle systems, the question arises whether a similar relation exists between Wasserstein gradient flows and stochastic particle systems. In the work presented in this thesis, such relation is studied for a number of systems. As explained in the introduction chapter, central to this research is the study of two types of large deviations for stochastic particle systems. The first type is related to the probability of the empirical measure of the particle system, after a fixed time step, conditioned on the initial empirical measure. The largedeviation rate provides a variational formulation of the transition between two macroscopic measures, in fixed time. This formulation can then be related to a discretetime formulation of a gradient flow, known as minimising movement. To this aim, a specific smalltime development of
the rate is used, based on the concept of Mosco or Gammaconvergence. The other type of large deviations concerns the probability of the trajectory of the empirical measure in a time interval. For these large deviations, the rate provides a variational formulation for the trajectory of macroscopic measures. For some systems, this rate can be related to a continuoustime formulation of gradient flows, known as an entropydissipation inequality. Chapter 2 serves as background, where a number of results from particle system theory and large deviations are proven in a general setting. In particular, the generator of the empirical measure is calculated, the manyparticle limit of the empirical measure is proven, and the discretetime large deviation principle is proven.
In Chapter 3, the discretetime largedeviation rate is studied for a system of independent
Brownian particles in a force field, which yields the FokkerPlanck equation in the manyparticle limit. Based on an estimate for the fundamental solution of the FokkerPlanck equation, it is proven that the smalltime development of the rate functional coincides with the minimising movement of the Wasserstein gradient flow, under the conjecture that a similar relation holds if there is no force field.
The FokkerPlanck equation is studied further in Chapter 4, but with a different technique. Here, an alternative formulation of the discretetime largedeviation rate is derived from the continuoustime largedeviation rate. With this alternative formulation, the smalltime development is proven in a much more general context, so that the conjecture of the previous chapter can indeed be dropped. To complete the discussion, it is mentioned that the continuoustime large deviations can be coupled to a Wasserstein gradient flow more directly, via an entropydissipation inequality. Chapter 5 discusses two related systems, whose evolutions are described by a system of reactiondiffusion equations, and by the diffusion equation with decay. In order to deal with the fact that the diffusion equation with decay is not massconserving, the decayed mass is added back to the system as decayed matter, which results in a system of reactiondiffusion equations. This allows for the construction of a system of independent particles whose probability evolves according to the reactiondiffusion equations. Similar to the previous chapters, a smalltime development of the discretetime largedeviation rate is proven. It turns out that in general the resulting functional can not be associated with a minimising movement formulation of a Wasserstein gradient flow. Nevertheless, it is proven that this functional defines a variational scheme that approximates the system of reactiondiffusion equations. As such, the functional, which involves entropy terms and
Wasserstein distances, can be interpreted as a generalisation of a minimising movement.
Chapter 6 is concerned with the effect of trivial Dirichlet boundary conditions on the discretetime rate functional and its smalltime development. Since the diffusion equation with trivial Dirichlet boundary conditions loses mass at the boundaries, the system is first transformed into a massconserving system by adding the amount of lost mass back to the system in two delta measures at the boundaries. This again allows for the construction of a corresponding microscopic particle system. For this particle system, the discretetime largedeviation rate is calculated, and its smalltime development is proven. It is still unclear whether the resulting functional can be used to define an variational approxiation scheme for the diffusion equation with Dirichlet boundary conditions.
In Chapter 7, general Markov chains on a finite state space are studied. The macroscopic equation is then a linear system of ordinary differential equations, and the microscopic particle system consists of independent Markovian particles on the finite state space. First, the discretetime rate functional for this particle system is calculated, and its smalltime asymptotic development is proven for a twostate system. Although the resulting function looks like a minimising movement, it is still unknown whether this function defines a variational formulation for the system of linear differential equations. Next, the continuoustime rate functional is calculated, at least formally, using the FengKurtz
formalism. It is then shown that the continoustime rate function can be rewritten into an entropydissipation inequality. However, this inequality may imply unphysical behaviour unless the Markov chain satisfies detailed balance. Finally, the used methods and techniques and the general results are evaluated in Chapter8. It is shown that one can expect to find a gradientflow structure via largedeviations of microscopic particle systems, if the macroscopic equation satisfies a generalised version of the detailed balance condition. Some important implications of detailed balance are
proven for both the discretetime approach, as well as for the continuoustime approach. The chapter ends with a discussion of the results for the different systems, and the general methods that are used in this thesis.
Originele taal2  Engels 

Kwalificatie  Doctor in de Filosofie 
Toekennende instantie 

Begeleider(s)/adviseur 

Datum van toekenning  21 feb 2013 
Plaats van publicatie  Eindhoven 
Uitgever  
Gedrukte ISBN's  9789038633299 
DOI's  
Status  Gepubliceerd  2013 
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Citeer dit
Renger, D. R. M. (2013). Microscopic interpretation of Wasserstein gradient flows. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR749143