The quantitative hydrodynamic limit
In this post I explain the main result of my recent preprint:
The quantitative hydrodynamic limit of the Kawasaki dynamics. Deniz Dizdar, Georg Menz, Felix Otto, Tianqi Wu. arXiv:1807.09850.
What is a hydrodynamic limit?
The hydrodynamic limit can be seen as a dynamic version of the law of large numbers (LLN).
Oversimplifying, the LLN states that a random object, i.e. the sum \(\sum_{i=1}^N X_i\) of iid. random variables \(X_i\), converges under right scaling limit to a deterministic object, i.e. the expectation \(\mathbb{E}[X_i]\): \(\frac{1}{N} \sum_{i=1}^N X_i \to \mathbb{E}[X_1]\) as \(N \to \infty\).
In the hydrodynamic limit, the situation is similar. A random object, i.e. a certain microscopic evolution \(X_t\) usually described by a stochastic differential equation, converges under the right scaling limit to a deterministic object, usually described by the solution \(\zeta_t\) of a partial differential equation.
Let us turn to the specifics of our model. We start with considering the microscopic dynamic \(X_t \in \mathbb{R}^N\) which is given by the so-called Kawasaki dynamic.
(1) \(dX_t = - A \nabla H (X_t) dt + \sqrt{2A} dB_t\).
For the details of the meaning of (1) let me refer to the above mentioned article. At the moment, it is sufficient to know that the Kawasaki dynamic describes an evolution of lattice system. The lattice is given by \(\Lambda:= \left\{ 1, \ldots, N \right\} \subset \mathbb{Z}\). At each site $i \in \Lambda$ there is attached a real-valued spin~\(x_i \in \mathbb{R}\). The state space of the system is therefore \(\mathbb{R}^N\). If the spins are \(\left\{ 0,1 \right\}\)-valued one would recover the Ising model. The energy of a state \(x \in \mathbb{R}^N\) is given by the Hamiltonian \(H: \mathbb{R}^N \to \mathbb{R}\). In our article there are no interactions between sites in the Hamiltonian
The first term on the right hand side of (1) causes the Kawasaki dynamic \(X_t\) to minimize the energy \(H(X_t)\) under presence of thermal noise which is given by the second term on the right hand side of (1). The term \(B_t\) denotes a \(N\)-dimensional standard Brownian motion. The matrix \(A\) is the second order difference operator on \(\Lambda\) with periodic boundary conditions. It causes that the Kawasaki dynamic conserves the mean spin of the system:
\(\frac{1}{N} \sum_{i=1}^N (X_t)_i = \frac{1}{N} \sum_{i=1}^N (X_0)_i = \mbox{const}\).
By the matrix \(A\) the Kawasaki dynamic is a "spin-exchange" dynamic. If one chooses \(A= \mbox{Id}\) as the identity matrix one recovers the Glauber dynamics, which is a "spin-flip" dynamic.
Here is a simulation of the Kawasaki dynamics without thermal noise:
Notice that the evolution tends to minimize the energy of the system and stays there.
This changes when adding thermal noise. Then, the dynamics rushes to the next energetic minimum and wiggles around that state.
Here, is a simulation of the full Kawasaki dynamics on a larger system:
Let us now turn to the macroscopic dynamics \(\zeta_t\). It is given by the solution of a non-linear heat equation:
(2) \(\frac{d}{dt} \zeta_t = \Delta \varphi'(\zeta_t)\).
The nonlinearity \(\varphi\) is determined by the Hamiltonian \(H\) of the microscopic system via the local Cramér theorem.
Deriving the hydrodynamic limit on a qualitative level is a classical topic and well understood. The main innovation in our work is that we deduce quantitative error estimates, which seems to be the first time in the literature. Quantitative statements are a lot more powerful and of fundamental importance for applications. Unfortunately, they are also a lot more difficult to deduce. Only quantitative statements allow to deduce error estimates, predict confidence intervals and estimate how long a computer has to run calculations.
The main result is the following explicit error estimate:
(3) \(\sup_{0 \leq t \leq T} \mathbb{E} \left[ |X_t - \zeta_t |_{H^{-1}}^2 \right] \leq C \mathbb{E} \left[ |X_0- \zeta_0 |_{H^{-1}}^2 \right] + \frac{C}{N^{\frac{2}{3}}} (1+T)\).
More comments on the estimate (3) will be given later.