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## Numerical integration of Hamiltonian mechanical systems

The methods for integrating field theories are widely inspired from the one of mechanics, and most of the observations valid in the latter case will remain valid. The aim of this section is to give an overview of the existing methods for numerically solving conservative ordinary differential equations (odes); keeping in mind our final goal, we shall carefully analyse if the methods can correctly handle the long-time dynamics of the problem. There is a broad variety of methods to tackle odes numerically, but, as already stated, here we focus on finite-difference methods. Besides, this presentation is not intended to be exhaustive; our objective is to bring out some general concepts, and we only introduce a restricted selection of the most known technics. This presentation is inspired from [75,79,80,91] where one can find a complete review on numerical methods in general and, in particular, on the subject of numerical integration of odes. First, two mechanical systems are introduced to be used as application examples. Secondly, we present the construction, as well as a proof of the main properties of some of the most renowned methods. Next, their strengths and weaknesses will be analysed, as well as their accuracy, on the two cases mentioned just above.

### A selection of other well-known methods

As mentioned earlier, there exist many numerical methods to integrate odes [75, 79, 80,91]. In the present section, we shall discuss two noteworthy classes of methods that can be constructed in such a way to be symplectic. On the one hand, the methods of the Runge – Kutta type and, on the other hand, the methods constructed from the Yoshida expansion of the evolution operator [70]. The aim of this section if only to give a sketch of these methods; it is absolutely not intended to be exhaustive. There are two reasons for that: firstly, because the class of the Runge – Kutta integrators is a wide subject, whose discussion should require an entire book. Secondly, because these two kinds of methods are not suitable for a generalization to Hamiltonian pdes. In fact they can be generalised to Hamiltonian pdes: many papers treat of Runge – Kutta high-order multi-symplectic integrators, and it would be possible to generalise the Yoshida expansion as well. However, this generalization is at the cost of treating space and time in a different way, manifestly breaking the covariance of the theory.

The importance of the covariance will be presented in the next sections but we shall carefully pay attention, and put a lot of efforts not to break this symmetry. Hence, treating space and time differently is not a satisfying discretisation approach.

#### The De Donder – Weyl Hamiltonian formulation of field theories

This section introduces the concept of multi-symplecticity through the framework of the De Donder – Weyl Hamiltonian formulation of field theories [19]. We first introduce the De Donder – Weyl formalism and the concept of Hamiltonian pdes. Next, we prove that the DW definition of the phase space is a multi-symplectic manifold, and we establish the conservation of the multi-symplectic structure under Hamiltonian flow. Afterwards, we discuss a possible issue in this fundamental structure and we present a way to address it. Finally, we shall discuss the definition and properties of the stress-energy tensor. This section is treated both in a general setup and with the example of the non-linear wave equation.

Most of the points presented here are just reminders except for two of them. Firstly, the link between the De Donder – Weyl formulation and multi-symplectic geometry is not so common (usually, the De Donder – Weyl formulation is treated through the formalism of the poly-symplectic geometry, see footnote 1 page 48). Secondly, as far as we know, the discussion of the degeneracy of the multi-symplectic structure, and especially its resolution in any dimension, is completely new. In fact, this issue was already discussed in [42] for the particular case of the non-linear wave equation in 1+1 dimensions. However, the argument they propose in this paper does not enable any generalisation to higher dimensions; the new argument we introduce here naturally extends in any dimension.

The De Donder – Weyl (DW) Hamiltonian formalism is a broad topic that is still not clearly understood; once again this presentation is not intended to be exhaustive. See [18–37] for further details.

**From Lagrangian to DW Hamiltonian formulation **

Let us start with a space-time Lorentzian manifold (a manifold endowed with a Lorentzian metric), M, of dimension D = 1 + d. We assume M to be non-dynamic Numerical integration of classical conservative field theories (ie the metric is not subject to an equation of motion) and flat1 with metric : diag (1;1; ;1). We parameterise M by the local coordinate system fxg, with @ := @=@x a basis of TM, and where 2 J0; dK. Next, we consider a field theory onM, described by the action S , where i is a collection of dynamic fields with i 2 J1;NK. This action originates in a Lagrangian density, L, which is assumed to depend only on the field and its first derivatives: S := Z dDx L i; @i .

**Table of contents :**

**I Numerical integration of classical conservative field theories **

1 Introduction and preliminaries

1.1 Introduction

1.2 Geometry preliminaries

Tangent space: vectors

Dual space

Cotangent space: co-vectors

Tensor product: tensors and p-forms

Metric and dual vectors

Gradient and exterior derivative

Interior product and Lie derivative

1.3 Reminders of Hamiltonian mechanics

Lagrangian, action and Euler – Lagrange equation of motion

Hamiltonian formulation

Phase space and symplectic structure

Poisson bracket

Canonical transformations: symplectic geometry

Liouville’s theore

Time-dependent Hamiltonian

1.4 Numerical integration of Hamiltonian mechanical systems

The harmonic oscillator and the simple pendulum

The Euler’s methods

Partitioned Euler

A selection of other well-known methods

Numerical results

**2 The De Donder – Weyl Hamiltonian formulation of field theories **

2.1 From Lagrangian to DW Hamiltonian formulation

Generic field theory

The non-linear wave equation example

2.2 Multi-symplectic structure

The multi-symplectic structure

Degeneracy

Conservation of the multi-symplectic structure

2.3 Stress-energy tensor and charges

The stress-energy tensor

Charges

2.4 Summary

**3 Multi-symplectic integrators **

3.1 Preliminaries

The (partitioned) Euler method

The centred box scheme

3.2 The lattice: sampling the space-time manifold

3.3 The numerical approximation scheme

Definition

Application to the 4 theory in 0 + 1 dimension

The 4 theory in 1 + 1 dimensions

3.4 Conservation properties

Leibniz’s product rule for quadratic forms

Preservation of Schwarz’s theorem

Exact conservation of the multi-symplectic structure

3.5 Conservation of the stress-energy tensor

Local approximate conservation of the stress-energy tensor

The 4 theory in 1 + 1 dimensions

Note on the possibility of an exact conservation of the stress-energy tensor

3.6 Motivation to use the light-cone coordinates

3.7 Alternative discretisation in dimensions higher than 1 + 1

**4 Application: the 4 theory in 1 + 1 dimensions **

4.1 The 4 theory in 1 + 1 dimensions

The equation of motion

Boundary and initial conditions

The stress-energy tensor, its conservation and the charges

The testing conditions

4.2 The Euler method

Sampling the space-time manifold

The Euler scheme

The energy and the stress-energy tensor

Energy conservation

4.3 The Boyanovsky – Destri – de Vega (BDdV) method

The lattice

Exact energy preserving approximation

The stress-energy tensor

Energy conservation

4.4 The msilcc method: a short review of properties

4.5 Numerical results

Influence of the non-linearity

long-time behaviour

Conclusion

Symmetry breaking potential

**5 Conclusio**

**II Critical percolation in ferromagnetic Ising spin models **

**1 Introduction and preliminaries**

1.1 Introduction

1.2 Some reminders of site percolation

1.3 Criticality in Schramm — Loewner evolution

**2 Ising models and observables **

2.1 Ising models

The bi-dimensional ferromagnetic Ising model

Glauber dynamics – kinetic Ising model

2.2 Observables

Space time correlation function and correlation length

Variance of the winding angle

Average occupancy rate

**3 Equilibrium behaviour **

**4 Instantaneous quenches**

4.1 Quench to T = 0

4.2 Quench to T = Tc

**5 Effects of a finite cooling rate **

5.1 The Kibble – Zurek mechanism

5.2 The out-of-equilibrium dynamics

5.3 behaviour at the Ising critical point

5.4 Dynamics before reaching the critical point

**6 Conclusions **

**Bibliography **