Space time correlation function and correlation length 

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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.

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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 

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