# Stochastic Evolution of Augmented Born–Infeld Equations

## Abstract

This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born–Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases, which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born–Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two appendices treat the Hamiltonian structures underlying these results.

## Keywords

Geometric mechanics Stochastic processes Uncertainty quantification Fluid dynamics Electromagnetic fields## Mathematics Subject Classification

37H10 37J15 60H10## 1 Introduction

Physics is an observational science. Hence, one may be led to consider how the modern stochastic methods for uncertainty quantification and data assimilation currently being developed for large-scale observational sciences such as weather forecasting and climate change might be applied in foundational classical physics models, such as Euler’s fluid vorticity equations and Maxwell’s electromagnetic field equations. One might also wonder what mathematical differences may arise in the approaches for quantifying uncertainty in two such different foundational models, one concerning swirling fluids and the other concerning electromagnetic waves propagating in a vacuum. In addition, one might wonder about the role of mathematical structure in the formulation of stochastic methods of uncertainty quantification for two such different models.

We will address these questions here by comparing the stochastic equations developed for quantifying uncertainty in the nonlinear dynamics of the ideal Euler fluid equations with the corresponding stochastic equations for the Born–Infeld electromagnetic field equations. Of course, the physics of these two models is fundamentally different. The deterministic Born–Infeld model arose in quantum field theory (QFT) and is closely related to string theory. Its QFT origins are discussed, for example, in Bialynicki-Birula (1983, 1992), and its relationship to string theory is discussed in Polchinski (1998) and Gibbons (2001). Its analytical properties have been reviewed recently in Kiessling (2011). Its parallels with hydrodynamics are discussed in Arik et al. (1989), Brenier (2002, 2004) and Brenier and Yong (2005).

The theories being discussed here all share the same conceptual framework. Born–Infeld electromagnetism, string theory and ideal fluid dynamics are all Hamiltonian theories, whose symmetries enable reduction to variables that are invariant under a Lie group. In addition, the Born–Infeld field equations imply augmented equations for energy and momentum conservation that are reminiscent of conservation laws in fluid dynamics, as shown, for example, in Arik et al. (1989) and Brenier (2002).

The shared Hamiltonian structure of the *augmented Born–Infeld* (ABI) equations and ideal fluid dynamics will put the Hamiltonian approach discussed here for introducing stochastic uncertainty quantification methods into a common framework. Here, we will concentrate on introducing stochasticity by using the parallels between the Born–Infeld field equations and hydrodynamics. As a result, the stochastic version will turn out to conserve the deterministic Born–Infeld energy. The preservation of other analytical properties of the Born–Infeld field equations under the addition of this type of stochasticity will be explored elsewhere. The possibilities for applying this structure-preserving stochastic Hamiltonian approach to estimating the unknown effects of unobserved degrees of freedom and quantifying uncertainty in string theory will also be explored elsewhere.

The aim of this paper is to determine how the association of the Born–Infeld equations for nonlinear electromagnetism with their augmented hydrodynamic counterparts discussed in Brenier (2002) will inform us about how to add noise to the evolution of the Born–Infeld displacement flux and magnetic flux. The approach will rely on via the variational and Hamiltonian method introduced for hydrodynamics in Holm (2015), which adds noise geometrically, by a canonical transformation. In particular, stochasticity will be introduced by making the electromagnetic flux fields \({\mathbf {D} }\) and \({\mathbf {B} }\) evolve under a Stratonovich stochastic flow, to be transported by stochastic vector fields carrying spatial statistical correlation information, via the Lie-derivative operation of vector fields on two forms \({\mathbf {D} }\) and \({\mathbf {B} }\). The transport obtained via the action of these spatially correlated stochastic vector fields will be implemented as a canonical transformation generated by a familiar momentum map from the electromagnetic field variables to the hydrodynamics variables; namely, the Poynting vector, \({\mathbf {P} }:={\mathbf {D} }\times {\mathbf {B} }\).

This paper compares the effects on the equations of motion of introducing stochasticity as cylindrical Stratonovich noise (Bismut 1981) into the Hamiltonian formulations of either Euler’s equations for fluid vorticity, or the Born–Infeld electromagnetic field equations (Born and Infeld 1934). This may seem like an unlikely comparison. However, because of an intriguing hydrodynamic analogue for electromagnetic waves, the comparison turns out to be closer than one might have thought at first glance.

*fixed in space*, while, in contrast, \(\mathrm{d}{\mathbf {S} }\) and \(\mathrm{d}{\mathbf {x} }\) may be interpreted as

*moving with the fluid*in Eq. (1.1), because of the Lie derivative operation, \(\mathcal {L} _{u}\).

Comparisons between Euler’s fluid equations and Maxwell’s field equations have been an intriguing issue in the physics literature ever since the mid-nineteenth century. For a recent historical survey of these comparisons, see Siegel (1991). For further mathematical relations between hydrodynamics and the Born–Infeld model, see Brenier (2004) and Brenier and Yong (2005). For an in-depth, special relativistic treatment which includes interactions of Maxwell fields with fluid dynamics, see Holm (1987).

The present paper will offer yet another aspect of these comparisons, by investigating how the introduction of stochasticity, representing various types of uncertainty, will affect the evolutionary operators in the two sets of equations.

Euler’s fluid equations and Maxwell’s field equations both arise via Hamiltonian reduction by symmetry. On the one hand, Eulerian fluid dynamics possesses relabelling symmetry, which allows reduction by symmetry in transforming from Lagrangian to Eulerian fluid variables. Namely, the Eulerian fluid variables are invariant under relabelling of the Lagrangian fluid particles. On the other hand, classical electromagnetic theory possesses gauge symmetry, which allows reduction by transforming from the potentials to the fields, the latter being invariant under gauge transformations of the potentials (Weyl symmetry). The Hamiltonian structures resulting from these two types of symmetry reduction are quite different. However, they each result in a map from canonical field variables to a momentum variable taking values in the space of 1-form densities, dual to vector fields with respect to \(L^2\) pairing. This property of sharing a *momentum map* from the canonical field variables to a momentum density will provide an avenue for introducing a stochastic vector field into both models by using their shared Hamiltonian structure.

### 1.1 Plan

In the remainder of the paper, Sect. 2 sketches the method of Holm (2015) for introducing Stratonovich noise into Hamiltonian dynamics of nonlinear field theories by using momentum maps dual to vector fields. This approach is illustrated by comparing its results for two apparently different theories, namely ideal fluid dynamics in Sect. 2 and Born–Infeld electromagnetism in Sect. 3. The main part of the paper concludes and summarise the results in Sect. 4. Lagrangian and Hamiltonian variational formulations of ABI are provided in Appendix A. The corresponding results for the high-field MHD limit of the ABI equations are discussed in Appendix B.

The fundamentals of the Hamiltonian structures for the two theories are reviewed from first principles for fluid dynamics in Holm et al. (1998) and for the Born–Infeld theory in Appendices A and B. These appendices derive the connection between the Hamiltonian structures for the Born–Infeld equations (canonical Poisson bracket) and ideal hydrodynamics (Lie–Poisson bracket). Appendix A does this in general, and Appendix B discusses the high-field limit. A connection to magnetohydrodynamics is also revealed in the high-field limit discussed in Appendix B.

The Poynting vector momentum map is the key to understanding the analogy between electromagnetism and hydrodynamics. To augment the Hamiltonian operator for Born–Infeld electromagnetism to include the Poynting vector, we follow a mathematical approach introduced in Krishnaprasad and Marsden (1987) for deriving the dynamics of a rigid body with flexible attachments. This approach leads to a compound Poisson structure that may be written as the sum of a canonical structure and Lie–Poisson structure obtained from a cotangent lift momentum map in which variations are applied independently. This augmented Poisson bracket for the Born–Infeld theory provides a fundamental explanation of the hydrodynamic analogy for electromagnetism. Namely, the cotangent lift momentum map to the Poynting vector, corresponding to Lie transformations of the canonical electromagnetic variables by smooth vector fields, leads via Krishnaprasad and Marsden (1987) to an augmented Poisson bracket for electromagnetism which satisfies the same semidirect-product Lie–Poisson bracket relations as those found in ideal continuum dynamics (Holm et al. 1998).

## 2 Stochastic Fluid Dynamics

*Stratonovich*stochastic, divergence-free vector field,

For in-depth treatments of cylindrical noise, see Pardoux (2007) and Schaumlöffel (1988). In our case, the \({\varvec{\xi } }_{i}({{\mathbf {y} }}_{t})\), \(i=1,2,\dots ,N\), appearing in the stochastic vector field in (2.2) comprise *N* prescribed, time-independent, divergence-free vectors which ideally may be obtained from data measured at fixed points \({\mathbf {x} }\) along the Lagrangian path \({{\mathbf {y} }}_{t}\). For example, one may take the \({\varvec{\xi } }_{i}({\mathbf {x} })\) to be Empirical Orthogonal Functions (EOFs), which are eigenvectors of the velocity–velocity correlation tensor for a certain measured flow with stationary statistics (Hannachi et al. 2007). The \({\varvec{\xi } }_{i}({\mathbf {x} })\) may also be obtained numerically by comparisons of Lagrangian trajectories at fine and coarse space and timescales (Cotter et al. 2018a, b).

It may not be surprising that the variational introduction of cylindrical Stratonovich noise into Euler’s fluid equation proposed in Holm (2015) for fluids has simply introduced an additional, stochastic vector field \(\sum _{i} {\varvec{\xi } }_{i}({\mathbf {x} }) \circ \mathrm{d}W^{i}_{t}\) into Eq. (2.2), which augments the Lie transport in Eq. (2.1) in the Eulerian representation, while preserving its Hamiltonian geometric structure and many of its analytical properties. After all, the essence of Euler fluid dynamics is Lie transport (Holm et al. 1998). However, it might be more surprising if the variational introduction of noise into the Born–Infeld electromagnetic field equations turned out to introduce the same sort of stochastic Lie transport, for example, in the displacement current. Investigating this issue and explaining it will be our concern for the remainder of the paper.

## 3 Augmented Born–Infeld (ABI) Equations

### 3.1 Deterministic Born–Infeld Equations

### Remark 1

*(Conservation laws*, (Brenier 2004

*)*The dynamical equations for the energy density \({\mathcal {H} }\) and the Poynting vector \({\mathbf {P} }\) (momentum density, also energy flux density) may be written in conservative form, as

*augmented Born–Infeld (ABI) equations*in Brenier (2004) and Brenier and Yong (2005).

### Remark 2

*(Hydrodynamic analogy)*From their equations in (3.8), we see that \({\mathcal {H} }\) is a scalar density, while \({\mathbf {P} }\) is a 1-form density. Thus, we may write these equations in a more geometric form, reminiscent of hydrodynamics, upon introducing the vector notation \({\mathbf {v} }={\mathbf {P} }/{\mathcal {H} }\), \({\varvec{\gamma } }:= {\mathbf {D} }/{\mathcal {H} }\) and \({\varvec{\beta } }:= {\mathbf {B} }/{\mathcal {H} }\). Equation (3.8) then may be written equivalently as

*v*, whose components are given by \(v={\mathbf {v} }\cdot \nabla =v^j\partial _j\). Namely,

*c*(

*v*) moving with the velocity \({\mathbf {v} }({\mathbf {x} },t)\), to obtain

### Remark 3

The Born–Infeld equations (3.1) comprise a nonlinear deformation of Maxwell’s equations. Indeed, Maxwell’s equations may be recovered from the variational equations (3.5) for appropriately small magnitudes \(|{\mathbf {D} }|\ll 1\) and \(|{\mathbf {B} }|\ll 1\), for which \({\delta {\mathbb {H} }} /{\delta {\mathbf {D} }} \rightarrow {\mathbf {D} }\) and \({\delta {\mathbb {H} }}/{\delta {\mathbf {B} }}\rightarrow {\mathbf {B} }\). Conservation equations analogous to (3.8) and circulation equations analogous to (3.11) and (3.12) also hold for the Maxwell case.

To provide a geometric motivation for the fluid dynamics interpretation via the circulation theorem (3.11) for the deterministic augmented Born–Infeld equations in (3.8), we introduce a bit of standard terminology from geometric mechanics.

### Definition 4

*G*is a Lie group which acts on a configuration manifold

*Q*and, hence, on its canonical phase space \(T^*Q\) by cotangent lifts. The corresponding momentum map

*J*(

*q*,

*p*) from the canonical phase space \(T^*Q\) to the dual \(\mathfrak {g}^*\) of the Lie algebra \(\mathfrak {g}\) of Lie group

*G*is given by Marsden and Ratiu (1994) and Holm (2011)

*q*, \(\Phi _\xi (q)\) the infinitesimal transformation of

*q*by

*G*and natural pairings \(\langle \,\cdot ,\cdot \,\rangle _{\mathfrak {g}}: \mathfrak {g}^*\times \mathfrak {g} \rightarrow \mathbb {R}\) and \(\langle \langle \,\cdot ,\cdot \,\rangle \rangle _{TQ}: T^*Q\times TQ \rightarrow \mathbb {R}\).

In the Born–Infeld case, we replace the Lie group *G* in the definition above by the diffeomorphisms \(\mathrm{Diff}(\mathbb {R}^3)\). We then define the canonical phase space \(T^*\mathcal {D}\) as the set of pairs \(({\mathbf {D} },{\mathbf {A} })\) whose canonical Poisson bracket is given in (3.3) and take the infinitesimal transformation \(\Phi _\xi (q)\) to be \(-\mathcal {L}_\xi D\), namely (minus) the Lie derivative of the closed 2-forms \(D=({\mathbf {D} }\cdot \mathrm{d}{\mathbf {S} })\) by the divergence-free vector fields, \(\xi :={\varvec{\xi } }\cdot \nabla \in \mathfrak {X}(\mathbb {R}^3)\) with \(\mathrm{div}{\varvec{\xi } }({\mathbf {x} })=0\). In terms of these variables, we find the following.

### Theorem 5

The 1-form density \(P:={\mathbf {P} }\cdot \mathrm{d}{\mathbf {x} }\otimes d^3x\) with components given by the Poynting vector \({\mathbf {P} }={\mathbf {D} }\times {\mathbf {B} }\) defines a cotangent lift momentum map, \(T^*\mathcal {D} \rightarrow \mathfrak {X}^*\), from the canonical phase space \(T^*\mathcal {D}\) identified with the set of pairs \(({\mathbf {D} },{\mathbf {A} })\) whose canonical Poisson bracket is given in (3.1), to the dual space \(\mathfrak {X}^*(\mathbb {R}^3)\) of the smooth vector fields \(\mathfrak {X}(\mathbb {R}^3)\) with respect to the \(L^2\) pairing.

### Proof

### Remark 6

### 3.2 Stochastic Born–Infeld Equations

*N*Brownian motions \(\mathrm{d}W^i_t\), \(i=1,2,\dots ,N\), each interacting with the dynamical variables \(({\mathbf {D} },{\mathbf {B} })\) via its own Hamiltonian amplitude, \(h_i({\mathbf {D} },{\mathbf {B} })\) (Bismut 1981).

### Remark 7

Notice that the inclusion of the stochastic term (3.19) into the Hamiltonian density in (3.18) has introduced explicit dependence on time and space coordinates in the total Hamiltonian. Consequently, the conservation laws in (3.8) for the deterministic Born–Infeld total energy \(\int {\mathcal {H} }d^3x\) and total momentum \(\int {\mathbf {P} }d^3x\) may no longer apply in the stochastic case; see, however, Remark 11. Moreover, the stochastic Born–Infeld equations in (3.20) are no longer Lorentz invariant, although this was to be expected, because loss of explicit Lorentz invariance arises, in general, when casting Lorentz invariant dynamics into the Hamiltonian formalism.

### Remark 8

The stochastic part of the Born–Infeld Hamiltonian density in (3.19) is the integrand in the first line of (3.14). Thus, the Stratonovich noise in (3.18) has been coupled to the deterministic Born–Infeld field theory through the momentum map in (3.14) corresponding to the Poynting vector, \(P=A\diamond D = {\mathbf {D} }\times {\mathbf {B} }\cdot \mathrm{d}{\mathbf {x} }\otimes d^3x\), which is a 1-form density.

#### 3.2.1 Stratonovich Form

#### 3.2.2 Itô Form

When dealing with cylindrical noise, the spatial coordinates are treated merely as parameters. That is, one may regard the cylindrical noise process as a finite-dimensional stochastic process parametrized by \({\mathbf {x} }\) (the spatial coordinates). In this regard, the Stratonovich equation makes analytical sense pointwise, for each fixed \({\mathbf {x} }\). Once this is agreed, then the transformation to Itô by the standard method also makes sense pointwise in space. For more details, see Pardoux (2007) and Schaumlöffel (1988).

### Remark 9

*(Application to the stochastic Maxwell equations in the weak-field limit)*The reduction of the stochastic Born–Infeld equations in (3.20) and (3.25) to the linear stochastic Maxwell equations occurs in the weak-field limit via replacing the Born–Infeld Hamiltonian density \({\mathcal {H} }\) in (3.4) by the Maxwell energy density

## 4 Conclusion

In conclusion, we have seen that the association of the Born–Infeld equations (3.1) with their augmented hydrodynamic counterparts in (3.9) has informed us via the method introduced for hydrodynamics in Holm (2015) how to add noise to the evolution of the Born–Infeld displacement flux and magnetic flux, so as to preserve its energy \({\mathcal {H} }\) in (3.9) as well as to implement the addition of noise geometrically as a process of moving into a stochastic frame of motion. Namely, the flux fields \({\mathbf {D} }\) and \({\mathbf {B} }\) evolve under a Stratonovich stochastic *flow*, being Lie-transported by a sum of stochastic vector fields \(\xi _i({\mathbf {x} })\) carrying spatial correlation information, via the Lie-derivative operation \(\sum _i {\mathcal {L} }_{\xi _i \circ \mathrm{d}W^i_t}(\cdot )\).

Section 2 concluded via a variational approach to stochastic fluid dynamics that noise enhanced Lie transport of the fluid vorticity in Eq. (2.2). For fluids, this conclusion seemed intuitive. On the other hand, it seemed less intuitive to find in Sect. 3 that noise would induce Lie transport of the electromagnetic fields \({\mathbf {D} }\) and \({\mathbf {B} }\) as 2-forms along random paths generated by the vector fields associated with the spatial correlations of the stochasticity. The correspondence between vorticity 2-forms and electromagnetic flux 2-forms has attracted attention at least since Maxwell and W. Thompson. It turned out that the hydrodynamic analogue equations derived in (3.9) provided the key to understanding why the introduction of noise into the Born–Infeld equations should appear as stochastic Lie transport. The hydrodynamic analogue for the Born–Infeld equations is summarised in the following theorem.

### Theorem 10

### Remark 11

The first equation in (4.1) of Theorem 10 implies that the Born–Infeld energy Hamiltonian \({\mathbb {H} }\) in (3.4) for the deterministic augmented Born–Infeld equations (3.7) *remains conserved* after introducing the stochastic Hamiltonian density in (3.19) which pairs the noise with the Poynting vector.

The second equation in (4.1) of Theorem 10 shows that the hydrodynamic analogy of the ABI equations via the Kelvin circulation theorem for the deterministic case in (3.11) persists for the stochasticity introduced here.

### Proof

### Remark 12

Inserting the second equation in (4.3) into the second equation in (4.2) implies that conservation of total momentum \(\int {\mathbf {P} }\,d^3x\) obtained in Eq. (3.8) for the deterministic ABI *does not persist* in the stochastic case, unless \(\partial _k \xi ^j =0\), i.e., unless the amplitude of the noise is constant.

### Remark 13

*(Fluid interpretation of stochastic augmented Born–Infeld equations*(3.22) & (4.5)

*)*The stochastic ABI equations may be expressed in the same Lie–Poisson Hamiltonian form as for the deterministic case, which is introduced Appendix A, as

### 4.1 The Role of the Momentum Map

So, why did the introduction of noise in both fluid dynamics and the Born–Infeld equations result in the same effect of introducing stochastic Lie transport into the motion equations for both particles and fields? The answer lies in the momentum map which relates the Hamiltonian structures of the two theories. Namely, although their Poisson brackets are different, the two theories are related by the hydrodynamics analogy in (3.9) comprising the momentum map given by the definition of the Poynting vector \({\mathbf {P} }={\mathbf {D} }\times {\mathbf {B} }\), regarded as a momentum 1-form density.

Indeed, as we show in Appendix A, the Born–Infeld equations (3.7) for the \({\mathbf {B} }\) and \({\mathbf {D} }\) fields, augmented by the local conservation laws (3.8) for the Poynting momentum \({\mathbf {P} }\), together comprise Hamiltonian dynamics on a Poisson manifold \(\mathfrak {X}^*(\mathbb {R}^3)\times T^*\mathcal {C}^\infty (\mathbb {R}^3)\), whose Poisson structure is given by the sum of the canonical Poisson bracket for the electromagnetic fields, plus a Lie–Poisson bracket which is dual in the sense of \(L^2\) pairing to the semidirect-product Lie algebra \(\mathfrak {X}\,\circledS \,(\Lambda ^1\otimes \Lambda ^1)\), with dual coordinates \(P\in \mathfrak {X}^*\), \(B\in \Lambda ^2\) and \(D\in \Lambda ^2\). This sum of a canonical Poisson bracket and a semidirect-product Lie–Poisson bracket derives from the definitions of the magnetic field flux \(B:=dA\) and the cotangent lift momentum map \(P=D\diamond A\) in (3.14). We refer to such augmented Poisson structures as KM brackets, after Krishnaprasad and Marsden (1987). The KM bracket for ABI derived in Appendix A reveals why the introduction of stochasticity by adding to the deterministic Hamiltonian the stochastic term \(\langle P,\xi (x) \rangle \circ \mathrm{d}W_t\) in Eq. (3.21) simply introduces a Lie derivative stochastic transport term in the resulting SPDE. In particular, introducing stochasticity in the transport of the Poynting vector momentum density in the ABI corresponds to introducing stochastic transport in both the displacement current and magnetic induction rate for the original Born–Infeld field equations.

### 4.2 Applications

For both fluids and electromagnetic fields, the modified Hamiltonian which added the variational noise contribution was constructed by pairing eigenvector fields ostensibly describing the spatial correlations of the stochasticity in the data as cylindrical noise, with the momentum 1-form density variable for either the fluid or the fields. The infinitesimal Hamiltonian flow generated by the \(L^2\) pairing of the momentum 1-form density with the sum of Stratonovich cylindrical noise terms with eigenvector fields derived from the spatial correlations of the data in both cases turned out to be an infinitesimal stochastic diffeomorphism. In fact, the infinitesimal map which resulted from the original Poisson structure was a sum of Lie derivatives with respect to the correlation eigenvector fields \({\varvec{\xi } }_i({\mathbf {x} })\) for each component of the cylindrical noise.

Thus, we conclude that uncertainty quantification for Hamiltonian systems can be based on stochastic Hamiltonian flows that are obtained from coupling the momentum map for the deterministic system with the sum over stochastic Stratonovich Brownian motions, formulated as cylindrical noise terms for each fixed spatial correlation eigenvector, as determined from the data being simulated.

The application of the stochastic fluid dynamics treated here for quantifying uncertainty will depend crucially on determining the spatial correlations of the cylindrical noise represented by the eigenvectors \({\varvec{\xi } }_i({\mathbf {x} })\) in Theorem 10. Extensive examples in fluid dynamics of how to obtain the spatial structure \({\varvec{\xi } }_i({\mathbf {x} })\) of the cylindrical noise and thereby quantify uncertainty in low-resolution numerical simulations by comparing them to high-resolution results for the same problem are given in Cotter et al. (2018a) and Cotter et al. (2018b) for fluid dynamics simulations. These results should also be helpful examples for applying similar methods to the Born–Infeld equations.

## Notes

### Acknowledgements

I am grateful to Y. Brenier for illuminating discussions of the Born–Infeld model. I am grateful to D. O. Crisan, V. Putkaradze, T. S. Ratiu and C. Tronci, for encouraging and incisive remarks in the course of this work. I’d also like to thank the anonymous referees for their constructive suggestions. Finally, the author is also grateful to be partially supported by the European Research Council Advanced Grant 267382 FCCA and EPSRC Standard Grant EP/N023781/1.

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