# Computation of the generating functional using the Martin-Siggia-Rose formalism

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## Wetting of disordered susbtrates

When a liquid is dropped on a solid substrate it can either spread and form a film that covers the solid (total wetting) or form a meniscus (partial wetting). In the latter case, the line where the liquid surface meets the solid is called contact line, wetting front or triple line. It has been shown independently by Joanny and de Gennes  and by Pomeau and Vannimenus  that the contact line can be described as an elastic line with long-range elasticity arising from the surface tension of the meniscus. In this section I reproduce the computation from Joanny and de Gennes of the elastic energy associated to a small deformation of the line, as it is useful to understand the assumptions under which the form of the elastic energy holds (this has also been done in the PhD Thesis ). Then I present an experiment and discuss results about the roughness exponent.

### Derivation of the elastic energy of a contact line

For partial wetting the meniscus is characterized by the contact angle between the surface of the solid and the liquid/gaz interface (see figure 2.6 left). The contact line is submitted to forces coming from the surface tensions SG, SL and LG of the solid/gaz, solid/liquid and liquid/gaz interfaces respectively. At equilibrium the resulting component of the total force parallel to the solid surface must be nul. This gives the Young-Dupr´e relation : LG cos() = SG − SL .

#### Elastoplasticity and the yielding transition

In the depinning transition, an elastic interface starts to flow above a finite force threshold fc. A behavior that is similar in many respects is shared by some amorphous solids. In contrast to crystalline solids, amorphous solids have no internal structure. Typical exemples of such materials are foams, mayonnaise or whipped cream. When left at rest they are solids : they do not flow and keep their initial shape. But one can make them flow easily by applying a sufficiently large shear stress, in which case they behave more like liquids. See  for a recent review on the deformation and flow of amorphous solids. As for the depinning the onset of the flow arises at a finite value of the stress, called yield stress because the solid yields. These materials are named yield stress solids and the transition from solid to liquid behaviour is called the yielding transition.
The deformation of the solid under the shear stress occurs through local shear transformations (also named plastic events) where the particles rearrange locally. This rearrangement reduces the local stress which is redistributed elastically in the rest of the material. The redistributed stress can trigger new plastic events, possibly leading to an avalanche of plastic events and a macroscopic deformation of the material. Formally the local deformation from the initial configuration is measured by the local plastic strain (x), where x 2 Rd is the internal coordinate of the solid. It can be seen as a d-dimensional interface in d+1 dimension, as illustrated in figure 2.9. As the depinning, the yielding transition is characterized by a set of critical exponents characterizing a diverging length scale, the avalanche size distribution or the strain rate versus stress curve (see right of figure 2.9 for the latter). The dynamics of the plastic strain is governed by : @t (x) = + Z G(x − y) (y) + dis 􀀀 x, (x) .

Existing methods to characterize the transition

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The determination of the roughness exponent has been the focus of many works. It is easily accessible in experiments where a direct observation of the front is possible such as the crack and wetting front experiments presented in sections 2.2 and 2.3. Perturbative dimensional expansion within the FRG provided analytical predictions [11, 12]. It was determined with high accuracy for different models using numerical simulations [31, 90]. I discussed in the previous chapter the evolution of the measured values of in the crack and wetting front experiments and the agreement with the theoretical values. In this section I present the different methods to determine . There are three widely used definition of . Two of them are equivalent while the third one does not allow to measure an exponent > 1.
Average width of the interface The first definition, used in simulations , wetting front  and crack front experiments [52, 53], relies on the scaling of the width of portions of the interface and was already mentioned in section 1.1. Considering a portion of the interface of size ` we can formally define its width as the mean quadratic displacement over this portion : W2 u (`) = D􀀀 u − hui 2E = hu2i − hui2.

Avalanche size and duration exponents

Once avalanches are defined, the statistics of the size and duration can be computed over a large number of avalanches and the probability distributions P(S) and P(T) can be experimentally measured. The critical exponents and ˜ are then determined by fitting the power law part of the distributions. This method has been applied with success in Barkhausen noise experiments  (already discussed in section 2.1). The experiments yield two distinct sets of values, depending on the material used. The experimental values were found to be in agreement with numerical values obtained from simulations of two-dimensional (d = 2) interfaces with different elastic interactions. The first set of values ( = 1.5±0.05., ˜ = 2±0.2) correspond to LR elasticity with = 1 while the second one ( = 1.27±0.03, ˜ = 1.5±0.1) is consistent with SR elasticity. This showed that there are two distinct universality classes within the Barkhausen noise experiments.

Introduction
1 Disordered elastic systems and the depinning transition
1.1 Phenomenology of the depinning transition
1.2 Equation of motion
1.3 The Larkin length and the upper critical dimension
1.4 Scaling relations between exponents
1.5 Middleton theorems
1.6 Conclusion
2 Experimental realisations
2.1 The Barkhausen noise
2.2 Crack front propagation
2.3 Wetting of disordered susbtrates
2.4 Earthquakes
2.5 Elastoplasticity and the yielding transition
2.6 Conclusion
3 Assessing the Universality Class of the transition
3.1 Existing methods to characterize the transition
3.2 A novel method based on the universal scaling of the local velocity field
3.3 Conjecture for the correlation functions in plasticity
3.4 Conclusion
4 Cluster Statistics
4.1 Introduction of the observables and various critical exponents
4.2 Recall of previous results
4.3 Cluster statistics
4.4 Statistics of gaps and avalanche diameter
4.5 Tables of exponents
4.6 Conclusion
5 Mean-Field models
5.1 What is mean-field ?
5.2 Fully-connected and ABBM models
5.3 Introduction of the Brownian force model
5.4 Insights into the long-range instanton equation with a local source
5.5 Conclusion
Summary and perspectives
Appendix
A Fourier transform of the elastic force
B Computation of the elastic coefficients for numerical simulations
C Analysis of the experimental data
D Computation of the generating functional using the Martin-Siggia-Rose formalism
Bibliography

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