Non-perturbative Renormalization Group Approach to Some Out-of-Equilibrium Systems: Diffusive Epidemic Process and Fully Developed Turbulence (Springer Theses)

Supervisor’s Foreword
Acknowledgements
Contents
1 General Introduction
	References
2 Universal Behaviors in the Diffusive Epidemic Process and in Fully Developed Turbulence
	2.1 The Absorbing Phase Transition in the Diffusive Epidemic Process
		2.1.1 Directed Percolation
		2.1.2 Diffusive Epidemic Process
	2.2 Breaking of Scale Invariance in Fully Developed Turbulence
		2.2.1 The Navier–Stokes Equation and Scale-Invariance  in Turbulence
		2.2.2 The Phenomenon of Intermittency in Turbulence
		2.2.3 Time Dependence of Correlation Functions in Turbulence
		2.2.4 The Question of Intermittency in the Direct Cascade of 2D Turbulence
	References
3 Introduction to Non-perturbative Renormalization Group for Out-of-Equilibrium Field Theories
	3.1 Notations
	3.2 Response Field Formalism for Langevin Equation
	3.3 Statistical Physics and Mean-Field Theories
		3.3.1 Free Theories and Saddle-Point Methods
		3.3.2 The Effective Action
		3.3.3 Corrections to the Mean-Field Approximation
	3.4 Introduction to the Non-perturbative Renormalization Group
		3.4.1 The Wilson Renormalization Group
		3.4.2 The Regulator and the Wetterich Equation
		3.4.3 Fixed Point Solutions of the Flow and Scale Invariance
		3.4.4 Running Scaling Dimensions and Dimensionless Quantities
	3.5 Causality and Itô Prescription in NPRG
	3.6 Ward Identities and Dualities in NPRG
	References
4 Study of the Absorbing Phase Transition in DEP
	4.1 The Field Theories of DEP and DP-C
		4.1.1 Response Field Action for DP-C
		4.1.2 Coherent Field Action for DEP
		4.1.3 Response Field Action for DEP
	4.2 Symmetries, Ward Identities and Exact Results for DEP and DP-C
		4.2.1 Symmetries of the DP-C Action
		4.2.2 Symmetries of the Response Field Action of DEP and Equivalence with DP-C
		4.2.3 Symmetries of the Coherent Field Action of DEP
	4.3 Modified Local Potential Approximation for DEP  and DP-C
		4.3.1 The Zeroth Order of the Derivative Expansion
		4.3.2 Choice of Φ0 as a Minimum Configuration
		4.3.3 The Litim Θ Regulator
		4.3.4 Truncation of the Potential
	4.4 Results of the Numerical Integration of the DP-C Flow
	4.5 Integration of the DEP Flow and Shortfalls of the LPA\'
	References
5 Breaking of Scale Invariance in Correlation Functions of Turbulence
	5.1 The Field Theory of the Stochastic Navier–Stokes Equation
		5.1.1 SNS Action in the Velocity Formulation
		5.1.2 Interpretation of ΔmathcalS as a Regulator
		5.1.3 Stream Function Formulation in 2-D
	5.2 Symmetries and Extended Symmetries of SNS
		5.2.1 Extended Symmetries in the Velocity Formulation
		5.2.2 Extended Symmetries of the Stream Function Action
	5.3 Ward Identities for the Field Theory of SNS
		5.3.1 Ward Identities in the Velocities Formulation
		5.3.2 Ward Identities for the SNS Field Theory in 2-D
	5.4 Expansion at Large Wave-Number of the RG Flow Equation
	5.5 Leading Order at Unequal Time in 2- and 3-D
		5.5.1 Solution for the 2-Point Functions in 3-D
		5.5.2 Form of the Solution for Generic Correlation Functions in 3-D
	5.6 Large Wave-Number Expansion in the Stream Function Formulation
		5.6.1 Leading Order of the Flow Equation at Unequal Times
		5.6.2 Next-to-Leading Order of the Flow Equation
	References
6 General Conclusion
	6.1 Summary
	6.2 Prospects
	References
Appendix A Master Equation, Generating Function and Mean-Field Equations for Reaction-Diffusion Processes
Appendix B Out of Equilibrium Field Theories and NPRG
B.1 The Martin-Siggia-Rose, Janssen, de Dominicis Formalism
B.2 The Prescription Θ(0)=0 in Perturbation Theory
Appendix C Mappings to Field Theories
C.1 The Doi-Peliti Construction
C.2 Gardiner\'s Poisson Representation
Appendix D Consequence of the Duality Identity
Appendix E LPA\' for DEP and DP-C
E.1 Proof of the Accidental Non-renormalization within LPA\'
E.1.1 Pertubative Proof
E.1.2 Non-renormalization of the barΨΨ Factor of the LPA\' Potential
E.1.3 Accidental Non-renormalization of the LPA\' Kinetic Part
E.2 Derivation of the NPRG Flow of the Couplings
Appendix F Large Wave-Number Expansion of the RG Flow Equation of SNS
F.1 Ward Identities for the Vertex Functions
F.1.1 Ward Identities in Velocity Formulation
F.1.2 Ward Identities in the Stream Function Formulation
F.1.2.1 Time-Gauged Galilean Identity
F.1.2.2 Time-Gauged Rotation Identity
F.2 Form of the Flow Equation of Correlation Functions in the Large Wave-Number Regime
F.3 Leading Order in the Velocity Formulation
F.3.1  Leading Order: Two-Point Function
F.3.2 Leading Order: General Case
F.4 Solution of the Fixed-Point Equations
F.4.1 Small Time Delays
F.4.2 Large Time Delays
F.5 Next-to-Leading Order Term in the Stream Function Formulation
F.6 Next-to-Leading Order Terms at Equal Times
F.6.1 Vanishing of the Contribution 1-3 and 3-1
F.6.2 Uncrossed Derivatives Contribution for the Flow of the Two-Point Function
F.6.3 General Proof of the Closure of the Uncrossed Derivatives